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+SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS
+NICOLA DE NITTI AND FLORIAN SCHWEIGER
+Abstract. This work is concerned with fractional Gaussian fields, i.e. Gaussian fields whose covari-
+ance operator is given by the inverse fractional Laplacian (−∆)−s (where, in particular, we include
+the case s > 1). We define a lattice discretization of these fields and show that their scaling limits –
+with respect to the optimal Besov space topology – are the original continuous fields. As a byproduct,
+in dimension d < 2s, we prove the convergence in distribution of the maximum of the fields. A key
+tool in the proof is a sharp error estimate for the natural finite difference scheme for (−∆)s under
+minimal regularity assumptions, which is also of independent interest.
+Contents
+1.
+Introduction
+1
+1.1.
+Fractional Gaussian Fields
+1
+1.2.
+Finite difference schemes for fractional operators
+3
+1.3.
+Main results
+4
+1.4.
+Future work
+6
+2.
+Fractional polyharmonic Gaussian fields
+7
+2.1.
+The (continuous) fractional Gaussian field
+7
+2.2.
+The (discrete) fractional Gaussian field
+9
+3.
+Rigorous estimates for the finite difference scheme
+10
+4.
+Proofs of the scaling limits
+13
+4.1.
+Scaling limit in the space of distributions
+13
+4.2.
+Scaling limit in Besov, Sobolev and Hölder spaces
+15
+Appendix A.
+Technical lemmas
+21
+A.1.
+Discretization and restriction
+21
+A.2.
+Discrete inequalities
+21
+Appendix B.
+Fractional Gaussian Fields via eigenfunctions
+23
+Acknowledgments
+25
+References
+25
+1. Introduction
+1.1. Fractional Gaussian Fields. Fractional Gaussian fields form a natural one-parameter family
+of Gaussian interface models.
+For a fixed parameter s ≥ 0, the s-fractional Gaussian field is the
+Gaussian field whose covariance operator is (−∆)−s, the inverse of the fractional Laplacian of order
+s. We emphasize right away that we do not assume s ∈ [0, 1], and in fact our main interest is in the
+polyharmonic case s > 1. A purely formal and non-rigorous way to define the s-fractional Gaussian
+field on some domain Ω ⊂ Rd is to set
+P(dϕ) = 1
+Z exp
+�
+−1
+2
+�
+Ω
+ϕ(x)((−∆)sϕ)(x) dx
+�
+dϕ.
+(1.1)
+This cannot be taken as a rigorous definition, as dϕ refers to the Lebesgue measure on the infinite-
+dimensional space RΩ, which does not exist.
+1
+arXiv:2301.13781v1  [math.PR]  31 Jan 2023
+
+2
+N. DE NITTI AND F. SCHWEIGER
+(a) s = 0 (white noise)
+(b) s = 0.5
+(c) s = 1 (Gaussian free field)
+(d) s = 1.5
+(e) s = 2 (membrane model)
+(f) s = 2.5
+(g) s = 3
+(h) s = 3.5
+(i) s = 4
+Figure 1.1. Surface plots of discrete fractional polyharmonic Gaussian fields on Ω :=
+(0, 1)2 ∩
+1
+100Z2 with zero boundary conditions. These discrete random functions are
+linearly interpolated. See also [15, Fig. 1.1] for further numerical experiments.
+There are also other issues with (1.1): namely, one needs to decide how to define (−∆)s for functions
+ϕ: Ω → R and (closely related to that issue) one needs to decide on boundary values of ϕ. For these
+questions, we have a clear answer, though. We take 0 boundary values (i.e., we take ϕ to be extended
+by 0 to the whole Rd), and we let (−∆)s be the fractional Laplacian on the full space Rd, which is
+defined by using the Fourier transform1. That is, for any ξ ∈ Rd,
+F [(−∆)su] (ξ) = |ξ|2sF[u](ξ)
+with
+F[u](ξ) =
+�
+Rd e−iξ·xu(x) dx.
+These choices are natural from a probabilistic point of view, as we will explain in Remark 2.2, and
+they can be implemented to provide a rigorous meaning to (1.1), for example, as a probability measure
+on the space of tempered distributions. This is discussed in detail in the excellent survey [15] and in
+Section 2.1 we recall the points which are important for us.
+1An equivalent hypersingular integral formulation of the polyharmonic fractional operator of order s ∈ (0, m), for
+any m ∈ N, is given by
+(−∆)su(x) := Cd,m,s
+�
+Rd
+�m
+j=−m(−1)j
+� 2m
+m − j
+�
+u(x + jy)
+|y|d+2s
+dy,
+where
+Cd,m,s :=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+22sΓ(n/2 + s)
+πn/2Γ(−s)
+�
+�
+m
+�
+j=1
+(−1)j
+�
+2m
+m − j
+�
+j2s
+�
+�
+−1
+if s ∈ (0, m)\N,
+22sΓ(n/2 + s)s!
+2πn/2
+�
+�
+m
+�
+j=2
+(−1)j
+�
+2m
+m − j
+�
+j2s ln j
+�
+�
+−1
+if s ∈ (0, m) ∩ N.
+We refer to [1] and references therein for further information on the theory of higher-order fractional Laplacians.
+
+SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS
+3
+The main goal of the present work is to define a discrete version ϕh of the s-fractional Gaussian field
+ϕ on a lattice Ωh = Ω ∩ hZd and to study its properties. It is not immediately obvious how one should
+define this discrete version, but we will argue that our definition is quite natural. Certainly, one would
+expect that in the limit h → 0 the discrete s-fractional Gaussian field converges in law, with respect
+to a suitable topology, to the (continuous) s-fractional Gaussian field. Our main result is that, with
+our definition of a s-fractional Gaussian field, this convergence holds in a rather strong sense, namely
+in law with respect to the topology of Besov spaces for the optimal range of parameters.
+Similar problems have been studied before for specific values of s. If s = 1, the field is the Gaussian
+free field and the convergence of the discrete Gaussian free field to its continuous variant is folklore
+(see [20, Section 4] for related results). The proof relies on the fact that covariances of the discrete
+Gaussian free field can be represented using simple random walk, which in the scaling limit becomes
+Brownian motion. The case 0 ≤ s < 1 is addressed in [15, Section 12]2, and the proof of the scaling
+limit follows a similar strategy as for the case s = 1, just with the 2s-stable Lévy process taking the
+place of Brownian motion.
+These results for s ≤ 1 all rely on some form of a random walk representation. For s > 1 and for our
+choice of boundary values, there is no such random walk representation and so proofs become much
+more difficult.3 The only existing result in this regime is for s = 2, where the s-fractional Gaussian field
+is the so-called membrane model. In [7], it was proven that this field is the scaling limit of its discrete
+version. The main ingredient in the proof were estimates for finite difference schemes for (−∆)2 from
+[22], and estimates for its Green’s function from [16].
+Thus, previous work was restricted to s ∈ [0, 1] ∪ {2}, while our results cover the entire range
+s ∈ [0, ∞). Even in the case s ∈ [0, 1] ∪ {2}, our results improve upon the previous work. Namely, the
+convergence in [15, Section 12] is with respect to the topology of distributions and the convergence in
+[7] is with respect to the topology of some negative Sobolev space (for non-optimal parameters). As
+an easy corollary of our result with respect to the Besov-space topology, one obtains convergence with
+respect to the Sobolev-topology and also with respect to the Hölder topology (both with the optimal
+range of parameters).
+Our method of proof uses estimates for finite difference schemes like [7], but of a different flavor. In-
+stead of the estimate from [22] used in [7] (that needs Ck-regularity of the function to be approximated
+by the scheme), we establish an estimate that needs only minimal regularity assumptions (essentially
+just Hs+ε-regularity for some ε > 0). This is the main technical result of the paper and we will discuss
+it and its context next.
+1.2. Finite difference schemes for fractional operators. There is a close relation between discrete
+versions of Gaussian fields and finite difference approximations of the corresponding operator. Indeed,
+if we want to define a lattice version of (1.1) that is suitably close to (1.1) itself, then we need a lattice
+approximation of (−∆)s; the better this approximation, the closer the resulting lattice field will be to
+its continuous version.
+Before discussing finite difference schemes, let us mention that there has been work on finite element
+approaches to the fractional Laplacian (at least for s ≤ 1). We cannot cover the whole body of relevant
+literature here, but we refer to the very recent survey [3].
+Let us now turn to finite difference schemes. The subject of finite difference schemes for the fractional
+Laplacian (−∆)s has been studied before and various schemes have been proposed. However, the main
+focus has been on the case s < 1 and often also d = 1. We refer to the survey [13] and the references
+therein for an overview.
+The main challenge when constructing a finite difference scheme for the
+2There, a definition of the discrete FGF that is slightly different from ours is used; the proof, however, should apply
+to all reasonable discretizations including ours.
+3If one uses another definition of the fractional Gaussian field in terms of spectral powers of the ordinary Laplacian
+(the so-called eigenfunction FGF from [15, Section 9]), one retains a random walk representations and it is comparably
+easy to establish a scaling limit. In fact, in [2], this is done not in the lattice case, but in the more complicated case of
+a Sierpinski gasket.
+
+4
+N. DE NITTI AND F. SCHWEIGER
+fractional Laplacian is that it is given by convolution with a singular integral kernel, and a naive
+discretization of this kernel might not capture its behavior near the singularity.
+Our preferred way to construct a finite difference scheme arises naturally when working in Fourier
+space. Let us consider first the usual Laplacian, i.e. the case s = 1. Its symbol is |ξ|2 and its standard
+finite difference approximation (the 2d + 1-point stencil in dimension d) has symbol
+(1.2)
+Mh(ξ)2 :=
+d
+�
+j=1
+4
+h2 sin2
+�ξjh
+2
+�
+.
+So, for the fractional Laplacian (−∆)s with symbol |ξ|2s, a natural way to define a finite difference
+scheme is to take the finite difference operator with symbol Mh(ξ)2s. That is, we define
+Fh [(−∆h)suh] (ξ) = Mh(ξ)2sFh[uh](ξ)
+with
+Fh[uh](ξ) = hd �
+x∈hZd
+e−iξ·xuh(x).
+We can also use Mh(ξ)2s as a continuous Fourier multiplier and thereby understand (−∆h)s also as a
+continuous operator. This is consistent with the previous definitions, as pointed out in Lemma A.1.
+This scheme for s ≤ 1 (but general d) has already been studied in [12] (and the special case d = 1
+already in [13, Section 4.2] and, in more detail, in [6]) and has many desirable properties. First of all,
+for s ∈ N, we recover the standard schemes for polyharmonic Laplacians. We also have the property
+that (−∆h)s(−∆h)s′ = (−∆h)s+s′. Moreover, while for other schemes the accuracy often degenerates
+as s ↗ 1, our scheme has accuracy h2 uniformly in s (as follows from Theorem 1.3). In Remark 3.2
+below, we comment on how this scheme might work in practice.
+Now that we have chosen our scheme, let us discuss rigorous estimates for its approximation quality.
+In the literature on finite difference schemes, it is common to derive pointwise estimates on the error
+under a strong regularity assumption (Ck or Ck,α for a large enough k). In fact, for d ∈ {1, 2} and
+s ≤ 1, there are two such results in the literature: in [6], pointwise estimates for the approximation
+error for functions in Hölder spaces (at least C0,2s+ε) are shown and, in [12], such pointwise estimates
+are shown under the assumption that the 2s + ε-th derivative has integrable Fourier transform (which,
+roughly speaking, again corresponds to C0,2s+ε).
+However, as already mentioned in the previous subsection, our interest is more in estimates under
+low-regularity assumptions, i.e. in a Sobolev scale. To the best of our knowledge, such estimates are
+new (even in the case d = 1, s < 1). For the case of the Laplacian or Bilaplacian, though, such results
+are classical (see the textbook [14] or a recent refinement for the Bilaplacian in [19]), and our result is
+inspired by the latter. However, the proof is quite different. Namely, the proof of [19, Theorem 2.3]
+relied on the Bramble-Hilbert lemma and thereby used that s ∈ N. In our general setting, we use a
+different approach, based on the Poisson summation formula and a lengthy estimate of various error
+terms in Fourier space.
+1.3. Main results. Let us now state our main results more precisely. We consider the discrete FGF
+ϕh and the continuous FGF ϕ, as introduced informally in Section 1.1. As the rigorous definitions are
+quite technical, we postpone them to Sections 2.2 and 2.1, respectively.
+We claim that the scaling limit of ϕh in an appropriate sense is ϕ. However, ϕh is defined only
+on hZd, so we need to interpolate it to a function on Rd first. For that purpose, we fix a compactly
+supported function Θ ∈ S(Rd) with
+�
+Rd Θ(x) dx = 1 and define Θh(x) =
+1
+hd Θ
+� x
+h
+�
+.
+Using Θh, we can define the interpolated field
+Ihϕh(x) :=
+�
+y∈hZd
+hdϕh(y)Θh(x − y)
+as a random element of S′(Rd).
+Some of the results below also hold if Θ is just a tempered distribution (and, in fact, in [7] only the
+choice Θ = δ0 was used). However, if we hope to find a scaling limit in some Banach space of optimal
+
+SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS
+5
+regularity, we need to consider more regular Θ (as otherwise Ihϕh might not even be an element of
+the Banach space in question); so, to avoid unneccessarily complicated notations, we directly assume
+that Θ is a measurable function.
+As a first result, we claim that, for any Θ chosen as above, the interpolated fields Ihϕh converge in
+the sense of distributions. Note that our definition of ϕh is made in such a way that we do not need
+to rescale it with some power of h to obtain a scaling limit. Indeed, we have the following result.
+Theorem 1.1 (Scaling limit in the space of distributions). Let Ω ⊂ Rd be a bounded domain with
+Lipschitz boundary, and let s ≥ 0.
+Let Θ be a compactly supported function with integral 1. Then Ihϕh converges in law with respect to
+the topology of S′(Rd) to ϕ. That is, for any f ∈ S(Rd), the random variable (Ihϕh, f)L2
+h(hZd)converges
+in law to (ϕ, f)L2(Rd).
+In Theorem 1.1, we established a scaling limit in the space of distributions. However, as we discuss
+in detail in Section 2, the continuous FGF is defined not just as a distribution-valued random variable,
+but actually has a certain Besov-, Sobolev- and Hölder-regularity. Hence, it is natural that we can
+take the scaling limit of the ϕh also in these spaces. In order to do so, however, we need some further
+assumptions on the regularization Θ (otherwise the interpolated field Ihϕh might not even be an
+element of the space). The result now is the following.
+Theorem 1.2 (Scaling limits in Sobolev and Hölder Spaces). Let Ω ⊂ Rd be a bounded domain with
+Lipschitz boundary and let s ≥ 0. Let Θ be a compactly supported function with integral 1, and suppose
+that there is some k with k > s + d
+2 such that
+(1.3)
+|FΘ(ξ)| ≤ C
+(�d
+j=1 sin2(ξj))k/2
+|ξ|k
+for all ξ.
+Let s′ < s − d
+2, p, q ∈ [1, ∞]. Then the interpolated fields Ihϕh converge in law with respect to the
+topology of ˆBs′
+p,q(Rd) to ϕ.
+Moreover, for any fixed bounded domain ˆΩ with Lipschitz boundary such that Ω ⋐ ˆΩ, Ihϕh are
+supported in ˆΩ for h sufficiently small. For any s′ < s − d
+2, the interpolated fields Ihϕh converge in
+law with respect to the topology of ˙Hs′(ˆΩ) to ϕ. In addition, if H := s − d
+2 > 0, m = ⌈H⌉ − 1, and
+0 < α < H − m, then Ihϕh converges in law with respect to the topology of Cm,α(Rd) to ϕ.
+Here, ˆBs′
+p,q(Rd) is, up to a minor technicality that we again discuss in Section 2, equal to the standard
+Besov space Bs′
+p,q(Rd).
+Several remarks are in order. First of all, a convenient example of a function satisfying (1.3) for
+some k ∈ N is given by the centered B-spline of order k (see, e.g., [14, Section 1.9.4] for a summary of
+their properties).
+Next, the restriction to s′ < s − d
+2 cannot be avoided as the continuous FGF is not in ˆBs−d/2
+p,q
+for
+any p, q. Thus, the range of allowed s′ is optimal.
+Regarding the convergence in Sobolev spaces, we cannot expect convergence with respect to the
+topology of ˙Hs′(Ω) for the simple reason that, because of the mollification, Ihϕh need not have zero
+boundary values outside of Ω.
+A fundamental step in the proof of the results above is establishing the following error estimate for
+a fractional Poisson equation (which is of interest in itself). Our goal is to compare the solutions of
+(−∆)su = f and of (−∆h)suh = f and we will estimate the error u − uh in the (discrete) energy norm
+∥ · ∥ ˙Hs
+h. As we work under minimal regularity assumptions on u, the precise result is somewhat more
+technical. Namely, in general u and f might not be continuous functions and so it is not clear how to
+restrict them to the lattice. We circumvent this by introducing two additional mollifiers.
+The result then takes the following shape.
+Theorem 1.3 (Error estimate on the discrete approximation). Let Ω ⊂ Rd be an open bounded set
+with Lipschitz boundary. Let Θ: Rd → R and θ: Rd → R be mollifiers that are compactly supported,
+
+6
+N. DE NITTI AND F. SCHWEIGER
+symmetric around 0, and have integral 1. Furthermore, let us assume that there exist k, l ≥ 0 such
+that
+|FΘ(ξ)| ≤ C
+(�d
+j=1 sin2(ξj))k/2
+|ξ|k
+,
+|Fθ(ξ)| ≤ C 1
+|ξ|l ,
+and define Θh(x) =
+1
+hd Θ
+� x
+h
+�
+, θh(x) =
+1
+hd θ
+� x
+h
+�
+.
+Let 0 < s < t and let u ∈ Ht(Rd) be the solution of
+�
+(−∆)su(x) = f(x),
+x ∈ Ω,
+u(x) = 0,
+x ∈ Rd \ Ω,
+for some f ∈ Ht−2s(Rd), and uh : hZd → R be the solution of
+�
+(−∆h)suh(x) = Θh ∗ f(x),
+x ∈ hZd ∩ Ω,
+uh(x) = 0,
+x ∈ hZd \ Ω.
+Then, if h ≤ 1, k ≥ s, k > d
+2 + 2s − t, l > d
+2 − t, and t − s ≤ 2, we have the estimate
+∥θh ∗ u − uh∥ ˙Hs
+h(hZd) ≤ Cht−s∥u∥ ˙Ht(Rd),
+where C > 0 does not depend on h.
+Here (as in the rest of the paper) C denotes some generic constant that might change from line to
+line, but is always independent of h.
+Let us give some explanations regarding the linear constraints on the parameters in this result. The
+most important constraint is t − s ≤ 2. It arises from the fact that the proposed finite difference
+scheme is of second order (see [13, Section 5.2]), so that the accurary of our scheme saturates at h2.
+The condition k > d
+2 + 2s − t is needed in order for Θh ∗ f to be continuous (so that it has a well-
+defined restriction to the lattice). Similarly, the condition l > d
+2 − t is needed in order for θh ∗ u to be
+continuous.
+As mentioned in Section 1.2, this is the first rigorous estimate for a finite difference scheme for
+(−∆)s under low regularity assumptions. For finite elements, a result that is similar in spirit can
+be found in [3, Theorem 2.6]. There an estimate for piecewise linear finite elements is shown that is
+similar to our result (albeit with the additional restriction t − s ≤ 1
+2 instead of t − s ≤ 2). The method
+of proof is very different.
+1.4. Future work. The most well-studied discrete Gaussian interface model is certainly the discrete
+Gaussian free field (corresponding to s = 1 in our notation). In recent years there has been a lot
+of activity to extend results known for the discrete Gaussian free field to other discrete (Gaussian or
+non-Gaussian) interface models, and this work is a first step to include the discrete FGFs in the latter
+class.
+Let us highlight one such question, namely regarding the maximum of the field. In case of the
+discrete Gaussian free field, this is well-understood. In the subcritical dimension (d = 1), the field
+is nothing but a random walk bridge, so it is easy to see that the rescaled maximum converges to a
+non-degenerate random variable. In supercritical dimensions (d ≥ 3), correlations decay so rapidly
+that the maximum behaves as if the field values were independent [5]. The most interesting case is the
+critical case, d = 2, where the field is log-correlated and obtains the typical second-order correction
+[4].
+These results have already been extended to the case of the membrane model (corresponding to
+s = 2). The subcritical case (d ≤ 3) was studied in [7] using results from [16], the supercritical case
+in [5], and finally the critical case in [19]. An important tool in the latter proof was an estimate for
+finite difference schemes very similar to the one in Theorem 1.3.
+
+SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS
+7
+For general s, it is very likely that similar results hold true. In fact, in the subcritical case d < 2s,
+convergence of the rescaled maximum is a straightforward corollary of Theorem 1.2.
+Corollary 1.4 (Convergence of the maximum for d < 2s). Let d < 2s, Ω ⊂ Rd be a fixed bounded
+domain with Lipschitz boundary, and consider the family ϕh of s-FGF on Ωh = Ω∩hZd as h → 0. Then
+the random variables maxx∈Ωh ϕh(x) converge in distribution to a non-degenerate random variable.
+While this corollary covers the subcritical case, the critical case (d = 2s) and the supercritical case
+(d > 2s) remain open, and we hope to address them in the future.
+In particular, a study of the
+critical case would be very interesting, as most existing examples of discrete log-correlated fields in the
+literature are in d = 2 or some other even dimension while the 3
+2-discrete FGF, for instance, would be
+a natural example of a log-correlated field in odd dimensions.
+2. Fractional polyharmonic Gaussian fields
+In this section, we give precise definitions for the continuous and discrete FGF. For the continuous
+FGF, we follow [15]. The major difference is that we only require Lipschitz continuity of the boundary
+of our domain Ω (and hence our results cover in particular the important special case Ω = (0, 1)d).
+Because of this, several functional-analytic statements require extra attention and we give precise
+references for the results we use.
+2.1. The (continuous) fractional Gaussian field. We first fix our conventions for the Fourier
+transform, and then use it to define some relevant function spaces.
+For a function u: Rd → R, we let F[u](ξ): Rd → R, defined by
+F[u](ξ) =
+�
+Rd e−iξ·xu(x) dx,
+be its continuous Fourier transform. Then, we have the Fourier inversion formula,
+u(x) = F−1[F[u]](x) =
+1
+(2π)d
+�
+Rd eiξ·xF[u](ξ) dξ,
+and Plancherel’s theorem,
+�
+Rd |u(x)|2 dx =
+1
+(2π)d
+�
+Rd |F[u](ξ)|2 dξ.
+We can also define the Sobolev norms
+∥u∥2
+˙Hs(Rd) =
+�
+Rd |ξ|2s|F[u](ξ)|2 dξ,
+∥u∥2
+Hs(Rd) =
+�
+Rd(1 + |ξ|2)s|F[u](ξ)|2 dξ,
+where |ξ|2 is the Fourier multiplier of the Laplacian −∆.
+Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary and let S′(Rd) be the space of tempered
+distributions on Rd. In what follows, we collect some results on fractional Sobolev spaces on Ω. If Ω
+has a smooth boundary, all of them are well-known (and [23, Chapter 4] is a comprehensive reference).
+If Ω has merely Lipschitz boundary, the situation is slightly more complicated and we rely on the
+reference [24]. A brief version of some of these results is also contained in [15, Section 4.1], but some
+of them are not made explicit there.
+For s ≥ 0, let ˙˜Hs(Ω) be the closure of C∞
+c (Ω) with respect to the norm ∥· ∥ ˙Hs(Rd)
+4 and let ˙H−s(Ω)
+be its dual space. The space ˙H−s(Ω) can alternatively be described as follows. Let
+∥u∥ ˙H−s(Ω) =
+inf
+v∈ ˙H−s(Rd)
+u=v in Ω
+∥v∥ ˙H−s(Rd)
+4In [15] this space is denoted Hs
+0(Ω). However, more commonly ˙Hs
+0(Ω) is defined as the closure of C∞
+c (Ω) with respect
+to the norm ∥ · ∥ ˙Hs(Ω), while our space ˙˜Hs(Ω) is equal to the Lions-Magenes space (which is also denoted by ˙Hs
+00(Ω)).
+The two spaces are different whenever s ∈ N + 1
+2 . Our notation is based on the one in [23, Chapter 4].
+
+8
+N. DE NITTI AND F. SCHWEIGER
+for u ∈ S(Rd). Then if S(Ω) is the quotient of S(Rd) under the equivalence relation that identifies
+functions when they agree in Ω, we have that ˙H−s(Ω) is the closure of S(Ω) with respect to the norm
+∥ · ∥ ˙H−s(Ω) (this follows from [24, Theorem 3.5 (i)] upon observing that our spaces ˙˜Hs(Ω), for s ≥ 0,
+and ˙H−s(Ω), for s < 0, are equal to Triebel’s ¯Bs
+2,2(Ω) by [24, Proposition 3.1]).
+From the Lax-Milgram lemma (and the fact that C∞
+c (Ω) is dense in ˙˜Hs(Ω)) we also obtain that
+(−∆)s is an isometry from ˙˜Hs(Ω) to ˙H−s(Ω).
+For convenience (and with a slight abuse of notation) we define
+˙Hs(Ω) =
+� ˙˜Hs(Ω)
+if s ≥ 0,
+˙Hs(Ω)
+if s < 0.
+This scale of Hilbert spaces has various desirable properties. For any s < t, the embedding from ˙Ht(Ω)
+to ˙Hs(Ω) is compact (cf. [24, Theorem 2.7]). Even more importantly, the spaces form an interpolation
+scale with respect to complex (or equivalently real) interpolation (cf. [24, Theorem 3.5 (iv)]).
+In [15, Section 4.2], the continuous FGF is defined as a probability measure P on S′(Rd). More
+precisely, it is defined such that when ϕ is distributed according to P, then for every Schwartz function
+f ∈ S(Rd) we have that (ϕ, f) is a centered Gaussian with variance ∥f∥2
+˙H−s(Ω). By [15, Theorem 2.3
+and Proposition 2.4], this property defines P as a probability measure on S′(Rd) uniquely.
+Let us remark that one can one alternatively define the FGF as a random sum of eigenfunctions of
+(−∆)s. We give details on this in Appendix B.
+The regularity of ϕ is best measured in Besov spaces. For s′ ∈ R, p, q ∈ [1, ∞], we let ∥ · ∥Bs′
+p,q(Rd)
+be the usual Besov norm (defined, e.g., via Littlewood-Paley decomposition or via wavelets; see,
+for example, [23, Chapter 2]) and let ˆBs′
+p,q(Rd) be the closure of C∞
+c (Rd) with respect to the norm
+∥ · ∥Bs′
+p,q(Rd)
+5.
+Then we have the following regularity results for ϕ.
+Proposition 2.1 (Regularity of the FGF). Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary
+and let s ≥ 0. For any s′ < s − d
+2 and p, q ∈ [1, ∞], the FGF on Ω is P-almost surely an element of
+ˆBs′
+p,q(Rd).
+In particular, for any s′ < s − d
+2 the FGF on Ω is P-almost surely an element of ˙Hs′(Ω).
+Moreover, if H := s − d
+2 > 0, then the FGF on Ω is also P-almost surely an element of Cm,α
+loc (Rd)
+for m = ⌈H⌉ − 1 and any 0 < α < H − m.
+Proof. The Besov regularity could be shown using the tightness criterion in Lemma 4.1 below applied
+to the constant sequence ϕ(m) = ϕ. However, according to Theorem 1.2 we have the much stronger
+statement that the FGF is the limit (with respect to the ˆBs′
+p,q(Rd)-topology) of the discrete fractional
+Gaussian fields, suitably interpolated; so we do not give details for the proof of the Besov regularity
+here.
+It is well-known that ˆBs′
+2,2(Rd) = Hs′(Rd) and ˆBs′
+∞,∞(Rd) �→ Cs′(Rd), where Cs′(Rd) is the Hölder-
+Zygmund space, which embeds into the classical Hölder space C⌊s′′⌋,s′′−⌊s′′⌋ for any 0 < s′′ < s′ (see
+[24, Section 2.1]). These results together with the fact that the FGF is supported in Ω easily imply
+the Sobolev and Hölder regularity results in the proposition.
+Let us remark that the Sobolev regularity alternatively follows from the fact the random series defin-
+ing ˜ϕ converges in ˙Hs′(Ω) almost surely, while the Hölder regularity also follows from [15, Proposition
+6.2 and Theorem 8.3]6.
+□
+5Note that the Besov space Bs′
+p,q(Rd) commonly defined as the set of all tempered distributions for which ∥·∥Bs′
+p,q(Rd)
+is finite. Clearly ˆBs′
+p,q(Rd) ⊂ Bs′
+p,q(Rd), and the inclusion is strict if p = ∞ or q = ∞.
+6N.B. There is a typo in the statement of [15, Proposition 6.2]: it should read H − k instead of H − ⌈H⌉.
+
+SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS
+9
+2.2. The (discrete) fractional Gaussian field. Our definition of the discrete FGF follows the one
+of the continuous FGF as closely as possibly. Let us again begin by fixing our conventions for discrete
+Fourier transforms and discrete function spaces.
+For a function uh : hZd → R, we let Fh[uh](ξ): Rd → R, defined by
+Fh[uh](ξ) = hd �
+x∈hZd
+e−iξ·xuh(x),
+be its discrete Fourier transform (note that this function is 2π
+h -periodic). Then we have the discrete
+Fourier inversion formula,
+uh(x) = F−1
+h [Fh[uh]](x) =
+1
+(2π)d
+�
+(− π
+h , π
+h)
+d eiξ·xFh[uh](ξ) dξ,
+and Plancherel’s theorem,
+hd �
+x∈hZd
+|uh(x)|2 =
+1
+(2π)d
+�
+(− π
+h , π
+h)
+d |Fh[uh](ξ)|2.
+We can define the discrete Sobolev norms
+∥uh∥2
+˙Hs
+h(hZd) =
+�
+(− π
+h , π
+h)
+d Mh(ξ)2s|Fh[uh](ξ)|2 dξ,
+∥uh∥2
+Hs
+h(hZd) =
+�
+(− π
+h , π
+h)
+d(1 + Mh(ξ)2)s|Fh[uh](ξ)|2 dξ,
+where Mh(ξ)2 := �d
+j=1
+4
+h2 sin2 �
+ξjh
+2
+�
+is the discrete Fourier multiplier of the discrete Laplacian.
+Let Ω be as before and let Ωh = Ω ∩ hZd. Similarly as in the continuous setting, we define the
+space ˙˜Hs(Ωh) as the space of functions hZd → R that vanish outside of Ωh (equipped with the norm
+induced by ∥ · ∥ ˙Hs
+h(hZd)). We let ˙H−s(Ωh) be its dual space, and define
+˙Hs(Ω) =
+� ˙˜Hs
+h(Ωh)
+if s ≥ 0,
+˙Hs
+h(Ωh)
+if s < 0.
+We define the discrete FGF as a probability measure on ˙˜Hs(Ωh). More precisely, we consider the
+measure
+Ph( dϕh) = 1
+Zh
+exp
+�
+−1
+2∥ϕh∥2
+˙Hs
+h(hZd)
+� �
+x∈Ωh
+dϕh(x)
+�
+x∈hZd\Ωh
+δ0( dϕh(x))
+= 1
+Zh
+exp
+�
+−1
+2
+�
+x∈Ωh
+hdϕh(x)(−∆h)sϕh(x)
+� �
+x∈Ωh
+dϕh(x)
+�
+x∈hZd\Ωh
+δ0( dϕh(x)).
+This is a well-defined Gaussian measure with mean 0, and variance
+Eh(ϕh, fh)2
+L2
+h(hZ)d = ∥fh∥2
+˙H−s
+h
+(Ωh)
+(2.1)
+for any fh : hZd → R.
+Remark 2.2 (Boundary values). Let us comment on our choice of boundary values. The main advantage
+of our definition is the fact that it is consistent with projections. Namely, let Ω ⊂ ˜Ω be open sets
+and consider the discrete FGF ˜ϕh on ˜Ωh. Then, the restriction of ˜ϕh to Ωh is equal in distribution to
+the sum of the (−∆h)s-harmonic extension of ˜ϕh from hZd \ Ωh to Ωh and of an independent discrete
+FGF ϕh on Ωh 7. In particular, even if we had started with a field without boundary values (i.e., with
+7If s = 1, this reduces to the familiar domain Markov property for the discrete Gaussian free field: the field in a
+subdomain is equal in distribution to the harmonic extension of its boundary values plus an independent zero-boundary
+field.
+
+10
+N. DE NITTI AND F. SCHWEIGER
+˜Ω = Rd), then looking at the field on a subset naturally leads to consider fields with zero boundary
+values outside that subset.
+3. Rigorous estimates for the finite difference scheme
+In this section, we present the proof of Theorem 1.3. As mentioned in Section 1.2, the proof of
+the analogous statement for s = 2 in [19, Theorem 2.3] was based on the Bramble-Hilbert lemma to
+estimate various error terms. Thus it relied on the fact that (−∆h)2 (and hence the finite difference
+scheme) is local in that case.
+In the generic case s ̸∈ N, however, (−∆h)s is not local, and so this proof strategy can no longer
+be applied. Instead, we use the fact that both (−∆)s and (−∆h)s are defined via Fourier multipliers
+and directly estimate all relevant error terms in Fourier space. However, this requires extra care as we
+need to switch from discrete Fourier space to continuous Fourier space at some point. In fact, we need
+a way to compare Fh and F. Fortunately, the following Poisson-type summation formula enables us
+to do so easily.
+Lemma 3.1 (Poisson-type summation formula). Suppose that g: Rd → R is a Schwartz function.
+Then we have the identity
+Fh[g](ξ) =
+�
+ζ∈ 2π
+h Zd
+F[g](ξ + ζ).
+Proof. By the Poisson summation formula (see, e.g., [18, Chapter 4.4]), for any Schwartz function f
+we have
+hd �
+x∈hZd
+f(x) =
+�
+ζ∈ 2π
+h Zd
+F[f](ζ).
+Applying this to f(x) = e−iξ·xg(x), we find
+hd �
+x∈hZd
+e−iξ·xg(x) =
+�
+ζ∈ 2π
+h Zd
+F[e−iξ·g](ζ) =
+�
+ζ∈ 2π
+h Zd
+F[g](ξ + ζ),
+which implies the claim.
+□
+Using Lemma 3.1, we can now turn to the proof of our estimate on finite difference schemes.
+Proof of Theorem 1.3. By density, we can assume that u is smooth; then, in particular, u is a Schwartz
+function and F[u] is a Schwartz function as well. Therefore, all integrals and sums below will be well-
+defined.
+Step 1: Representation of the error. From the definitions we have
+(3.1)
+∥u − uh∥ ˙Hs
+h(hZd) = ∥(−∆h)s(θh ∗ u − uh)∥ ˙H−s
+h
+(Ωh) =
+inf
+v : hZd→R
+v=(−∆h)s(θh∗u−uh) in Ωh
+∥v∥ ˙H−s
+h
+(hZd).
+Using Lemma A.1, we can also rewrite, for x ∈ Ωh,
+(−∆h)s(θh ∗ u − uh)(x)
+= (−∆h)s(θh ∗ u)(x) − Θh ∗ f(x)
+= (−∆h)s(θh ∗ u)(x) − Θh ∗ (−∆)su(x)
+=
+1
+(2π)d
+�
+(− π
+h , π
+h)
+d eiξ·xMh(ξ)2sFh[θh ∗ u](ξ) dξ −
+1
+(2π)d
+�
+Rd eiξ·x|ξ|2sF[Θh](ξ)F[u](ξ) dξ
+= I1 + I2 + I3 + I4 + I5,
+where
+I1(x) :=
+1
+(2π)d
+�
+(− π
+h , π
+h)
+d eiξ·xMh(ξ)2s (Fh[θh ∗ u](ξ) − F[θ ∗ u](ξ)) dξ,
+I2(x) :=
+1
+(2π)d
+�
+(− π
+h , π
+h)
+d eiξ·xMh(ξ)2s (F[θh ∗ u](ξ) − F[u](ξ)) dξ,
+
+SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS
+11
+I3(x) :=
+1
+(2π)d
+�
+(− π
+h , π
+h)
+d eiξ·x (1 − F[Θh](ξ)) Mh(ξ)2sF[u](ξ) dξ,
+I4(x) :=
+1
+(2π)d
+�
+(− π
+h , π
+h)
+d eiξ·x �
+Mh(ξ)2s − |ξ|2s�
+F[Θh](ξ)F[u](ξ) dξ,
+I5(x) := −
+1
+(2π)d
+�
+Rd\(− π
+h , π
+h)
+d eiξ·x|ξ|2sF[Θh](ξ)F[u](ξ) dξ.
+To conclude the proof, we need to show that
+∥Ij∥ ˙H−s
+h
+(hZd) ≤ Cht−s∥u∥ ˙Ht(Rd)
+holds for each j ∈ {1, 2, 3, 4, 5}.
+Step 2: Estimate of I2. Directly from the definition, we see that
+Fh[I2](ξ) = Mh(ξ)2s (F[θh ∗ u](ξ) − F[u](ξ)) = Mh(ξ)2s (F[θh](ξ) − 1) F[u](ξ).
+As F[θh](ξ) = F[θ](hξ), our assumption on θ implies that F[θh](ξ) ≤
+C
+hl|ξ|l ≤ C for ξ ∈
+�
+− π
+h, π
+h
+�d.
+Therefore
+∥I2∥2
+˙H−s
+h
+(hZd) =
+�
+(− π
+h , π
+h)
+d Mh(ξ)−2s|Fh[I2](ξ)|2 dξ
+=
+�
+(− π
+h , π
+h)
+d Mh(ξ)2s |F[θh](ξ) − 1|2 |F[u](ξ)|2 dξ
+≤ C
+�
+(− π
+h , π
+h)
+d |ξ|2s|F[u](ξ)|2 dξ
+≤ C
+�
+(− π
+h , π
+h)
+d |ξ|2(t−s)h2(t−s)|F[u](ξ)|2 dξ
+≤ Ch2(t−s)∥u∥2
+˙Ht(Rd).
+Step 3: Estimate of I3. The estimate of I3 is quite similar: again, we have that
+Fh[I3](ξ) = (1 − F[Θh](ξ)) Mh(ξ)2sF[u](ξ).
+The assumptions on Θ imply that FΘ(0) = 1 and ∇FΘ(0) = 1. Furthermore, FΘ is a Schwartz
+function. So, by Taylor’s theorem, there is a constant C such that |1 − FΘ(ξ)| ≤ C|ξ|2. This implies
+|1 − F[Θh](ξ)| ≤ Ch2|ξ|2. We can now estimate
+∥I3∥2
+˙H−s
+h
+(hZd) =
+�
+(− π
+h , π
+h)
+d Mh(ξ)−2s|Fh[I3](ξ)|2 dξ
+=
+�
+(− π
+h , π
+h)
+d Mh(ξ)2s(1 − F[Θh](ξ))2|F[u](ξ)|2 dξ
+≤ C
+�
+(− π
+h , π
+h)
+d |ξ|2sh4|ξ|4|F[u](ξ)|2 dξ
+≤ C
+�
+(− π
+h , π
+h)
+d |ξ|2(t−s)h2(t−s)|F[u](ξ)|2 dξ
+≤ Ch2(t−s)∥u∥2
+˙Ht(Rd).
+Step 4:
+Estimate of I4.
+The argument for I4 is very similar to that for I3:
+we use that
+��Mh(ξ)2s − |ξ|2s�� ≤ Ch2|ξ|2 and proceed as for I3.
+Step 5: Estimate of I1. From the definition and Lemma 3.1, we have that
+Fh[I1](ξ) = Mh(ξ)2s (Fh[θh ∗ u](ξ) − F[θh ∗ u](ξ))
+
+12
+N. DE NITTI AND F. SCHWEIGER
+= Mh(ξ)2s
+�
+ζ∈ 2π
+h Zd\{0}
+F[θh ∗ u](ξ + ζ)
+= Mh(ξ)2s
+�
+ζ∈ 2π
+h Zd\{0}
+F[θh](ξ + ζ)F[u](ξ + ζ).
+The Cauchy-Schwartz inequality then yields
+∥I1∥2
+˙H−s
+h
+(hZd) =
+�
+(− π
+h , π
+h)
+d Mh(ξ)−2s|Fh[I1](ξ)|2 dξ
+=
+�
+(− π
+h , π
+h)
+d Mh(ξ)2s
+������
+�
+ζ∈ 2π
+h Zd\{0}
+F[θh](ξ + ζ)F[u](ξ + ζ)
+������
+2
+dξ
+≤
+�
+(− π
+h , π
+h)
+d Mh(ξ)2s
+�
+�
+�
+ζ∈ 2π
+h Zd\{0}
+|ξ + ζ|2t|F[u](ξ + ζ)|2
+�
+�
+�
+�
+�
+ζ∈ 2π
+h Zd\{0}
+F[θh](ξ + ζ)
+|ξ + ζ|2t
+�
+� dξ.
+We know that F[θh](ξ + ζ) ≤
+C
+hl|ξ+ζ|l . As 2(t + l) > d, we can bound
+sup
+ξ∈(− π
+h , π
+h)
+d
+�
+ζ∈ 2π
+h Zd\{0}
+F[θh](ξ + ζ)
+|ξ + ζ|2t
+≤
+sup
+ξ∈(− π
+h , π
+h)
+d
+�
+ζ∈ 2π
+h Zd\{0}
+1
+h2l|ξ + ζ|2(t+l)
+≤ C
+�
+ζ∈ 2π
+h Zd\{0}
+1
+h2l|ζ|2(t+l)
+≤ Ch2t
+and deduce
+∥I1∥2
+˙H−s
+h
+(hZd) ≤ C 1
+h2s h2t
+�
+(− π
+h , π
+h)
+d
+�
+ζ∈ 2π
+h Zd\{0}
+|ξ + ζ|2t|F[u](ξ + ζ)|2 dξ
+≤ Ch2(t−s)
+�
+Rd |ξ2t|F[u](ξ)|2 dξ
+≤ Ch2(t−s)∥u∥2
+˙Ht(Rd).
+Step 6: Estimate of I5. We see that
+F[I5](ξ) = |ξ|2sF[Θh](ξ)F[u](ξ)χRd\(− π
+h , π
+h)
+d(ξ),
+where χA is the indicator function of the set A. Lemma 3.1 then implies that, for ξ ∈
+�
+− π
+h, π
+h
+�d,
+Fh[I5](ξ) =
+�
+ζ∈ 2π
+h Zd
+F[I5](ξ + ζ)
+=
+�
+ζ∈ 2π
+h Zd
+|ξ + ζ|2sF[Θh](ξ + ζ)F[u](ξ)χRd\(− π
+h , π
+h)
+d(ξ + ζ)
+=
+�
+ζ∈ 2π
+h Zd\{0}
+|ξ + ζ|2sF[Θh](ξ + ζ)F[u](ξ + ζ)
+and therefore (recalling that Mh(ξ) is 2π
+h -periodic)
+∥I5∥2
+˙H−s
+h
+(hZd)
+=
+�
+(− π
+h , π
+h)
+d Mh(ξ)−2s|Fh[I5](ξ)|2 dξ
+
+SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS
+13
+=
+�
+(− π
+h , π
+h)
+d Mh(ξ)−2s
+������
+�
+ζ∈ 2π
+h Zd\{0}
+|ξ + ζ|2sF[Θh](ξ + ζ)F[u](ξ + ζ)
+������
+2
+dξ
+=
+�
+(− π
+h , π
+h)
+d
+������
+�
+ζ∈ 2π
+h Zd\{0}
+|ξ + ζ|2s
+Mh(ξ + ζ)s F[Θh](ξ + ζ)F[u](ξ + ζ)
+������
+2
+dξ
+≤
+�
+(− π
+h , π
+h)
+d
+�
+�
+�
+ζ∈ 2π
+h Zd\{0}
+|ξ + ζ|2t|F[u](ξ + ζ)|2
+�
+�
+�
+�
+�
+ζ∈ 2π
+h Zd\{0}
+|F[Θh](ξ + ζ)|2
+Mh(ξ + ζ)2s|ξ + ζ|2(t−2s)
+�
+� dξ.
+Note that F[Θh](ξ) = F[Θ](hξ) and so |F[Θh](ξ)| ≤ C
+(�d
+j=1 sin2(hξj))k/2
+hk|ξ|k
+; moreover,
+(�d
+j=1 sin2(hξj))1/2
+Mh(ξ)
+≤
+Ch. Since k ≥ s, Mh(ξ + ζ)2s is controlled by the sin-terms from |F[Θh](ξ + ζ)|2 and so we can bound
+sup
+ξ∈(− π
+h , π
+h)
+d
+�
+ζ∈ 2π
+h Zd\{0}
+|F[Θh](ξ + ζ)|2
+Mh(ξ + ζ)2s|ξ + ζ|2t−4s ≤ C
+sup
+ξ∈(− π
+h , π
+h)
+d
+�
+ζ∈ 2π
+h Zd\{0}
+h2s
+h2k|ξ + ζ|2(t+k−2s)
+≤ C
+�
+ζ∈ 2π
+h Zd\{0}
+1
+h2(k−s)|ζ|2(t+k−2s)
+≤ Ch2(t−s),
+where we used the fact that 2(t + k − 2s) > d. Hence,
+∥I5∥2
+˙H−s
+h
+(hZd) ≤ Ch2(t−s)
+�
+(− π
+h , π
+h)
+d
+�
+ζ∈ 2π
+h Zd\{0}
+|ξ + ζ|2t|F[u](ξ + ζ)|2 dξ
+≤ Ch2(t−s)
+�
+Rd |ξ|2t|F[u](ξ)|2 dξ
+≤ Ch2(t−s)∥u∥2
+˙Ht(Rd).
+This completes the proof.
+□
+Remark 3.2 (Usage of the finite difference scheme). So far we have not said much regarding the
+practical applications of the finite difference scheme in Theorem 1.3. It would go beyond the scope of
+this work to report on some practical experiments, but let us make a few comments.
+In order to use the scheme to approximate a solution of (−∆)su = f, a first challenge is to compute
+the entries of ((−∆h)s)x,y∈Ωh. Even using the translation-invariance of (−∆h)s, we need to compute
+O
+� 1
+h2
+�
+entries, where each is given as a singular integral. This is quite costly, but avoids introduction
+of additional error. In fact, the pictures in Figure 1.1 were computed using this method.
+If one is willing to accept an additional error term, then a more efficient way to compute an
+approximation to the entries of ((−∆h)s)x,y∈Ωh was suggested in [12]: choose a parameter h′ ≤ h, and
+approximate the integral over
+�
+− π
+h, π
+h
+�d appearing in the definition of (−∆h)s by a Riemann sum on
+a lattice of width h′
+h . The advantage is that this Riemann sum can be computed very efficiently using
+the fast Fourier transform. Moreover, in [12, Section 4.2], it is suggested that this should lead to an
+additional error of order O
+�
+h′d+2s
+h2s
+�
+. In other words, if we choose h′ ≤ h(2s+2)/(2s+d), the error should
+be of order h2 and thus not bigger than the error in Theorem 1.3. While the error estimate in [12,
+Section 4.2] is not rigorous, it should be possible to give a full proof.
+4. Proofs of the scaling limits
+4.1. Scaling limit in the space of distributions. With Theorem 1.3 in hand, we are ready to
+prove that ϕ is indeed the scaling limit of the ϕh. First, we study the scaling limit in the space of
+distributions, Theorem 1.1.
+
+14
+N. DE NITTI AND F. SCHWEIGER
+Proof of Theorem 1.1. Step 1: Characterization of the convergence. Let us consider some f ∈ S(Rd).
+Both (Ihϕh, f)L2(Rd) and (ϕ, f)L2(Rd) are centered Gaussian random variables and so it suffices to
+prove that their variances converge. We have that
+Eh(Ihϕh, f)2 = Eh
+�
+�
+�
+Rd
+�
+y∈hZd
+hdϕh(y)Θh(x − y)f(x) dx
+�
+�
+2
+= Eh
+�
+� �
+y∈hZd
+hdϕh(y)
+�
+Rd Θh(x − y)f(x) dx
+�
+�
+2
+= Eh(ϕh, Θh ∗ f)2
+L2
+h(hZd)
+= ∥Θh ∗ f∥2
+˙H−s
+h
+(Ωh)
+(4.1)
+and so we only need to prove that
+(4.2)
+lim
+h→∞ ∥Θh ∗ f∥2
+˙H−s
+h
+(Ωh) = ∥f∥2
+˙H−s(Ω).
+Step 2: Representation of the error. For each h > 0, let uh : hZd → R be the solution of
+�
+(−∆h)suh(x) = Θh ∗ f(x),
+x ∈ hZd ∩ Ω,
+uh(x) = 0,
+x ∈ hZd \ Ω
+and let u ∈ Hs(Rd) be the solution of
+�
+(−∆)su(x) = f(x),
+x ∈ Ω,
+u(x) = 0,
+x ∈ Rd \ Ω.
+Moreover, let ˜Θ, ˜θ be functions satisfying the assumptions of Theorem 1.3 with k := max
+� d
+2 + s, s
+�
+and l := max
+� d
+2 − s, 0
+�
+; for example, let us take ˜Θ to be a B-spline of order ⌈k⌉ and ˜θ any smooth
+mollifier. Then, let us define ˜Θh and ˜θh as before and let ˜uh : hZd → R be the solution of
+�
+(−∆h)s˜uh(x) = ˜Θh ∗ f(x),
+x ∈ hZd ∩ Ω,
+˜uh(x) = 0,
+x ∈ hZd \ Ω.
+Then, we can write
+∥Θh ∗ f∥2
+˙H−s
+h
+(Ωh) − ∥f∥2
+˙H−s(Ω) = (Θh ∗ f, uh)L2
+h(hZd) − (f, u)L2(Rd)
+= J1 + J2 + J3 + J4 + J5,
+(4.3)
+where
+J1 := (Θh ∗ f, uh)L2
+h(hZd) − (Θh ∗ f, ˜uh)L2
+h(hZd),
+J2 := (Θh ∗ f, ˜uh)L2
+h(hZd) − (Θh ∗ f, ˜θh ∗ u)L2
+h(hZd),
+J3 := (Θh ∗ f, ˜θh ∗ u)L2
+h(hZd) − (f, ˜θh ∗ u)L2
+h(hZd),
+J4 := (f, ˜θh ∗ u)L2
+h(hZd) − (f, ˜θh ∗ u)L2(Rd),
+J5 := (f, ˜θh ∗ u)L2(Rd) − (f, u)L2(Rd).
+We need to show that Ji → 0 as h → 0. This implies (4.2), as required. The most important term is
+J2, for which we shall need to use Theorem 1.3; the other terms will be straightforward to control.
+Step 3: Estimate of J2. Let t > s be a constant to be chosen later. Our choices of k, l ensure that
+the assumptions of Theorem 1.3 are all satisfied. Theorem 1.3 and the discrete Poincaré inequality
+(see Lemma A.2) then imply that
+J2 = (Θh ∗ f, ˜uh − ˜θh ∗ u)L2
+h(hZd)
+
+SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS
+15
+≤ ∥Θh ∗ f∥L2
+h(hZd)∥˜uh − ˜θh ∗ u∥L2
+h(hZd)
+≤ C∥Θh ∗ f∥L2
+h(hZd)∥˜uh − ˜θh ∗ u∥ ˙Hs
+h(hZd)
+≤ C∥f∥L∞(Rd)ht−s∥u∥ ˙Ht(Rd).
+For t−s small enough (depending on s and Ω), Lemma A.4 implies that we have Ht-regularity-estimates
+on Ω and hence in particular ∥u∥ ˙Ht(Rd) < ∞. Thus J2 → 0 as h → 0.
+Step 3: Estimate of J1, J3, J4, J5. For J1, using again the discrete Poincaré inequality, we estimate
+J1 = (Θh ∗ f, uh − ˜uh)L2
+h(hZd)
+≤ ∥Θh ∗ f∥L2
+h(hZ)d∥uh − ˜uh∥L2
+h(hZd)
+≤ C∥f∥L∞(Rd)∥uh − ˜uh∥ ˙Hs
+h(Ωh)
+≤ C∥f∥L∞(Rd)∥Θh ∗ f − ˜Θh ∗ f∥ ˙H−s
+h
+(Ωh)
+≤ C∥f∥L∞(Rd)∥Θh ∗ f − ˜Θh ∗ f∥L2
+h(Ωh)
+≤ Ch∥f∥L∞(Rd)∥∇f∥L∞(Rd);
+where the right-hand side tends to 0 as h → 0. The same argument also applies to J3.
+For J5, it suffices to observe that ˜θh ∗ u tends to u in L2(Rd). Finally, for J4, we use the fact that
+˜θh ∗ u is continuous (and thus f · (˜θh ∗ u) is continuous), and so
+lim
+h→0(f, ˜θh ∗ u)L2
+h(hZd) = (f, u)L2(Rd)
+as a Riemann sum.
+□
+4.2. Scaling limit in Besov, Sobolev and Hölder spaces. We now turn to the proof of the scaling
+limit in Besov spaces (which then implies the result in Sobolev and Hölder spaces as well). As we have
+already established convergence of the fields in the space of distributions, the main challenge is to
+prove tightness in Besov spaces. To this end, we use a very convenient criterion from [9]. As in our
+case we do not need to worry about boundary issues, we do not the full generality of that criterion.
+Let us state the version that we will use.
+Lemma 4.1 (Tightness criterion). Let r ∈ N and let ˆΩ ⊂ Rd be an open bounded set. Then there
+exist functions f, (gj)2d−1
+j=1
+∈ Cr
+c (Rd) such that, for any multi-index m ∈ Nd with |m| < r and any
+j ∈ {1, . . . , 2d − 1}, we have
+(4.4)
+�
+Rd xmgj(x) dx = 0
+and such that the following statement holds. Let (φn)n∈N be a family of random linear forms on Cr
+c (Rd)
+with support in ˆΩ. Let t, t′ ∈ R with t < t′, |t|, |t′| < r and let p ∈ [1, ∞), q ∈ [1, ∞]. Let us suppose
+that there exists a constant C such that
+(4.5)
+lim sup
+n→∞
+sup
+x∈Rd (E |⟨φn, f(· − x)⟩|p)1/p < ∞
+and
+(4.6)
+lim sup
+n→∞
+sup
+k∈N
+sup
+x∈Rd
+max
+1≤j≤2d−1 (E |⟨φn, gj(2a(· − x))⟩|p)1/p ≤
+C
+2a(d+t′) .
+Then the family (φn)n∈N is tight in ˆBt
+p,q(Rd). If t < t′ − d
+p, it is also tight in ˆBt
+∞,q(Rd).
+Proof. This is essentially [9, Theorem 2.30]. There a local version of the theorem is given. The global
+version presented here is obtained by choosing U = Rd, ˆΩ ⊂ K1 ⊂ K2 ⊂ . . . such that already K1 is far
+larger than ˆΩ, k1 = k2 = . . . = 0 and observing that, for functions with uniformly compact support,
+the local and global Besov spaces agree.
+
+16
+N. DE NITTI AND F. SCHWEIGER
+We also used lim supn→∞ instead of supn∈N in (4.5) and (4.6), but this clearly does not make a
+difference. Finally, the assertion that
+�
+Rd xmgj(x) dx = 0 is stated in [9, Equation (2.2)].
+□
+Proof of Theorem 1.2. Step 1: Simplifications. It suffices to prove tightness of Ihϕh in the corre-
+sponding spaces; the convergence then follows easily from Theorem 1.1 by the same argument as in
+[7, Proof of Theorem 3.11]. In order to prove tightness, we will apply Lemma 4.1. We fix some open
+bounded set ˆΩ ⋑ Ω and note that for h small enough Ihϕh is supported in ˆΩ. Let us fix some r ∈ N
+with r >
+��s − d
+2
+�� and let f, (gj) be as in the lemma. We claim that, for any p′ < ∞,
+lim sup
+h→0
+sup
+x∈Rd Eh
+���(Ihϕh, f(· − x))Lp′(Rd)
+���
+p′
+< ∞,
+(4.7)
+lim sup
+h→0
+sup
+a∈N
+sup
+x∈Rd
+max
+1≤j≤2d−1 Eh
+���(Ihϕh, gj(2a(· − x)))Lp′(Rd)
+���
+p′
+≤
+C
+2a(d+2s) .
+(4.8)
+Once we have verified this, Lemma 4.1 (with t′ = s − d
+2) directly implies tightness in ˆBs′
+p,q(Rd) for any
+p ∈ [1, ∞), q ∈ [1, ∞], and choosing p′ sufficiently large such that s′ < t′ − d
+p′ we cover the case p = ∞
+as well. Once we know tightness in Besov spaces, the tightness in Sobolev- and Hölder spaces follows
+directly from Besov embedding.
+Regarding (4.5) and (4.6), we can make some immediate simplifications. First of all, it suffices to
+check the two estimates for p′ ∈ 2N (the result for other p′ then follows from Jensen’s inequality).
+In addition, as ϕh is a Gaussian random variable, all even functions of linear functionals of ϕh are
+controlled by its second moment. This means that we only need to consider p′ = 2. That is, we
+actually only need to verify that
+lim sup
+h→0
+sup
+x∈Rd Eh
+��(Ihϕh, f(· − x))L2(Rd)
+��2 < ∞,
+(4.9)
+lim sup
+h→0
+sup
+k∈N
+sup
+x∈Rd
+max
+1≤j≤2d−1 Eh
+��(Ihϕh, gj(2a(· − x)))L2(Rd)
+��2 ≤
+C
+22a(d+t′) =
+C
+2a(d+2s) .
+(4.10)
+The estimate (4.10) is the crucial one. So we give its proof in detail, and then explain how to prove
+(4.9) as well.
+Step 2: Proof of (4.10). Let us fix some h ≤ 1, a ∈ N, x ∈ Rd, and abbreviate ˜g(a)
+j
+(y) := gj(−2ay).
+A computation similar to the one in (4.1) shows that
+Eh
+��(Ihϕh, gj(2a(· − x)))L2(Rd)
+��2 = Eh
+������
+�
+y∈Rd
+�
+z∈hZd
+hdϕh(z)Θh(y − z)gj(2a(y − x)) dy
+������
+2
+= Eh
+�
+� �
+z∈hZd
+hdϕh(z)
+�
+y∈Rd Θh(y − z) ˜f (a)
+j
+(x − y) dy
+�
+�
+2
+= Eh
+�
+ϕh, (Θh ∗ ˜g(a)
+j
+)(x − ·)
+�2
+L2
+h(Ωh)
+= ∥(Θh ∗ ˜g(a)
+j
+)(x − ·)∥2
+˙H−s
+h
+(Ωh)
+≤ ∥(Θh ∗ ˜g(a)
+j
+)(x − ·)∥2
+˙H−s
+h
+(hZd).
+(4.11)
+We estimate the right-hand side of (4.11) by arguing in Fourier space (similarly as in the proof of
+Theorem 1.3). Namely, using Lemma 3.1 and the fact that the Fourier transform of a convolution is
+the product of the Fourier transforms, we compute
+Eh
+��(Ihϕh, gj(2a(· − x)))L2(Rd)
+��2
+≤
+�
+(− π
+h , π
+h)
+d Mh(ξ)−2s|Fh[(Θh ∗ ˜g(a)
+j
+)(x − ·)](ξ)|2 dξ
+
+SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS
+17
+=
+�
+(− π
+h , π
+h)
+d Mh(ξ)−2s
+������
+�
+ζ∈ 2π
+h Zd
+F[(Θh ∗ ˜g(a)
+j
+)(x − ·)](ξ + ζ)
+������
+2
+dξ
+=
+�
+(− π
+h , π
+h)
+d Mh(ξ)−2s
+������
+�
+ζ∈ 2π
+h Zd
+F[(Θh(x − ·)](ξ + ζ)F[˜g(a)
+j
+(x − ·)](ξ + ζ)
+������
+2
+dξ.
+Next, we fix some t with d
+2 < t < k − s and use the Cauchy-Schwarz inequality (as in the proof of
+Theorem 1.3) to rewrite this as
+Eh
+��(Ihϕh, gj(2a(· − x)))L2(Rd)
+��2
+≤
+�
+(− π
+h , π
+h)
+d Mh(ξ)−2s
+�
+� �
+ζ∈ 2π
+h Zd
+� 1
+h + |ξ + ζ|
+�2t
+|F[Θh(x − ·)](ξ + ζ)|2|F[˜g(a)
+j
+(x − ·)](ξ + ζ)|2
+�
+�
+×
+�
+� �
+ζ∈ 2π
+h Zd
+1
+� 1
+h + |ξ + ζ|
+�2t
+�
+� dξ.
+(4.12)
+Observing that
+sup
+ξ∈(− π
+h , π
+h)
+d
+�
+ζ∈ 2π
+h Zd
+1
+� 1
+h + |ξ + ζ|
+�2t ≤ C
+�
+ζ∈ 2π
+h Zd
+h2t
+(1 + h|ζ|)2t ≤ Ch2t
+as well as the fact that Mh(ξ) is 2π
+h -periodic, we can rewrite (4.12) as
+Eh
+��(Ihϕh, gj(2a(· − x)))L2(Rd)
+��2
+≤ Ch2t
+�
+(− π
+h , π
+h)
+d Mh(ξ + ζ)−2s
+�
+ζ∈ 2π
+h Zd
+� 1
+h + |ξ + ζ|
+�2t
+|F[Θh(x − ·)](ξ + ζ)|2|F[˜g(a)
+j
+(x − ·)](ξ + ζ)|2 dξ
+= Ch2t
+�
+Rd Mh(ξ)−2s
+� 1
+h + |ξ|
+�2t
+|F[Θh(x − ·)](ξ)|2|F[˜g(a)
+j
+(x − ·)](ξ)|2 dξ.
+(4.13)
+After all these manipulations, we have rewritten the term to be estimated as an integral involving the
+absolute values of the Fourier transforms of Θh, ˜g(a)
+j
+. To complete the proof, we use our assumptions
+on Θh, ˜g(a)
+j
+to bound these Fourier transforms.
+Regarding Θh, we know that F[Θh](ξ) = F[Θ](hξ); so assumption (1.3) implies that |F[Θh](ξ)| ≤
+C
+(�d
+j=1 sin2(hξj))k/2
+hk|ξ|k
+and
+(4.14)
+|F[Θh(x·)](ξ)| ≤ C
+(�d
+j=1 sin2(hξj))k/2
+hk|ξ|k
+.
+Regarding ˜g(a)
+j
+, we first note that F[˜g(a)
+j
+](ξ) = F[gj(−2a·)](ξ) =
+1
+2ad F[gj]
+�
+− ξ
+2a
+�
+. As gj ∈ Cr
+c (Rd), we
+know that F[gj] is smooth and decays at least like
+1
+|ξ|r as ξ → ∞. On the other hand, the moments of
+gj up to order r − 1 vanish by (4.4) and so ∇mF[gj](0) = 0 for any m ≤ r − 1. By Taylor’s theorem,
+this implies |F[gj](ξ)| ≤ C|ξ|r. Altogether, we conclude that |F[gj](ξ)| ≤ C
+|ξ|r
+(1+|ξ|)2r and thus also
+|F[˜g(a)
+j
+](ξ)| ≤ C 1
+2ad
+��� ξ
+2a
+���
+r
+�
+1 +
+��� ξ
+2a
+���
+�2r =
+C|ξ|r
+2a(d+r)(1 + 2−a|ξ|)2r
+
+18
+N. DE NITTI AND F. SCHWEIGER
+and
+(4.15)
+|F[˜g(a)
+j
+(x − ·)](ξ)| ≤
+C|ξ|r
+2a(d+r)(1 + 2−a|ξ|)2r .
+Returning to (4.13), we obtain that
+Eh
+��(Ihϕh, gj(2a(· − x)))L2(Rd)
+��2
+≤ Ch2t
+�
+Rd
+h2s
+�d
+j=1 sin2(hξj))s
+(1 + h|ξ|)2t
+h2t
+�����
+(�d
+j=1 sin2(hξj))k/2
+hk|ξ|k
+�����
+2 ����
+|ξ|r
+2a(d+r)(1 + 2−a|ξ|)2r
+����
+2
+dξ
+≤ C h2(s−k)
+22a(d+r)
+�
+Rd
+(1 + h|ξ|)2t(�d
+j=1 sin2(hξj))k−s
+|ξ|2(k−r)(1 + 2−a|ξ|)4r
+dξ.
+(4.16)
+In particular, the integrand decays like
+1
+|ξ|2(k+r−t) and so our assumptions t < k − s and r >
+��s − d
+2
+�� ≥
+d
+2 − s ensure its integrability.
+We distinguish the two cases whether h < 2−a or h ≥ 2−a. In the former case, we can bound the
+integral on the right-hand side of (4.16) as
+�
+Rd
+(1 + h|ξ|)2t(�d
+j=1 sin2(hξj))k−s
+|ξ|2(k−r)(1 + 2−a|ξ|)4r
+dξ
+≤ C
+�
+|ξ|≤2a
+1 · (h|ξ|)2(k−s)
+|ξ|2(k−r) · 1
+dξ + C
+�
+2a<|ξ|≤1/h
+1 · (h|ξ|)2(k−s)
+|ξ|2(k−r) · (2−a|ξ|)4r + C
+�
+|ξ|>1/h
+(h|ξ|)2t · 1
+|ξ|2(k−r)(2−a|ξ|)4r dξ
+≤ C2adh2(k−s)22a(r−s) + C 1
+hd 24arh2(k−s)h2(r+s) + C 1
+hd 24arh2th2(k+r−t)
+≤ C2a(d+2r−2s)h2(k−s) �
+1 + 2a(2r+2s−d)h2r+2s−d + 2a(2r+2s−d)h2r+2s−d�
+≤ C2a(d+2r−2s)h2(k−s),
+where in the last step we used that 2ah < 1 and 2r + 2s − d > 0. In case h ≥ 2−a, we can similarly
+estimate the integral by
+�
+Rd
+(1 + h|ξ|)2t(�d
+j=1 sin2(hξj))k−s
+|ξ|2(k−r)(1 + 2−a|ξ|)4r
+dξ
+≤ C
+�
+|ξ|≤1/h
+1 · (h|ξ|)2(k−s)
+|ξ|2(k−r) · 1
+dξ + C
+�
+1/h<|ξ|≤2a
+(h|ξ|)2t · 1
+|ξ|2(k−r) · 1 + C
+�
+|ξ|>2a
+(h|ξ|)2t · 1
+|ξ|2(k−r)(2−a|ξ|)4r dξ
+≤ C 1
+hd h2(k−s)h2(s−r) + C2adh2t22a(−k+r+t) + C2ad24arh2t22a(−k−r+t)
+≤ C2a(d+2r−2s)h2(k−s) �
+2a(−d−2r+2s)h−d−2r+2s + 22a(−k+s+t)h2(−k+s+t + 22a(−k+s+t)h2(−k+s+t)�
+≤ C2a(d+2r−2s)h2(k−s).
+Thus, in any case, the integral on the right-hand side of (4.16) is bounded by C2a(d+2r−2s)h2(k−s).
+Using this, (4.16) implies (4.10).
+Step 3: Proof of (4.9). The proof of (4.9) is similar. One difference is that we no longer need to
+prove decay of the term in question, only boundedness. On the other hand, the function f does not
+satisfy a moment bound like (4.4) and so we have less control over the behavior of F[f] near 0. To
+deal with the latter problem, we will use the Poincaré inequality on a suitable bounded domain ˜˜Ωh
+right in the beginning of the argument to replace the term
+1
+Mh(ξ)2s with
+1
+(1+Mh(ξ)2)s and thereby make
+sure there is no singularity at 0.
+
+SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS
+19
+In more detail, let us fix some bounded domain ˜Ω ⋑ ˆΩ such that supp(f(x − ·)) ⊂ ˜Ω whenever
+x ∈ ˆΩ. Then (4.9) vanishes for x ̸∈ ˜Ω, and we can restrict attention to x ∈ ˜Ω. Let us fix yet another
+bounded domain ˜˜Ω such that supp(f(x − ·)) ⊂ ˜˜Ω when x ∈ ˜Ω, and let ˜˜Ωh = ˜˜Ω ∩ hZd.
+We also abbreviate ˜f(y) = f(−y). Arguing as for (4.10) we find that, for x ∈ ˜Ω and h ≤ 1,
+Eh
+��(Ihϕh, f(2a(· − x)))L2(Rd)
+��2 ≤ ∥(Θh ∗ ˜f (a))(x − ·)∥2
+˙H−s
+h
+(Ωh) ≤ ∥(Θh ∗ ˜f (a))(x − ·)∥2
+˙H−s
+h
+(˜˜Ωh).
+Since supp(f(x−·))∗Ωh ⊂ ˜˜Ω, we use Poincaré’s inequality on ˜˜Ωh to deduce that the ∥·∥ ˙H−s
+h
+(˜˜Ωh)-norm
+is bounded by a multiple of the ∥ · ∥H−s
+h
+(˜˜Ωh)-norm. Using this, we can now continue with a calculation
+similar as the one for (4.10) to obtain
+Eh
+��(Ihϕh, f(2a(· − x)))L2(Rd)
+��2
+≤ ∥(Θh ∗ ˜f)(x − ·)∥2
+H−s
+h
+(˜˜Ωh)
+≤ ∥(Θh ∗ ˜f)(x − ·)∥2
+H−s
+h
+(hZd)
+=
+�
+(− π
+h , π
+h)
+d(1 + Mh(ξ)2)−s
+������
+�
+ζ∈ 2π
+h Zd
+F[(Θh(x − ·)](ξ + ζ)F[ ˜f(x − ·)](ξ + ζ)
+������
+2
+dξ
+≤ Ch2t
+�
+Rd(1 + Mh(ξ)2)−s
+� 1
+h + |ξ|
+�2t
+|F[(Θh(x − ·)](ξ)|2|F[ ˜f(x − ·)](ξ)|2 dξ.
+Using the bound (4.14) for F[Θh] as well as the estimate
+|F[ ˜f(x − ·)](ξ)| ≤
+C
+(1 + |ξ|)r
+(the analogue of (4.15)), we obtain that
+Eh
+��(Ihϕh, f(2a(· − x)))L2(Rd)
+��2 ≤ Ch2(s−k)
+�
+Rd
+(1 + h|ξ|)2t(�d
+j=1 sin2(hξj))k
+(h2 + (�d
+j=1 sin2(hξj))2)s|ξ|2k(1 + |ξ|)2r dξ
+and (splitting the integral into three integrals over |ξ| ≤ 1, 1 < |ξ| ≤ 1/h, and 1/h < |ξ|) we see as
+before that the right-hand side is indeed bounded by a constant.
+□
+Finally, let us give the argument for convergence of the maximum of the subcritical discrete FGF.
+Some technicalities arise because Theorem 1.2 applies to Ihϕh while we are interested in ϕh itself. So
+we need to argue that the regularity of Ihϕh implies that ϕh is necessarily close to Ihϕh.
+Proof of Corollary 1.4. Step 1: Consequences of Theorem 1.2. Let k = d > s + d
+2 and take Θ to be a
+product of one-dimensional B-splines of order k, i.e.
+F[Θ](ξ) =
+d
+�
+j=1
+�sin(ξ)
+ξ
+�k
+.
+This Θ is a compactly supported non-negative mollifier that satisfies the assumptions of Theorem 1.2.
+Moreover, let us fix some α with 0 < α < min
+�
+s − d
+2, 1
+�
+. Theorem 1.2 implies that Ihϕh converges to
+ϕ in law with respect to the topology of C0,α(Rd). This directly implies that the maximum of Ihϕh
+converges in distribution to the maximum of ϕ. Therefore, it suffices to prove that the maximum of
+ϕh is close enough to the maximum of Ihϕh in the sense that
+(4.17)
+max
+x∈Ωh ϕh(x) − max
+y∈Rd Ihϕh(y) → 0
+in probability as h → 0.
+
+20
+N. DE NITTI AND F. SCHWEIGER
+Step 2: Regularity of ϕh. In order to prove (4.17), we need to quantify the regularity of ϕh. The
+idea here is that if ϕh oscillates a lot, then also Ihϕh oscillates a lot and hence have large C0,α-norm,
+which is unlikely. In making this rigorous, we use our choice of Θ, which simplifies some calculations.
+The function Θh has support precisely
+�
+− hk
+2 , hk
+2
+�d and is piecewise a polynomial of degree at most
+k − 1 in each variable.
+Let us take x ∈ hZd and consider an arbitrary fh : hZd → R. Then Ihfh ↾x+(0,h/2)2 is a polynomial of
+degree at most k−1 in each variable, which depends precisely on the values of fh in x+
+�
+− hk
+2 , − h(k+1)
+2
+�
+∩
+hZd.
+The space of polynomials of degree at most k − 1 in each variable is an R-vector space of dimension
+exactly kd. The same holds true for the space of functions from x +
+�
+− hk
+2 , − h(k+1)
+2
+�
+∩ hZd to R.
+This means that Ihfh induces a linear map between two finite-dimensional vector spaces of the same
+dimension. By standard properties of B-splines, this map is surjective and hence, in fact, bijective.
+As all norms on a finite-dimensional R-vector space are equivalent, we conclude that
+max
+y∈x+(− hk
+2 ,− h(k+1)
+2
+)∩hZd fh(y) ≤ C
+max
+z∈x+(0,h/2)d Ihfh(z)
+and (quotienting out constant functions) also
+max
+y,y′∈x+(− hk
+2 ,− h(k+1)
+2
+)∩hZd |fh(y) − fh(y′)|
+≤ C
+max
+z,z′∈x+(0,h/2)d Ihfh(z) − Ihfh(z′) ≤ Chα∥Ihfh∥C0,α(x+(0,h/2)d).
+The constant in the latter estimate is independent of x. This means that we actually obtain
+(4.18)
+max
+y,y′∈hZd
+|y−y′|y∞≤kh
+|fh(y) − fh(y′)| ≤ Chα∥Ihfh∥C0,α(Rd).
+We know that Ihϕh converges in C0,α(Rd) and so it is, in particular, tight in that space. This means
+that, if we define
+EM =
+�
+[Ihϕh]C0,α(Rd)
+�
+,
+then limM→∞ limh→0 P(EM) = 1. On the other hand, (4.18) implies that, on the event EM, we have
+(4.19)
+max
+y,y′∈hZd
+|y−y′|y∞≤kh
+|ϕh(y) − ϕh(y′)| ≤ CMhα.
+This is the desired regularity estimate for ϕh.
+Step 3: Completion of the proof. Our specific choice of Θ has the property that �
+x∈hZd hdΘh(y −
+x) = 1 for any y ∈ Rd. This means that Ihϕh(y) is a convex combination of the ϕh(x) with |x− y|∞ <
+hk
+2 , and so we have
+max
+x∈Ωh ϕh(x) ≥ max
+y∈Rd Ihϕh(y),
+which implies the lower bound in (4.17). For the upper bound we need to use (4.19). As Ihϕh(y) is
+a convex combination of the ϕh(x) with |x − y|∞ < hk
+2 , (4.19) implies that on the event EM we have
+|ϕh(x) − Ihϕ(x)| ≤ CMhα for any x ∈ hZd. Therefore, on the event EM we have
+max
+x∈Ωh ϕh(x) ≤ max
+x∈Ωh Ihϕh(x) + CMhα ≤ max
+y∈Rd Ihϕh(y) + CMhα.
+Putting these considerations together, we conclude that
+lim
+M→∞ lim
+h→0 P
+�
+max
+y∈Rd Ihϕh(y) ≤ max
+x∈Ωh ϕh(x) ≤ max
+y∈Rd Ihϕh(y) + CMhα
+�
+,
+which yields (4.17).
+□
+
+SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS
+21
+Appendix A. Technical lemmas
+In this appendix, we provide the proof of several technical results that have been used throughout
+the paper.
+A.1. Discretization and restriction. We start by proving that the applications of restricting to
+hZd and applying (−∆h)s commute.
+Lemma A.1 (Discretization and restriction). Let u: Rd → R be a Schwartz function. Then, restricting
+to hZd and applying (−∆h)s commute: i.e.,
+((−∆h)su)↾hZd= (−∆h)s (u↾hZd) .
+This allows us to be rather careless about when we restrict functions to hZd. In fact, we will omit
+writing ↾hZd when (because of Lemma A.1) there is no ambiguity.
+Proof. The crucial fact here is that Mh(ξ) is 2π
+h -periodic. Using this, we compute that, for x ∈ hZd,
+((−∆h)su) (x) =
+�
+Rd Mh(ξ)2sF[u](ξ) dξ
+=
+�
+ζ∈ 2π
+h Zd
+�
+(− π
+h , π
+h)
+d Mh(ξ + ζ)2sF[u](ξ + ζ) dξ
+=
+�
+(− π
+h , π
+h)
+d Mh(ξ)2s
+�
+ζ∈ 2π
+h Zd
+F[u](ξ + ζ) dξ.
+Using Lemma 3.1, we can rewrite this as
+((−∆h)su) (x) =
+�
+(− π
+h , π
+h)
+d Mh(ξ)2sFh[u](ξ) dξ
+= (−∆h)s (u↾hZd) ,
+which is what we wanted to show.
+□
+A.2. Discrete inequalities. Let us state the discrete Poincaré inequality that we used in the proof.
+Lemma A.2. Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary and let s > 0. Then, there
+exists a constant C such that, for any uh : hZd → R that vanishes outside of Ωh, we have
+∥uh∥L2(Ωh) ≤ C∥uh∥ ˙Hs
+h(Ω).
+Proof. We shall present a discrete version of the proof in [8, Theorem 3.7]. The key idea is to use
+Plancherel’s theorem and split low and high frequencies as follows:
+∥uh∥2
+L2(Ω) =
+�
+Bϵ(0)
+|Fhuh(ξ)|2 dξ +
+�
+(−π/h,π/h)d\Bϵ(0)
+|Fhuh(ξ)|2 dξ
+=: I1 + I2,
+where ϵ > 0 is to be fixed later on.
+Step 1. Low-frequencies. For the low-frequency part, I1, Hölder’s inequality yields
+|Fhuh(ξ)| ≤ ∥uh∥L1(Ωh) ≤ |Ωh|1/2hd/2∥uh∥L2(Ωh),
+where we used the notation
+∥uh∥Lp
+h(Ωh) :=
+� �
+x∈Ωh
+hd|uh|p
+� 1
+p
+,
+p ∈ [1, +∞).
+Therefore, we have
+I1 ≤ ϵdB1(0)|Ωh|hd∥uh∥2
+L2
+h(Ωh).
+
+22
+N. DE NITTI AND F. SCHWEIGER
+Step 2. High-frequencies. For the high-frequency part, I2, we compute
+�
+(−π/h,π/h)d\Bϵ(0)
+|Fhuh(ξ)|2 dξ =
+�
+(−π/h,π/h)d\Bϵ(0)
+Mh(ξ)2s|Fhuh(ξ)|2
+Mh(ξ)2s
+dξ
+≤ ϵ−2s∥(−∆h)s/2uh∥2
+L2
+h(Rd).
+Step 3. Conclusion. Choosing 0 < ϵ < (|Ωh|hd|B1(0)|)−1/d, we conclude
+∥uh∥L2(Ωh) ≤
+ϵ−s
+�
+1 − ϵd|Ωh|hd|B1(0)|
+∥uh∥Hs
+h(Ω).
+By considering a square of side L ≥ diam(Ω) (containing Ωh), we deduce that |Ωh| ≤ C
+hd (note that is
+holds even if h ≫ diam(Ω)). This means that Ωh|hd|B1(0)| ≤ C, and so we can make a choice of ε > 0
+independent of h. This concludes the proof.
+□
+Remark A.3 (Generalized Poincaré inequality). Let s ≥ t ≥ 0. Arguing as in [8, Theorem 1.5], Lemma
+A.2 also implies that, for u ∈ ˜Hs(Ω), there exists a constant c = c(d, Ω, s) > 0 such that
+∥(−∆h)t/2uh∥L2(Ωh) ≤ c∥(−∆h)s/2uh∥L2(Ωh)).
+Indeed,
+∥(−∆h)t/2uh∥L2(Ω) = ∥uh∥ ˙Ht
+h(Ω) ≤ ∥uh∥Ht
+h(Ω) ≤ ∥uh∥Hs
+h(Ω) ≤ 2
+s+1
+2 (∥uh∥L2(Ωh) + ∥uh∥ ˙Hs
+h(Ω))
+≤ 2
+s+1
+2 (c∥(−∆h)s/2uh∥L2(Ωh) + ∥(−∆h)s/2uh∥L2(Ωh))
+= c∥(−∆h)s/2uh∥L2(Ωh).
+We also used the fact that solutions of the Dirichlet problem for (−∆)s have a little bit of additional
+regularity.
+Lemma A.4. Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary and let s ≥ 0. Then, there
+exists κ0 > 0 with the following property. If 0 ≤ κ ≤ κ0, then, for each f ∈ H−s+κ(Ω), there exists
+a unique u ∈ ˙Hs+κ(Ω) such that (−∆)su = f in the sense of distributions; moreover, we have the
+estimate
+∥u∥ ˙Hs+κ(Ω) ≤ Cκ∥f∥ ˙H−s+κ(Ω)
+for a constant Cκ depending only on κ.
+Let us remark that according to [3, Theorem 2.3] one can take any κ0 < 1
+2 here. The argument,
+however, is rather complicated; so we prefer to present an easy perturbative argument that gives
+existence of some κ0 > 0 (which is enough for our purposes).
+Proof. We adapt the argument used in [19, Theorem 3.3] for the biharmonic operator to the fractional
+case.
+We first show that the claimed estimate holds for κ = 0. To do so, we test the equation with u and
+deduce
+∥(−∆)s/2u∥2
+L2(Ω) = (u, (−∆)su)L2(Ω) = (u, f)L2(Ω) ≤ ∥u∥ ˙Hs(Ω)∥f∥ ˙H−s(Ω).
+Using Poincaré’s inequality, we see that indeed
+∥u∥ ˙Hs(Ω) ≤ Cκ∥f∥ ˙H−s(Ω)
+To show that we also can take some κ > 0, we use a stability result for analytic families
+of operators on Banach spaces:
+The spaces
+˙Hs(Ω) form an interpolation family with respect to
+complex interpolation; thus, by [21, Proposition 4.1], the set of those α for which the operator
+(−∆)s :
+˙Hα(Ω) →
+˙Hα−2s(Ω) has a bounded inverse is open. We have seen that this set contains
+s, so the existence of κ0 as in the statement of the theorem follows.
+□
+
+SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS
+23
+Appendix B. Fractional Gaussian Fields via eigenfunctions
+In this appendix, we shall present an alternate description of the continuous FGF. As remarked in
+Section 2.1, (−∆)s is an isometry from ˙Hs(Ω) to ˙H−s(Ω). Its inverse, restricted to L2(Ω), is a positive-
+definitive compact operator on L2(Ω); so, by the spectral theorem, there exists an orthonormal basis
+(v1, v2, . . .) of L2(Ω) consisting of eigenfunctions of (−∆)s with associated eigenvalues 0 < λ1 ≤ λ2 ≤
+. . .. Let Xj be a collection of independent standard Gaussians, and let ˜ϕ be the random variable
+˜ϕ = �∞
+j=1
+Xj
+√
+λj vj.
+According to Lemma B.1 below, this sum converges almost surely in ˙Hs′(Ω) ⊂ ˙Hs′(Rd) for any
+s′ < s − d
+2. Therefore, ˜ϕ is a well-defined random variable on ˙Hs′(Ω) ⊂ ˙Hs′(Rd). Every element of
+˙Hs′(Rd) induces an element of S′(Rd) and so we can think of ˜ϕ as a random element of S′(Rd). Again,
+according to Lemma B.1, for any f ∈ S(Rd) we have that ( ˜ϕ, f) is a centered Gaussian with variance
+∥f∥ ˙H−s(Ω). This means that ˜ϕ has the law P on S′(Rd) and so we can identify ϕ and ˜ϕ.
+Let us present the aforementioned lemma.
+Lemma B.1. Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary. Let s ≥ 0 and s′ < s − d
+2 be
+arbitrary.
+(i) The series
+˜ϕ :=
+∞
+�
+j=1
+Xj
+�
+λj
+vj
+converges almost surely in ˙Hs′(Ω).
+(ii) For any f ∈ S(Rd), we have
+E( ˜ϕ, f)2 = ∥f∥2
+˙H−s(Ω).
+For the proof, we need a sharp estimate on the eigenfunction expansion of a function.
+Lemma B.2. Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary. Let s ≥ 0 and s′ ≤ s be
+arbitrary. Then, for any f ∈ ˙Hs′(Ω), we have
+(B.1)
+∥f∥2
+˙Hs′(Ω) ≤ C
+∞
+�
+j=1
+λs′/s
+j
+(f, vj)2
+L2(Rd).
+Note that we make no claim about the ˙Hs′-regularity for s′ > s.
+Proof. For s′ ∈ {−s, 0, s} the estimate (B.1) follows directly from the definition. We next claim that
+(B.1) holds whenever −s ≤ s′ ≤ s. To see this, we adapt the argument in [17, Corollary 1]. Namely
+take first 0 < s′ < s. Consider (−∆)s restricted to functions in
+˙Hs(Ω) and let ((−∆)s)s′/s
+N
+be its
+(spectral) s′
+s -th power. Explicitly,
+((−∆)s)s′/s
+N
+f =
+∞
+�
+j=1
+λs′/s
+j
+(f, vj)L2(Rd)vj
+Note that, if we define ˙Hs′(Ω) to be the space of functions in L2(Ω) such that this quantity is finite,
+then the domain of ((−∆)s)s′/s
+N
+is exactly ˙Hs′(Ω). According to the theory of interpolation of fractional
+powers of self-adjoint operators (see, e.g., [23, Section 1.18.10]), the Hilbert spaces
+˙Hs′(Ω) form an
+interpolation scale. However, we know that the same holds true for the Hilbert spaces ˙Hs′(Ω), and
+moreover ˙Hs′(Ω) = ˙Hs′(Ω) (with equivalent norms) for s′ ∈ {0, s}, and so we have actually have this
+equality for any s′ with 0 ≤ s′ ≤ s. So, for 0 ≤ s′ ≤ s, there is some C > 0 such that
+1
+C
+∞
+�
+j=1
+λs′/s
+j
+(f, vj)2
+L2(Rd) ≤ ∥f∥2
+˙Hs′(Ω) ≤ C
+∞
+�
+j=1
+λs′/s
+j
+(f, vj)2
+L2(Rd)
+
+24
+N. DE NITTI AND F. SCHWEIGER
+By duality, the same holds true for −s ≤ s′ ≤ 0. Putting these considerations together, we obtain a
+statement even stronger than (B.1).
+It remains to study the case that s′ < −s. We proceed inductively. Let us suppose that we know
+that (B.1) holds for s′ ≥ −(2k − 1)s, for some k ∈ N, and let us consider some s′ with −(2k + 1)s ≤
+s′ ≤ −(2k − 1)s. We have that
+∥f∥2
+˙Hs′(Ω) =
+inf
+g∈ ˙Hs′(Rd)
+f=g in Ω
+∥g∥2
+˙H−s(Rd).
+Let u be such that
+�
+(−∆)su(x) = f(x),
+x ∈ Ω,
+u(x) = 0,
+x ∈ Rd \ Ω.
+We can choose g = (−∆)su and obtain, using the induction hypothesis, that
+∥f∥2
+˙Hs′(Ω) ≤ ∥(−∆)su∥2
+˙Hs′(Rd)
+≤ ∥u∥2
+˙Hs′+2s(Rd)
+≤ C
+∞
+�
+j=1
+λ(s′+2s)/s
+j
+(u, vj)2
+L2(Rd)
+= C
+∞
+�
+j=1
+λ(s′+2s)/s
+j
+�
+u, (−∆)svj
+λj
+�2
+L2(Rd)
+= C
+∞
+�
+j=1
+λs′/s
+j
+((−∆)su, vj)2
+L2(Rd)
+= C
+∞
+�
+j=1
+λs′/s
+j
+(f, vj)2
+L2(Rd) .
+This completes the induction step.
+□
+Proof of Lemma B.1. Claim (i). By the Hilbert-space-valued version of Kolmogorov’s two series the-
+orem (see e.g. [11, Corollary on p. 386]), the series
+∞
+�
+j=1
+Xj
+�
+λj
+vj
+converges almost surely in ˙Hs′(Ω) if
+∞
+�
+j=1
+�����
+1
+�
+λj
+uj
+�����
+2
+˙Hs′(Ω)
+< ∞.
+From Lemma B.2, we know in particular that
+∥vj∥2
+˙Hs′(Ω) ≤ λs′/s
+j
+.
+Moreover, by Weyl’s law for the operator (−∆)s restricted to ˙Hs(Ω) (as follows, e.g., from the main
+result of [10]), we have that
+λj ≍ j2s/d.
+Therefore,
+∞
+�
+j=1
+�����
+1
+�
+λj
+vj
+�����
+2
+˙Hs′(Ω)
+≤
+∞
+�
+j=1
+λs′/s−1
+j
+
+SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS
+25
+≤
+∞
+�
+j=1
+(cj)2s/d·(s′/s−1) ≤ C
+∞
+�
+j=1
+j2(s′−s)/d,
+and this sum is indeed convergent if s′ − s < − d
+2.
+Claim (ii). Let f ∈ S(Rd). The functions vj are by definition orthonormal in L2(Ω), and the Xj
+are independent. So we can calculate that
+E( ˜ϕ, f)2 = E
+�
+�
+∞
+�
+j=1
+Xj
+�
+λj
+(vj, f)
+�
+�
+2
+=
+∞
+�
+j=1
+1
+λj
+(vj, f)2 =
+�
+�f,
+∞
+�
+j=1
+1
+λj
+(vj, f)vj
+�
+�
+=
+�
+f, (−∆)−sf
+�
+= ∥f∥2
+H−s(Ω).
+□
+Acknowledgments
+N. De Nitti is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le
+loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). He has been
+supported by the Alexander von Humboldt Foundation and by the TRR-154 project of the Deutsche
+Forschungsgemeinschaft (DFG, German Research Foundation).
+F. Schweiger is supported by the Foreign Postdoctoral Fellowship Program of the Israel Academy
+of Sciences and Humanities, and partially by ISF grant No. 421/20.
+We thank O. Zeitouni and E. Zuazua for helpful comments on the topic of this work.
+References
+[1] N. Abatangelo. Higher-order fractional Laplacians:
+An overview. Bruno Pini Mathematical Analysis Seminar,
+12(1):53–80, 2021.
+[2] F. Baudoin and L. Chen. Dirichlet fractional Gaussian fields on the Sierpinski gasket and their discrete graph
+approximations. ArXiv:2201.03970, 2022.
+[3] J. P. Borthagaray, W. Li, and R. H. Nochetto. Fractional elliptic problems on Lipschitz domains: Regularity and
+approximation. ArXiv:2212.14070, 2022.
+[4] M. Bramson, J. Ding, and O. Zeitouni. Convergence in law of the maximum of the two-dimensional discrete Gaussian
+free field. Comm. Pure Appl. Math., 69(1):62–123, 2016.
+[5] A. Chiarini, A. Cipriani, and R. S. Hazra. Extremes of some Gaussian random interfaces. J. Stat. Phys., 165(3):521–
+544, 2016.
+[6] O. Ciaurri, L. Roncal, P. R. Stinga, J. L. Torrea, and J. L. Varona. Nonlocal discrete diffusion equations and the
+fractional discrete Laplacian, regularity and applications. Adv. Math., 330:688–738, 2018.
+[7] A. Cipriani, B. Dan, and R. S. Hazra. The scaling limit of the membrane model. Ann. Probab., 47(6):3963–4001,
+2019.
+[8] G. Covi, K. Mönkkönen, and J. Railo. Unique continuation property and Poincaré inequality for higher order
+fractional Laplacians with applications in inverse problems. Inverse Probl. Imaging, 15(4):641–681, 2021.
+[9] M. Furlan and J.-C. Mourrat. A tightness criterion for random fields, with application to the Ising model. Electron.
+J. Probab., 22:Paper No. 97, 29, 2017.
+[10] L. Geisinger. A short proof of Weyl’s law for fractional differential operators. J. Math. Phys., 55(1):011504, 7, 2014.
+[11] I. I. Gikhman and A. V. Skorokhod. The theory of stochastic processes. I. Classics in Mathematics. Springer-Verlag,
+Berlin, 2004. Translated from the Russian by S. Kotz, Reprint of the 1974 edition.
+[12] Z. Hao, Z. Zhang, and R. Du. Fractional centered difference scheme for high-dimensional integral fractional Lapla-
+cian. J. Comput. Phys., 424:Paper No. 109851, 17, 2021.
+[13] Y. Huang and A. Oberman. Finite difference methods for fractional Laplacians. ArXiv:1611.00164, 2016.
+[14] B. S. Jovanović and E. Süli. Analysis of finite difference schemes, volume 46 of Springer Series in Computational
+Mathematics. Springer, London, 2014. For linear partial differential equations with generalized solutions.
+
+26
+N. DE NITTI AND F. SCHWEIGER
+[15] A. Lodhia, S. Sheffield, X. Sun, and S. S. Watson. Fractional Gaussian fields: a survey. Probab. Surv., 13:1–56,
+2016.
+[16] S. Müller and F. Schweiger. Estimates for the Green’s function of the discrete bilaplacian in dimensions 2 and 3.
+Vietnam J. Math., 47(1):133–181, 2019.
+[17] R. Musina and A. I. Nazarov. On fractional Laplacians. Comm. Partial Differential Equations, 39(9):1780–1790,
+2014.
+[18] M. A. Pinsky. Introduction to Fourier analysis and wavelets, volume 102 of Graduate Studies in Mathematics.
+American Mathematical Society, Providence, RI, 2009. Reprint of the 2002 original.
+[19] F. Schweiger. The maximum of the four-dimensional membrane model. Ann. Probab., 48(2):714–741, 2020.
+[20] S. Sheffield. Gaussian free fields for mathematicians. Probab. Theory Related Fields, 139(3-4):521–541, 2007.
+[21] A. Tabacco Vignati and M. Vignati. Spectral theory and complex interpolation. J. Funct. Anal., 80(2):383–397,
+1988.
+[22] V. Thomée. Elliptic difference operators and Dirichlet’s problem. Contributions to Differential Equations, 3:301–324,
+1964.
+[23] H. Triebel. Interpolation theory, function spaces, differential operators, volume 18 of North-Holland Mathematical
+Library. North-Holland Publishing Co., Amsterdam-New York, 1978.
+[24] H. Triebel. Function spaces in Lipschitz domains and on Lipschitz manifolds. Characteristic functions as pointwise
+multipliers. Rev. Mat. Complut., 15(2):475–524, 2002.
+(N. De Nitti) Friedrich-Alexander-Universität Erlangen-Nürnberg, Department of Data Science, Chair
+for Dynamics, Control and Numerics (Alexander von Humboldt Professorship), Cauerstr.
+11, 91058
+Erlangen, Germany.
+Email address: nicola.de.nitti@fau.de
+(F. Schweiger) Weizmann Institute of Science, Department of Mathematics, Rehovot 7610001, Israel.
+Email address: florian.schweiger@weizmann.ac.il
+
diff --git a/1dFST4oBgHgl3EQfWjig/content/tmp_files/load_file.txt b/1dFST4oBgHgl3EQfWjig/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..a9e841f9ed8ecd0b36b350c0ce28a26fd082af8e
--- /dev/null
+++ b/1dFST4oBgHgl3EQfWjig/content/tmp_files/load_file.txt
@@ -0,0 +1,1062 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf,len=1061
+page_content='SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS  NICOLA DE NITTI AND FLORIAN SCHWEIGER  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' This work is concerned with fractional Gaussian fields, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Gaussian fields whose covari-  ance operator is given by the inverse fractional Laplacian (−∆)−s (where, in particular, we include  the case s > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' We define a lattice discretization of these fields and show that their scaling limits –  with respect to the optimal Besov space topology – are the original continuous fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' As a byproduct,  in dimension d < 2s, we prove the convergence in distribution of the maximum of the fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' A key  tool in the proof is a sharp error estimate for the natural finite difference scheme for (−∆)s under  minimal regularity assumptions, which is also of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Contents  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Introduction  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Fractional Gaussian Fields  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Finite difference schemes for fractional operators  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Main results  4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Future work  6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Fractional polyharmonic Gaussian fields  7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  The (continuous) fractional Gaussian field  7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  The (discrete) fractional Gaussian field  9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Rigorous estimates for the finite difference scheme  10  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Proofs of the scaling limits  13  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Scaling limit in the space of distributions  13  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Scaling limit in Besov, Sobolev and Hölder spaces  15  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Technical lemmas  21  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Discretization and restriction  21  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Discrete inequalities  21  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Fractional Gaussian Fields via eigenfunctions  23  Acknowledgments  25  References  25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Introduction  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Fractional Gaussian Fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Fractional Gaussian fields form a natural one-parameter family  of Gaussian interface models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  For a fixed parameter s ≥ 0, the s-fractional Gaussian field is the  Gaussian field whose covariance operator is (−∆)−s, the inverse of the fractional Laplacian of order  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' We emphasize right away that we do not assume s ∈ [0, 1], and in fact our main interest is in the  polyharmonic case s > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' A purely formal and non-rigorous way to define the s-fractional Gaussian  field on some domain Ω ⊂ Rd is to set  P(dϕ) = 1  Z exp  �  −1  2  �  Ω  ϕ(x)((−∆)sϕ)(x) dx  �  dϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1)  This cannot be taken as a rigorous definition, as dϕ refers to the Lebesgue measure on the infinite-  dimensional space RΩ, which does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='13781v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='PR]  31 Jan 2023  2  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' DE NITTI AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' SCHWEIGER  (a) s = 0 (white noise)  (b) s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='5  (c) s = 1 (Gaussian free field)  (d) s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='5  (e) s = 2 (membrane model)  (f) s = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='5  (g) s = 3  (h) s = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='5  (i) s = 4  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Surface plots of discrete fractional polyharmonic Gaussian fields on Ω :=  (0, 1)2 ∩  1  100Z2 with zero boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' These discrete random functions are  linearly interpolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' See also [15, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1] for further numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  There are also other issues with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1): namely, one needs to decide how to define (−∆)s for functions  ϕ: Ω → R and (closely related to that issue) one needs to decide on boundary values of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' For these  questions, we have a clear answer, though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' We take 0 boundary values (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=', we take ϕ to be extended  by 0 to the whole Rd), and we let (−∆)s be the fractional Laplacian on the full space Rd, which is  defined by using the Fourier transform1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' That is, for any ξ ∈ Rd,  F [(−∆)su] (ξ) = |ξ|2sF[u](ξ)  with  F[u](ξ) =  �  Rd e−iξ·xu(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  These choices are natural from a probabilistic point of view, as we will explain in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2, and  they can be implemented to provide a rigorous meaning to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1), for example, as a probability measure  on the space of tempered distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' This is discussed in detail in the excellent survey [15] and in  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1 we recall the points which are important for us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  1An equivalent hypersingular integral formulation of the polyharmonic fractional operator of order s ∈ (0, m), for  any m ∈ N, is given by  (−∆)su(x) := Cd,m,s  �  Rd  �m  j=−m(−1)j  � 2m  m − j  �  u(x + jy)  |y|d+2s  dy,  where  Cd,m,s :=  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  22sΓ(n/2 + s)  πn/2Γ(−s)  �  �  m  �  j=1  (−1)j  �  2m  m − j  �  j2s  �  �  −1  if s ∈ (0, m)\\N,  22sΓ(n/2 + s)s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  2πn/2  �  �  m  �  j=2  (−1)j  �  2m  m − j  �  j2s ln j  �  �  −1  if s ∈ (0, m) ∩ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  We refer to [1] and references therein for further information on the theory of higher-order fractional Laplacians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS  3  The main goal of the present work is to define a discrete version ϕh of the s-fractional Gaussian field  ϕ on a lattice Ωh = Ω ∩ hZd and to study its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' It is not immediately obvious how one should  define this discrete version, but we will argue that our definition is quite natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Certainly, one would  expect that in the limit h → 0 the discrete s-fractional Gaussian field converges in law, with respect  to a suitable topology, to the (continuous) s-fractional Gaussian field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Our main result is that, with  our definition of a s-fractional Gaussian field, this convergence holds in a rather strong sense, namely  in law with respect to the topology of Besov spaces for the optimal range of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Similar problems have been studied before for specific values of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' If s = 1, the field is the Gaussian  free field and the convergence of the discrete Gaussian free field to its continuous variant is folklore  (see [20, Section 4] for related results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The proof relies on the fact that covariances of the discrete  Gaussian free field can be represented using simple random walk, which in the scaling limit becomes  Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The case 0 ≤ s < 1 is addressed in [15, Section 12]2, and the proof of the scaling  limit follows a similar strategy as for the case s = 1, just with the 2s-stable Lévy process taking the  place of Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  These results for s ≤ 1 all rely on some form of a random walk representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' For s > 1 and for our  choice of boundary values, there is no such random walk representation and so proofs become much  more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3 The only existing result in this regime is for s = 2, where the s-fractional Gaussian field  is the so-called membrane model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' In [7], it was proven that this field is the scaling limit of its discrete  version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The main ingredient in the proof were estimates for finite difference schemes for (−∆)2 from  [22], and estimates for its Green’s function from [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Thus, previous work was restricted to s ∈ [0, 1] ∪ {2}, while our results cover the entire range  s ∈ [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Even in the case s ∈ [0, 1] ∪ {2}, our results improve upon the previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Namely, the  convergence in [15, Section 12] is with respect to the topology of distributions and the convergence in  [7] is with respect to the topology of some negative Sobolev space (for non-optimal parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' As  an easy corollary of our result with respect to the Besov-space topology, one obtains convergence with  respect to the Sobolev-topology and also with respect to the Hölder topology (both with the optimal  range of parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Our method of proof uses estimates for finite difference schemes like [7], but of a different flavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' In-  stead of the estimate from [22] used in [7] (that needs Ck-regularity of the function to be approximated  by the scheme), we establish an estimate that needs only minimal regularity assumptions (essentially  just Hs+ε-regularity for some ε > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' This is the main technical result of the paper and we will discuss  it and its context next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Finite difference schemes for fractional operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' There is a close relation between discrete  versions of Gaussian fields and finite difference approximations of the corresponding operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Indeed,  if we want to define a lattice version of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1) that is suitably close to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1) itself, then we need a lattice  approximation of (−∆)s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' the better this approximation, the closer the resulting lattice field will be to  its continuous version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Before discussing finite difference schemes, let us mention that there has been work on finite element  approaches to the fractional Laplacian (at least for s ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' We cannot cover the whole body of relevant  literature here, but we refer to the very recent survey [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Let us now turn to finite difference schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The subject of finite difference schemes for the fractional  Laplacian (−∆)s has been studied before and various schemes have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' However, the main  focus has been on the case s < 1 and often also d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' We refer to the survey [13] and the references  therein for an overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  The main challenge when constructing a finite difference scheme for the  2There, a definition of the discrete FGF that is slightly different from ours is used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' the proof, however, should apply  to all reasonable discretizations including ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  3If one uses another definition of the fractional Gaussian field in terms of spectral powers of the ordinary Laplacian  (the so-called eigenfunction FGF from [15, Section 9]), one retains a random walk representations and it is comparably  easy to establish a scaling limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' In fact, in [2], this is done not in the lattice case, but in the more complicated case of  a Sierpinski gasket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  4  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' DE NITTI AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' SCHWEIGER  fractional Laplacian is that it is given by convolution with a singular integral kernel, and a naive  discretization of this kernel might not capture its behavior near the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Our preferred way to construct a finite difference scheme arises naturally when working in Fourier  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let us consider first the usual Laplacian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' the case s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Its symbol is |ξ|2 and its standard  finite difference approximation (the 2d + 1-point stencil in dimension d) has symbol  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2)  Mh(ξ)2 :=  d  �  j=1  4  h2 sin2  �ξjh  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  So, for the fractional Laplacian (−∆)s with symbol |ξ|2s, a natural way to define a finite difference  scheme is to take the finite difference operator with symbol Mh(ξ)2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' That is, we define  Fh [(−∆h)suh] (ξ) = Mh(ξ)2sFh[uh](ξ)  with  Fh[uh](ξ) = hd �  x∈hZd  e−iξ·xuh(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  We can also use Mh(ξ)2s as a continuous Fourier multiplier and thereby understand (−∆h)s also as a  continuous operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' This is consistent with the previous definitions, as pointed out in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  This scheme for s ≤ 1 (but general d) has already been studied in [12] (and the special case d = 1  already in [13, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2] and, in more detail, in [6]) and has many desirable properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' First of all,  for s ∈ N, we recover the standard schemes for polyharmonic Laplacians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' We also have the property  that (−∆h)s(−∆h)s′ = (−∆h)s+s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Moreover, while for other schemes the accuracy often degenerates  as s ↗ 1, our scheme has accuracy h2 uniformly in s (as follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' In Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2  below, we comment on how this scheme might work in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Now that we have chosen our scheme, let us discuss rigorous estimates for its approximation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  In the literature on finite difference schemes, it is common to derive pointwise estimates on the error  under a strong regularity assumption (Ck or Ck,α for a large enough k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' In fact, for d ∈ {1, 2} and  s ≤ 1, there are two such results in the literature: in [6], pointwise estimates for the approximation  error for functions in Hölder spaces (at least C0,2s+ε) are shown and, in [12], such pointwise estimates  are shown under the assumption that the 2s + ε-th derivative has integrable Fourier transform (which,  roughly speaking, again corresponds to C0,2s+ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  However, as already mentioned in the previous subsection, our interest is more in estimates under  low-regularity assumptions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' in a Sobolev scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' To the best of our knowledge, such estimates are  new (even in the case d = 1, s < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' For the case of the Laplacian or Bilaplacian, though, such results  are classical (see the textbook [14] or a recent refinement for the Bilaplacian in [19]), and our result is  inspired by the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' However, the proof is quite different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Namely, the proof of [19, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3]  relied on the Bramble-Hilbert lemma and thereby used that s ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' In our general setting, we use a  different approach, based on the Poisson summation formula and a lengthy estimate of various error  terms in Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let us now state our main results more precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' We consider the discrete FGF  ϕh and the continuous FGF ϕ, as introduced informally in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' As the rigorous definitions are  quite technical, we postpone them to Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  We claim that the scaling limit of ϕh in an appropriate sense is ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' However, ϕh is defined only  on hZd, so we need to interpolate it to a function on Rd first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' For that purpose, we fix a compactly  supported function Θ ∈ S(Rd) with  �  Rd Θ(x) dx = 1 and define Θh(x) =  1  hd Θ  � x  h  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Using Θh, we can define the interpolated field  Ihϕh(x) :=  �  y∈hZd  hdϕh(y)Θh(x − y)  as a random element of S′(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Some of the results below also hold if Θ is just a tempered distribution (and, in fact, in [7] only the  choice Θ = δ0 was used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' However, if we hope to find a scaling limit in some Banach space of optimal  SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS  5  regularity, we need to consider more regular Θ (as otherwise Ihϕh might not even be an element of  the Banach space in question);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' so, to avoid unneccessarily complicated notations, we directly assume  that Θ is a measurable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  As a first result, we claim that, for any Θ chosen as above, the interpolated fields Ihϕh converge in  the sense of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Note that our definition of ϕh is made in such a way that we do not need  to rescale it with some power of h to obtain a scaling limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Indeed, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1 (Scaling limit in the space of distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let Ω ⊂ Rd be a bounded domain with  Lipschitz boundary, and let s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Let Θ be a compactly supported function with integral 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Then Ihϕh converges in law with respect to  the topology of S′(Rd) to ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' That is, for any f ∈ S(Rd), the random variable (Ihϕh, f)L2  h(hZd)converges  in law to (ϕ, f)L2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  In Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1, we established a scaling limit in the space of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' However, as we discuss  in detail in Section 2, the continuous FGF is defined not just as a distribution-valued random variable,  but actually has a certain Besov-, Sobolev- and Hölder-regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Hence, it is natural that we can  take the scaling limit of the ϕh also in these spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' In order to do so, however, we need some further  assumptions on the regularization Θ (otherwise the interpolated field Ihϕh might not even be an  element of the space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The result now is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2 (Scaling limits in Sobolev and Hölder Spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let Ω ⊂ Rd be a bounded domain with  Lipschitz boundary and let s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let Θ be a compactly supported function with integral 1, and suppose  that there is some k with k > s + d  2 such that  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3)  |FΘ(ξ)| ≤ C  (�d  j=1 sin2(ξj))k/2  |ξ|k  for all ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Let s′ < s − d  2, p, q ∈ [1, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Then the interpolated fields Ihϕh converge in law with respect to the  topology of ˆBs′  p,q(Rd) to ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Moreover, for any fixed bounded domain ˆΩ with Lipschitz boundary such that Ω ⋐ ˆΩ, Ihϕh are  supported in ˆΩ for h sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' For any s′ < s − d  2, the interpolated fields Ihϕh converge in  law with respect to the topology of ˙Hs′(ˆΩ) to ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' In addition, if H := s − d  2 > 0, m = ⌈H⌉ − 1, and  0 < α < H − m, then Ihϕh converges in law with respect to the topology of Cm,α(Rd) to ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Here, ˆBs′  p,q(Rd) is, up to a minor technicality that we again discuss in Section 2, equal to the standard  Besov space Bs′  p,q(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Several remarks are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' First of all, a convenient example of a function satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3) for  some k ∈ N is given by the centered B-spline of order k (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=', [14, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='4] for a summary of  their properties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Next, the restriction to s′ < s − d  2 cannot be avoided as the continuous FGF is not in ˆBs−d/2  p,q  for  any p, q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Thus, the range of allowed s′ is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Regarding the convergence in Sobolev spaces, we cannot expect convergence with respect to the  topology of ˙Hs′(Ω) for the simple reason that, because of the mollification, Ihϕh need not have zero  boundary values outside of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  A fundamental step in the proof of the results above is establishing the following error estimate for  a fractional Poisson equation (which is of interest in itself).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Our goal is to compare the solutions of  (−∆)su = f and of (−∆h)suh = f and we will estimate the error u − uh in the (discrete) energy norm  ∥ · ∥ ˙Hs  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' As we work under minimal regularity assumptions on u, the precise result is somewhat more  technical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Namely, in general u and f might not be continuous functions and so it is not clear how to  restrict them to the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' We circumvent this by introducing two additional mollifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  The result then takes the following shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3 (Error estimate on the discrete approximation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let Ω ⊂ Rd be an open bounded set  with Lipschitz boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let Θ: Rd → R and θ: Rd → R be mollifiers that are compactly supported,  6  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' DE NITTI AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' SCHWEIGER  symmetric around 0, and have integral 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Furthermore, let us assume that there exist k, l ≥ 0 such  that  |FΘ(ξ)| ≤ C  (�d  j=1 sin2(ξj))k/2  |ξ|k  ,  |Fθ(ξ)| ≤ C 1  |ξ|l ,  and define Θh(x) =  1  hd Θ  � x  h  �  , θh(x) =  1  hd θ  � x  h  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Let 0 < s < t and let u ∈ Ht(Rd) be the solution of  �  (−∆)su(x) = f(x),  x ∈ Ω,  u(x) = 0,  x ∈ Rd \\ Ω,  for some f ∈ Ht−2s(Rd), and uh : hZd → R be the solution of  �  (−∆h)suh(x) = Θh ∗ f(x),  x ∈ hZd ∩ Ω,  uh(x) = 0,  x ∈ hZd \\ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Then, if h ≤ 1, k ≥ s, k > d  2 + 2s − t, l > d  2 − t, and t − s ≤ 2, we have the estimate  ∥θh ∗ u − uh∥ ˙Hs  h(hZd) ≤ Cht−s∥u∥ ˙Ht(Rd),  where C > 0 does not depend on h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Here (as in the rest of the paper) C denotes some generic constant that might change from line to  line, but is always independent of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Let us give some explanations regarding the linear constraints on the parameters in this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The  most important constraint is t − s ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' It arises from the fact that the proposed finite difference  scheme is of second order (see [13, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2]), so that the accurary of our scheme saturates at h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  The condition k > d  2 + 2s − t is needed in order for Θh ∗ f to be continuous (so that it has a well-  defined restriction to the lattice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Similarly, the condition l > d  2 − t is needed in order for θh ∗ u to be  continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  As mentioned in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2, this is the first rigorous estimate for a finite difference scheme for  (−∆)s under low regularity assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' For finite elements, a result that is similar in spirit can  be found in [3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' There an estimate for piecewise linear finite elements is shown that is  similar to our result (albeit with the additional restriction t − s ≤ 1  2 instead of t − s ≤ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The method  of proof is very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The most well-studied discrete Gaussian interface model is certainly the discrete  Gaussian free field (corresponding to s = 1 in our notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' In recent years there has been a lot  of activity to extend results known for the discrete Gaussian free field to other discrete (Gaussian or  non-Gaussian) interface models, and this work is a first step to include the discrete FGFs in the latter  class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Let us highlight one such question, namely regarding the maximum of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' In case of the  discrete Gaussian free field, this is well-understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' In the subcritical dimension (d = 1), the field  is nothing but a random walk bridge, so it is easy to see that the rescaled maximum converges to a  non-degenerate random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' In supercritical dimensions (d ≥ 3), correlations decay so rapidly  that the maximum behaves as if the field values were independent [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The most interesting case is the  critical case, d = 2, where the field is log-correlated and obtains the typical second-order correction  [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  These results have already been extended to the case of the membrane model (corresponding to  s = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The subcritical case (d ≤ 3) was studied in [7] using results from [16], the supercritical case  in [5], and finally the critical case in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' An important tool in the latter proof was an estimate for  finite difference schemes very similar to the one in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS  7  For general s, it is very likely that similar results hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' In fact, in the subcritical case d < 2s,  convergence of the rescaled maximum is a straightforward corollary of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='4 (Convergence of the maximum for d < 2s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let d < 2s, Ω ⊂ Rd be a fixed bounded  domain with Lipschitz boundary, and consider the family ϕh of s-FGF on Ωh = Ω∩hZd as h → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Then  the random variables maxx∈Ωh ϕh(x) converge in distribution to a non-degenerate random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  While this corollary covers the subcritical case, the critical case (d = 2s) and the supercritical case  (d > 2s) remain open, and we hope to address them in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  In particular, a study of the  critical case would be very interesting, as most existing examples of discrete log-correlated fields in the  literature are in d = 2 or some other even dimension while the 3  2-discrete FGF, for instance, would be  a natural example of a log-correlated field in odd dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Fractional polyharmonic Gaussian fields  In this section, we give precise definitions for the continuous and discrete FGF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' For the continuous  FGF, we follow [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The major difference is that we only require Lipschitz continuity of the boundary  of our domain Ω (and hence our results cover in particular the important special case Ω = (0, 1)d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Because of this, several functional-analytic statements require extra attention and we give precise  references for the results we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The (continuous) fractional Gaussian field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' We first fix our conventions for the Fourier  transform, and then use it to define some relevant function spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  For a function u: Rd → R, we let F[u](ξ): Rd → R, defined by  F[u](ξ) =  �  Rd e−iξ·xu(x) dx,  be its continuous Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Then, we have the Fourier inversion formula,  u(x) = F−1[F[u]](x) =  1  (2π)d  �  Rd eiξ·xF[u](ξ) dξ,  and Plancherel’s theorem,  �  Rd |u(x)|2 dx =  1  (2π)d  �  Rd |F[u](ξ)|2 dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  We can also define the Sobolev norms  ∥u∥2  ˙Hs(Rd) =  �  Rd |ξ|2s|F[u](ξ)|2 dξ,  ∥u∥2  Hs(Rd) =  �  Rd(1 + |ξ|2)s|F[u](ξ)|2 dξ,  where |ξ|2 is the Fourier multiplier of the Laplacian −∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary and let S′(Rd) be the space of tempered  distributions on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' In what follows, we collect some results on fractional Sobolev spaces on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' If Ω  has a smooth boundary, all of them are well-known (and [23, Chapter 4] is a comprehensive reference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  If Ω has merely Lipschitz boundary, the situation is slightly more complicated and we rely on the  reference [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' A brief version of some of these results is also contained in [15, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1], but some  of them are not made explicit there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  For s ≥ 0, let ˙˜Hs(Ω) be the closure of C∞  c (Ω) with respect to the norm ∥· ∥ ˙Hs(Rd)  4 and let ˙H−s(Ω)  be its dual space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The space ˙H−s(Ω) can alternatively be described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let  ∥u∥ ˙H−s(Ω) =  inf  v∈ ˙H−s(Rd)  u=v in Ω  ∥v∥ ˙H−s(Rd)  4In [15] this space is denoted Hs  0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' However, more commonly ˙Hs  0(Ω) is defined as the closure of C∞  c (Ω) with respect  to the norm ∥ · ∥ ˙Hs(Ω), while our space ˙˜Hs(Ω) is equal to the Lions-Magenes space (which is also denoted by ˙Hs  00(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  The two spaces are different whenever s ∈ N + 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Our notation is based on the one in [23, Chapter 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  8  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' DE NITTI AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' SCHWEIGER  for u ∈ S(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Then if S(Ω) is the quotient of S(Rd) under the equivalence relation that identifies  functions when they agree in Ω, we have that ˙H−s(Ω) is the closure of S(Ω) with respect to the norm  ∥ · ∥ ˙H−s(Ω) (this follows from [24, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='5 (i)] upon observing that our spaces ˙˜Hs(Ω), for s ≥ 0,  and ˙H−s(Ω), for s < 0, are equal to Triebel’s ¯Bs  2,2(Ω) by [24, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  From the Lax-Milgram lemma (and the fact that C∞  c (Ω) is dense in ˙˜Hs(Ω)) we also obtain that  (−∆)s is an isometry from ˙˜Hs(Ω) to ˙H−s(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  For convenience (and with a slight abuse of notation) we define  ˙Hs(Ω) =  � ˙˜Hs(Ω)  if s ≥ 0,  ˙Hs(Ω)  if s < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  This scale of Hilbert spaces has various desirable properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' For any s < t, the embedding from ˙Ht(Ω)  to ˙Hs(Ω) is compact (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' [24, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Even more importantly, the spaces form an interpolation  scale with respect to complex (or equivalently real) interpolation (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' [24, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='5 (iv)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  In [15, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2], the continuous FGF is defined as a probability measure P on S′(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' More  precisely, it is defined such that when ϕ is distributed according to P, then for every Schwartz function  f ∈ S(Rd) we have that (ϕ, f) is a centered Gaussian with variance ∥f∥2  ˙H−s(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' By [15, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3  and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='4], this property defines P as a probability measure on S′(Rd) uniquely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Let us remark that one can one alternatively define the FGF as a random sum of eigenfunctions of  (−∆)s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' We give details on this in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  The regularity of ϕ is best measured in Besov spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' For s′ ∈ R, p, q ∈ [1, ∞], we let ∥ · ∥Bs′  p,q(Rd)  be the usual Besov norm (defined, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=', via Littlewood-Paley decomposition or via wavelets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' see,  for example, [23, Chapter 2]) and let ˆBs′  p,q(Rd) be the closure of C∞  c (Rd) with respect to the norm  ∥ · ∥Bs′  p,q(Rd)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Then we have the following regularity results for ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1 (Regularity of the FGF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary  and let s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' For any s′ < s − d  2 and p, q ∈ [1, ∞], the FGF on Ω is P-almost surely an element of  ˆBs′  p,q(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  In particular, for any s′ < s − d  2 the FGF on Ω is P-almost surely an element of ˙Hs′(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Moreover, if H := s − d  2 > 0, then the FGF on Ω is also P-almost surely an element of Cm,α  loc (Rd)  for m = ⌈H⌉ − 1 and any 0 < α < H − m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The Besov regularity could be shown using the tightness criterion in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1 below applied  to the constant sequence ϕ(m) = ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' However, according to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2 we have the much stronger  statement that the FGF is the limit (with respect to the ˆBs′  p,q(Rd)-topology) of the discrete fractional  Gaussian fields, suitably interpolated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' so we do not give details for the proof of the Besov regularity  here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  It is well-known that ˆBs′  2,2(Rd) = Hs′(Rd) and ˆBs′  ∞,∞(Rd) �→ Cs′(Rd), where Cs′(Rd) is the Hölder-  Zygmund space, which embeds into the classical Hölder space C⌊s′′⌋,s′′−⌊s′′⌋ for any 0 < s′′ < s′ (see  [24, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' These results together with the fact that the FGF is supported in Ω easily imply  the Sobolev and Hölder regularity results in the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Let us remark that the Sobolev regularity alternatively follows from the fact the random series defin-  ing ˜ϕ converges in ˙Hs′(Ω) almost surely, while the Hölder regularity also follows from [15, Proposition  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2 and Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3]6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  □  5Note that the Besov space Bs′  p,q(Rd) commonly defined as the set of all tempered distributions for which ∥·∥Bs′  p,q(Rd)  is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Clearly ˆBs′  p,q(Rd) ⊂ Bs′  p,q(Rd), and the inclusion is strict if p = ∞ or q = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  6N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' There is a typo in the statement of [15, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2]: it should read H − k instead of H − ⌈H⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS  9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The (discrete) fractional Gaussian field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Our definition of the discrete FGF follows the one  of the continuous FGF as closely as possibly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let us again begin by fixing our conventions for discrete  Fourier transforms and discrete function spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  For a function uh : hZd → R, we let Fh[uh](ξ): Rd → R, defined by  Fh[uh](ξ) = hd �  x∈hZd  e−iξ·xuh(x),  be its discrete Fourier transform (note that this function is 2π  h -periodic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Then we have the discrete  Fourier inversion formula,  uh(x) = F−1  h [Fh[uh]](x) =  1  (2π)d  �  (− π  h , π  h)  d eiξ·xFh[uh](ξ) dξ,  and Plancherel’s theorem,  hd �  x∈hZd  |uh(x)|2 =  1  (2π)d  �  (− π  h , π  h)  d |Fh[uh](ξ)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  We can define the discrete Sobolev norms  ∥uh∥2  ˙Hs  h(hZd) =  �  (− π  h , π  h)  d Mh(ξ)2s|Fh[uh](ξ)|2 dξ,  ∥uh∥2  Hs  h(hZd) =  �  (− π  h , π  h)  d(1 + Mh(ξ)2)s|Fh[uh](ξ)|2 dξ,  where Mh(ξ)2 := �d  j=1  4  h2 sin2 �  ξjh  2  �  is the discrete Fourier multiplier of the discrete Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Let Ω be as before and let Ωh = Ω ∩ hZd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Similarly as in the continuous setting, we define the  space ˙˜Hs(Ωh) as the space of functions hZd → R that vanish outside of Ωh (equipped with the norm  induced by ∥ · ∥ ˙Hs  h(hZd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' We let ˙H−s(Ωh) be its dual space, and define  ˙Hs(Ω) =  � ˙˜Hs  h(Ωh)  if s ≥ 0,  ˙Hs  h(Ωh)  if s < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  We define the discrete FGF as a probability measure on ˙˜Hs(Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' More precisely, we consider the  measure  Ph( dϕh) = 1  Zh  exp  �  −1  2∥ϕh∥2  ˙Hs  h(hZd)  � �  x∈Ωh  dϕh(x)  �  x∈hZd\\Ωh  δ0( dϕh(x))  = 1  Zh  exp  �  −1  2  �  x∈Ωh  hdϕh(x)(−∆h)sϕh(x)  � �  x∈Ωh  dϕh(x)  �  x∈hZd\\Ωh  δ0( dϕh(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  This is a well-defined Gaussian measure with mean 0, and variance  Eh(ϕh, fh)2  L2  h(hZ)d = ∥fh∥2  ˙H−s  h  (Ωh)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1)  for any fh : hZd → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2 (Boundary values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let us comment on our choice of boundary values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The main advantage  of our definition is the fact that it is consistent with projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Namely, let Ω ⊂ ˜Ω be open sets  and consider the discrete FGF ˜ϕh on ˜Ωh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Then, the restriction of ˜ϕh to Ωh is equal in distribution to  the sum of the (−∆h)s-harmonic extension of ˜ϕh from hZd \\ Ωh to Ωh and of an independent discrete  FGF ϕh on Ωh 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' In particular, even if we had started with a field without boundary values (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=', with  7If s = 1, this reduces to the familiar domain Markov property for the discrete Gaussian free field: the field in a  subdomain is equal in distribution to the harmonic extension of its boundary values plus an independent zero-boundary  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  10  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' DE NITTI AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' SCHWEIGER  ˜Ω = Rd), then looking at the field on a subset naturally leads to consider fields with zero boundary  values outside that subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Rigorous estimates for the finite difference scheme  In this section, we present the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' As mentioned in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2, the proof of  the analogous statement for s = 2 in [19, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3] was based on the Bramble-Hilbert lemma to  estimate various error terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Thus it relied on the fact that (−∆h)2 (and hence the finite difference  scheme) is local in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  In the generic case s ̸∈ N, however, (−∆h)s is not local, and so this proof strategy can no longer  be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Instead, we use the fact that both (−∆)s and (−∆h)s are defined via Fourier multipliers  and directly estimate all relevant error terms in Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' However, this requires extra care as we  need to switch from discrete Fourier space to continuous Fourier space at some point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' In fact, we need  a way to compare Fh and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Fortunately, the following Poisson-type summation formula enables us  to do so easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1 (Poisson-type summation formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Suppose that g: Rd → R is a Schwartz function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Then we have the identity  Fh[g](ξ) =  �  ζ∈ 2π  h Zd  F[g](ξ + ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' By the Poisson summation formula (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=', [18, Chapter 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='4]), for any Schwartz function f  we have  hd �  x∈hZd  f(x) =  �  ζ∈ 2π  h Zd  F[f](ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Applying this to f(x) = e−iξ·xg(x), we find  hd �  x∈hZd  e−iξ·xg(x) =  �  ζ∈ 2π  h Zd  F[e−iξ·g](ζ) =  �  ζ∈ 2π  h Zd  F[g](ξ + ζ),  which implies the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  □  Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1, we can now turn to the proof of our estimate on finite difference schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' By density, we can assume that u is smooth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' then, in particular, u is a Schwartz  function and F[u] is a Schwartz function as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Therefore, all integrals and sums below will be well-  defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Step 1: Representation of the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' From the definitions we have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1)  ∥u − uh∥ ˙Hs  h(hZd) = ∥(−∆h)s(θh ∗ u − uh)∥ ˙H−s  h  (Ωh) =  inf  v : hZd→R  v=(−∆h)s(θh∗u−uh) in Ωh  ∥v∥ ˙H−s  h  (hZd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' we can also rewrite,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' for x ∈ Ωh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  (−∆h)s(θh ∗ u − uh)(x)  = (−∆h)s(θh ∗ u)(x) − Θh ∗ f(x)  = (−∆h)s(θh ∗ u)(x) − Θh ∗ (−∆)su(x)  =  1  (2π)d  �  (− π  h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' π  h)  d eiξ·xMh(ξ)2sFh[θh ∗ u](ξ) dξ −  1  (2π)d  �  Rd eiξ·x|ξ|2sF[Θh](ξ)F[u](ξ) dξ  = I1 + I2 + I3 + I4 + I5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  where  I1(x) :=  1  (2π)d  �  (− π  h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' π  h)  d eiξ·xMh(ξ)2s (Fh[θh ∗ u](ξ) − F[θ ∗ u](ξ)) dξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  I2(x) :=  1  (2π)d  �  (− π  h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' π  h)  d eiξ·xMh(ξ)2s (F[θh ∗ u](ξ) − F[u](ξ)) dξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS  11  I3(x) :=  1  (2π)d  �  (− π  h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' π  h)  d eiξ·x (1 − F[Θh](ξ)) Mh(ξ)2sF[u](ξ) dξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  I4(x) :=  1  (2π)d  �  (− π  h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' π  h)  d eiξ·x �  Mh(ξ)2s − |ξ|2s�  F[Θh](ξ)F[u](ξ) dξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  I5(x) := −  1  (2π)d  �  Rd\\(− π  h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' π  h)  d eiξ·x|ξ|2sF[Θh](ξ)F[u](ξ) dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  To conclude the proof, we need to show that  ∥Ij∥ ˙H−s  h  (hZd) ≤ Cht−s∥u∥ ˙Ht(Rd)  holds for each j ∈ {1, 2, 3, 4, 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Step 2: Estimate of I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Directly from the definition, we see that  Fh[I2](ξ) = Mh(ξ)2s (F[θh ∗ u](ξ) − F[u](ξ)) = Mh(ξ)2s (F[θh](ξ) − 1) F[u](ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  As F[θh](ξ) = F[θ](hξ), our assumption on θ implies that F[θh](ξ) ≤  C  hl|ξ|l ≤ C for ξ ∈  �  − π  h, π  h  �d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Therefore  ∥I2∥2  ˙H−s  h  (hZd) =  �  (− π  h , π  h)  d Mh(ξ)−2s|Fh[I2](ξ)|2 dξ  =  �  (− π  h , π  h)  d Mh(ξ)2s |F[θh](ξ) − 1|2 |F[u](ξ)|2 dξ  ≤ C  �  (− π  h , π  h)  d |ξ|2s|F[u](ξ)|2 dξ  ≤ C  �  (− π  h , π  h)  d |ξ|2(t−s)h2(t−s)|F[u](ξ)|2 dξ  ≤ Ch2(t−s)∥u∥2  ˙Ht(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Step 3: Estimate of I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The estimate of I3 is quite similar: again, we have that  Fh[I3](ξ) = (1 − F[Θh](ξ)) Mh(ξ)2sF[u](ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  The assumptions on Θ imply that FΘ(0) = 1 and ∇FΘ(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Furthermore, FΘ is a Schwartz  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' So, by Taylor’s theorem, there is a constant C such that |1 − FΘ(ξ)| ≤ C|ξ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' This implies  |1 − F[Θh](ξ)| ≤ Ch2|ξ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' We can now estimate  ∥I3∥2  ˙H−s  h  (hZd) =  �  (− π  h , π  h)  d Mh(ξ)−2s|Fh[I3](ξ)|2 dξ  =  �  (− π  h , π  h)  d Mh(ξ)2s(1 − F[Θh](ξ))2|F[u](ξ)|2 dξ  ≤ C  �  (− π  h , π  h)  d |ξ|2sh4|ξ|4|F[u](ξ)|2 dξ  ≤ C  �  (− π  h , π  h)  d |ξ|2(t−s)h2(t−s)|F[u](ξ)|2 dξ  ≤ Ch2(t−s)∥u∥2  ˙Ht(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Step 4:  Estimate of I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  The argument for I4 is very similar to that for I3:  we use that  ��Mh(ξ)2s − |ξ|2s�� ≤ Ch2|ξ|2 and proceed as for I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Step 5: Estimate of I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' From the definition and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1, we have that  Fh[I1](ξ) = Mh(ξ)2s (Fh[θh ∗ u](ξ) − F[θh ∗ u](ξ))  12  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' DE NITTI AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' SCHWEIGER  = Mh(ξ)2s  �  ζ∈ 2π  h Zd\\{0}  F[θh ∗ u](ξ + ζ)  = Mh(ξ)2s  �  ζ∈ 2π  h Zd\\{0}  F[θh](ξ + ζ)F[u](ξ + ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  The Cauchy-Schwartz inequality then yields  ∥I1∥2  ˙H−s  h  (hZd) =  �  (− π  h , π  h)  d Mh(ξ)−2s|Fh[I1](ξ)|2 dξ  =  �  (− π  h , π  h)  d Mh(ξ)2s  ������  �  ζ∈ 2π  h Zd\\{0}  F[θh](ξ + ζ)F[u](ξ + ζ)  ������  2  dξ  ≤  �  (− π  h , π  h)  d Mh(ξ)2s  �  �  �  ζ∈ 2π  h Zd\\{0}  |ξ + ζ|2t|F[u](ξ + ζ)|2  �  �  �  �  �  ζ∈ 2π  h Zd\\{0}  F[θh](ξ + ζ)  |ξ + ζ|2t  �  � dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  We know that F[θh](ξ + ζ) ≤  C  hl|ξ+ζ|l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' As 2(t + l) > d, we can bound  sup  ξ∈(− π  h , π  h)  d  �  ζ∈ 2π  h Zd\\{0}  F[θh](ξ + ζ)  |ξ + ζ|2t  ≤  sup  ξ∈(− π  h , π  h)  d  �  ζ∈ 2π  h Zd\\{0}  1  h2l|ξ + ζ|2(t+l)  ≤ C  �  ζ∈ 2π  h Zd\\{0}  1  h2l|ζ|2(t+l)  ≤ Ch2t  and deduce  ∥I1∥2  ˙H−s  h  (hZd) ≤ C 1  h2s h2t  �  (− π  h , π  h)  d  �  ζ∈ 2π  h Zd\\{0}  |ξ + ζ|2t|F[u](ξ + ζ)|2 dξ  ≤ Ch2(t−s)  �  Rd |ξ2t|F[u](ξ)|2 dξ  ≤ Ch2(t−s)∥u∥2  ˙Ht(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Step 6: Estimate of I5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' We see that  F[I5](ξ) = |ξ|2sF[Θh](ξ)F[u](ξ)χRd\\(− π  h , π  h)  d(ξ),  where χA is the indicator function of the set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1 then implies that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' for ξ ∈  �  − π  h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' π  h  �d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Fh[I5](ξ) =  �  ζ∈ 2π  h Zd  F[I5](ξ + ζ)  =  �  ζ∈ 2π  h Zd  |ξ + ζ|2sF[Θh](ξ + ζ)F[u](ξ)χRd\\(− π  h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' π  h)  d(ξ + ζ)  =  �  ζ∈ 2π  h Zd\\{0}  |ξ + ζ|2sF[Θh](ξ + ζ)F[u](ξ + ζ)  and therefore (recalling that Mh(ξ) is 2π  h -periodic)  ∥I5∥2  ˙H−s  h  (hZd)  =  �  (− π  h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' π  h)  d Mh(ξ)−2s|Fh[I5](ξ)|2 dξ  SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS  13  =  �  (− π  h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' π  h)  d Mh(ξ)−2s  ������  �  ζ∈ 2π  h Zd\\{0}  |ξ + ζ|2sF[Θh](ξ + ζ)F[u](ξ + ζ)  ������  2  dξ  =  �  (− π  h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' π  h)  d  ������  �  ζ∈ 2π  h Zd\\{0}  |ξ + ζ|2s  Mh(ξ + ζ)s F[Θh](ξ + ζ)F[u](ξ + ζ)  ������  2  dξ  ≤  �  (− π  h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' π  h)  d  �  �  �  ζ∈ 2π  h Zd\\{0}  |ξ + ζ|2t|F[u](ξ + ζ)|2  �  �  �  �  �  ζ∈ 2π  h Zd\\{0}  |F[Θh](ξ + ζ)|2  Mh(ξ + ζ)2s|ξ + ζ|2(t−2s)  �  � dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Note that F[Θh](ξ) = F[Θ](hξ) and so |F[Θh](ξ)| ≤ C  (�d  j=1 sin2(hξj))k/2  hk|ξ|k  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' moreover,  (�d  j=1 sin2(hξj))1/2  Mh(ξ)  ≤  Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Since k ≥ s, Mh(ξ + ζ)2s is controlled by the sin-terms from |F[Θh](ξ + ζ)|2 and so we can bound  sup  ξ∈(− π  h , π  h)  d  �  ζ∈ 2π  h Zd\\{0}  |F[Θh](ξ + ζ)|2  Mh(ξ + ζ)2s|ξ + ζ|2t−4s ≤ C  sup  ξ∈(− π  h , π  h)  d  �  ζ∈ 2π  h Zd\\{0}  h2s  h2k|ξ + ζ|2(t+k−2s)  ≤ C  �  ζ∈ 2π  h Zd\\{0}  1  h2(k−s)|ζ|2(t+k−2s)  ≤ Ch2(t−s),  where we used the fact that 2(t + k − 2s) > d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Hence,  ∥I5∥2  ˙H−s  h  (hZd) ≤ Ch2(t−s)  �  (− π  h , π  h)  d  �  ζ∈ 2π  h Zd\\{0}  |ξ + ζ|2t|F[u](ξ + ζ)|2 dξ  ≤ Ch2(t−s)  �  Rd |ξ|2t|F[u](ξ)|2 dξ  ≤ Ch2(t−s)∥u∥2  ˙Ht(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2 (Usage of the finite difference scheme).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' So far we have not said much regarding the  practical applications of the finite difference scheme in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' It would go beyond the scope of  this work to report on some practical experiments, but let us make a few comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  In order to use the scheme to approximate a solution of (−∆)su = f, a first challenge is to compute  the entries of ((−∆h)s)x,y∈Ωh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Even using the translation-invariance of (−∆h)s, we need to compute  O  � 1  h2  �  entries, where each is given as a singular integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' This is quite costly, but avoids introduction  of additional error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' In fact, the pictures in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1 were computed using this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  If one is willing to accept an additional error term, then a more efficient way to compute an  approximation to the entries of ((−∆h)s)x,y∈Ωh was suggested in [12]: choose a parameter h′ ≤ h, and  approximate the integral over  �  − π  h, π  h  �d appearing in the definition of (−∆h)s by a Riemann sum on  a lattice of width h′  h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The advantage is that this Riemann sum can be computed very efficiently using  the fast Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Moreover, in [12, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2], it is suggested that this should lead to an  additional error of order O  �  h′d+2s  h2s  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' In other words, if we choose h′ ≤ h(2s+2)/(2s+d), the error should  be of order h2 and thus not bigger than the error in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' While the error estimate in [12,  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2] is not rigorous, it should be possible to give a full proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Proofs of the scaling limits  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Scaling limit in the space of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' With Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3 in hand, we are ready to  prove that ϕ is indeed the scaling limit of the ϕh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' First, we study the scaling limit in the space of  distributions, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  14  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' DE NITTI AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' SCHWEIGER  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Step 1: Characterization of the convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let us consider some f ∈ S(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Both (Ihϕh, f)L2(Rd) and (ϕ, f)L2(Rd) are centered Gaussian random variables and so it suffices to  prove that their variances converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' We have that  Eh(Ihϕh, f)2 = Eh  �  �  �  Rd  �  y∈hZd  hdϕh(y)Θh(x − y)f(x) dx  �  �  2  = Eh  �  � �  y∈hZd  hdϕh(y)  �  Rd Θh(x − y)f(x) dx  �  �  2  = Eh(ϕh, Θh ∗ f)2  L2  h(hZd)  = ∥Θh ∗ f∥2  ˙H−s  h  (Ωh)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1)  and so we only need to prove that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2)  lim  h→∞ ∥Θh ∗ f∥2  ˙H−s  h  (Ωh) = ∥f∥2  ˙H−s(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Step 2: Representation of the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' For each h > 0, let uh : hZd → R be the solution of  �  (−∆h)suh(x) = Θh ∗ f(x),  x ∈ hZd ∩ Ω,  uh(x) = 0,  x ∈ hZd \\ Ω  and let u ∈ Hs(Rd) be the solution of  �  (−∆)su(x) = f(x),  x ∈ Ω,  u(x) = 0,  x ∈ Rd \\ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Moreover, let ˜Θ, ˜θ be functions satisfying the assumptions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3 with k := max  � d  2 + s, s  �  and l := max  � d  2 − s, 0  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' for example, let us take ˜Θ to be a B-spline of order ⌈k⌉ and ˜θ any smooth  mollifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Then, let us define ˜Θh and ˜θh as before and let ˜uh : hZd → R be the solution of  �  (−∆h)s˜uh(x) = ˜Θh ∗ f(x),  x ∈ hZd ∩ Ω,  ˜uh(x) = 0,  x ∈ hZd \\ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Then, we can write  ∥Θh ∗ f∥2  ˙H−s  h  (Ωh) − ∥f∥2  ˙H−s(Ω) = (Θh ∗ f, uh)L2  h(hZd) − (f, u)L2(Rd)  = J1 + J2 + J3 + J4 + J5,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3)  where  J1 := (Θh ∗ f, uh)L2  h(hZd) − (Θh ∗ f, ˜uh)L2  h(hZd),  J2 := (Θh ∗ f, ˜uh)L2  h(hZd) − (Θh ∗ f, ˜θh ∗ u)L2  h(hZd),  J3 := (Θh ∗ f, ˜θh ∗ u)L2  h(hZd) − (f, ˜θh ∗ u)L2  h(hZd),  J4 := (f, ˜θh ∗ u)L2  h(hZd) − (f, ˜θh ∗ u)L2(Rd),  J5 := (f, ˜θh ∗ u)L2(Rd) − (f, u)L2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  We need to show that Ji → 0 as h → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' This implies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The most important term is  J2, for which we shall need to use Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' the other terms will be straightforward to control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Step 3: Estimate of J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let t > s be a constant to be chosen later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Our choices of k, l ensure that  the assumptions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3 are all satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3 and the discrete Poincaré inequality  (see Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2) then imply that  J2 = (Θh ∗ f, ˜uh − ˜θh ∗ u)L2  h(hZd)  SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS  15  ≤ ∥Θh ∗ f∥L2  h(hZd)∥˜uh − ˜θh ∗ u∥L2  h(hZd)  ≤ C∥Θh ∗ f∥L2  h(hZd)∥˜uh − ˜θh ∗ u∥ ˙Hs  h(hZd)  ≤ C∥f∥L∞(Rd)ht−s∥u∥ ˙Ht(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  For t−s small enough (depending on s and Ω), Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='4 implies that we have Ht-regularity-estimates  on Ω and hence in particular ∥u∥ ˙Ht(Rd) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Thus J2 → 0 as h → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Step 3: Estimate of J1, J3, J4, J5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' For J1, using again the discrete Poincaré inequality, we estimate  J1 = (Θh ∗ f, uh − ˜uh)L2  h(hZd)  ≤ ∥Θh ∗ f∥L2  h(hZ)d∥uh − ˜uh∥L2  h(hZd)  ≤ C∥f∥L∞(Rd)∥uh − ˜uh∥ ˙Hs  h(Ωh)  ≤ C∥f∥L∞(Rd)∥Θh ∗ f − ˜Θh ∗ f∥ ˙H−s  h  (Ωh)  ≤ C∥f∥L∞(Rd)∥Θh ∗ f − ˜Θh ∗ f∥L2  h(Ωh)  ≤ Ch∥f∥L∞(Rd)∥∇f∥L∞(Rd);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  where the right-hand side tends to 0 as h → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The same argument also applies to J3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  For J5, it suffices to observe that ˜θh ∗ u tends to u in L2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Finally, for J4, we use the fact that  ˜θh ∗ u is continuous (and thus f · (˜θh ∗ u) is continuous), and so  lim  h→0(f, ˜θh ∗ u)L2  h(hZd) = (f, u)L2(Rd)  as a Riemann sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Scaling limit in Besov, Sobolev and Hölder spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' We now turn to the proof of the scaling  limit in Besov spaces (which then implies the result in Sobolev and Hölder spaces as well).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' As we have  already established convergence of the fields in the space of distributions, the main challenge is to  prove tightness in Besov spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' To this end, we use a very convenient criterion from [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' As in our  case we do not need to worry about boundary issues, we do not the full generality of that criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Let us state the version that we will use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1 (Tightness criterion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let r ∈ N and let ˆΩ ⊂ Rd be an open bounded set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Then there  exist functions f, (gj)2d−1  j=1  ∈ Cr  c (Rd) such that, for any multi-index m ∈ Nd with |m| < r and any  j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' , 2d − 1}, we have  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='4)  �  Rd xmgj(x) dx = 0  and such that the following statement holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let (φn)n∈N be a family of random linear forms on Cr  c (Rd)  with support in ˆΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let t, t′ ∈ R with t < t′, |t|, |t′| < r and let p ∈ [1, ∞), q ∈ [1, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let us suppose  that there exists a constant C such that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='5)  lim sup  n→∞  sup  x∈Rd (E |⟨φn, f(· − x)⟩|p)1/p < ∞  and  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='6)  lim sup  n→∞  sup  k∈N  sup  x∈Rd  max  1≤j≤2d−1 (E |⟨φn, gj(2a(· − x))⟩|p)1/p ≤  C  2a(d+t′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Then the family (φn)n∈N is tight in ˆBt  p,q(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' If t < t′ − d  p, it is also tight in ˆBt  ∞,q(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' This is essentially [9, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' There a local version of the theorem is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The global  version presented here is obtained by choosing U = Rd, ˆΩ ⊂ K1 ⊂ K2 ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' such that already K1 is far  larger than ˆΩ, k1 = k2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' = 0 and observing that, for functions with uniformly compact support,  the local and global Besov spaces agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  16  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' DE NITTI AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' SCHWEIGER  We also used lim supn→∞ instead of supn∈N in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='5) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='6), but this clearly does not make a  difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Finally, the assertion that  �  Rd xmgj(x) dx = 0 is stated in [9, Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  □  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Step 1: Simplifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' It suffices to prove tightness of Ihϕh in the corre-  sponding spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' the convergence then follows easily from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1 by the same argument as in  [7, Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' In order to prove tightness, we will apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' We fix some open  bounded set ˆΩ ⋑ Ω and note that for h small enough Ihϕh is supported in ˆΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let us fix some r ∈ N  with r >  ��s − d  2  �� and let f, (gj) be as in the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' We claim that, for any p′ < ∞,  lim sup  h→0  sup  x∈Rd Eh  ���(Ihϕh, f(· − x))Lp′(Rd)  ���  p′  < ∞,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='7)  lim sup  h→0  sup  a∈N  sup  x∈Rd  max  1≤j≤2d−1 Eh  ���(Ihϕh, gj(2a(· − x)))Lp′(Rd)  ���  p′  ≤  C  2a(d+2s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='8)  Once we have verified this, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1 (with t′ = s − d  2) directly implies tightness in ˆBs′  p,q(Rd) for any  p ∈ [1, ∞), q ∈ [1, ∞], and choosing p′ sufficiently large such that s′ < t′ − d  p′ we cover the case p = ∞  as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Once we know tightness in Besov spaces, the tightness in Sobolev- and Hölder spaces follows  directly from Besov embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Regarding (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='5) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='6), we can make some immediate simplifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' First of all, it suffices to  check the two estimates for p′ ∈ 2N (the result for other p′ then follows from Jensen’s inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  In addition, as ϕh is a Gaussian random variable, all even functions of linear functionals of ϕh are  controlled by its second moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' This means that we only need to consider p′ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' That is, we  actually only need to verify that  lim sup  h→0  sup  x∈Rd Eh  ��(Ihϕh, f(· − x))L2(Rd)  ��2 < ∞,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='9)  lim sup  h→0  sup  k∈N  sup  x∈Rd  max  1≤j≤2d−1 Eh  ��(Ihϕh, gj(2a(· − x)))L2(Rd)  ��2 ≤  C  22a(d+t′) =  C  2a(d+2s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='10)  The estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='10) is the crucial one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' So we give its proof in detail, and then explain how to prove  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='9) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Step 2: Proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let us fix some h ≤ 1, a ∈ N, x ∈ Rd, and abbreviate ˜g(a)  j  (y) := gj(−2ay).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  A computation similar to the one in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1) shows that  Eh  ��(Ihϕh, gj(2a(· − x)))L2(Rd)  ��2 = Eh  ������  �  y∈Rd  �  z∈hZd  hdϕh(z)Θh(y − z)gj(2a(y − x)) dy  ������  2  = Eh  �  � �  z∈hZd  hdϕh(z)  �  y∈Rd Θh(y − z) ˜f (a)  j  (x − y) dy  �  �  2  = Eh  �  ϕh, (Θh ∗ ˜g(a)  j  )(x − ·)  �2  L2  h(Ωh)  = ∥(Θh ∗ ˜g(a)  j  )(x − ·)∥2  ˙H−s  h  (Ωh)  ≤ ∥(Θh ∗ ˜g(a)  j  )(x − ·)∥2  ˙H−s  h  (hZd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='11)  We estimate the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='11) by arguing in Fourier space (similarly as in the proof of  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Namely, using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1 and the fact that the Fourier transform of a convolution is  the product of the Fourier transforms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' we compute  Eh  ��(Ihϕh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' gj(2a(· − x)))L2(Rd)  ��2  ≤  �  (− π  h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' π  h)  d Mh(ξ)−2s|Fh[(Θh ∗ ˜g(a)  j  )(x − ·)](ξ)|2 dξ  SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS  17  =  �  (− π  h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' π  h)  d Mh(ξ)−2s  ������  �  ζ∈ 2π  h Zd  F[(Θh ∗ ˜g(a)  j  )(x − ·)](ξ + ζ)  ������  2  dξ  =  �  (− π  h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' π  h)  d Mh(ξ)−2s  ������  �  ζ∈ 2π  h Zd  F[(Θh(x − ·)](ξ + ζ)F[˜g(a)  j  (x − ·)](ξ + ζ)  ������  2  dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Next, we fix some t with d  2 < t < k − s and use the Cauchy-Schwarz inequality (as in the proof of  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3) to rewrite this as  Eh  ��(Ihϕh, gj(2a(· − x)))L2(Rd)  ��2  ≤  �  (− π  h , π  h)  d Mh(ξ)−2s  �  � �  ζ∈ 2π  h Zd  � 1  h + |ξ + ζ|  �2t  |F[Θh(x − ·)](ξ + ζ)|2|F[˜g(a)  j  (x − ·)](ξ + ζ)|2  �  �  ×  �  � �  ζ∈ 2π  h Zd  1  � 1  h + |ξ + ζ|  �2t  �  � dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='12)  Observing that  sup  ξ∈(− π  h , π  h)  d  �  ζ∈ 2π  h Zd  1  � 1  h + |ξ + ζ|  �2t ≤ C  �  ζ∈ 2π  h Zd  h2t  (1 + h|ζ|)2t ≤ Ch2t  as well as the fact that Mh(ξ) is 2π  h -periodic, we can rewrite (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='12) as  Eh  ��(Ihϕh, gj(2a(· − x)))L2(Rd)  ��2  ≤ Ch2t  �  (− π  h , π  h)  d Mh(ξ + ζ)−2s  �  ζ∈ 2π  h Zd  � 1  h + |ξ + ζ|  �2t  |F[Θh(x − ·)](ξ + ζ)|2|F[˜g(a)  j  (x − ·)](ξ + ζ)|2 dξ  = Ch2t  �  Rd Mh(ξ)−2s  � 1  h + |ξ|  �2t  |F[Θh(x − ·)](ξ)|2|F[˜g(a)  j  (x − ·)](ξ)|2 dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='13)  After all these manipulations, we have rewritten the term to be estimated as an integral involving the  absolute values of the Fourier transforms of Θh, ˜g(a)  j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' To complete the proof, we use our assumptions  on Θh, ˜g(a)  j  to bound these Fourier transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Regarding Θh, we know that F[Θh](ξ) = F[Θ](hξ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' so assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3) implies that |F[Θh](ξ)| ≤  C  (�d  j=1 sin2(hξj))k/2  hk|ξ|k  and  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='14)  |F[Θh(x·)](ξ)| ≤ C  (�d  j=1 sin2(hξj))k/2  hk|ξ|k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Regarding ˜g(a)  j  , we first note that F[˜g(a)  j  ](ξ) = F[gj(−2a·)](ξ) =  1  2ad F[gj]  �  − ξ  2a  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' As gj ∈ Cr  c (Rd), we  know that F[gj] is smooth and decays at least like  1  |ξ|r as ξ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' On the other hand, the moments of  gj up to order r − 1 vanish by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='4) and so ∇mF[gj](0) = 0 for any m ≤ r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' By Taylor’s theorem,  this implies |F[gj](ξ)| ≤ C|ξ|r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Altogether, we conclude that |F[gj](ξ)| ≤ C  |ξ|r  (1+|ξ|)2r and thus also  |F[˜g(a)  j  ](ξ)| ≤ C 1  2ad  ��� ξ  2a  ���  r  �  1 +  ��� ξ  2a  ���  �2r =  C|ξ|r  2a(d+r)(1 + 2−a|ξ|)2r  18  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' DE NITTI AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' SCHWEIGER  and  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='15)  |F[˜g(a)  j  (x − ·)](ξ)| ≤  C|ξ|r  2a(d+r)(1 + 2−a|ξ|)2r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Returning to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='13), we obtain that  Eh  ��(Ihϕh, gj(2a(· − x)))L2(Rd)  ��2  ≤ Ch2t  �  Rd  h2s  �d  j=1 sin2(hξj))s  (1 + h|ξ|)2t  h2t  �����  (�d  j=1 sin2(hξj))k/2  hk|ξ|k  �����  2 ����  |ξ|r  2a(d+r)(1 + 2−a|ξ|)2r  ����  2  dξ  ≤ C h2(s−k)  22a(d+r)  �  Rd  (1 + h|ξ|)2t(�d  j=1 sin2(hξj))k−s  |ξ|2(k−r)(1 + 2−a|ξ|)4r  dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='16)  In particular, the integrand decays like  1  |ξ|2(k+r−t) and so our assumptions t < k − s and r >  ��s − d  2  �� ≥  d  2 − s ensure its integrability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  We distinguish the two cases whether h < 2−a or h ≥ 2−a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' In the former case, we can bound the  integral on the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='16) as  �  Rd  (1 + h|ξ|)2t(�d  j=1 sin2(hξj))k−s  |ξ|2(k−r)(1 + 2−a|ξ|)4r  dξ  ≤ C  �  |ξ|≤2a  1 · (h|ξ|)2(k−s)  |ξ|2(k−r) · 1  dξ + C  �  2a<|ξ|≤1/h  1 · (h|ξ|)2(k−s)  |ξ|2(k−r) · (2−a|ξ|)4r + C  �  |ξ|>1/h  (h|ξ|)2t · 1  |ξ|2(k−r)(2−a|ξ|)4r dξ  ≤ C2adh2(k−s)22a(r−s) + C 1  hd 24arh2(k−s)h2(r+s) + C 1  hd 24arh2th2(k+r−t)  ≤ C2a(d+2r−2s)h2(k−s) �  1 + 2a(2r+2s−d)h2r+2s−d + 2a(2r+2s−d)h2r+2s−d�  ≤ C2a(d+2r−2s)h2(k−s),  where in the last step we used that 2ah < 1 and 2r + 2s − d > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' In case h ≥ 2−a, we can similarly  estimate the integral by  �  Rd  (1 + h|ξ|)2t(�d  j=1 sin2(hξj))k−s  |ξ|2(k−r)(1 + 2−a|ξ|)4r  dξ  ≤ C  �  |ξ|≤1/h  1 · (h|ξ|)2(k−s)  |ξ|2(k−r) · 1  dξ + C  �  1/h<|ξ|≤2a  (h|ξ|)2t · 1  |ξ|2(k−r) · 1 + C  �  |ξ|>2a  (h|ξ|)2t · 1  |ξ|2(k−r)(2−a|ξ|)4r dξ  ≤ C 1  hd h2(k−s)h2(s−r) + C2adh2t22a(−k+r+t) + C2ad24arh2t22a(−k−r+t)  ≤ C2a(d+2r−2s)h2(k−s) �  2a(−d−2r+2s)h−d−2r+2s + 22a(−k+s+t)h2(−k+s+t + 22a(−k+s+t)h2(−k+s+t)�  ≤ C2a(d+2r−2s)h2(k−s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Thus, in any case, the integral on the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='16) is bounded by C2a(d+2r−2s)h2(k−s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Using this, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='16) implies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Step 3: Proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='9) is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' One difference is that we no longer need to  prove decay of the term in question, only boundedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' On the other hand, the function f does not  satisfy a moment bound like (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='4) and so we have less control over the behavior of F[f] near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' To  deal with the latter problem, we will use the Poincaré inequality on a suitable bounded domain ˜˜Ωh  right in the beginning of the argument to replace the term  1  Mh(ξ)2s with  1  (1+Mh(ξ)2)s and thereby make  sure there is no singularity at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS  19  In more detail, let us fix some bounded domain ˜Ω ⋑ ˆΩ such that supp(f(x − ·)) ⊂ ˜Ω whenever  x ∈ ˆΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='9) vanishes for x ̸∈ ˜Ω, and we can restrict attention to x ∈ ˜Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let us fix yet another  bounded domain ˜˜Ω such that supp(f(x − ·)) ⊂ ˜˜Ω when x ∈ ˜Ω, and let ˜˜Ωh = ˜˜Ω ∩ hZd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  We also abbreviate ˜f(y) = f(−y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Arguing as for (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='10) we find that, for x ∈ ˜Ω and h ≤ 1,  Eh  ��(Ihϕh, f(2a(· − x)))L2(Rd)  ��2 ≤ ∥(Θh ∗ ˜f (a))(x − ·)∥2  ˙H−s  h  (Ωh) ≤ ∥(Θh ∗ ˜f (a))(x − ·)∥2  ˙H−s  h  (˜˜Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Since supp(f(x−·))∗Ωh ⊂ ˜˜Ω, we use Poincaré’s inequality on ˜˜Ωh to deduce that the ∥·∥ ˙H−s  h  (˜˜Ωh)-norm  is bounded by a multiple of the ∥ · ∥H−s  h  (˜˜Ωh)-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Using this, we can now continue with a calculation  similar as the one for (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='10) to obtain  Eh  ��(Ihϕh, f(2a(· − x)))L2(Rd)  ��2  ≤ ∥(Θh ∗ ˜f)(x − ·)∥2  H−s  h  (˜˜Ωh)  ≤ ∥(Θh ∗ ˜f)(x − ·)∥2  H−s  h  (hZd)  =  �  (− π  h , π  h)  d(1 + Mh(ξ)2)−s  ������  �  ζ∈ 2π  h Zd  F[(Θh(x − ·)](ξ + ζ)F[ ˜f(x − ·)](ξ + ζ)  ������  2  dξ  ≤ Ch2t  �  Rd(1 + Mh(ξ)2)−s  � 1  h + |ξ|  �2t  |F[(Θh(x − ·)](ξ)|2|F[ ˜f(x − ·)](ξ)|2 dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Using the bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='14) for F[Θh] as well as the estimate  |F[ ˜f(x − ·)](ξ)| ≤  C  (1 + |ξ|)r  (the analogue of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='15)), we obtain that  Eh  ��(Ihϕh, f(2a(· − x)))L2(Rd)  ��2 ≤ Ch2(s−k)  �  Rd  (1 + h|ξ|)2t(�d  j=1 sin2(hξj))k  (h2 + (�d  j=1 sin2(hξj))2)s|ξ|2k(1 + |ξ|)2r dξ  and (splitting the integral into three integrals over |ξ| ≤ 1, 1 < |ξ| ≤ 1/h, and 1/h < |ξ|) we see as  before that the right-hand side is indeed bounded by a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  □  Finally, let us give the argument for convergence of the maximum of the subcritical discrete FGF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Some technicalities arise because Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2 applies to Ihϕh while we are interested in ϕh itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' So  we need to argue that the regularity of Ihϕh implies that ϕh is necessarily close to Ihϕh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Step 1: Consequences of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let k = d > s + d  2 and take Θ to be a  product of one-dimensional B-splines of order k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  F[Θ](ξ) =  d  �  j=1  �sin(ξ)  ξ  �k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  This Θ is a compactly supported non-negative mollifier that satisfies the assumptions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Moreover, let us fix some α with 0 < α < min  �  s − d  2, 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2 implies that Ihϕh converges to  ϕ in law with respect to the topology of C0,α(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' This directly implies that the maximum of Ihϕh  converges in distribution to the maximum of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Therefore, it suffices to prove that the maximum of  ϕh is close enough to the maximum of Ihϕh in the sense that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='17)  max  x∈Ωh ϕh(x) − max  y∈Rd Ihϕh(y) → 0  in probability as h → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  20  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' DE NITTI AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' SCHWEIGER  Step 2: Regularity of ϕh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' In order to prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='17), we need to quantify the regularity of ϕh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The  idea here is that if ϕh oscillates a lot, then also Ihϕh oscillates a lot and hence have large C0,α-norm,  which is unlikely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' In making this rigorous, we use our choice of Θ, which simplifies some calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  The function Θh has support precisely  �  − hk  2 , hk  2  �d and is piecewise a polynomial of degree at most  k − 1 in each variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Let us take x ∈ hZd and consider an arbitrary fh : hZd → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Then Ihfh ↾x+(0,h/2)2 is a polynomial of  degree at most k−1 in each variable, which depends precisely on the values of fh in x+  �  − hk  2 , − h(k+1)  2  �  ∩  hZd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  The space of polynomials of degree at most k − 1 in each variable is an R-vector space of dimension  exactly kd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The same holds true for the space of functions from x +  �  − hk  2 , − h(k+1)  2  �  ∩ hZd to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  This means that Ihfh induces a linear map between two finite-dimensional vector spaces of the same  dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' By standard properties of B-splines, this map is surjective and hence, in fact, bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  As all norms on a finite-dimensional R-vector space are equivalent, we conclude that  max  y∈x+(− hk  2 ,− h(k+1)  2  )∩hZd fh(y) ≤ C  max  z∈x+(0,h/2)d Ihfh(z)  and (quotienting out constant functions) also  max  y,y′∈x+(− hk  2 ,− h(k+1)  2  )∩hZd |fh(y) − fh(y′)|  ≤ C  max  z,z′∈x+(0,h/2)d Ihfh(z) − Ihfh(z′) ≤ Chα∥Ihfh∥C0,α(x+(0,h/2)d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  The constant in the latter estimate is independent of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' This means that we actually obtain  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='18)  max  y,y′∈hZd  |y−y′|y∞≤kh  |fh(y) − fh(y′)| ≤ Chα∥Ihfh∥C0,α(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  We know that Ihϕh converges in C0,α(Rd) and so it is, in particular, tight in that space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' This means  that, if we define  EM =  �  [Ihϕh]C0,α(Rd)  �  ,  then limM→∞ limh→0 P(EM) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' On the other hand, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='18) implies that, on the event EM, we have  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='19)  max  y,y′∈hZd  |y−y′|y∞≤kh  |ϕh(y) − ϕh(y′)| ≤ CMhα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  This is the desired regularity estimate for ϕh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Step 3: Completion of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Our specific choice of Θ has the property that �  x∈hZd hdΘh(y −  x) = 1 for any y ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' This means that Ihϕh(y) is a convex combination of the ϕh(x) with |x− y|∞ <  hk  2 , and so we have  max  x∈Ωh ϕh(x) ≥ max  y∈Rd Ihϕh(y),  which implies the lower bound in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' For the upper bound we need to use (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' As Ihϕh(y) is  a convex combination of the ϕh(x) with |x − y|∞ < hk  2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='19) implies that on the event EM we have  |ϕh(x) − Ihϕ(x)| ≤ CMhα for any x ∈ hZd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Therefore, on the event EM we have  max  x∈Ωh ϕh(x) ≤ max  x∈Ωh Ihϕh(x) + CMhα ≤ max  y∈Rd Ihϕh(y) + CMhα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Putting these considerations together, we conclude that  lim  M→∞ lim  h→0 P  �  max  y∈Rd Ihϕh(y) ≤ max  x∈Ωh ϕh(x) ≤ max  y∈Rd Ihϕh(y) + CMhα  �  ,  which yields (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  □  SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS  21  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Technical lemmas  In this appendix, we provide the proof of several technical results that have been used throughout  the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Discretization and restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' We start by proving that the applications of restricting to  hZd and applying (−∆h)s commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1 (Discretization and restriction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let u: Rd → R be a Schwartz function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Then, restricting  to hZd and applying (−∆h)s commute: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=',  ((−∆h)su)↾hZd= (−∆h)s (u↾hZd) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  This allows us to be rather careless about when we restrict functions to hZd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' In fact, we will omit  writing ↾hZd when (because of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1) there is no ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The crucial fact here is that Mh(ξ) is 2π  h -periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Using this, we compute that, for x ∈ hZd,  ((−∆h)su) (x) =  �  Rd Mh(ξ)2sF[u](ξ) dξ  =  �  ζ∈ 2π  h Zd  �  (− π  h , π  h)  d Mh(ξ + ζ)2sF[u](ξ + ζ) dξ  =  �  (− π  h , π  h)  d Mh(ξ)2s  �  ζ∈ 2π  h Zd  F[u](ξ + ζ) dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1, we can rewrite this as  ((−∆h)su) (x) =  �  (− π  h , π  h)  d Mh(ξ)2sFh[u](ξ) dξ  = (−∆h)s (u↾hZd) ,  which is what we wanted to show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  □  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Discrete inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let us state the discrete Poincaré inequality that we used in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary and let s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Then, there  exists a constant C such that, for any uh : hZd → R that vanishes outside of Ωh, we have  ∥uh∥L2(Ωh) ≤ C∥uh∥ ˙Hs  h(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' We shall present a discrete version of the proof in [8, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The key idea is to use  Plancherel’s theorem and split low and high frequencies as follows:  ∥uh∥2  L2(Ω) =  �  Bϵ(0)  |Fhuh(ξ)|2 dξ +  �  (−π/h,π/h)d\\Bϵ(0)  |Fhuh(ξ)|2 dξ  =: I1 + I2,  where ϵ > 0 is to be fixed later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Low-frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' For the low-frequency part, I1, Hölder’s inequality yields  |Fhuh(ξ)| ≤ ∥uh∥L1(Ωh) ≤ |Ωh|1/2hd/2∥uh∥L2(Ωh),  where we used the notation  ∥uh∥Lp  h(Ωh) :=  � �  x∈Ωh  hd|uh|p  � 1  p  ,  p ∈ [1, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Therefore, we have  I1 ≤ ϵdB1(0)|Ωh|hd∥uh∥2  L2  h(Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  22  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' DE NITTI AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' SCHWEIGER  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' High-frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' For the high-frequency part, I2, we compute  �  (−π/h,π/h)d\\Bϵ(0)  |Fhuh(ξ)|2 dξ =  �  (−π/h,π/h)d\\Bϵ(0)  Mh(ξ)2s|Fhuh(ξ)|2  Mh(ξ)2s  dξ  ≤ ϵ−2s∥(−∆h)s/2uh∥2  L2  h(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Choosing 0 < ϵ < (|Ωh|hd|B1(0)|)−1/d, we conclude  ∥uh∥L2(Ωh) ≤  ϵ−s  �  1 − ϵd|Ωh|hd|B1(0)|  ∥uh∥Hs  h(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  By considering a square of side L ≥ diam(Ω) (containing Ωh), we deduce that |Ωh| ≤ C  hd (note that is  holds even if h ≫ diam(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' This means that Ωh|hd|B1(0)| ≤ C, and so we can make a choice of ε > 0  independent of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  □  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3 (Generalized Poincaré inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let s ≥ t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Arguing as in [8, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='5], Lemma  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2 also implies that, for u ∈ ˜Hs(Ω), there exists a constant c = c(d, Ω, s) > 0 such that  ∥(−∆h)t/2uh∥L2(Ωh) ≤ c∥(−∆h)s/2uh∥L2(Ωh)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Indeed,  ∥(−∆h)t/2uh∥L2(Ω) = ∥uh∥ ˙Ht  h(Ω) ≤ ∥uh∥Ht  h(Ω) ≤ ∥uh∥Hs  h(Ω) ≤ 2  s+1  2 (∥uh∥L2(Ωh) + ∥uh∥ ˙Hs  h(Ω))  ≤ 2  s+1  2 (c∥(−∆h)s/2uh∥L2(Ωh) + ∥(−∆h)s/2uh∥L2(Ωh))  = c∥(−∆h)s/2uh∥L2(Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  We also used the fact that solutions of the Dirichlet problem for (−∆)s have a little bit of additional  regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary and let s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Then, there  exists κ0 > 0 with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' If 0 ≤ κ ≤ κ0, then, for each f ∈ H−s+κ(Ω), there exists  a unique u ∈ ˙Hs+κ(Ω) such that (−∆)su = f in the sense of distributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' moreover, we have the  estimate  ∥u∥ ˙Hs+κ(Ω) ≤ Cκ∥f∥ ˙H−s+κ(Ω)  for a constant Cκ depending only on κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Let us remark that according to [3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3] one can take any κ0 < 1  2 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The argument,  however, is rather complicated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' so we prefer to present an easy perturbative argument that gives  existence of some κ0 > 0 (which is enough for our purposes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' We adapt the argument used in [19, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='3] for the biharmonic operator to the fractional  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  We first show that the claimed estimate holds for κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' To do so, we test the equation with u and  deduce  ∥(−∆)s/2u∥2  L2(Ω) = (u, (−∆)su)L2(Ω) = (u, f)L2(Ω) ≤ ∥u∥ ˙Hs(Ω)∥f∥ ˙H−s(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Using Poincaré’s inequality, we see that indeed  ∥u∥ ˙Hs(Ω) ≤ Cκ∥f∥ ˙H−s(Ω)  To show that we also can take some κ > 0, we use a stability result for analytic families  of operators on Banach spaces:  The spaces  ˙Hs(Ω) form an interpolation family with respect to  complex interpolation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' thus, by [21, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1], the set of those α for which the operator  (−∆)s :  ˙Hα(Ω) →  ˙Hα−2s(Ω) has a bounded inverse is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' We have seen that this set contains  s, so the existence of κ0 as in the statement of the theorem follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  □  SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS  23  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Fractional Gaussian Fields via eigenfunctions  In this appendix, we shall present an alternate description of the continuous FGF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' As remarked in  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1, (−∆)s is an isometry from ˙Hs(Ω) to ˙H−s(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Its inverse, restricted to L2(Ω), is a positive-  definitive compact operator on L2(Ω);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' so, by the spectral theorem, there exists an orthonormal basis  (v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=') of L2(Ω) consisting of eigenfunctions of (−∆)s with associated eigenvalues 0 < λ1 ≤ λ2 ≤  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='. Let Xj be a collection of independent standard Gaussians, and let ˜ϕ be the random variable  ˜ϕ = �∞  j=1  Xj  √  λj vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  According to Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1 below, this sum converges almost surely in ˙Hs′(Ω) ⊂ ˙Hs′(Rd) for any  s′ < s − d  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Therefore, ˜ϕ is a well-defined random variable on ˙Hs′(Ω) ⊂ ˙Hs′(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Every element of  ˙Hs′(Rd) induces an element of S′(Rd) and so we can think of ˜ϕ as a random element of S′(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Again,  according to Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1, for any f ∈ S(Rd) we have that ( ˜ϕ, f) is a centered Gaussian with variance  ∥f∥ ˙H−s(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' This means that ˜ϕ has the law P on S′(Rd) and so we can identify ϕ and ˜ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Let us present the aforementioned lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let s ≥ 0 and s′ < s − d  2 be  arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  (i) The series  ˜ϕ :=  ∞  �  j=1  Xj  �  λj  vj  converges almost surely in ˙Hs′(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  (ii) For any f ∈ S(Rd), we have  E( ˜ϕ, f)2 = ∥f∥2  ˙H−s(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  For the proof, we need a sharp estimate on the eigenfunction expansion of a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let s ≥ 0 and s′ ≤ s be  arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Then, for any f ∈ ˙Hs′(Ω), we have  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1)  ∥f∥2  ˙Hs′(Ω) ≤ C  ∞  �  j=1  λs′/s  j  (f, vj)2  L2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Note that we make no claim about the ˙Hs′-regularity for s′ > s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' For s′ ∈ {−s, 0, s} the estimate (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1) follows directly from the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' We next claim that  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1) holds whenever −s ≤ s′ ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' To see this, we adapt the argument in [17, Corollary 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Namely  take first 0 < s′ < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Consider (−∆)s restricted to functions in  ˙Hs(Ω) and let ((−∆)s)s′/s  N  be its  (spectral) s′  s -th power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Explicitly,  ((−∆)s)s′/s  N  f =  ∞  �  j=1  λs′/s  j  (f, vj)L2(Rd)vj  Note that, if we define ˙Hs′(Ω) to be the space of functions in L2(Ω) such that this quantity is finite,  then the domain of ((−∆)s)s′/s  N  is exactly ˙Hs′(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' According to the theory of interpolation of fractional  powers of self-adjoint operators (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=', [23, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='10]), the Hilbert spaces  ˙Hs′(Ω) form an  interpolation scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' However, we know that the same holds true for the Hilbert spaces ˙Hs′(Ω), and  moreover ˙Hs′(Ω) = ˙Hs′(Ω) (with equivalent norms) for s′ ∈ {0, s}, and so we have actually have this  equality for any s′ with 0 ≤ s′ ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' So, for 0 ≤ s′ ≤ s, there is some C > 0 such that  1  C  ∞  �  j=1  λs′/s  j  (f, vj)2  L2(Rd) ≤ ∥f∥2  ˙Hs′(Ω) ≤ C  ∞  �  j=1  λs′/s  j  (f, vj)2  L2(Rd)  24  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' DE NITTI AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' SCHWEIGER  By duality, the same holds true for −s ≤ s′ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Putting these considerations together, we obtain a  statement even stronger than (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  It remains to study the case that s′ < −s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' We proceed inductively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let us suppose that we know  that (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1) holds for s′ ≥ −(2k − 1)s, for some k ∈ N, and let us consider some s′ with −(2k + 1)s ≤  s′ ≤ −(2k − 1)s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' We have that  ∥f∥2  ˙Hs′(Ω) =  inf  g∈ ˙Hs′(Rd)  f=g in Ω  ∥g∥2  ˙H−s(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Let u be such that  �  (−∆)su(x) = f(x),  x ∈ Ω,  u(x) = 0,  x ∈ Rd \\ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  We can choose g = (−∆)su and obtain, using the induction hypothesis, that  ∥f∥2  ˙Hs′(Ω) ≤ ∥(−∆)su∥2  ˙Hs′(Rd)  ≤ ∥u∥2  ˙Hs′+2s(Rd)  ≤ C  ∞  �  j=1  λ(s′+2s)/s  j  (u, vj)2  L2(Rd)  = C  ∞  �  j=1  λ(s′+2s)/s  j  �  u, (−∆)svj  λj  �2  L2(Rd)  = C  ∞  �  j=1  λs′/s  j  ((−∆)su, vj)2  L2(Rd)  = C  ∞  �  j=1  λs′/s  j  (f, vj)2  L2(Rd) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  This completes the induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  □  Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Claim (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' By the Hilbert-space-valued version of Kolmogorov’s two series the-  orem (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' [11, Corollary on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' 386]), the series  ∞  �  j=1  Xj  �  λj  vj  converges almost surely in ˙Hs′(Ω) if  ∞  �  j=1  �����  1  �  λj  uj  �����  2  ˙Hs′(Ω)  < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  From Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='2, we know in particular that  ∥vj∥2  ˙Hs′(Ω) ≤ λs′/s  j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Moreover, by Weyl’s law for the operator (−∆)s restricted to ˙Hs(Ω) (as follows, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=', from the main  result of [10]), we have that  λj ≍ j2s/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Therefore,  ∞  �  j=1  �����  1  �  λj  vj  �����  2  ˙Hs′(Ω)  ≤  ∞  �  j=1  λs′/s−1  j  SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS  25  ≤  ∞  �  j=1  (cj)2s/d·(s′/s−1) ≤ C  ∞  �  j=1  j2(s′−s)/d,  and this sum is indeed convergent if s′ − s < − d  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Claim (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Let f ∈ S(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The functions vj are by definition orthonormal in L2(Ω), and the Xj  are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' So we can calculate that  E( ˜ϕ, f)2 = E  �  �  ∞  �  j=1  Xj  �  λj  (vj, f)  �  �  2  =  ∞  �  j=1  1  λj  (vj, f)2 =  �  �f,  ∞  �  j=1  1  λj  (vj, f)vj  �  �  =  �  f, (−∆)−sf  �  = ∥f∥2  H−s(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  □  Acknowledgments  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' De Nitti is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le  loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' He has been  supported by the Alexander von Humboldt Foundation and by the TRR-154 project of the Deutsche  Forschungsgemeinschaft (DFG, German Research Foundation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Schweiger is supported by the Foreign Postdoctoral Fellowship Program of the Israel Academy  of Sciences and Humanities, and partially by ISF grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' 421/20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  We thank O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Zeitouni and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Zuazua for helpful comments on the topic of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  References  [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Abatangelo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Higher-order fractional Laplacians:  An overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Bruno Pini Mathematical Analysis Seminar,  12(1):53–80, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  [2] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Baudoin and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Dirichlet fractional Gaussian fields on the Sierpinski gasket and their discrete graph  approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' ArXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='03970, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Borthagaray, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Li, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Nochetto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Fractional elliptic problems on Lipschitz domains: Regularity and  approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' ArXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='14070, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Bramson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Ding, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Zeitouni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Convergence in law of the maximum of the two-dimensional discrete Gaussian  free field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=', 69(1):62–123, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Chiarini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Cipriani, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Hazra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Extremes of some Gaussian random interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=', 165(3):521–  544, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  [6] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Ciaurri, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Roncal, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Stinga, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Torrea, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Varona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Nonlocal discrete diffusion equations and the  fractional discrete Laplacian, regularity and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=', 330:688–738, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Cipriani, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Dan, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Hazra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The scaling limit of the membrane model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=', 47(6):3963–4001,  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  [8] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Covi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Mönkkönen, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Railo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Unique continuation property and Poincaré inequality for higher order  fractional Laplacians with applications in inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Inverse Probl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Imaging, 15(4):641–681, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Furlan and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Mourrat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' A tightness criterion for random fields, with application to the Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=', 22:Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' 97, 29, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  [10] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Geisinger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' A short proof of Weyl’s law for fractional differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=', 55(1):011504, 7, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  [11] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Gikhman and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Skorokhod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The theory of stochastic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Classics in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Springer-Verlag,  Berlin, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Translated from the Russian by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Kotz, Reprint of the 1974 edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  [12] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Hao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Zhang, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Fractional centered difference scheme for high-dimensional integral fractional Lapla-  cian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=', 424:Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' 109851, 17, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  [13] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Huang and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Oberman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Finite difference methods for fractional Laplacians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' ArXiv:1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='00164, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  [14] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Jovanović and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Süli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Analysis of finite difference schemes, volume 46 of Springer Series in Computational  Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Springer, London, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' For linear partial differential equations with generalized solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  26  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' DE NITTI AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' SCHWEIGER  [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Lodhia, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Sheffield, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Sun, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Watson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Fractional Gaussian fields: a survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=', 13:1–56,  2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Müller and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Schweiger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Estimates for the Green’s function of the discrete bilaplacian in dimensions 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Vietnam J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=', 47(1):133–181, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  [17] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Musina and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Nazarov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' On fractional Laplacians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Partial Differential Equations, 39(9):1780–1790,  2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  [18] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Pinsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Introduction to Fourier analysis and wavelets, volume 102 of Graduate Studies in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  American Mathematical Society, Providence, RI, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Reprint of the 2002 original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  [19] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Schweiger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' The maximum of the four-dimensional membrane model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=', 48(2):714–741, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  [20] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Sheffield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Gaussian free fields for mathematicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Theory Related Fields, 139(3-4):521–541, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Tabacco Vignati and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Vignati.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Spectral theory and complex interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=', 80(2):383–397,  1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  [22] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Thomée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Elliptic difference operators and Dirichlet’s problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Contributions to Differential Equations, 3:301–324,  1964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  [23] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Triebel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Interpolation theory, function spaces, differential operators, volume 18 of North-Holland Mathematical  Library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' North-Holland Publishing Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=', Amsterdam-New York, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  [24] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Triebel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Function spaces in Lipschitz domains and on Lipschitz manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Characteristic functions as pointwise  multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Complut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=', 15(2):475–524, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' De Nitti) Friedrich-Alexander-Universität Erlangen-Nürnberg, Department of Data Science, Chair  for Dynamics, Control and Numerics (Alexander von Humboldt Professorship), Cauerstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  11, 91058  Erlangen, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Email address: nicola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='de.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='nitti@fau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='de  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content=' Schweiger) Weizmann Institute of Science, Department of Mathematics, Rehovot 7610001, Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='  Email address: florian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='schweiger@weizmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFST4oBgHgl3EQfWjig/content/2301.13781v1.pdf'}
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+arXiv:2301.00029v1  [math-ph]  31 Dec 2022
+Generalized conformal maps as classical symmetries of
+Yang-Mills fields
+Edward B. Baker III∗
+January 3, 2023
+We show that a class of previously defined maps, called causal and
+self-dual morphisms, form classical symmetries of Yang-Mills fields
+in four complex dimensions. These maps generalize conformal trans-
+formations, and admit a nonlocal pullback connection that preserves
+the equations of the theory. First it is shown that self-dual mor-
+phisms form symmetries of the anti-self-dual Yang-Mills equations
+under this pullback. Then a supersymmetric generalization of causal
+morphisms is defined which preserves solutions of the field equations
+for N=3 supersymmetric Yang-Mills theory. As a special case, this
+implies that a modified definition of causal morphisms form sym-
+metries for the ordinary Yang-Mills field equations.
+1
+Introduction
+Hidden symmetries have played an important role in the study of Yang-Mills (YM) theory.
+As an example, the anti-self dual Yang Mills (ASDYM) equations have an infinite class
+of hidden symmetries which bear some resemblance to the infinite-dimensional conformal
+group in two dimensions [1][2][3]. In addition, an extended conformal symmetry called
+dual superconformal invariance has been uncovered in the study of N=4 supersymmetric
+Yang-Mills (SYM) theory [4][5], leading to an infinite dimensional Yangian symmetry [6].
+This and other advances have led to powerful tools for the study of N=4 SYM.
+Many of these results can be understood best with the use of twistor and ambitwistor
+methods, which have been used extensively in the study of Yang-Mills fields. For example,
+the Penrose-Ward correspondence reformulates the ASDYM equations in Twistor space,
+∗edwardbaker86@gmail.com
+1
+
+which leads to the ADHM construction of instantons [7][8][9]. This construction was gen-
+eralized to a geometric formulation of the Yang-Mills field equations in ambitwistor space,
+with a natural interpretation in superspace [10][11][12][13]. More recently, twistor and am-
+bitwistor methods have been used in string theory to understand Yang-Mills scattering
+amplitudes and their properties [14][15].
+In this paper we investigate a previously defined class of generalized maps [16] in
+the context of Yang-Mills theory. These maps are motivated by twistor and ambitwistor
+theory, and are called self-dual and causal morphisms, respectively. Under certain as-
+sumptions, one can define a non-local pullback connection under these transformations
+that preserves integrability on certain subspaces. In the case of self-dual morphisms, the
+maps preserve integrability on self-dual planes which imply that they are symmetries of
+the ASDYM equations. A supersymmetric generalization of causal morphisms is then
+developed which preserves integrability on super null lines, implying that these maps are
+symmetries of the N=3 SYM field equations. This fact is used to show that a modified
+version of causal morphisms are symmetries of the YM field equations as a special case.
+It is likely that some of these symmetries are related to known hidden symmetries for
+the different cases, but characterizing these relationships will be left as a topic of future
+investigation.
+2
+Self-dual morphisms as symmetries of ASDYM
+Self-dual morphisms were introduced in a previous paper, where they were defined using
+maps on null surfaces [16]. Here we provide a self contained summary of these results,
+using different but equivalent definitions. To begin, define the twistor correspondence
+space F = C4 × CP1 with the usual double fibration [17][18]
+C4
+π1
+←− F
+π2
+−→ PT ,
+(1)
+where PT = CP3 is the projective twistor space of C4. Now define a self-dual embedding
+as a totally null holomorphic embedding χ : C2 → C4, which means that vectors at a
+point t ∈ C2 are mapped under χ⋆ to vectors of the form vα ˙α = λα˜λ ˙α for ˜λ ˙α fixed. We
+will call the image of such a map a self-dual surface. If ˜λ is independent of t then this
+surface maps to a self-dual plane (or α-plane) Z. Furthermore, for any point on a self-dual
+embedding there is a tangent α-plane passing through χ(t) that is characterized by ˜λ(t).
+For brevity, we say that χ is tangent to ˜λ at t. Now define the self-dual prolongation
+jsχ : C2 → F by jsχ = (χ, ˜λ), where dependence on t is suppressed. The prolongation
+satisfies a contact condition, that χ is tangent to ˜λ for all t ∈ C2. Conversely, given a
+surface ψ : C2 → F, we say that it satisfies the contact condition if ψ = jsχ for some
+self-dual embedding χ. A map f : F → F is said to preserve the contact condition if
+2
+
+f ◦ jsχ satisfies the contact condition for any χ.1 We then define
+Definition 1. A self-dual morphism is a holomorphic map f : F → F which preserves
+the contact condition.
+Defined in this way, a self-dual morphism naturally induces maps on self-dual embed-
+dings and self-dual planes
+Definition 2. Given a self-dual morphism f and a self-dual embedding χ, define the
+contraction map f⌟χ := π1 ◦ f ◦ jsχ. Furthermore, for a self-dual plane Z tangent to ˜λ,
+define f⌟Z : Z → C4 by f⌟Z(x) = π1 ◦ f(x, ˜λ) where x ∈ Z.
+This map on surfaces was the starting point for the definitions in the previous paper,
+and the two definitions are equivalent.
+Now consider a GL(n, C) connection with vector potential A satisfying the ASDYM
+equations on MC = C4. Given a self-dual morphism f, there is a natural definition for
+a pullback connection f ∗A. To see this, first restrict to a self-dual plane Z, which can
+be parameterized linearly by coordinates on C2. The contraction map f⌟Z then gives
+a self-dual embedding, and the pullback connection (f⌟Z)∗A is integrable on Z because
+the curvature of A vanishes on the self-dual planes tangent to f⌟Z as a consequence
+of ASDYM. By varying Z this allows us to define the bundle of parallel sections on
+the twistor space, and to use the Penrose-Ward procedure to define a connection on
+the pullback bundle, which defines the pullback connection f ∗A and gives a solution of
+ASDYM. This requires that the bundle of parallel sections is trivial for points x ∈ C4,
+which will be shown with an explicit construction of f ∗A.
+For the construction, consider two points x1, x2 ∈ Z and their images yi = f⌟Z(xi).
+Define a Wilson line for the pullback connection by
+W ∗
+Z(x1, x2) = Wf⌟Z(y1, y2) = P exp
+��
+γ
+Aµdxµ
+�
+.
+(2)
+Here the path of integration is any path γ confined to the image of f⌟Z starting at y1
+and ending at y2. The Wilson line is independent of path due to the integrability of the
+connection on self-dual surfaces. We can then define the patching matrix used in the
+Penrose-Ward correspondence by
+G = W ∗
+Z(q, p) = ˜HH−1
+(3)
+where
+H = W ∗
+Z(p, x),
+˜H = W ∗
+Z(q, x).
+(4)
+1These constructions are all assumed to be local and defined in some neighborhood, but are written
+globally for ease of notation.
+3
+
+Here p and q are the points of intersection between Z and self-dual planes P and Q with
+twistor coordinates ˆP = (0, 0, 1, 0) and ˆQ = (0, 0, 0, 1), as defined in the usual patching
+construction. The patching matrix descends to the twistor space, and the splitting formula
+guarantees that the bundle is trivial, so the bundle satisfies the conditions of the Penrose-
+Ward correspondence. We therefore can recover a self-dual pullback connection f ∗A. This
+all assumes that the integrals are non-singular, which depends on the details of the maps,
+connections and domains under consideration.
+To gain intuition for this construction, it is useful to derive an explicit formula for f ∗A.
+To this end, restrict to a self-dual plane Z and use the formula ˜λ ˙αf ∗Aα ˙α = H−1˜λ ˙α∂α ˙αH
+from the Penrose-Ward correspondence, in addition to equation (4), to find
+v · f ∗A|x= (f⌟Z)∗v · A|f⌟Z(x),
+x ∈ Z, v ∈ TxZ.
+(5)
+By varying the self-dual plane through a fixed point x this gives the value for null vectors
+v ∈ TxC4, and the value on other vectors can be recovered from linearity of the connection.
+Linearity and self-duality follow from the Penrose-Ward construction, but it is instructive
+to derive these results directly from equation (5), which can be expanded as
+(f⌟Z)∗v · A|f⌟Z(x)= vα ˙α�∂yβ ˙β
+∂xα ˙α Aβ ˙β
+����
+f⌟Z(x), x ∈ Z, v ∈ TxZ,
+(6)
+where yβ ˙β = f⌟Z(x)β ˙β. This equation is degree one in both λ and ˜λ and globally holo-
+morphic on CP 1 × CP 1, and so is linear by Liouville’s theorem. Self-duality is equivalent
+to the connection being integrable on self-dual planes, which follows from using equa-
+tion (5) to show that [v1 · D∗, v2 · D⋆] = 0, where v1,2 ∈ TxZ are linearly independent
+vectors tangent to a self-dual plane Z. These arguments generalize well to the N = 3
+supersymmetric case.
+3
+Super causal morphisms and N=3 SYM
+In this section we will define a super causal morphism, which is an extension of the pre-
+viously defined causal morphisms to superspace. The supersymmetric generalization is
+useful because of an interpretation of the N=3 SYM field equations as an integrability
+condition on supersymmetric null lines [10][12]. This interpretation allows for a gener-
+alization of the arguments of the previous section to the N = 3 SYM field equations.
+Furthermore, solutions of the usual YM field equations are special cases of the supersym-
+metric solutions, and this will be used to show that a modified version of causal morphisms
+are also symmetries of the ordinary YM field equations.
+The definition of super causal morphisms follows closely to the previous definitions.
+To begin, consider the superspace C4|4N with coordinates zA = (xα ˙α, θiα, ˜θ ˙α
+j ) and super-
+symmetry generators qiα =
+∂
+∂θiα + i˜θ ˙α
+i
+∂
+∂xα ˙α and ˜qi
+˙α =
+∂
+∂˜θ ˙α
+i + iθiα
+∂
+∂xα ˙α. The lightlike lines
+4
+
+through z ∈ C4|4N and tangent to (λ, ˜λ) are generated by fermionic translation operators
+Ti = λαqαi and ˜T i = ˜λ ˙α˜qi
+˙α which satisfy the algebra
+{Ti, Tj} = { ˜T i, ˜T j} = 0,
+{Ti, ˜T j} = 2iδj
+i D,
+(7)
+where D = λα˜λ ˙α∂α ˙α. Define the correspondence space (F 6|4N, π1, C4|4N) as the bundle
+over superspace whose fiber at a point z is the set of super null lines that intersect z.
+These fibers are isomorphic to CP 1 ×CP 1, corresponding to the projective spinors λ and
+˜λ that generate bosonic translations along a given null line.
+The super null lines described above have one complex dimension and 2N fermionic
+dimensions, which can be parameterized by coordinates σ = (s, ξi, ˜ξj) ∈ C1|2N.
+The
+supersymmetry generators on this line are ∂s, qi = ∂ξi + i˜ξi∂s and ˜qi = ∂˜ξi + iξi∂s. More
+generally, we can also consider a super null curve, which is a morphism χ : C1|2N → C4|4N
+whose pushforward χ∗ takes the form
+(∂s, qi, ˜qk) → (∂, Mj
+i tj, ˜
+Mk
+l ˜tl),
+(8)
+at every point σ, where the operators (∂, tj, ˜tl) are defined for a super null line that is
+tangent to χ at χ(σ). As with the self-dual case, we can then define the prolongation
+jcχ : C1|2N → F 6|4N by jcχ = (χ, λ, ˜λ), and given a curve ψ : C1|2N → F 6|4N we say that
+it satisfies the contact condition if ψ = jcχ for some nonsingular null curve χ. As before,
+a map f : F 6|4N → F 6|4N preserves the contact condition if f ◦ jcχ satisfies the contact
+condition for any χ. Furthermore, for a super null line L tangent to (λ, ˜λ), we also define
+f⌟L(z) = π1 ◦ f(z, λ, ˜λ) ∀z ∈ L as before.
+In the supersymmetric case there is an extra consideration necessary to ensure inte-
+grability of the pullback connection. To see this, consider a morphism f : F 6|4N → F 6|4N
+which preserves the contact condition. Given a super null line L, we want to demand
+that the supersymmetry relations (7) are preserved under (f⌟L)∗. By construction, this
+pushforward takes the form of equation (8), where the coordinates on L are also chosen
+to satisfy the supersymmetry relations. To preserve these relations, we must demand
+Mi
+j ˜
+Mj
+k = δi
+k.
+(9)
+This condition must be satisfied for every super null line L, which is the condition that
+must be satisfied for integrability.
+We can now define
+Definition 3. A super causal morphism is a holomorphic map f : F 6|4N → F 6|4N which
+preserves the contact condition, and preserves the supersymmetry relations (7) under
+(f⌟L)⋆ for tangent vectors to any super null line L.
+Now consider an N=3 supersymmetric Yang-Mills field satisfying the field equations
+on C4|12. This field can be defined by a superconnection characterized by a one form Φ
+with components ΦA = (ωiα, ˜ωi
+˙α, Aα ˙α), which defines covariant derivative operators
+Qiα = qiα + ωiα, ˜Qi
+˙α = qi
+˙α + ˜ωi
+˙α, Dα ˙α = ∂α ˙α + Aα ˙α.
+(10)
+5
+
+The field equations are equivalent to integrability on super null lines [12], which for a
+given line L tangent to vα ˙α = λα˜λ ˙α are given by equations (7) for the translation operators
+Ti = λαQiα, ˜T i = ˜λ ˙α ˜Qi
+˙α and D = λα˜λ ˙αDα ˙α.
+Now, given a super causal morphism f, we can define the pullback connection f ∗Φ
+similarly to the self-dual case.
+To do so, consider a super null line L, and the super
+null curve f⌟L it generates. The bundle of parallel sections on these super null lines
+then generates a pullback connection in the usual manner. To calculate this pullback
+connection, we can directly generalize equation (5) to
+v · f ∗Φ|z= (f⌟L)∗v · Φ|f⌟L(z),
+z ∈ L, v ∈ TxL.
+(11)
+As in the previous section, linearity of the pullback connection follows from a variant
+of Liouville’s theorem. Integrability on lines follows from writing (7) for the pullback
+translation operators defined on L, and then using (11) and the assumption that Φ is
+integrable on lines.
+This implies that super causal morphisms are symmetries of the
+N = 3 SYM field equations.
+4
+Reduction to Yang-Mills field equations
+The geometric interpretation of the YM field equations using field extensions can be
+naturally understood by viewing these equations as a special case of the N=3 SYM field
+equations for a Yang-Mills super multiplet with the scalar and spinor fields set to zero
+[10][12][13][19]. In a similar spirit, it is possible to use a modified definition of causal
+morphisms to generate an N=3 super causal morphism which preserves the property that
+the scalar and spinor fields equal zero, thus forming a symmetry of the YM field equations.
+A causal morphism can be defined as an N = 0 super causal morphism, or a map
+f : G → G that preserves the contact condition, where G = F 4|0 is the usual ambitwistor
+correspondence space. Given such a function, we can construct an extended morphism
+ˆf : F 6|4N → F 6|4N given by
+ˆf(g, θiα, ˜θ ˙α
+j ) = (f(g), [V −1]α
+βθiβ, [ ˜V −1] ˙α
+˙β ˜θ
+˙β
+j ), g ∈ G,
+(12)
+where V , ˜V are invertible matrix functions of g. Now restrict to a super null line L tangent
+to a null vector vα ˙α = λα˜λ ˙α. Along L, the supersymmetry relations (7) are preserved if
+and only if
+vβ ˙βV α
+β ˜V ˙α
+˙β = ((f⌟L0)∗v)α ˙α,
+(13)
+where L0 is the bosonic projection of L. The existence of holomorphically varying matrix
+functions V and ˜V satisfying this condition is the extra modification necessary to extend
+a causal morphism f to ˆf, and will be assumed.
+The above construction yields a symmetry of the N = 3 SYM field equations, but we
+must also show that solutions of the YM field equations, with scalar and spinor fields set
+6
+
+to zero, are preserved by these extended causal morphisms. To do so, we will use two
+results proved by Harnad et. al. [13]. In that paper, theorem 3.3 characterizes the form
+of the superconnection induced from a solution of the YM field equations when embedded
+as a gauge fixed N=3 SYM connection, which is
+ωiα = ˜θ ˙α
+i hα ˙α(xβ ˙β, τ β ˙β),
+˜ωi
+α = θiα˜hα ˙α(xβ ˙β, τ β ˙β),
+(14)
+where τ β ˙β = �
+i θiβ ˜θ
+˙β
+i , Aα ˙α = Aα ˙α(xβ ˙β, τ β ˙β), and the gauge condition is θiαωiα+˜θi
+˙α˜ω ˙α
+i = 0.
+Conversely, corollary 4.3 shows that a gauge-fixed connection that is integrable on super
+null lines and takes the above form corresponds to an N = 3 extended solution of the YM
+field equations.
+Based on these considerations, showing that the extended causal morphisms preserve
+the form of equation (14) and the gauge condition implies that they preserve solutions
+of the YM field equations. To show this, restrict to a super null line L and use (11) to
+compute the pullback connection of (14), which gives
+λα ˆf ∗ωiα(z) = ˜θ
+˙β
+i λα �
+V β
+α hβ ˙β(x′, τ ′)
+�
+,
+˜λ ˙α ˆf ∗˜ωi
+˙α(z) = θiβ˜λ ˙α �
+˜V
+˙β
+˙α ˜hβ ˙β(x′, τ ′)
+�
+,
+(15)
+where x′, τ ′ are evaluated at ˆf⌟L(z).
+Now consider two points z1, z2 ∈ C4|4N with
+x1 = x2 and τ1 = τ2, lying on two parallel lines. Under ˆf⌟L, τ transforms as τ α ˙α →
+[V −1]α
+β[ ˜V −1] ˙α
+˙βτ β ˙β, so the quantities in parentheses are the same for these two points and
+parallel lines, but the line can be varied, so this is true for any (λ, ˜λ). We therefore see
+that the connection has the form of equation (14), as desired. Furthermore, the gauge
+condition can be written τ α ˙α(hα ˙α + ˜hα ˙α) = 0. For τ α ˙α proportional to a null vector this
+condition is preserved, but this must be true for any null line, so by linearity the gauge
+condition is preserved. This implies that the pullback connection corresponds to an N = 3
+extended solution of the YM field equations.
+5
+Discussion
+We have shown that self-dual, N = 3 super causal and causal morphisms yield symmetries
+of the ASDYM, N = 3 SYM and YM field equations, respectively. To further understand
+these symmetries, it will be necessary to classify their solutions and to investigate their
+action on concrete examples of YM fields. Some partial results were found in the previous
+paper [16], where examples of self-dual morphisms were constructed from holomorphic
+endomorphisms of twistor space. A method was also developed to construct causal mor-
+phisms from these self-dual morphisms. Although these constructions provide examples
+of solutions, it will be important to find a more complete classification. In particular, it is
+likely that there are more general examples than those constructed from endomorphisms
+7
+
+of the twistor space, or super ambitwistor space, which could be analogous to holomorphic
+functions that preserve the real line in two dimensions. This preliminary interpretation
+is based on the CR ambitwistor space used in [20], but will require further investigation
+to make precise. It will also be important to understand how these maps are related to
+other well known hidden symmetries for these equations.
+There are many additional avenues of further research. Here the action of these maps
+was only considered for classical Yang-Mills fields, but the ultimate goal is to further
+understand the quantum theory.
+Furthermore, it will be interesting to consider how
+gravitational fields transform under these maps. In this vein, one could define a causal
+manifold with coordinate transformations that are morphisms of these types, in analogy
+to the definition of Riemann surfaces for holomorphic functions. Due to the nonlocal
+nature of these maps, the theory could lead to interesting new mathematics.
+References
+[1] L. Dolan, A new symmetry group of real self-dual yang-mills theory,
+Physics Letters B 113 (1982) 387.
+[2] L.-L. Chau, G. Mo-Lin, A. Sinha and W. Yong-Shi, Hidden-symmetry algebra for
+the self-dual yang-mills equation, Physics Letters B 121 (1983) 391.
+[3] A.D. Popov, Self-dual yang-mills: Symmetries and moduli space,
+Reviews in Mathematical Physics 11 (1999) 1091.
+[4] L. Mason and D. Skinner, Dual superconformal invariance, momentum twistors and
+grassmannians, Journal of High Energy Physics 2009 (2009) 045.
+[5] N. Arkani-Hamed, F. Cachazo and C. Cheung, The grassmannian origin of dual
+superconformal invariance, Journal of High Energy Physics 2010 (2010) .
+[6] J. Drummond, J. Henn and J. Plefka, Yangian symmetry of scattering amplitudes
+in n = 4 super yang-mills theory, Journal of High Energy Physics 2009 (2009) 046.
+[7] R.S. Ward, On self-dual gauge fields, Physics Letters A 61 (1977) 81.
+[8] M. Atiyah, N. Hitchin, V. Drinfeld and Y. Manin, Construction of instantons,
+Physics Letters A 65 (1978) 185 .
+[9] M. Atiyah, A. nazionale dei Lincei and S. normale superiore (Italy), Geometry of
+Yang-Mills fields, Lezioni fermiane, Scuola normale superiore (1979).
+[10] E. Witten, An interpretation of classical yang-mills theory,
+Physics Letters B 77 (1978) 394 .
+8
+
+[11] J. Isenberg, P.B. Yasskin and P.S. Green, Non-self-dual gauge fields,
+Physics Letters B 78 (1978) 462 .
+[12] J. Harnad, J. Hurtubise, M. Legare and S. Shnider, Constraint equations and field
+equations in supersymmetric n = 3 yang-mills theory,
+Nuclear Physics B 256 (1985) 609.
+[13] J. Harnad, J. Hurtubise and S. Shnider, Supersymmetric yang-mills equations and
+supertwistors, Annals of Physics 193 (1989) 40.
+[14] E. Witten, Perturbative gauge theory as a string theory in twistor space,
+Communications in Mathematical Physics 252 (2004) 189.
+[15] M. Atiyah, M. Dunajski and L.J. Mason, Twistor theory at fifty: from contour
+integrals to twistor strings, Proceedings of the Royal Society A: Mathematical,
+Physical and Engineering Sciences 473 (2017) 20170530.
+[16] E.B. Baker, Causal and self-dual morphisms in four complex dimensions, 2022.
+10.48550/ARXIV.2203.07952.
+[17] M. Dunajski, Solitons, Instantons, and Twistors, Oxford Graduate Texts in
+Mathematics, OUP Oxford (2010).
+[18] R. Ward and R. Wells, Twistor Geometry and Field Theory, Cambridge
+Monographs on Mathematical Physics, Cambridge University Press (1991).
+[19] M. Eastwood, Supersymmetry, twistors, and the yang-mills equations, Transactions
+of the American Mathematical Society 301 (1987) 615.
+[20] L. Mason and D. Skinner, An ambitwistor yang–mills lagrangian,
+Physics Letters B 636 (2006) 60.
+9
+
diff --git a/2NAyT4oBgHgl3EQfPvY8/content/tmp_files/load_file.txt b/2NAyT4oBgHgl3EQfPvY8/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf,len=209
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='00029v1  [math-ph]  31 Dec 2022  Generalized conformal maps as classical symmetries of  Yang-Mills fields  Edward B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Baker III∗  January 3, 2023  We show that a class of previously defined maps, called causal and  self-dual morphisms, form classical symmetries of Yang-Mills fields  in four complex dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' These maps generalize conformal trans-  formations, and admit a nonlocal pullback connection that preserves  the equations of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' First it is shown that self-dual mor-  phisms form symmetries of the anti-self-dual Yang-Mills equations  under this pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Then a supersymmetric generalization of causal  morphisms is defined which preserves solutions of the field equations  for N=3 supersymmetric Yang-Mills theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' As a special case, this  implies that a modified definition of causal morphisms form sym-  metries for the ordinary Yang-Mills field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  1  Introduction  Hidden symmetries have played an important role in the study of Yang-Mills (YM) theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  As an example, the anti-self dual Yang Mills (ASDYM) equations have an infinite class  of hidden symmetries which bear some resemblance to the infinite-dimensional conformal  group in two dimensions [1][2][3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' In addition, an extended conformal symmetry called  dual superconformal invariance has been uncovered in the study of N=4 supersymmetric  Yang-Mills (SYM) theory [4][5], leading to an infinite dimensional Yangian symmetry [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  This and other advances have led to powerful tools for the study of N=4 SYM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  Many of these results can be understood best with the use of twistor and ambitwistor  methods, which have been used extensively in the study of Yang-Mills fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' For example,  the Penrose-Ward correspondence reformulates the ASDYM equations in Twistor space,  ∗edwardbaker86@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='com  1  which leads to the ADHM construction of instantons [7][8][9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' This construction was gen-  eralized to a geometric formulation of the Yang-Mills field equations in ambitwistor space,  with a natural interpretation in superspace [10][11][12][13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' More recently, twistor and am-  bitwistor methods have been used in string theory to understand Yang-Mills scattering  amplitudes and their properties [14][15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  In this paper we investigate a previously defined class of generalized maps [16] in  the context of Yang-Mills theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' These maps are motivated by twistor and ambitwistor  theory, and are called self-dual and causal morphisms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Under certain as-  sumptions, one can define a non-local pullback connection under these transformations  that preserves integrability on certain subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' In the case of self-dual morphisms, the  maps preserve integrability on self-dual planes which imply that they are symmetries of  the ASDYM equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' A supersymmetric generalization of causal morphisms is then  developed which preserves integrability on super null lines, implying that these maps are  symmetries of the N=3 SYM field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' This fact is used to show that a modified  version of causal morphisms are symmetries of the YM field equations as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  It is likely that some of these symmetries are related to known hidden symmetries for  the different cases, but characterizing these relationships will be left as a topic of future  investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  2  Self-dual morphisms as symmetries of ASDYM  Self-dual morphisms were introduced in a previous paper, where they were defined using  maps on null surfaces [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Here we provide a self contained summary of these results,  using different but equivalent definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' To begin, define the twistor correspondence  space F = C4 × CP1 with the usual double fibration [17][18]  C4  π1  ←− F  π2  −→ PT ,  (1)  where PT = CP3 is the projective twistor space of C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Now define a self-dual embedding  as a totally null holomorphic embedding χ : C2 → C4, which means that vectors at a  point t ∈ C2 are mapped under χ⋆ to vectors of the form vα ˙α = λα˜λ ˙α for ˜λ ˙α fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' We  will call the image of such a map a self-dual surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' If ˜λ is independent of t then this  surface maps to a self-dual plane (or α-plane) Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Furthermore, for any point on a self-dual  embedding there is a tangent α-plane passing through χ(t) that is characterized by ˜λ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  For brevity, we say that χ is tangent to ˜λ at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Now define the self-dual prolongation  jsχ : C2 → F by jsχ = (χ, ˜λ), where dependence on t is suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' The prolongation  satisfies a contact condition, that χ is tangent to ˜λ for all t ∈ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Conversely, given a  surface ψ : C2 → F, we say that it satisfies the contact condition if ψ = jsχ for some  self-dual embedding χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' A map f : F → F is said to preserve the contact condition if  2  f ◦ jsχ satisfies the contact condition for any χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='1 We then define  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' A self-dual morphism is a holomorphic map f : F → F which preserves  the contact condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  Defined in this way, a self-dual morphism naturally induces maps on self-dual embed-  dings and self-dual planes  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Given a self-dual morphism f and a self-dual embedding χ, define the  contraction map f⌟χ := π1 ◦ f ◦ jsχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Furthermore, for a self-dual plane Z tangent to ˜λ,  define f⌟Z : Z → C4 by f⌟Z(x) = π1 ◦ f(x, ˜λ) where x ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  This map on surfaces was the starting point for the definitions in the previous paper,  and the two definitions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  Now consider a GL(n, C) connection with vector potential A satisfying the ASDYM  equations on MC = C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Given a self-dual morphism f, there is a natural definition for  a pullback connection f ∗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' To see this, first restrict to a self-dual plane Z, which can  be parameterized linearly by coordinates on C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' The contraction map f⌟Z then gives  a self-dual embedding, and the pullback connection (f⌟Z)∗A is integrable on Z because  the curvature of A vanishes on the self-dual planes tangent to f⌟Z as a consequence  of ASDYM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' By varying Z this allows us to define the bundle of parallel sections on  the twistor space, and to use the Penrose-Ward procedure to define a connection on  the pullback bundle, which defines the pullback connection f ∗A and gives a solution of  ASDYM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' This requires that the bundle of parallel sections is trivial for points x ∈ C4,  which will be shown with an explicit construction of f ∗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  For the construction, consider two points x1, x2 ∈ Z and their images yi = f⌟Z(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  Define a Wilson line for the pullback connection by  W ∗  Z(x1, x2) = Wf⌟Z(y1, y2) = P exp  ��  γ  Aµdxµ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  (2)  Here the path of integration is any path γ confined to the image of f⌟Z starting at y1  and ending at y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' The Wilson line is independent of path due to the integrability of the  connection on self-dual surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' We can then define the patching matrix used in the  Penrose-Ward correspondence by  G = W ∗  Z(q, p) = ˜HH−1  (3)  where  H = W ∗  Z(p, x),  ˜H = W ∗  Z(q, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  (4)  1These constructions are all assumed to be local and defined in some neighborhood, but are written  globally for ease of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  3  Here p and q are the points of intersection between Z and self-dual planes P and Q with  twistor coordinates ˆP = (0, 0, 1, 0) and ˆQ = (0, 0, 0, 1), as defined in the usual patching  construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' The patching matrix descends to the twistor space, and the splitting formula  guarantees that the bundle is trivial, so the bundle satisfies the conditions of the Penrose-  Ward correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' We therefore can recover a self-dual pullback connection f ∗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' This  all assumes that the integrals are non-singular, which depends on the details of the maps,  connections and domains under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  To gain intuition for this construction, it is useful to derive an explicit formula for f ∗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  To this end, restrict to a self-dual plane Z and use the formula ˜λ ˙αf ∗Aα ˙α = H−1˜λ ˙α∂α ˙αH  from the Penrose-Ward correspondence, in addition to equation (4), to find  v · f ∗A|x= (f⌟Z)∗v · A|f⌟Z(x),  x ∈ Z, v ∈ TxZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  (5)  By varying the self-dual plane through a fixed point x this gives the value for null vectors  v ∈ TxC4, and the value on other vectors can be recovered from linearity of the connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  Linearity and self-duality follow from the Penrose-Ward construction, but it is instructive  to derive these results directly from equation (5), which can be expanded as  (f⌟Z)∗v · A|f⌟Z(x)= vα ˙α�∂yβ ˙β  ∂xα ˙α Aβ ˙β  ����  f⌟Z(x), x ∈ Z, v ∈ TxZ,  (6)  where yβ ˙β = f⌟Z(x)β ˙β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' This equation is degree one in both λ and ˜λ and globally holo-  morphic on CP 1 × CP 1, and so is linear by Liouville’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Self-duality is equivalent  to the connection being integrable on self-dual planes, which follows from using equa-  tion (5) to show that [v1 · D∗, v2 · D⋆] = 0, where v1,2 ∈ TxZ are linearly independent  vectors tangent to a self-dual plane Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' These arguments generalize well to the N = 3  supersymmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  3  Super causal morphisms and N=3 SYM  In this section we will define a super causal morphism, which is an extension of the pre-  viously defined causal morphisms to superspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' The supersymmetric generalization is  useful because of an interpretation of the N=3 SYM field equations as an integrability  condition on supersymmetric null lines [10][12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' This interpretation allows for a gener-  alization of the arguments of the previous section to the N = 3 SYM field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  Furthermore, solutions of the usual YM field equations are special cases of the supersym-  metric solutions, and this will be used to show that a modified version of causal morphisms  are also symmetries of the ordinary YM field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  The definition of super causal morphisms follows closely to the previous definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  To begin, consider the superspace C4|4N with coordinates zA = (xα ˙α, θiα, ˜θ ˙α  j ) and super-  symmetry generators qiα =  ∂  ∂θiα + i˜θ ˙α  i  ∂  ∂xα ˙α and ˜qi  ˙α =  ∂  ∂˜θ ˙α  i + iθiα  ∂  ∂xα ˙α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' The lightlike lines  4  through z ∈ C4|4N and tangent to (λ, ˜λ) are generated by fermionic translation operators  Ti = λαqαi and ˜T i = ˜λ ˙α˜qi  ˙α which satisfy the algebra  {Ti, Tj} = { ˜T i, ˜T j} = 0,  {Ti, ˜T j} = 2iδj  i D,  (7)  where D = λα˜λ ˙α∂α ˙α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Define the correspondence space (F 6|4N, π1, C4|4N) as the bundle  over superspace whose fiber at a point z is the set of super null lines that intersect z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  These fibers are isomorphic to CP 1 ×CP 1, corresponding to the projective spinors λ and  ˜λ that generate bosonic translations along a given null line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  The super null lines described above have one complex dimension and 2N fermionic  dimensions, which can be parameterized by coordinates σ = (s, ξi, ˜ξj) ∈ C1|2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  The  supersymmetry generators on this line are ∂s, qi = ∂ξi + i˜ξi∂s and ˜qi = ∂˜ξi + iξi∂s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' More  generally, we can also consider a super null curve, which is a morphism χ : C1|2N → C4|4N  whose pushforward χ∗ takes the form  (∂s, qi, ˜qk) → (∂, Mj  i tj, ˜  Mk  l ˜tl),  (8)  at every point σ, where the operators (∂, tj, ˜tl) are defined for a super null line that is  tangent to χ at χ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' As with the self-dual case, we can then define the prolongation  jcχ : C1|2N → F 6|4N by jcχ = (χ, λ, ˜λ), and given a curve ψ : C1|2N → F 6|4N we say that  it satisfies the contact condition if ψ = jcχ for some nonsingular null curve χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' As before,  a map f : F 6|4N → F 6|4N preserves the contact condition if f ◦ jcχ satisfies the contact  condition for any χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Furthermore, for a super null line L tangent to (λ, ˜λ), we also define  f⌟L(z) = π1 ◦ f(z, λ, ˜λ) ∀z ∈ L as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  In the supersymmetric case there is an extra consideration necessary to ensure inte-  grability of the pullback connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' To see this, consider a morphism f : F 6|4N → F 6|4N  which preserves the contact condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Given a super null line L, we want to demand  that the supersymmetry relations (7) are preserved under (f⌟L)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' By construction, this  pushforward takes the form of equation (8), where the coordinates on L are also chosen  to satisfy the supersymmetry relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' To preserve these relations, we must demand  Mi  j ˜  Mj  k = δi  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  (9)  This condition must be satisfied for every super null line L, which is the condition that  must be satisfied for integrability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  We can now define  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' A super causal morphism is a holomorphic map f : F 6|4N → F 6|4N which  preserves the contact condition, and preserves the supersymmetry relations (7) under  (f⌟L)⋆ for tangent vectors to any super null line L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  Now consider an N=3 supersymmetric Yang-Mills field satisfying the field equations  on C4|12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' This field can be defined by a superconnection characterized by a one form Φ  with components ΦA = (ωiα, ˜ωi  ˙α, Aα ˙α), which defines covariant derivative operators  Qiα = qiα + ωiα, ˜Qi  ˙α = qi  ˙α + ˜ωi  ˙α, Dα ˙α = ∂α ˙α + Aα ˙α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  (10)  5  The field equations are equivalent to integrability on super null lines [12], which for a  given line L tangent to vα ˙α = λα˜λ ˙α are given by equations (7) for the translation operators  Ti = λαQiα, ˜T i = ˜λ ˙α ˜Qi  ˙α and D = λα˜λ ˙αDα ˙α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  Now, given a super causal morphism f, we can define the pullback connection f ∗Φ  similarly to the self-dual case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  To do so, consider a super null line L, and the super  null curve f⌟L it generates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' The bundle of parallel sections on these super null lines  then generates a pullback connection in the usual manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' To calculate this pullback  connection, we can directly generalize equation (5) to  v · f ∗Φ|z= (f⌟L)∗v · Φ|f⌟L(z),  z ∈ L, v ∈ TxL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  (11)  As in the previous section, linearity of the pullback connection follows from a variant  of Liouville’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Integrability on lines follows from writing (7) for the pullback  translation operators defined on L, and then using (11) and the assumption that Φ is  integrable on lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  This implies that super causal morphisms are symmetries of the  N = 3 SYM field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  4  Reduction to Yang-Mills field equations  The geometric interpretation of the YM field equations using field extensions can be  naturally understood by viewing these equations as a special case of the N=3 SYM field  equations for a Yang-Mills super multiplet with the scalar and spinor fields set to zero  [10][12][13][19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' In a similar spirit, it is possible to use a modified definition of causal  morphisms to generate an N=3 super causal morphism which preserves the property that  the scalar and spinor fields equal zero, thus forming a symmetry of the YM field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  A causal morphism can be defined as an N = 0 super causal morphism, or a map  f : G → G that preserves the contact condition, where G = F 4|0 is the usual ambitwistor  correspondence space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Given such a function, we can construct an extended morphism  ˆf : F 6|4N → F 6|4N given by  ˆf(g, θiα, ˜θ ˙α  j ) = (f(g), [V −1]α  βθiβ, [ ˜V −1] ˙α  ˙β ˜θ  ˙β  j ), g ∈ G,  (12)  where V , ˜V are invertible matrix functions of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Now restrict to a super null line L tangent  to a null vector vα ˙α = λα˜λ ˙α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Along L, the supersymmetry relations (7) are preserved if  and only if  vβ ˙βV α  β ˜V ˙α  ˙β = ((f⌟L0)∗v)α ˙α,  (13)  where L0 is the bosonic projection of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' The existence of holomorphically varying matrix  functions V and ˜V satisfying this condition is the extra modification necessary to extend  a causal morphism f to ˆf, and will be assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  The above construction yields a symmetry of the N = 3 SYM field equations, but we  must also show that solutions of the YM field equations, with scalar and spinor fields set  6  to zero, are preserved by these extended causal morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' To do so, we will use two  results proved by Harnad et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' In that paper, theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='3 characterizes the form  of the superconnection induced from a solution of the YM field equations when embedded  as a gauge fixed N=3 SYM connection, which is  ωiα = ˜θ ˙α  i hα ˙α(xβ ˙β, τ β ˙β),  ˜ωi  α = θiα˜hα ˙α(xβ ˙β, τ β ˙β),  (14)  where τ β ˙β = �  i θiβ ˜θ  ˙β  i , Aα ˙α = Aα ˙α(xβ ˙β, τ β ˙β), and the gauge condition is θiαωiα+˜θi  ˙α˜ω ˙α  i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  Conversely, corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='3 shows that a gauge-fixed connection that is integrable on super  null lines and takes the above form corresponds to an N = 3 extended solution of the YM  field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  Based on these considerations, showing that the extended causal morphisms preserve  the form of equation (14) and the gauge condition implies that they preserve solutions  of the YM field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' To show this, restrict to a super null line L and use (11) to  compute the pullback connection of (14), which gives  λα ˆf ∗ωiα(z) = ˜θ  ˙β  i λα �  V β  α hβ ˙β(x′, τ ′)  �  ,  ˜λ ˙α ˆf ∗˜ωi  ˙α(z) = θiβ˜λ ˙α �  ˜V  ˙β  ˙α ˜hβ ˙β(x′, τ ′)  �  ,  (15)  where x′, τ ′ are evaluated at ˆf⌟L(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  Now consider two points z1, z2 ∈ C4|4N with  x1 = x2 and τ1 = τ2, lying on two parallel lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Under ˆf⌟L, τ transforms as τ α ˙α →  [V −1]α  β[ ˜V −1] ˙α  ˙βτ β ˙β, so the quantities in parentheses are the same for these two points and  parallel lines, but the line can be varied, so this is true for any (λ, ˜λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' We therefore see  that the connection has the form of equation (14), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Furthermore, the gauge  condition can be written τ α ˙α(hα ˙α + ˜hα ˙α) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' For τ α ˙α proportional to a null vector this  condition is preserved, but this must be true for any null line, so by linearity the gauge  condition is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' This implies that the pullback connection corresponds to an N = 3  extended solution of the YM field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  5  Discussion  We have shown that self-dual, N = 3 super causal and causal morphisms yield symmetries  of the ASDYM, N = 3 SYM and YM field equations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' To further understand  these symmetries, it will be necessary to classify their solutions and to investigate their  action on concrete examples of YM fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Some partial results were found in the previous  paper [16], where examples of self-dual morphisms were constructed from holomorphic  endomorphisms of twistor space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' A method was also developed to construct causal mor-  phisms from these self-dual morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Although these constructions provide examples  of solutions, it will be important to find a more complete classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' In particular, it is  likely that there are more general examples than those constructed from endomorphisms  7  of the twistor space, or super ambitwistor space, which could be analogous to holomorphic  functions that preserve the real line in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' This preliminary interpretation  is based on the CR ambitwistor space used in [20], but will require further investigation  to make precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' It will also be important to understand how these maps are related to  other well known hidden symmetries for these equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  There are many additional avenues of further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Here the action of these maps  was only considered for classical Yang-Mills fields, but the ultimate goal is to further  understand the quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  Furthermore, it will be interesting to consider how  gravitational fields transform under these maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' In this vein, one could define a causal  manifold with coordinate transformations that are morphisms of these types, in analogy  to the definition of Riemann surfaces for holomorphic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Due to the nonlocal  nature of these maps, the theory could lead to interesting new mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  References  [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Dolan, A new symmetry group of real self-dual yang-mills theory,  Physics Letters B 113 (1982) 387.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Chau, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Mo-Lin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Sinha and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Yong-Shi, Hidden-symmetry algebra for  the self-dual yang-mills equation, Physics Letters B 121 (1983) 391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Popov, Self-dual yang-mills: Symmetries and moduli space,  Reviews in Mathematical Physics 11 (1999) 1091.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  [4] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Mason and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Skinner, Dual superconformal invariance, momentum twistors and  grassmannians, Journal of High Energy Physics 2009 (2009) 045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  [5] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Arkani-Hamed, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Cachazo and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Cheung, The grassmannian origin of dual  superconformal invariance, Journal of High Energy Physics 2010 (2010) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Drummond, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Henn and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Plefka, Yangian symmetry of scattering amplitudes  in n = 4 super yang-mills theory, Journal of High Energy Physics 2009 (2009) 046.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  [7] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Ward, On self-dual gauge fields, Physics Letters A 61 (1977) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Atiyah, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Hitchin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Drinfeld and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Manin, Construction of instantons,  Physics Letters A 65 (1978) 185 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Atiyah, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' nazionale dei Lincei and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' normale superiore (Italy), Geometry of  Yang-Mills fields, Lezioni fermiane, Scuola normale superiore (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  [10] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Witten, An interpretation of classical yang-mills theory,  Physics Letters B 77 (1978) 394 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  8  [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Isenberg, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Yasskin and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Green, Non-self-dual gauge fields,  Physics Letters B 78 (1978) 462 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  [12] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Harnad, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Hurtubise, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Legare and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Shnider, Constraint equations and field  equations in supersymmetric n = 3 yang-mills theory,  Nuclear Physics B 256 (1985) 609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Harnad, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Hurtubise and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Shnider, Supersymmetric yang-mills equations and  supertwistors, Annals of Physics 193 (1989) 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  [14] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Witten, Perturbative gauge theory as a string theory in twistor space,  Communications in Mathematical Physics 252 (2004) 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  [15] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Atiyah, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Dunajski and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Mason, Twistor theory at fifty: from contour  integrals to twistor strings, Proceedings of the Royal Society A: Mathematical,  Physical and Engineering Sciences 473 (2017) 20170530.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  [16] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Baker, Causal and self-dual morphisms in four complex dimensions, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='07952.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Dunajski, Solitons, Instantons, and Twistors, Oxford Graduate Texts in  Mathematics, OUP Oxford (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  [18] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Ward and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Wells, Twistor Geometry and Field Theory, Cambridge  Monographs on Mathematical Physics, Cambridge University Press (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  [19] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Eastwood, Supersymmetry, twistors, and the yang-mills equations, Transactions  of the American Mathematical Society 301 (1987) 615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  [20] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Mason and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Skinner, An ambitwistor yang–mills lagrangian,  Physics Letters B 636 (2006) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
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+1
+On the Mutual Information of Multi-RIS
+Assisted MIMO: From Operator-Valued Free
+Probability Aspect
+Zhong Zheng, Member, IEEE, Siqiang Wang, Zesong Fei, Senior Member, IEEE,
+Zhi Sun, Senior Member, IEEE, Jinhong Yuan, Fellow, IEEE
+Abstract
+The reconfigurable intelligent surface (RIS) is useful to effectively improve the coverage and data rate
+of end-to-end communications. In contrast to the well-studied coverage-extension use case, in this paper,
+multiple RIS panels are introduced, aiming to enhance the data rate of multi-input multi-output (MIMO)
+channels in presence of insufficient scattering. Specifically, via the operator-valued free probability theory,
+the asymptotic mutual information of the large-dimensional RIS-assisted MIMO channel is obtained
+under the Rician fading with Weichselberger’s correlation structure, in presence of both the direct and
+the reflected links. Although the mutual information of Rician MIMO channels scales linearly as the
+number of antennas and the signal-to-noise ratio (SNR) in decibels, numerical results show that it requires
+sufficiently large SNR, proportional to the Rician factor, in order to obtain the theoretically guaranteed
+linear improvement. This paper shows that the proposed multi-RIS deployment is especially effective to
+improve the mutual information of MIMO channels under the large Rician factor conditions. When the
+reflected links have similar arriving and departing angles across the RIS panels, a small number of RIS
+panels are sufficient to harness the spatial degree of freedom of the multi-RIS assisted MIMO channels.
+Z. Zheng, S. Wang, and Z. Fei are with the School of Information and Electronics, Beijing Institute of Technology, Beijing,
+China. Z. Sun is with the Department of Electronic Engineering, Tsinghua University, Beijing, China. J. Yuan is with the School
+of Electrical Engineering and Telecommunications, UNSW Sydney, Sydney, Australia.
+DRAFT
+arXiv:2301.12144v1  [cs.IT]  28 Jan 2023
+
+2
+Index Terms
+Reconfigurable intelligent surface, MIMO, Rician channel, mutual information, operator-valued free
+probability.
+I. INTRODUCTION
+In both the current and forthcoming generations of mobile communication systems, multi-input multi-
+output (MIMO) is one of the mainstream physical-layer techniques to improve the spectral efficiency
+and the reliability of the wireless communications [1]. In the favorable environments with rich scattering,
+MIMO is able to increase the achievable data rate linearly with the number of antennas [2]. However,
+when the wireless systems operate in higher frequencies with larger bandwidth, such as the millimeter
+wave and terahertz bands, the radio signals are easily attenuated due to absorption and blockage. In this
+case, the MIMO channels typically have only a few dominating propagation paths and/or limited angular
+spread, which causes rank deficiency in the channel matrix that significantly degrades the MIMO channel
+capacity [3].
+Recently, reconfigurable intelligent surface (RIS) has attracted substantial attentions and is foreseen
+to be an important component in the future communication systems [4]. A typical RIS consists of
+a large number of low-power integrated electronic circuits, which can be programmed to modify the
+electromagnetic properties of the incoming radio waves in the desired frequency band [5], such as the
+phase and amplitude of the reflected signals from each programmable circuit. Therefore, by deploying
+some RIS panels in the environment, the signal’s radiation pattern within the operating spectrum bands
+of the communication systems can be reconfigured to increase the number of independent paths with
+diversified angular spreads, thus increasing the rank of the MIMO channels. As an example, Fig. 1
+illustrates the transmissions between a base station (BS) and a user equipment (UE) in an urban canyon.
+In this scenario, without RIS deployment, the signals have to propagate through a scattering-limited area,
+where the direct propagation link F0 dominates the end-to-end channel, while other scattered/reflected
+components are severally attenuated by the building materials. In comparison, RIS panels are able to
+actively and effectively reflect the signals to increase the number of independent specular components,
+DRAFT
+
+3
+• • • 
+UE
+BS
+RIS K
+RIS 1
+F0
+F1
+FK
+G1
+GK
+Fig. 1.
+Multi-RIS assisted MIMO communications.
+resulting in a total number of K +1 propagation links, including the direct link F0 and K reflected links
+that consist of channels {Fk}1≤k≤K between BS and RIS panels and channels {Gk}1≤k≤K between RIS
+panels and UE.
+There exist a number of studies focusing on the performance evaluation of the end-to-end communi-
+cations assisted by a single or multiple RIS panels. When both the transmitter and receiver are equipped
+with a single antenna and a single RIS is deployed, the signal-to-noise-ratio (SNR) of such RIS-assisted
+single-input single-output (SISO) channel is proportional to the squared amplitude of the end-to-end
+effective channel. There are two typical theoretic frameworks to analyze the statistical properties of
+the SNR: One is based on the Meijer’s G- and Fox’s H-function systems [6], which result in exact
+but rather complicated expressions. The other is to match the moments of the effective channel with
+classical random variables [7]–[10]. In particular, when the direct link is blocked, the SNR distribution and
+the corresponding outage probability of the RIS-assisted communications are approximated by Gamma
+random variables, when the component channels are independently Rayleigh-faded [7], independently
+Rician-faded [8], correlated Rician-faded [9], and independently Nakagami-faded [10], respectively. When
+both the direct and the reflected links exist, the Gamma-approximated SNR distribution and the finite
+block-length rate of the RIS-assisted channel are obtained in [11], when all component channels are
+DRAFT
+
+4
+Rayleigh-faded.
+When multiple RIS panels are deployed in the network, the RIS panels either work in exhaustive mode
+to jointly assist the end-to-end communications [12], or work in opportunistic mode, where only the RIS
+with maximum channel gains is selected [13]. In the former case, the end-to-end SNR is approximated
+by a Gaussian random variable due to the central limit theorem. The SNR and the average symbol error
+probability are derived for the phase shift keying signaling. In the latter case, the SNR of each reflected
+link corresponding to one RIS panel is approximated as a Gaussian random variable and the order-statistics
+is then obtained for the optimally selected RIS-assisting channel. In [14], a comprehensive performance
+comparison of those two operation modes is provided, under the scenario of multi-RIS assisted SISO
+channels assuming Nakagami-faded direct and reflected links.
+In the case of MIMO systems, the available performance analysis of RIS-assisted communications is
+rather limited due to the challenge of understanding the statistical distribution of matrix-valued prop-
+agation channels, and results only exist for the single-RIS deployment. In [15], the authors consider
+the RIS-assisted MIMO communications, where the direct link is blocked and the reflected link is the
+concatenated Rayleigh-faded MIMO channels with single-sided correlation. The exact outage probability
+of such channel is derived by using the Mellin transform [16]. The result is expressed as the integration
+of product of multiple Meijer’s G-functions, which is difficult to solve in practice. When the reflected
+link is the concatenated millimeter wave MIMO channels assuming Saleh-Valenzuela model [17], an
+upper bound of the ergodic achievable rate is derived in [18] using majorization theory and Jensen’s
+inequality. In the RIS-assisted uplink multiple access channel, the asymptotic ergodic sum rate of the
+multi-user MIMO system is derived in [19] by using the replica method, assuming that the reflected links
+are Rician-faded MIMO channels with Kronecker’s correlation. In the same channel model as [19], the
+finite-SNR diversity-multiplexing tradeoff (DMT) of the RIS-assisted MIMO channel is analyzed in [20]
+by the martingale method. When both the direct and the reflected links exist, the asymptotic achievable
+rate of the single-RIS assisted MIMO channel is derived in [21] via replica method, assuming that all
+the component channels are Rician fading with Kronecker’s correlation and all the channel dimensions
+grow to infinity.
+DRAFT
+
+5
+Although the RIS-assisted MIMO communications have been investigated in [15], [18]–[21], the results
+therein are obtained for the single-RIS deployment under certain MIMO channel configurations. In
+contrast, this paper aims to provide the theoretic framework that analyzes the general multi-RIS assisted
+MIMO communications under arbitrary Rician fading with Weichselberger’s correlation structure [22].
+Such a model fits a wider range of realistic MIMO channels compared to the conventional Kronecker’s
+correlation structure. Based on the above system settings, we first embed the component MIMO channel
+matrices into a large block matrix. Then, an operator-valued probability space over the algebra of the
+constructed block matrices is defined, where the operator-valued Cauchy transform is defined and is shown
+to be closely related to the classic Cauchy transform of the channel Gram matrix. The operator-valued
+Cauchy transform is then derived by leveraging the freeness over the defined probability space and the
+additive free convolution machinery. Based on the obtained Cauchy transform, the probability distribution
+of the eigenvalue of the channel Gram matrix as well as the mutual information of the multi-RIS assisted
+MIMO channel can be calculated, which avoids time-consuming Monte Carlo simulations. Numerical
+results show that in presence of strong line-of-sight conditions, although the mutual information could
+scale linearly as the number of antennas and the SNRs (in decibels), the SNR has to be sufficiently large
+in order to exhibit such linear scaling law. On the other hand, deploying additional RIS panels could
+effectively improve the channel’s mutual information and thus, alleviate the SNR requirement.
+The rest of this article is organized as follows. The signal model, the channel model, and the mutual
+information of the MIMO channel under consideration are introduced in Section II. In Section III, the
+operator-valued probability space is introduced and the main result of the Cauchy transform of the channel
+Gram matrix is given. Numerical simulation results on the spectral distribution and the mutual information
+of the MIMO channels are in Section IV. Section V concludes the main findings of this article.
+Notations. Throughout the paper, vectors and matrices are represented by lower-case and upper-case
+bold-face letters, respectively. The complex column vector with length n is denoted as Cn. We use
+CN(0, A) to denote the zero-mean complex Gaussian vector with covariance matrix A and In is an
+n × n identity matrix. The superscript (·)† denotes the matrix conjugate-transpose operation and (·)T
+is matrix transpose. We denote Tr(A) as the trace of n × n matrix A. The notation E[·] denotes the
+DRAFT
+
+6
+expectation, and det(·) denotes the matrix determinant.
+II. SYSTEM MODEL
+A. Signal Model
+Consider a MIMO communication channel between a transmitter equipped with T antennas and a
+receiver equipped with R antennas. The transmissions are assisted by K RIS panels, which reflect the
+impinging signals via their reflecting elements and each RIS panel is equipped with Lk reflecting elements,
+1 ≤ k ≤ K. For notational simplicity, we define R = L0 and use these two symbols interchangeably.
+Denote the transmitted signal as x ∈ CT and the additive noise at the receiver as n ∈ CR. The received
+signal y ∈ CR is expressed as
+y =
+�
+F0 +
+K
+�
+k=1
+√ρkGkFk
+�
+x + n,
+(1)
+where the R × T matrix F0 denotes the direct channel between transmitter and receiver, the Lk × T
+matrix Fk denotes the channels between the transmitter and the k-th RIS panel, and 0 < ρk ≤ 1 denotes
+the relative channel gain of the k-th reflected channel via the k-th RIS, compared to the direct channel.
+The R × Lk matrix Gk denotes phase-shifted reflected channel between the k-th RIS and the receiver,
+modeled as
+Gk = RkΘk,
+(2)
+where the R×Lk matrix Rk denotes the channel coefficients, and the diagonal matrix Θk = diag
+�
+eiφk,1, . . . , eiφk,Lk�
+contains the phase-shifts of the reflecting elements, where 0 ≤ φk,l ≤ 2π denotes the phase-shift of the
+l-th element of the k-th RIS.
+We adopt the following assumptions on the signal and the channels:
+(A1) The signal x is Gaussian distributed with uniform power allocation, i.e., x ∼ CN(0T , PIT ), where
+P is the average power of the signals from each transmit antenna;
+(A2) The noise n is assumed to be a white Gaussian random vector with i.i.d. zero-mean entries, i.e.,
+n ∼ CN(0R, σ2IR), where σ2 denotes the variance of the noise;
+DRAFT
+
+7
+(A3) The channel coefficients {Fk}0≤k≤K and {Rk}1≤k≤K are block-faded, which keeps constant within
+the coherence time, while changing randomly and independently in the next coherence time. The
+phase shifts {Θk}1≤k≤K are assumed to be fixed.
+Note that without the direct link F0, channel models similar to (1) have been also studied in [3],
+[23], [24] for the keyhole channel, the Rayleigh-product channel, and the double-scattering channel,
+respectively, by using different theoretic techniques, which cannot be applied here.
+B. Channel Model
+In order to characterize the directivity and the spatial correlation of the channels between antenna
+arrays, we adopt the non-central Weichselberger’s MIMO model for each component links [22], such
+that
+Fk = Fk + �Fk = Fk + Uk(Mk ⊙ Xk)V†
+k,
+0 ≤ k ≤ K,
+(3)
+Gk = Gk + �Gk = Gk +
+1
+√rk
+Wk(Nk ⊙ Yk)S†
+k,
+1 ≤ k ≤ K,
+(4)
+where Fk and Gk are the fixed specular components of Fk and Gk, respectively. The random scattering
+components are captured by �Fk and �Gk, where Uk, Vk, Wk, and Sk are deterministic unitary matrices.
+The deterministic matrices Mk and Nk represent the variance profiles of �Fk and �Gk, respectively, each
+having non-negative real elements. The Lk × T random matrix Xk and the R × Lk random matrix Yk
+are i.i.d. complex Gaussian distributed with entries having zero mean and variance 1/T, i.e., [Xk]i,j ∼
+CN(0, 1/T) and [Yk]i,j ∼ CN(0, 1/T). We denote rk as the ratio between Lk and T, i.e., rk = Lk/T,
+1 ≤ k ≤ K. The operator ⊙ denotes the element-wise matrix multiplication. Note that the phase-shift
+matrix Θk, 1 ≤ k ≤ K, is also unitary and can be absorbed into the deterministic matrices Gk and
+Sk. The ratio between the power of fixed specular component and the random scattering component is
+defined as the Rician factor of the MIMO channel, i.e.,
+κ(F)
+k
+= ||Fk||2
+F
+E[||�F||2
+F]
+, and κ(G)
+k
+= ||Gk||2
+F
+E[|| �G||2
+F]
+,
+(5)
+where || · ||F denotes the Frobenius norm of a matrix.
+DRAFT
+
+8
+For the correlated MIMO channel Gk, the one-sided correlation function ηk(�C) = E[ �G†
+k �C �Gk] param-
+eterized by an Hermitian matrix �C is given by [22, Thm. 1] as
+ηk(�C) = E[ �G†
+k �C �Gk] = 1
+Lk
+SkΠk(�C)S†
+k,
+1 ≤ k ≤ K,
+(6)
+where the Lk × Lk diagonal matrix Πk(�C) contains the diagonal entries
+�
+Πk(�C)
+�
+i,i =
+R
+�
+j=1
+([Nk]j,i)2 �
+W†
+k �CWk
+�
+j,j ,
+1 ≤ i ≤ Lk.
+(7)
+The other one-sided correlation function �ηk(Ck) = E[ �GkCk �G†
+k] parameterized by Ck is given by
+�ηk(Ck) = E[ �GkCk �G†
+k] = 1
+Lk
+Wk �Πk(Ck)W†
+k,
+1 ≤ k ≤ K,
+(8)
+where the R × R diagonal matrix �Πk(Ck) contains the diagonal entries
+�
+�Πk(Ck)
+�
+i,i =
+Lk
+�
+j=1
+([Nk]i,j)2 �
+S†
+kCkSk
+�
+j,j ,
+1 ≤ i ≤ R.
+(9)
+Similarly, for 0 ≤ k ≤ K, the two parameterized one-sided correlation functions of the matrix �Fk are
+given by:
+ζk(Dk) = E[�F†
+kDk�Fk] = 1
+T VkΣk(Dk)V†
+k,
+(10)
+�ζk( �D) = E[�Fk �D�F†
+k] = 1
+T Uk �Σk( �D)U†
+k,
+(11)
+where the T × T diagonal matrix Σk(Dk) and the Lk × Lk diagonal matrix �Σk( �D) respectively contain
+the diagonal entries
+[Σk(Dk)]i,i =
+Lk
+�
+j=1
+([Mk]j,i)2 �
+U†
+kDkUk
+�
+j,j ,
+1 ≤ i ≤ T,
+(12)
+�
+�Σk( �D)
+�
+i,i =
+T
+�
+j=1
+([Mk]i,j)2 �
+V†
+k �DVk
+�
+j,j ,
+1 ≤ i ≤ Lk.
+(13)
+In addition, since the channels {Fk}0≤k≤K and {Gk}1≤k≤K are spatially separated, channels corre-
+spond to different links are assumed to be independent.
+DRAFT
+
+9
+C. Mutual Information of Multi-RIS MIMO Channel
+Due to the assumptions (A1)-(A3), the channel (1) is a Gaussian MIMO channel and its mutual
+information is given by the well-known Telatar’s formula [2] as
+I(γ) = log det
+�
+IR + γHH†�
+,
+(14)
+where γ = P/σ2 is the average SNR, and the end-to-end channel H is given by
+H = F0 +
+K
+�
+k=1
+√ρkGkFk.
+(15)
+The channel H can be factorized as the product of two matrices G and F as
+H = GF =
+�
+IR
+√ρ1G1
+. . .
+√ρKGK
+�
+�
+���������
+F0
+F1
+...
+FK
+�
+���������
+.
+(16)
+Denoting L = �K
+k=0 Lk, G =
+�
+IR
+√ρ1G1
+. . .
+√ρKGK
+�
+is a R × L block matrix and F =
+�
+FT
+0 , . . . , FT
+K
+�T is a L × T block matrices.
+Letting B = HH† = GFF†G†, the mutual information (14) can be rewritten as
+I(γ) = R VB(γ) = R
+ˆ ∞
+0
+log(1 + γt)fB(t)dt,
+(17)
+where VB(x) is the Shannon transform of the matrix B [25], and fB(t) is the probability density function
+(PDF) of the eigenvalue of B. Applying the relation between the Shannon transform and the corresponding
+Cauchy transform [25], the mutual information (17) can be rewritten as
+I(γ) = R
+ˆ γ
+0
+�1
+t + 1
+t2 GB
+�
+−1
+t
+��
+dt,
+(18)
+where GB(z) is the Cauchy transform of B and is defined as
+GB(z) =
+ˆ ∞
+0
+1
+z − tfB(t)dt = 1
+RTr ◦ E
+�
+(zI − B)−1�
+= τR
+�
+(zI − B)−1�
+.
+(19)
+Here, τR(X) is the composite function 1
+RTr◦E[X]. Note that the PDF fB(t) has an one-to-one mapping
+with the Cauchy transform GB(z) via the inverse transform
+fB(t) = − 1
+π lim
+ϵ→0 ℑ(GB(t + iϵ)),
+(20)
+DRAFT
+
+10
+where ℑ(·) denotes the imaginary part of the complex number. Therefore, the problem of finding the
+mutual information I(γ) and the PDF fB(t) are amount to finding the Cauchy transform GB(z) of
+product of matrices B = GFF†G†. In the next section, we will resort to a linearization trick and the
+operator-valued free probability theory to derive the expression of GB(z).
+III. ASYMPTOTIC EIGENVALUE DISTRIBUTION VIA OPERATOR-VALUED FREE PROBABILITY
+THEORY
+In the classic free probability theory, it is common to combine the Cauchy transform and the free
+multiplicative convolution to obtain the limiting spectral distribution of product of random matrices. For
+example, in [26], the limiting spectral distribution of the concatenated MIMO channels of the form
+� K
+�
+k=1
+Hk
+� � K
+�
+k=1
+Hk
+�†
+(21)
+is derived, where Hk is Nk × Nk−1 random matrix and has i.i.d. zero-mean entries, unitarily invariant,
+and independent of each other. Such assumptions, in the language of free probability theory, is equivalent
+to requiring freeness among families of random variables as specified below.
+Let A be a unital algebra and B ⊂ A be a unital subalgebra. For H ∈ A, a linear map EB[H] : A → B
+is a B-valued conditional expectation, if EB[b] = b for all b ∈ B, and EB[b1Hb2] = b1EB[H]b2 for all
+H ∈ A and b1, b2 ∈ B. Then, a B-valued probability space is denoted as (A, EB, B), consisting of B ⊂ A
+and the linear functional EB. In addition, let A1, . . . , AK be the subalgebras of A with B ⊂ Ak for all
+1 ≤ k ≤ K. We also let {Hk ∈ Ak, 1 ≤ k ≤ K} denote a family of operator-valued random variables,
+which are free with amalgamation over B according to the following definition.
+Definition 1. Let n be an arbitrary integer. The families of random variables {H1, . . . , HK} are free
+with amalgamation over B, if for every family of index {k1, . . . , kn} ⊂ {1, . . . , K} with k1 ̸= k2, . . . ,
+kn−1 ̸= kn, and every family of polynomials {P1, . . . , Pn} satisfying EB[Pj(Hkj)] = 0, j ∈ {1, . . . , n},
+we have EB
+��n
+j=1 Pj(Hkj)
+�
+= 0.
+In order to observe the freeness among families of random matrices {Hk, H†
+k}1≤k≤K in (21), we
+can construct the random variable Hk as Hk = HkH†
+k. Let C denote the algebra of complex random
+DRAFT
+
+11
+variables. We define Ak = MNk(C) as the algebra of Nk × Nk complex Hermitian matrices, B = C and
+the linear functional EB as EB =
+1
+Nk Tr ◦ E. Then, as specified in Definition 1, the asymptotic freeness
+among {Hk}1≤k≤K over C has been established in some of the classic free probability theory, such as
+in [28], which further enables one to apply free multiplicative convolution [26] to obtain the limiting
+spectral distribution of the concatenated MIMO channels.
+However, in the considered problem with B = GFF†G†, both G and F are non-central and with
+non-trivial spatial correlations, and thus, are not free over C in the classic free probability aspect. More
+precisely, GG† and FF† are not free with respect to the linear functional τR = 1
+RTr ◦ E. Yet, as will be
+shown in the remaining of this section, via a linearization trick, the random matrix B of interest can be
+embedded into a larger block matrix, which can be then separated as the sum of a deterministic matrix
+and a random matrix. Instead of invoking the classic freeness over C, we are able to elevate them as the
+operator-valued variables, which are shown to be asymptotically free in the operator-valued probability
+space. The limiting spectral distribution of their sum can be then obtained by using the operator-valued
+free additive convolution.
+A. Linearization Trick and Operator-Valued Probability Space
+Let n denote 2L + R + T and M = Mn(C) denote the algebra of n × n complex random matrices.
+Although the original formulation of B is in the form of product of two random matrices that are not free
+with respect to τR, we could instead construct a block matrix L ∈ M, whose operator-valued Cauchy
+transform can be properly defined and is directly related to the conventional Cauchy transform of B.
+By using the Anderson’s linearization trick as described in [30, Prop. 3.4], we can construct a block
+matrix L ∈ M as follows
+L =
+�
+��
+L(1,1)
+L(1,2)
+L(2,1)
+L(2,2)
+�
+�� =
+�
+���������
+0R×R
+0R×L
+0R×T
+G
+0L×R
+0L×L
+F
+−IL
+0T×R
+F†
+−IT
+0T×L
+G†
+−IL
+0L×T
+0L×L
+�
+���������
+,
+(22)
+DRAFT
+
+12
+where the matrix blocks
+�
+L(i,j)�
+correspond to the partitions shown on the right-hand-side (RHS) of (22).
+In addition, let us consider the sub-algebra D ⊂ M as the n×n block diagonal matrix. For each K ∈ D,
+it is defined as
+K = blkdiag
+�
+�C, D, �D, C
+�
+,
+(23)
+where �C is a R × R sub-matrix and �D is a T × T sub-matrix. The L × L block diagonal matrices C
+and D are defined as C = blkdiag {C0, . . . , CK} and D = blkdiag {D0, . . . , DK}, respectively, where
+Ck and Dk are Lk × Lk sub-matrices. In (23), all the involved sub-matrices �C, �D, {Ck}0≤k≤K, and
+{Dk}0≤k≤K are Hermitian matrices.
+For X ∈ M, we define X �C, X �D, {XCk}0≤k≤K, and {XDk}0≤k≤K as the sub-blocks of X, corre-
+sponding to the same diagonal sub-blocks �C, �D, {Ck}0≤k≤K, and {Dk}0≤k≤K in the matrix K. Then,
+we define the linear functional τD : M → D as
+τD(X) = id ◦ ED [X] ,
+(24)
+where id denotes the identity operator on a Hilbert space and the expectation ED [X] is defined as
+ED [X] =
+�
+���������
+E[X �C]
+E[XD]
+E[X �D]
+E[XC]
+�
+���������
+,
+(25)
+and E[XC] = blkdiag {E[XC0], . . . , E[XCK]}, E[XD] = blkdiag {E[XD0], . . . , E[XDK]}. Then, we
+can define an operator-valued probability space (M, τD, D). For the M-valued random variable L ∈
+(M, τD, D), its D-valued Cauchy transform is defined as
+GD
+L (Λ(z)) = id ◦ ED
+�
+(Λ(z) − L)−1�
+= τD
+�
+(Λ(z) − L)−1�
+,
+(26)
+where Λ(z) ∈ M denotes the n × n diagonal matrix as
+Λ(z) =
+�
+��
+zIR
+0R×(2L+T)
+0(2L+T)×R
+0(2L+T)×(2L+T)
+�
+�� .
+(27)
+DRAFT
+
+13
+By substituting (22) and (27) into (26) and invoking Lemma 2 in the Appendix A, we obtain
+GD
+L (Λ(z)) = id ◦ ED
+�
+��
+�
+zIR + L(1,2) �
+L(2,2)�−1 L(2,1)�−1
+−L(1,2) �
+zL(2,2) + L(2,1)L(1,2)�−1
+−
+�
+zL(2,2) + L(2,1)L(1,2)�−1 L(2,1)
+−
+�
+L(2,2) + z−1L(2,1)L(1,2)�−1
+�
+�� . (28)
+In particular, the upper-left block of (28) can be explicitly written as
+�
+zIR + L(1,2) �
+L(2,2)�−1
+L(2,1)
+�−1
+=
+�
+zIR − GFF†G†�−1
+.
+(29)
+Therefore, the Cauchy transform of B over C is related to the D-valued Cauchy transform of L as
+GB(z) = 1
+RTr
+��
+GD
+L (Λ(z))
+�(1,1)�
+,
+(30)
+where {·}(1,1) denotes the upper-left R × R matrix block.
+B. Operator-Valued Free Additive Convolution
+Let us introduce the following notations:
+G =
+�
+IR
+√ρ1G1
+. . .
+√ρKGK
+�
+,
+(31)
+�G =
+�
+0R
+√ρ1 �G1
+. . .
+√ρK �GK
+�
+,
+(32)
+F =
+�
+F
+T
+0
+. . .
+F
+T
+K
+�T
+,
+(33)
+�F =
+�
+�FT
+0
+. . .
+�FT
+K
+�T
+.
+(34)
+The linearization matrix L can be further expressed as
+L = L + �L,
+(35)
+where L and �L contain the deterministic and the random parts of L, respectively, and are given as follows:
+L =
+�
+���������
+G
+F
+−IL
+F
+†
+−IT
+G
+†
+−IL
+�
+���������
+,
+(36)
+DRAFT
+
+14
+�L =
+�
+���������
+�G
+�F
+�F†
+�G†
+�
+���������
+,
+(37)
+where we omit the all-zero matrix blocks.
+The advantage of working with L as well as its D-valued Cauchy transform GD
+L is that the elements
+of �L are monomials of �G, �G†, �F, and �F†, which are decoupled from each other. This is in contrast
+to the Cauchy transform of B over C, where the random variables are mixed together. Then, following
+similar steps as in [32], �L is shown to be an operator-valued semicircular variable and is free from the
+deterministic matrix L over D. Thus, the limiting spectral distribution of L can be determined by the
+operator-valued free additive convolution of L and �L, over the sub-algebra D, which are summarized in
+the following propositions.
+Proposition 1. The random variable �L is semicircular and is free from L over D.
+Proof: The proof of Proposition 1 is given in Appendix B.
+Due to Proposition 1, the operator-valued Cauchy transform of L in (30) can be calculated as the free
+additive convolution between �L and L, by using a subordination formula [30] as follows:
+GD
+L (Λ(z)) = GD
+L
+�
+Λ(z) − RD
+�L
+�
+GD
+L (Λ(z))
+��
+= ED
+��
+Λ(z) − RD
+�L
+�
+GD
+L (Λ(z))
+�
+− L
+�−1�
+,
+(38)
+where RD
+�L (·) denotes the operator-valued R-transform of L over D. Then, GB(z) can be determined by
+the following proposition.
+Proposition 2. The Cauchy transform of B, with z ∈ C+, is given by
+GB(z) = 1
+RTr
+��
+�Ψ(z) − GΞ(z)−1G
+†�−1�
+,
+(39)
+where
+Ξ(z) = Ψ(z) −
+�
+�Φ(z) − FΦ(z)−1F
+†�−1
+.
+(40)
+DRAFT
+
+15
+The matrix-valued function �Ψ(z), Ψ(z), �Φ(z), Φ(z) satisfy the following fixed-point equations
+�Ψ(z) = zIR −
+K
+�
+k=1
+�ηk(GCk(z)),
+(41)
+Ψ(z) = blkdiag
+�
+0R, −η1(G �C(z)), . . . , −ηK(G �C(z))
+�
+,
+(42)
+�Φ(z) = blkdiag
+�
+−�ζ0(G �D(z)), −�ζ1(G �D(z)), . . . , −�ζK(G �D(z))
+�
+,
+(43)
+Φ(z) = IT −
+K
+�
+k=0
+ζk(GDk(z)),
+(44)
+where blkdiag {A1, . . . , An} constructs a block diagonal matrix with square matrices A1, . . . , An being
+the diagonal blocks, and G �C(z), GCk(z), G �D(z), GDk(z) are given by
+G �C(z) =
+�
+�Ψ(z) − GΞ(z)−1G
+†�−1
+,
+(45)
+GCk(z) =
+��
+Ψ(z) − G
+† �Ψ(z)−1G −
+�
+�Φ(z) − FΦ(z)−1F
+†�−1�−1�
+k+1
+,
+1 ≤ k ≤ K,
+(46)
+G �D(z) =
+�
+Φ(z) − F
+†
+�
+�Φ(z) −
+�
+Ψ(z) − G
+† �Ψ(z)−1G
+�−1�−1
+F
+�−1
+,
+(47)
+GDk(z) =
+��
+�Φ(z) − FΦ(z)−1F
+† −
+�
+Ψ(z) − G
+† �Ψ(z)−1G
+�−1�−1�
+k+1
+,
+0 ≤ k ≤ K.
+(48)
+The notation {A}k+1 with n × n matrix A denotes the (k + 1)-th diagonal matrix block containing
+entries from �k−1
+i=0 Li + 1 to �k
+i=0 Li rows and columns of A.
+Proof: The proof of Proposition 2 is given in Appendix C.
+As indicated by Proposition 2, the Cauchy transform GB(z) as well as the matrix-valued functions
+�Ψ(z), Ψ(z), �Φ(z), Φ(z) can be determined by solving the fixed-point equations. The numerical value
+of GB(z) can be obtained by iterating the set of equations (41)-(44) and (45)-(48).
+IV. NUMERICAL RESULTS
+In this section, numerical simulations are conducted to study the spectral distribution of the RIS-assisted
+MIMO channel as well as its mutual information. In particular, we examine the impacts of the number of
+RIS panels, the number of antennas at the transceivers, and the Rician factor of the propagation channels
+DRAFT
+
+16
+on the mutual information. The mutual information I(γ) and the eigenvalue PDF fB(t) are calculated
+by (17) and (20), respectively, where the involved Cauchy transform GB(z) is given in Proposition 2. In
+each simulation case, the MIMO system without RIS deployment is included for comparison, i.e., K = 0,
+where the eigenvalue PDF and the Cauchy transform can be calculated by using existing result from [32,
+Thm. 2]. Each simulation curve is obtained by averaging over 106 independent channel realizations.
+In the simulations, the antenna elements of the transceivers and the reflecting elements of the RIS
+panels are arranged as the uniform planar arrays (UPAs). Denote T = T (H) × T (V ), R = R(H) × R(V ),
+and Lk = L(H)
+k
+× L(V )
+k
+, where the numbers with the superscripts H and V represent the numbers of
+elements aligned in the horizontal and vertical dimensions, respectively. The specular component of each
+channel is the line-of-sight propagation component between two uniform planar arrays (UPAs), i.e.,
+Fk = a
+�
+ϕ(F)
+k
+, ν(F)
+k
+, L(H)
+k
+, L(V )
+k
+�
+a† �
+θ(F)
+k
+, φ(F)
+k
+, T (H), T (V )�
+,
+0 ≤ k ≤ K,
+(49)
+Gk = a
+�
+ϕ(G)
+k
+, ν(G)
+k
+, R(H), R(V )�
+a† �
+θ(G)
+k
+, φ(G)
+k
+, L(H)
+k
+, L(V )
+k
+�
+,
+1 ≤ k ≤ K,
+(50)
+where θ(i)
+k
+and φ(i)
+k
+are the azimuth and elevation angles of the k-th departing UPA, while ϕ(i)
+k
+and ν(i)
+k
+are the azimuth and elevation angles of the k-th arriving UPA, i ∈ {F, G}. The function a(·) denotes
+the steering vector of an M × N UPA and is defined as
+a(α, β, M, N) =
+�
+1, . . . , eiπ(n sin(α) sin(β)+m cos(β)), . . . , eiπ((N−1) sin(α) sin(β)+(M−1) cos(β))�T
+,
+(51)
+where 0 ≤ m ≤ M − 1 and 0 ≤ n ≤ N − 1.
+Fig. 2 shows the empirical and asymptotic eigenvalue PDF of the RIS-assisted MIMO channels HH†,
+assuming the number of RIS panels is K = 0, 1, 2, and 4, respectively. In all the cases, the numbers of
+transmit and receive antennas are set to T = R = 64, and the number of reflecting elements in each RIS
+panel is set to 144. The channel statistics, such as Fk, Uk, Vk, Mk in (3), and Gk, Wk, Sk, Nk in
+(4) are randomly generated but fix for the Monte Carlo simulations. The numerical results show that the
+asymptotic PDF calculated by (20) provides an excellent approximation to the simulated PDF for all the
+considered parameter configurations. By increasing the number of deployed RIS panels, it is possible to
+increase the maximum eigenvalue, therefore, improve amplitude of the eigen-channels.
+DRAFT
+
+17
+0
+1
+2
+3
+4
+5
+0
+0.5
+1
+1.5
+PDF
+(a) K = 0
+0
+1
+2
+3
+4
+5
+6
+7
+8
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+PDF
+(b) K = 1
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+PDF
+(c) K = 2
+0
+2
+4
+6
+8
+10
+12
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+PDF
+(d) K = 4
+Fig. 2.
+Comparisons of empirical and asymptotic eigenvalue PDFs of the RIS-assisted MIMO channels HH† with different
+numbers of RIS panels. The numbers of transmit and receive antennas are set to T = R = 64, and the number of reflecting
+elements of each RIS panel is set to 144.
+In Fig. 3, we investigate the impacts of the SNR, the number of antennas, and the Rician factors
+on the mutual information of the RIS-assisted MIMO channels. Equal number of antennas is set at the
+transmitter and the receiver, where T = R = 4 in Fig. 3 (a) and T = R = 8 in Fig. 3 (b), respectively.
+The MIMO communication is assisted by K = 6 RIS panels, and each RIS panel is composed of
+16 reflecting elements. Compared to the direct link F0, the relative channel gains [ρ1, . . . , ρ6] in (15)
+corresponding to the reflected links are configured as [0.9, 0.8, 0.7, 0.5, 0.3, 0.1]. All the Rician factors
+are set equal as κ = κ(F)
+k
+= κ(G)
+k
+, where κ is set to 1, 10, or 100. In presence of non-degenerate random
+DRAFT
+
+18
+scattering components �Fk in (3) and �Gk in (4), the RIS-assisted MIMO channels are full-rank, and the
+mutual information at large SNR linearly increases as min{T, R}/10 log10(e) nats/s/Hz for every 1 dB
+SNR improvement, depicted as the dashed lines in Fig. 3. However, as the Rician factor becomes large,
+although the mutual information has the same scaling law, it requires larger SNR levels to exhibit the
+linear improvement. This is illustrated in the insets of Fig. 3. When the Rician factors are κ = 1, 10,
+and 100, the asymptotic mutual information has at least 5% deviation from the high-SNR scaling law at
+SNRs 20.8 dB, 27.5 dB, and 34.2 dB when T = R = 4, and at SNRs 23.7 dB, 29.8 dB, and 36.1 dB
+when T = R = 8, respectively. This is due to the fact that as the Rician factor increases, the random
+scattering components to maintain the rank of the channel have less contributions to the overall MIMO
+channels.
+To further investigate the impacts of the Rician factor on the mutual information of the MIMO channels,
+we plot Fig. 4 to show the mutual information as a function of κ, with the numbers of RIS panels K
+set to 0, 1, 2, and 4, respectively. The number of transmit and receive antennas are set to T = 16 and
+R = 8, while the performance of the MIMO system is evaluated at SNR γ = 10 dB. It is observed that
+when κ is less than 1, the mutual information can be improved as κ increases, while it monotonically
+decreases for κ > 1 in all the considered cases. When the number of RIS panels is larger, the mutual
+information degradation is less prominent as each RIS provides independent reflected link, which increases
+the richness of the MIMO channels.
+In Fig. 5, the impact of the numbers of RIS panels is investigated in more details, when the mutual
+information is evaluated for different transmit antennas T = 8, 16, 32, and 64. The number of receive
+antennas is fixed to R = 10, and each RIS panel has 8 reflecting elements. In this simulation setting, we
+consider the urban canyon communication scenario as depicted in Fig. 1, where the specular components
+of {Fk} channels and of {Gk} channels have relatively small angular variations. That is, in (49)
+and (50), we assume that the departing angles
+�
+θ(F)
+k
+, φ(F)
+k
+�
+0≤k≤K of the transmitter UPA and the
+arriving angles
+�
+ϕ(G)
+k
+, ν(G)
+k
+�
+1≤k≤K of the receiver UPA are uniformly and randomly generated in some
+fixed intervals having length 0.05π. The departing angles
+�
+θ(G)
+k
+, φ(G)
+k
+�
+1≤k≤K and the arriving angles
+DRAFT
+
+19
+0
+10
+20
+30
+40
+5
+10
+15
+20
+25
+30
+35
+40
+Mutual information (nats/s/Hz)
+Asymptotic MI
+High-SNR MI
+Simulation
+20
+25
+30
+35
+15
+16
+17
+18
+ = 1, 10, 100
+(a) T = R = 4
+0
+10
+20
+30
+40
+50
+0
+10
+20
+30
+40
+50
+60
+70
+80
+Mutual information (nats/s/Hz)
+Asymptotic MI
+High-SNR MI
+Simulation
+22
+24
+26
+28
+30
+32
+34
+36
+38
+26
+28
+30
+32
+34
+ = 1, 10, 100
+(b) T = R = 8
+Fig. 3.
+Mutual information of RIS-assisted MIMO channels at varying SNR γ, when the number of antennas at the transceivers
+is T = R = 4 in (a) and T = R = 8 in (b), respectively. In each case, the Rician factor of the component channels is set equal
+to κ = 1, 10, or 100. There are K = 6 deployed RIS panels, each of which has 16 reflecting elements. Insets show the 5%
+deviations of the asymptotic mutual information from the high-SNR scaling law.
+DRAFT
+
+20
+0
+2
+4
+6
+8
+10
+12
+14
+4
+6
+8
+10
+12
+14
+16
+Mutual information (nat/s/Hz)
+Fig. 4. Mutual information of RIS-assisted MIMO channel for varying Rician factor κ. The number of antennas at the transmitter
+and the receiver are T = 16 and R = 8, respectively, and each RIS panel has 8 reflecting elements. The SNR of the end-to-end
+channel is set to γ = 10 dB.
+�
+ϕ(F)
+k
+, ν(F)
+k
+�
+1≤k≤K of the RIS panels are randomly generated in some fixed intervals having length
+0.1π. As K increases, Fig. 5 shows that the mutual information first improves at a larger rate between
+0 ≤ K ≤ 5, and then becomes slower thereafter. This is due to the fact that the richness of the channels
+can be improved more efficiently when the number of reflected links is small. Since the angular ranges
+are restricted, the added RIS panels have similar reflected links that cannot provide additional richness.
+Therefore, it is less effective to deploy more RIS panels to improve the mutual information. Finally, as
+shown in Figs. 3-5, the mutual information calculated by (17) via the Cauchy transform (39) achieves a
+good agreement with the simulation in all the considered simulation cases, and thus, can be applied to
+evaluate the performance of the RIS-assisted MIMO channels.
+V. CONCLUSIONS
+This paper studies the information-theoretic data rate of the RIS-assisted MIMO systems, where
+multiple RIS panels are deployed to improve the scattering-limited MIMO channels. By using the
+DRAFT
+
+21
+0
+5
+10
+15
+6
+7
+8
+9
+10
+11
+12
+13
+Mutual information (nat/s/Hz)
+Fig. 5.
+Mutual information of RIS-assisted MIMO channel for varying numbers of RIS panels K. The number of receive
+antennas is R = 8, the number of elements in each RIS panel is 8, and the SNR of the channel is γ = 10 dB.
+operator-valued free probability theory, the Cauchy transform of the MIMO matrix is obtained using
+the general Rician MIMO model with Weichselberger’s correlation structure. Based on this result, the
+asymptotic eigenvalue distribution of the channel matrix as well as the mutual information of the MIMO
+channel are calculated, which closely match the corresponding simulation results for practical system
+configurations. Numerical results show that the additional reflected links created by the RIS panels can
+increase the range of eigenvalues of the channel matrix, which can be leveraged to improve the amplitude
+of the eigen-channels. In the MIMO communications, the negative impact of a large Rician factor on the
+mutual information can be partly alleviated by deploying more RIS panels. However, the performance
+improvement of the multi-RIS deployment slows down when the added reflected links have similar
+arriving and departing angles.
+DRAFT
+
+22
+APPENDIX A
+SOME USEFUL MATRIX INVERSION IDENTITIES
+For the sake of completeness, the following matrix inversion identities are summarized in Lemmas 1-3,
+which are repeatedly applied throughout this paper. For notational simplicity, in this appendix, we use
+italic bold symbols to define matrices, which are different from those used in the main sections.
+Lemma 1. (Woodbury matrix inversion identity [31, Eq. (0.7.4.1)].) Let A denote a m × m invertible
+matrix, D denote a k × k matrix, B and C denote m × k and k × m matrices, respectively. Then the
+following identity holds
+(A + BDC)−1 = A−1 − A−1B
+�
+D−1 + CA−1B
+�−1 CA−1.
+(52)
+Lemma 2. (2 × 2 block matrix inversion identity [31, Eq. (0.7.3.1)].) Let A, B, C, and D be defined
+as in Lemma 1, the inversion identity of the following 2 × 2 block matrix holds
+�
+��
+A
+B
+C
+D
+�
+��
+−1
+=
+�
+��
+A−1 + A−1B(D − CA−1B)−1CA−1
+−A−1B(D − CA−1B)−1
+−(D − CA−1B)−1CA−1
+(D − CA−1B)−1
+�
+��
+=
+�
+��
+(A − BD−1C)−1
+−A−1B(D − CA−1B)−1
+−(D − CA−1B)−1CA−1
+(D − CA−1B)−1
+�
+�� ,
+(53)
+where the second equality holds when D is also invertible.
+Lemma 3. (3 × 3 block matrix inversion identity.) Let the matrices E, F , G, H, J, K, L, M, and N
+be the conformable partitions of the following 3 × 3 block matrix X
+X =
+�
+������
+E
+F
+G
+H
+J
+K
+L
+M
+N
+�
+������
+.
+DRAFT
+
+23
+When E is invertible, the inversion of X is given by
+X−1 =
+�
+������
+E−1 + E−1(F A−1H + US−1V )E−1
+−E−1(F − US−1C)A−1
+−E−1US−1
+−A−1(H − BS−1V )E−1
+A−1 + A−1BS−1CA−1
+−A−1BS−1
+−S−1V E−1
+−S−1CA−1
+S−1
+�
+������
+,
+where
+A = J − HE−1F ,
+B = K − HE−1G,
+C = M − LE−1F ,
+D = N − LE−1G,
+(54)
+U = G − F A−1B,
+V = L − CA−1H,
+(55)
+S = D − CA−1B.
+(56)
+Proof: Apply Lemma 2 to the inversion of X that is partitioned into a 2 × 2 block matrix as
+X−1 =
+�
+������
+E
+F
+G
+H
+J
+K
+L
+M
+N
+�
+������
+−1
+=
+�
+��
+P
+Q
+R
+Z−1
+�
+�� ,
+(57)
+where the matrix blocks P , Q, and R are given by
+P = E−1 + E−1
+�
+F
+G
+�
+Z−1
+�
+��
+H
+L
+�
+�� E−1,
+(58)
+Q = −E−1
+�
+F
+G
+�
+Z−1,
+(59)
+R = −Z−1
+�
+��
+H
+L
+�
+�� E−1,
+(60)
+and the matrix Z is a 2 × 2 block matrix such that
+Z =
+�
+��
+J
+K
+M
+N
+�
+�� −
+�
+��
+H
+L
+�
+�� E−1
+�
+F
+G
+�
+=
+�
+��
+A
+B
+C
+D
+�
+�� ,
+(61)
+where A, B, C, and D are given in (54).
+Applying again Lemma 2 to the inversion of Z, we obtain
+Z−1 =
+�
+��
+A−1 + A−1BS−1CA−1
+−A−1BS−1
+−S−1CA−1
+S−1
+�
+�� ,
+(62)
+DRAFT
+
+24
+where S is given in (56). Substituting (62) into (58), P can be rewritten as
+P = E−1 + E−1
+�
+F A−1 − US−1CA−1
+US−1
+�
+�
+��
+H
+L
+�
+�� E−1
+= E−1 + E−1 �
+F A−1H + US−1V
+�
+E−1,
+(63)
+where U and V are defined in (55). Similarly, Q and R can be obtained as
+Q =
+�
+−E−1 �
+F − (G − F A−1B)S−1C
+�
+A−1
+−E−1 �
+G − F A−1B
+�
+S−1
+�
+=
+�
+−E−1 �
+F − US−1C
+�
+A−1
+−E−1US−1
+�
+,
+(64)
+R =
+�
+��
+−A−1HE−1 + A−1BS−1(L − CA−1H)E−1
+−S−1(L − CA−1H)E−1
+�
+�� =
+�
+��
+−A−1(H − BS−1V )E−1
+−S−1V E−1
+�
+�� .
+(65)
+Finally, substituting (62)-(65) into (57) completes the proof of Lemma 3.
+APPENDIX B
+PROOF OF PROPOSITION 1
+A random variable �L ∈ M is said to be D-valued semicircular if the free cumulant
+κD
+m(�Lb1, �Lb2, . . . , �Lbm−1, �L) = 0,
+(66)
+for all n ̸= 2, and all b1, . . . , bn−1 ∈ D. The free cumulant κD
+m is a mapping from Mm to D and we
+refer the reader to [29] for detailed explanations on this topic. The proof is followed by expanding �L
+into a sum of n × n matrices, such that
+�L =
+K
+�
+k=0
+�L(F)
+k
++
+K
+�
+k=1
+�L(G)
+k
+,
+(67)
+DRAFT
+
+25
+where the matrices �L(F)
+k
+and �L(G)
+k
+are given by
+�L(F)
+k
+=
+�
+���������
+0R×L
+�Fk
+�F†
+k
+0L×R
+�
+���������
+,
+(68)
+�L(G)
+k
+=
+�
+���������
+�Gk
+0L×T
+0T×L
+�G†
+k
+�
+���������
+,
+(69)
+where �Fk and �Gk are L × T and R × L matrices, respectively, and are given by
+�Fk =
+�
+0T×L0
+. . .
+�F†
+k
+. . .
+0T×LK
+�†
+,
+0 ≤ k ≤ K,
+(70)
+�Gk =
+�
+0R×R
+0R×L1
+. . .
+√ρk �Gk
+. . .
+0R×LK
+�
+,
+1 ≤ k ≤ K.
+(71)
+Recalling the definitions of �Fk and �Gk in (3) and (4), we have
+�L(F)
+k
+= A(F)
+k
+�X kA(F)†
+k
+,
+(72)
+�L(G)
+k
+= A(G)
+k
+�YkA(G)†
+k
+,
+(73)
+where the matrix �X k has the same structure as the block matrix �L(F)
+k
+in (68) while replacing �Fk in (70)
+with �Xk = Mk ⊙ Xk, and �Yk has the same structure as the block matrix �L(G)
+k
+in (69) while replacing
+�Gk in (71) with �Yk =
+1
+√rk Nk ⊙ Yk. The n × n matrices A(F)
+k
+and A(G)
+k
+are given by
+A(F)
+k
+=
+�
+��
+�Uk
+0(R+L)×(T+L)
+0(T+L)×(R+L)
+�Vk
+�
+�� ,
+(74)
+A(G)
+k
+=
+�
+��
+�
+Wk
+0(R+L)×(T+L)
+0(T+L)×(R+L)
+�Sk
+�
+�� ,
+(75)
+DRAFT
+
+26
+where �Uk, �Vk, �
+Wk, �Sk are deterministic diagonal block matrices and are given by
+�Uk = blkdiag(0R, 0L0, . . . , Uk, . . . , 0LK),
+(76)
+�Vk = blkdiag(Vk, 0L),
+(77)
+�
+Wk = blkdiag(Wk, 0L),
+(78)
+�Sk = blkdiag(0T , 0L0, . . . , Sk, . . . , 0LK).
+(79)
+Since { �X k}0≤k≤K, {�Yk}1≤k≤K are Wigner matrices and independent from each other, they are semi-
+circular and free over the sub-algebra Dn ⊂ M of n × n diagonal matrices. Then, following the same
+arguments as in [32, Appendix B], {�L(F)
+k
+}0≤k≤K and {�L(G)
+k
+}1≤k≤K are semicircular and free over sub-
+algebra of block diagonal matrices D. Therefore, the sum of �L(F)
+k
+and �L(G)
+k
+is also semicircular over D
+and is free from any deterministic matrix from M.
+APPENDIX C
+PROOF OF PROPOSITION 2
+Since �L is an operator-valued semicircular variable over D and �L are free from L over D, the
+limiting spectral distribution of L is a free additive convolution of the limiting spectral distributions of
+�L and L. Specifically, the operator-valued Cauchy transform GD
+L can be calculated via the subordination
+formula (38). Recall that the R-transform RD
+�L (·) is the free cumulant generating function of �L with the
+following formal power series expansion:
+RD
+�L (K) = κD
+1 ( �K) + κD
+2 (�LK, �L) + κD
+3 (�LK, �LK, �L) + · · · ,
+(80)
+where κD
+i denotes the i-th free cumulant of �L over D. In addition, since �L is semicircular over D, all
+its cumulants in (80) except κD
+2 are zero. Therefore, the R-transform RD
+�L (K) reduces to the covariance
+DRAFT
+
+27
+function of �L over D parameterized by K, i.e.,
+RD
+�L(K) = ED
+�
+�LK�L
+�
+=
+�
+���������
+�K
+k=1 �ηk(Ck)
+�ζ( �D)
+�K
+k=0 ζk(Dk)
+η(�C)
+�
+���������
+,
+(81)
+where �ζ( �D) = blkdiag
+�
+�ζ0( �D), . . . , �ζK( �D)
+�
+and η(�C) = blkdiag
+�
+0R, η1(�C), . . . , ηK(�C)
+�
+.
+Since GD
+L (Λ(z)) ∈ D, by same matrix partitioning as in (23), GD
+L (Λ(z)) is partitioned into
+GD
+L (Λ(z)) = blkdiag
+�
+G �C(z), GD(z), G �D(z), GC(z)
+�
+,
+(82)
+where GD(z) = blkdiag {GD0(z), . . . , GDK(z)} and GC(z) = blkdiag {0R, GC1(z), . . . , GCK(z)}. Note
+that the upper-left block
+�
+GD
+L (Λ(z))
+�(1,1) = G �C(z), which is then used to compute GB(z) = 1
+RTr(G �C(z)).
+By replacing K in (81) with GD
+L (Λ(z)) in (82), and substituting L and RD
+�L with (36) and (81),
+respectively, we obtain GD
+L (Λ(z)) as
+GD
+L (Λ(z)) =
+�
+���������
+G �C(z)
+GD(z)
+G �D(z)
+GC(z)
+�
+���������
+= ED
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�Ψ(z)
+0
+0
+−G
+0
+�Φ(z)
+−F
+IL
+0
+−F
+†
+Φ(z)
+0
+−G
+†
+IL
+0
+Ψ(z)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+−1
+,
+(83)
+where �Ψ(z), Ψ(z), �Φ(z), and Φ(z) are given in (41)-(44). By invoking Lemma 2 to the RHS of (83)
+and taking expectation over D, the matrix-valued function G �C(z) = A−1
+1 , and GD(z), G �D(z), GC(z) are
+DRAFT
+
+28
+the diagonal blocks of the matrix A−1
+2 , where A1 and A2 are given by
+A1 = �Ψ(z) −
+�
+0
+0
+G
+�
+�
+������
+�Φ(z)
+−F
+IL
+−F
+†
+Φ(z)
+0
+IL
+0
+Ψ(z)
+�
+������
+−1 �
+������
+0
+0
+G
+†
+�
+������
+,
+(84)
+A2 =
+�
+������
+�Φ(z)
+−F
+IL
+−F
+†
+Φ(z)
+0
+IL
+0
+Ψ(z)
+�
+������
+−
+�
+������
+0
+0
+G
+†
+�
+������
+�Ψ(z)−1
+�
+0
+0
+G
+�
+.
+(85)
+Applying Lemma 3, the RHS of (84) can be further derived as
+A1 = �Ψ(z) − GS−1G
+†,
+(86)
+where S = Ξ(z) and is calculated in (56) as
+S = Ξ(z) = Ψ(z) − �Φ(z)−1 − �Φ(z)−1F
+�
+Φ(z) − F
+† �Φ(z)−1F
+�−1
+F
+† �Φ(z)−1
+= Ψ(z) −
+�
+�Φ(z) − FΦ(z)−1F
+†�−1
+.
+(87)
+The second equality of (87) is obtained by applying Lemma 1. Then, (45) is established by combining
+(86) and (87).
+The inverse of A2 can be explicitly calculated via Lemma 3, where E = �Φ(z), F = H† = −F,
+G = L = IL, J = Φ(z), K = M = 0, and N = Ψ(z) − G
+† �Ψ(z)−1G. We further let T =
+Φ(z)−F
+† �Φ(z)−1F and �T = �Φ(z)−FΦ(z)−1F
+†. Then, the matrix-valued functions GD(z), G �D(z), and
+GC(z), being the diagonal blocks of A−1
+2 , are given by
+GD(z) = �Φ(z)−1 + �Φ(z)−1FT −1F
+† �Φ(z)−1 + �T −1 �
+N − �T −1�−1 �T −1,
+(88)
+G �D(z) = T −1 + T −1F
+† �Φ(z)−1 �
+N − �T −1�−1 �Φ(z)−1F T −1,
+(89)
+GC(z) =
+�
+N −
+�
+�Φ(z)−1 + �Φ(z)−1FT −1F
+† �Φ(z)−1��−1
+.
+(90)
+Finally, applying Lemma 1 to (88)-(90), we obtain GD(z), G �D(z), and GC(z) as in (46)-(48).
+DRAFT
+
+29
+REFERENCES
+[1] W. Saad, M. Bennis, and M. Chen, “A vision of 6G wireless systems: Applications, trends, technologies, and open research
+problems,” IEEE Netw., vol. 34, no. 3, pp. 134–142, May 2020.
+[2] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur. Trans. Telecommun., vol. 10, no. 6, pp. 585–595, 1999.
+[3] H. Shin and J. H. Lee, “Capacity of multiple-antenna fading channels: Spatial fading correlation, double scattering, and
+keyhole,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2636–2647, Oct. 2003.
+[4] C. Pan, H. Ren, K. Wang, J. F. Kolb, M. Elkashlan, M. Chen, M. Di Renzo, Y. Hao, J. Wang, and A. L. Swindlehurst,
+“Reconfigurable intelligent surfaces for 6G systems: Principles, applications, and research directions,” IEEE Commun.
+Mag., vol. 59, no. 6, pp. 14–20, June 2021.
+[5] Q. Wu and R. Zhang, “Towards smart and reconfigurable environment: Intelligent reflecting surface aided wireless network,”
+IEEE Commun. Mag., vol. 58, no. 1, pp. 106–112, Jan. 2020.
+[6] Y. Zhang, J. Zhang, M. Di Renzo, H. Xiao, and B. Ai, “Performance analysis of RIS-aided systems with practical phase
+shift and amplitude response,” IEEE Trans. Veh. Technol., vol. 70, no. 5, pp. 4501–4511, May 2021.
+[7] L. Yang, F. Meng, Q. Wu, D. B. Da Costa, and M.-S. Alouini, “Accurate closed-form approximations to channel distributions
+of RIS-aided wireless systems,” IEEE Wireless Commun. Lett., vol. 9, no. 11, pp. 1985–1989, Nov. 2020.
+[8] A. M. Salhab and M. H. Samuh, “Accurate performance analysis of reconfigurable intelligent surfaces over Rician fading
+channels,” IEEE Wireless Commun. Lett., vol. 10, no. 5, pp. 1051–1055, May 2021.
+[9] S. P. Dash, S. Joshi, and S. Aissa, “Envelope distribution of two correlated complex Gaussian random variables and
+application to the performance evaluation of RIS-assisted communications,” IEEE Commun. Lett., vol. 26, no. 9, pp.
+2018-2022, Sept. 2022.
+[10] Z. Xie, W. Yi, X. Wu, Y. Liu, and A. Nallanathan, “Downlink multi-RIS aided transmission in backhaul limited networks,”
+IEEE Wireless Commun. Lett., vol. 11, no. 7, pp. 1458–1462, July 2022.
+[11] R. Hashemi, S. Ali, N. H. Mahmood, and M. Latva-aho, “Average rate and error probability analysis in short packet
+communications over RIS-aided URLLC systems,” IEEE Trans. Veh. Technol., vol. 70, no. 10, pp. 10320–10334, Oct.
+2021.
+[12] I. Yildirim, A. Uyrus, and E. Basar, “Modeling and analysis of reconfigurable intelligent surfaces for indoor and outdoor
+applications in future wireless networks,” IEEE Trans. Commun., vol. 69, no. 2, pp. 1290–1301, Feb. 2021.
+[13] L. Yang, Y. Yang, D. B. Da Costa, and I. Trigui, “Outage probability and capacity scaling law of multiple RIS-aided
+networks,” IEEE Wireless Commun. Lett., vol. 10, no. 2, pp. 256–260, Feb. 2021.
+[14] T. N. Do, G. Kaddoum, T. L. Nguyen, D. B. Da Costa, and Z. J. Haas, “Multi-RIS-aided wireless systems: Statistical
+characterization and performance analysis,” IEEE Trans. Commun., vol. 69, no. 12, pp. 8641–8658, Dec. 2021.
+[15] Z. Shi, H. Wang, Y. Fu, G. Yang, S. Ma, and F. Gao, “Outage analysis of reconfigurable intelligent surface aided MIMO
+communications with statistical CSI,” IEEE Trans. Wireless Commun., vol. 21, no. 2, pp. 823–839, Feb. 2022.
+DRAFT
+
+30
+[16] B. D. Carter and M. D. Springer, “The distribution of products, quotients and powers of independent H-function variates,”
+SIAM J. Applied Maths., vol. 33, no. 4, pp. 542–558, Dec. 1977.
+[17] O. E. Ayach, S. Rajagopal, S. Abu-Surra, Z. Pi, and R. W. Heath, “Spatially sparse precoding in millimeter wave MIMO
+systems,” IEEE Trans. Wireless Commun., vol. 13, no. 3, pp. 1499–1513, Mar. 2014.
+[18] R. Li, S. Sun, Y. Chen, C. Han, and M. Tao, “Ergodic achievable rate analysis and optimization of RIS-assisted millimeter-
+wave MIMO communication systems,” IEEE Trans. Wireless Commun., in press.
+[19] K. Xu, J. Zhang, X. Yang, S. Ma, and G. Yang, “On the sum-rate of RIS-assisted MIMO multiple-access channels over
+spatially correlated Rician fading,” IEEE Trans. Commun., vol. 69, no. 12, pp. 8228–8241, Dec. 2021.
+[20] X. Zhang, X. Yu, and S. H. Song, “Outage probability and finite-SNR DMT analysis for IRS-aided MIMO systems: How
+large IRSs need to be?,” IEEE J. Sel. Top. Signal Process., vol. 16, no. 5, pp. 1070–1085, Aug. 2022.
+[21] J. Zhang, J. Liu, S. Ma, C.-K. Wen, and S. Jin, “Large system achievable rate analysis of RIS-assisted MIMO wireless
+communication with statistical CSIT,” IEEE Trans. Wireless Commun., vol. 20, no. 9, pp. 5572–5585, Sept. 2021.
+[22] W. Weichselberger, M. Herdin, H. ¨Ozcelik, and E. Bonek, “A stochastic MIMO channel model with joint correlation of
+both link ends,” IEEE Trans. Wireless Commun., vol. 5, no. 1, pp. 90-100, Jan. 2006.
+[23] Z. Zheng, L. Wei, R. Speicher, R. R. M¨uller, J. H¨am¨al¨ainen, and J. Corander, “Asymptotic analysis of Rayleigh product
+channels: A free probability approach,” IEEE Trans. Inf. Theory, vol. 63, no. 3, pp. 1731–1745, Mar. 2017.
+[24] R. R. M¨uller, “A random matrix model of communication via antenna arrays,” IEEE Trans. Inf. Theory, vol. 48, no. 9, pp.
+2495–2506, Sept. 2002.
+[25] R. Couillet and M. Debbah, Random Matrix Methods for Wireless Communications. Cambridge: Cambridge University
+Press, 2011.
+[26] R. R. M¨uller, “On the asymptotic eigenvalue distribution of concatenated vector-valued fading channels,” IEEE Trans. Inf.
+Theory, vol. 48, no. 7, pp. 2086-2091, July 2002.
+[27] F. Benaych-Georges, “Rectangular random matrices, related free entropy and free Fischer’s information,” J. Operator Th.,
+vol. 62, no. 2, pp. 371-419, 2009.
+[28] M. Capitaine and C. Donati-Martin, “Strong asymptotic freeness for Wigner and Wishart matrices,” Indiana Univ. Math.
+J., vol. 56, no. 2, pp. 767-803, 2007.
+[29] J. A. Mingo and R. Speicher, Free Probability and Random Matrices. New York: Springer, 2017.
+[30] S. T. Belinschi, T. Mai, and R. Speicher, “Analytic subordination theory of operator-valued free additive convolution and
+the solution of a general random matrix problem,” J. Reine Angew. Math. (Crelles J.), vol. 2017, no. 732, pp. 21–53, Apr.
+2017.
+[31] R. A. Horn and C. R. Johnson, Matrix Analysis. New York: Cambridge University Press, 2013.
+[32] A.-A. Lu, X. Gao, and C. Xiao, “Free deterministic equivalents for the analysis of MIMO multiple access channel,” IEEE
+Trans. Inf. Theory, vol. 62, no. 8, pp. 4604-4629, Aug. 2016.
+DRAFT
+
diff --git a/3NFLT4oBgHgl3EQfri9A/content/tmp_files/load_file.txt b/3NFLT4oBgHgl3EQfri9A/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..c18a40d0dfb28a20047bf91ff63cff520759a104
--- /dev/null
+++ b/3NFLT4oBgHgl3EQfri9A/content/tmp_files/load_file.txt
@@ -0,0 +1,857 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf,len=856
+page_content='1  On the Mutual Information of Multi-RIS  Assisted MIMO: From Operator-Valued Free  Probability Aspect  Zhong Zheng, Member, IEEE, Siqiang Wang, Zesong Fei, Senior Member, IEEE,  Zhi Sun, Senior Member, IEEE, Jinhong Yuan, Fellow, IEEE  Abstract  The reconfigurable intelligent surface (RIS) is useful to effectively improve the coverage and data rate  of end-to-end communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' In contrast to the well-studied coverage-extension use case, in this paper,  multiple RIS panels are introduced, aiming to enhance the data rate of multi-input multi-output (MIMO)  channels in presence of insufficient scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Specifically, via the operator-valued free probability theory,  the asymptotic mutual information of the large-dimensional RIS-assisted MIMO channel is obtained  under the Rician fading with Weichselberger’s correlation structure, in presence of both the direct and  the reflected links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Although the mutual information of Rician MIMO channels scales linearly as the  number of antennas and the signal-to-noise ratio (SNR) in decibels, numerical results show that it requires  sufficiently large SNR, proportional to the Rician factor, in order to obtain the theoretically guaranteed  linear improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' This paper shows that the proposed multi-RIS deployment is especially effective to  improve the mutual information of MIMO channels under the large Rician factor conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' When the  reflected links have similar arriving and departing angles across the RIS panels, a small number of RIS  panels are sufficient to harness the spatial degree of freedom of the multi-RIS assisted MIMO channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Zheng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Wang, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Fei are with the School of Information and Electronics, Beijing Institute of Technology, Beijing,  China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Sun is with the Department of Electronic Engineering, Tsinghua University, Beijing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Yuan is with the School  of Electrical Engineering and Telecommunications, UNSW Sydney, Sydney, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  DRAFT  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='12144v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='IT]  28 Jan 2023  2  Index Terms  Reconfigurable intelligent surface, MIMO, Rician channel, mutual information, operator-valued free  probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' INTRODUCTION  In both the current and forthcoming generations of mobile communication systems, multi-input multi-  output (MIMO) is one of the mainstream physical-layer techniques to improve the spectral efficiency  and the reliability of the wireless communications [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' In the favorable environments with rich scattering,  MIMO is able to increase the achievable data rate linearly with the number of antennas [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' However,  when the wireless systems operate in higher frequencies with larger bandwidth, such as the millimeter  wave and terahertz bands, the radio signals are easily attenuated due to absorption and blockage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' In this  case, the MIMO channels typically have only a few dominating propagation paths and/or limited angular  spread, which causes rank deficiency in the channel matrix that significantly degrades the MIMO channel  capacity [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Recently, reconfigurable intelligent surface (RIS) has attracted substantial attentions and is foreseen  to be an important component in the future communication systems [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' A typical RIS consists of  a large number of low-power integrated electronic circuits, which can be programmed to modify the  electromagnetic properties of the incoming radio waves in the desired frequency band [5], such as the  phase and amplitude of the reflected signals from each programmable circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Therefore, by deploying  some RIS panels in the environment, the signal’s radiation pattern within the operating spectrum bands  of the communication systems can be reconfigured to increase the number of independent paths with  diversified angular spreads, thus increasing the rank of the MIMO channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' As an example, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 1  illustrates the transmissions between a base station (BS) and a user equipment (UE) in an urban canyon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  In this scenario, without RIS deployment, the signals have to propagate through a scattering-limited area,  where the direct propagation link F0 dominates the end-to-end channel, while other scattered/reflected  components are severally attenuated by the building materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' In comparison, RIS panels are able to  actively and effectively reflect the signals to increase the number of independent specular components,  DRAFT  3  • •  UE  BS  RIS K  RIS 1  F0  F1  FK  G1  GK  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Multi-RIS assisted MIMO communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  resulting in a total number of K +1 propagation links, including the direct link F0 and K reflected links  that consist of channels {Fk}1≤k≤K between BS and RIS panels and channels {Gk}1≤k≤K between RIS  panels and UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  There exist a number of studies focusing on the performance evaluation of the end-to-end communi-  cations assisted by a single or multiple RIS panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' When both the transmitter and receiver are equipped  with a single antenna and a single RIS is deployed, the signal-to-noise-ratio (SNR) of such RIS-assisted  single-input single-output (SISO) channel is proportional to the squared amplitude of the end-to-end  effective channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' There are two typical theoretic frameworks to analyze the statistical properties of  the SNR: One is based on the Meijer’s G- and Fox’s H-function systems [6], which result in exact  but rather complicated expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The other is to match the moments of the effective channel with  classical random variables [7]–[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' In particular, when the direct link is blocked, the SNR distribution and  the corresponding outage probability of the RIS-assisted communications are approximated by Gamma  random variables, when the component channels are independently Rayleigh-faded [7], independently  Rician-faded [8], correlated Rician-faded [9], and independently Nakagami-faded [10], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' When  both the direct and the reflected links exist, the Gamma-approximated SNR distribution and the finite  block-length rate of the RIS-assisted channel are obtained in [11], when all component channels are  DRAFT  4  Rayleigh-faded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  When multiple RIS panels are deployed in the network, the RIS panels either work in exhaustive mode  to jointly assist the end-to-end communications [12], or work in opportunistic mode, where only the RIS  with maximum channel gains is selected [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' In the former case, the end-to-end SNR is approximated  by a Gaussian random variable due to the central limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The SNR and the average symbol error  probability are derived for the phase shift keying signaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' In the latter case, the SNR of each reflected  link corresponding to one RIS panel is approximated as a Gaussian random variable and the order-statistics  is then obtained for the optimally selected RIS-assisting channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' In [14], a comprehensive performance  comparison of those two operation modes is provided, under the scenario of multi-RIS assisted SISO  channels assuming Nakagami-faded direct and reflected links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  In the case of MIMO systems, the available performance analysis of RIS-assisted communications is  rather limited due to the challenge of understanding the statistical distribution of matrix-valued prop-  agation channels, and results only exist for the single-RIS deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' In [15], the authors consider  the RIS-assisted MIMO communications, where the direct link is blocked and the reflected link is the  concatenated Rayleigh-faded MIMO channels with single-sided correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The exact outage probability  of such channel is derived by using the Mellin transform [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The result is expressed as the integration  of product of multiple Meijer’s G-functions, which is difficult to solve in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' When the reflected  link is the concatenated millimeter wave MIMO channels assuming Saleh-Valenzuela model [17], an  upper bound of the ergodic achievable rate is derived in [18] using majorization theory and Jensen’s  inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' In the RIS-assisted uplink multiple access channel, the asymptotic ergodic sum rate of the  multi-user MIMO system is derived in [19] by using the replica method, assuming that the reflected links  are Rician-faded MIMO channels with Kronecker’s correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' In the same channel model as [19], the  finite-SNR diversity-multiplexing tradeoff (DMT) of the RIS-assisted MIMO channel is analyzed in [20]  by the martingale method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' When both the direct and the reflected links exist, the asymptotic achievable  rate of the single-RIS assisted MIMO channel is derived in [21] via replica method, assuming that all  the component channels are Rician fading with Kronecker’s correlation and all the channel dimensions  grow to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  DRAFT  5  Although the RIS-assisted MIMO communications have been investigated in [15], [18]–[21], the results  therein are obtained for the single-RIS deployment under certain MIMO channel configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' In  contrast, this paper aims to provide the theoretic framework that analyzes the general multi-RIS assisted  MIMO communications under arbitrary Rician fading with Weichselberger’s correlation structure [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Such a model fits a wider range of realistic MIMO channels compared to the conventional Kronecker’s  correlation structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Based on the above system settings, we first embed the component MIMO channel  matrices into a large block matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Then, an operator-valued probability space over the algebra of the  constructed block matrices is defined, where the operator-valued Cauchy transform is defined and is shown  to be closely related to the classic Cauchy transform of the channel Gram matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The operator-valued  Cauchy transform is then derived by leveraging the freeness over the defined probability space and the  additive free convolution machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Based on the obtained Cauchy transform, the probability distribution  of the eigenvalue of the channel Gram matrix as well as the mutual information of the multi-RIS assisted  MIMO channel can be calculated, which avoids time-consuming Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Numerical  results show that in presence of strong line-of-sight conditions, although the mutual information could  scale linearly as the number of antennas and the SNRs (in decibels), the SNR has to be sufficiently large  in order to exhibit such linear scaling law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' On the other hand, deploying additional RIS panels could  effectively improve the channel’s mutual information and thus, alleviate the SNR requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  The rest of this article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The signal model, the channel model, and the mutual  information of the MIMO channel under consideration are introduced in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' In Section III, the  operator-valued probability space is introduced and the main result of the Cauchy transform of the channel  Gram matrix is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Numerical simulation results on the spectral distribution and the mutual information  of the MIMO channels are in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Section V concludes the main findings of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Throughout the paper, vectors and matrices are represented by lower-case and upper-case  bold-face letters, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The complex column vector with length n is denoted as Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' We use  CN(0, A) to denote the zero-mean complex Gaussian vector with covariance matrix A and In is an  n × n identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The superscript (·)† denotes the matrix conjugate-transpose operation and (·)T  is matrix transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' We denote Tr(A) as the trace of n × n matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The notation E[·] denotes the  DRAFT  6  expectation, and det(·) denotes the matrix determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' SYSTEM MODEL  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Signal Model  Consider a MIMO communication channel between a transmitter equipped with T antennas and a  receiver equipped with R antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The transmissions are assisted by K RIS panels, which reflect the  impinging signals via their reflecting elements and each RIS panel is equipped with Lk reflecting elements,  1 ≤ k ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' For notational simplicity, we define R = L0 and use these two symbols interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Denote the transmitted signal as x ∈ CT and the additive noise at the receiver as n ∈ CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The received  signal y ∈ CR is expressed as  y =  �  F0 +  K  �  k=1  √ρkGkFk  �  x + n,  (1)  where the R × T matrix F0 denotes the direct channel between transmitter and receiver, the Lk × T  matrix Fk denotes the channels between the transmitter and the k-th RIS panel, and 0 < ρk ≤ 1 denotes  the relative channel gain of the k-th reflected channel via the k-th RIS, compared to the direct channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  The R × Lk matrix Gk denotes phase-shifted reflected channel between the k-th RIS and the receiver,  modeled as  Gk = RkΘk,  (2)  where the R×Lk matrix Rk denotes the channel coefficients, and the diagonal matrix Θk = diag  �  eiφk,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , eiφk,Lk�  contains the phase-shifts of the reflecting elements, where 0 ≤ φk,l ≤ 2π denotes the phase-shift of the  l-th element of the k-th RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  We adopt the following assumptions on the signal and the channels:  (A1) The signal x is Gaussian distributed with uniform power allocation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=', x ∼ CN(0T , PIT ), where  P is the average power of the signals from each transmit antenna;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (A2) The noise n is assumed to be a white Gaussian random vector with i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' zero-mean entries, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=',  n ∼ CN(0R, σ2IR), where σ2 denotes the variance of the noise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  DRAFT  7  (A3) The channel coefficients {Fk}0≤k≤K and {Rk}1≤k≤K are block-faded, which keeps constant within  the coherence time, while changing randomly and independently in the next coherence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The  phase shifts {Θk}1≤k≤K are assumed to be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Note that without the direct link F0, channel models similar to (1) have been also studied in [3],  [23], [24] for the keyhole channel, the Rayleigh-product channel, and the double-scattering channel,  respectively, by using different theoretic techniques, which cannot be applied here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Channel Model  In order to characterize the directivity and the spatial correlation of the channels between antenna  arrays, we adopt the non-central Weichselberger’s MIMO model for each component links [22], such  that  Fk = Fk + �Fk = Fk + Uk(Mk ⊙ Xk)V†  k,  0 ≤ k ≤ K,  (3)  Gk = Gk + �Gk = Gk +  1  √rk  Wk(Nk ⊙ Yk)S†  k,  1 ≤ k ≤ K,  (4)  where Fk and Gk are the fixed specular components of Fk and Gk, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The random scattering  components are captured by �Fk and �Gk, where Uk, Vk, Wk, and Sk are deterministic unitary matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  The deterministic matrices Mk and Nk represent the variance profiles of �Fk and �Gk, respectively, each  having non-negative real elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The Lk × T random matrix Xk and the R × Lk random matrix Yk  are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' complex Gaussian distributed with entries having zero mean and variance 1/T, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=', [Xk]i,j ∼  CN(0, 1/T) and [Yk]i,j ∼ CN(0, 1/T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' We denote rk as the ratio between Lk and T, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=', rk = Lk/T,  1 ≤ k ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The operator ⊙ denotes the element-wise matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Note that the phase-shift  matrix Θk, 1 ≤ k ≤ K, is also unitary and can be absorbed into the deterministic matrices Gk and  Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The ratio between the power of fixed specular component and the random scattering component is  defined as the Rician factor of the MIMO channel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=',  κ(F)  k  = ||Fk||2  F  E[||�F||2  F]  , and κ(G)  k  = ||Gk||2  F  E[|| �G||2  F]  ,  (5)  where || · ||F denotes the Frobenius norm of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  DRAFT  8  For the correlated MIMO channel Gk, the one-sided correlation function ηk(�C) = E[ �G†  k �C �Gk] param-  eterized by an Hermitian matrix �C is given by [22, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 1] as  ηk(�C) = E[ �G†  k �C �Gk] = 1  Lk  SkΠk(�C)S†  k,  1 ≤ k ≤ K,  (6)  where the Lk × Lk diagonal matrix Πk(�C) contains the diagonal entries  �  Πk(�C)  �  i,i =  R  �  j=1  ([Nk]j,i)2 �  W†  k �CWk  �  j,j ,  1 ≤ i ≤ Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (7)  The other one-sided correlation function �ηk(Ck) = E[ �GkCk �G†  k] parameterized by Ck is given by  �ηk(Ck) = E[ �GkCk �G†  k] = 1  Lk  Wk �Πk(Ck)W†  k,  1 ≤ k ≤ K,  (8)  where the R × R diagonal matrix �Πk(Ck) contains the diagonal entries  �  �Πk(Ck)  �  i,i =  Lk  �  j=1  ([Nk]i,j)2 �  S†  kCkSk  �  j,j ,  1 ≤ i ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (9)  Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' for 0 ≤ k ≤ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' the two parameterized one-sided correlation functions of the matrix �Fk are  given by:  ζk(Dk) = E[�F†  kDk�Fk] = 1  T VkΣk(Dk)V†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (10)  �ζk( �D) = E[�Fk �D�F†  k] = 1  T Uk �Σk( �D)U†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (11)  where the T × T diagonal matrix Σk(Dk) and the Lk × Lk diagonal matrix �Σk( �D) respectively contain  the diagonal entries  [Σk(Dk)]i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='i =  Lk  �  j=1  ([Mk]j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='i)2 �  U†  kDkUk  �  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  1 ≤ i ≤ T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (12)  �  �Σk( �D)  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='i =  T  �  j=1  ([Mk]i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='j)2 �  V†  k �DVk  �  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  1 ≤ i ≤ Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (13)  In addition, since the channels {Fk}0≤k≤K and {Gk}1≤k≤K are spatially separated, channels corre-  spond to different links are assumed to be independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  DRAFT  9  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Mutual Information of Multi-RIS MIMO Channel  Due to the assumptions (A1)-(A3), the channel (1) is a Gaussian MIMO channel and its mutual  information is given by the well-known Telatar’s formula [2] as  I(γ) = log det  �  IR + γHH†�  ,  (14)  where γ = P/σ2 is the average SNR, and the end-to-end channel H is given by  H = F0 +  K  �  k=1  √ρkGkFk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (15)  The channel H can be factorized as the product of two matrices G and F as  H = GF =  �  IR  √ρ1G1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  √ρKGK  �  �  ���������  F0  F1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  FK  �  ���������  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (16)  Denoting L = �K  k=0 Lk, G =  �  IR  √ρ1G1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  √ρKGK  �  is a R × L block matrix and F =  �  FT  0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , FT  K  �T is a L × T block matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Letting B = HH† = GFF†G†, the mutual information (14) can be rewritten as  I(γ) = R VB(γ) = R  ˆ ∞  0  log(1 + γt)fB(t)dt,  (17)  where VB(x) is the Shannon transform of the matrix B [25], and fB(t) is the probability density function  (PDF) of the eigenvalue of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Applying the relation between the Shannon transform and the corresponding  Cauchy transform [25], the mutual information (17) can be rewritten as  I(γ) = R  ˆ γ  0  �1  t + 1  t2 GB  �  −1  t  ��  dt,  (18)  where GB(z) is the Cauchy transform of B and is defined as  GB(z) =  ˆ ∞  0  1  z − tfB(t)dt = 1  RTr ◦ E  �  (zI − B)−1�  = τR  �  (zI − B)−1�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (19)  Here, τR(X) is the composite function 1  RTr◦E[X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Note that the PDF fB(t) has an one-to-one mapping  with the Cauchy transform GB(z) via the inverse transform  fB(t) = − 1  π lim  ϵ→0 ℑ(GB(t + iϵ)),  (20)  DRAFT  10  where ℑ(·) denotes the imaginary part of the complex number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Therefore, the problem of finding the  mutual information I(γ) and the PDF fB(t) are amount to finding the Cauchy transform GB(z) of  product of matrices B = GFF†G†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' In the next section, we will resort to a linearization trick and the  operator-valued free probability theory to derive the expression of GB(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' ASYMPTOTIC EIGENVALUE DISTRIBUTION VIA OPERATOR-VALUED FREE PROBABILITY  THEORY  In the classic free probability theory, it is common to combine the Cauchy transform and the free  multiplicative convolution to obtain the limiting spectral distribution of product of random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' For  example, in [26], the limiting spectral distribution of the concatenated MIMO channels of the form  � K  �  k=1  Hk  � � K  �  k=1  Hk  �†  (21)  is derived, where Hk is Nk × Nk−1 random matrix and has i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' zero-mean entries, unitarily invariant,  and independent of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Such assumptions, in the language of free probability theory, is equivalent  to requiring freeness among families of random variables as specified below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Let A be a unital algebra and B ⊂ A be a unital subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' For H ∈ A, a linear map EB[H] : A → B  is a B-valued conditional expectation, if EB[b] = b for all b ∈ B, and EB[b1Hb2] = b1EB[H]b2 for all  H ∈ A and b1, b2 ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Then, a B-valued probability space is denoted as (A, EB, B), consisting of B ⊂ A  and the linear functional EB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' In addition, let A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , AK be the subalgebras of A with B ⊂ Ak for all  1 ≤ k ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' We also let {Hk ∈ Ak, 1 ≤ k ≤ K} denote a family of operator-valued random variables,  which are free with amalgamation over B according to the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Let n be an arbitrary integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The families of random variables {H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , HK} are free  with amalgamation over B, if for every family of index {k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , kn} ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , K} with k1 ̸= k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' ,  kn−1 ̸= kn, and every family of polynomials {P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , Pn} satisfying EB[Pj(Hkj)] = 0, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , n},  we have EB  ��n  j=1 Pj(Hkj)  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  In order to observe the freeness among families of random matrices {Hk, H†  k}1≤k≤K in (21), we  can construct the random variable Hk as Hk = HkH†  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Let C denote the algebra of complex random  DRAFT  11  variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' We define Ak = MNk(C) as the algebra of Nk × Nk complex Hermitian matrices, B = C and  the linear functional EB as EB =  1  Nk Tr ◦ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Then, as specified in Definition 1, the asymptotic freeness  among {Hk}1≤k≤K over C has been established in some of the classic free probability theory, such as  in [28], which further enables one to apply free multiplicative convolution [26] to obtain the limiting  spectral distribution of the concatenated MIMO channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  However, in the considered problem with B = GFF†G†, both G and F are non-central and with  non-trivial spatial correlations, and thus, are not free over C in the classic free probability aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' More  precisely, GG† and FF† are not free with respect to the linear functional τR = 1  RTr ◦ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Yet, as will be  shown in the remaining of this section, via a linearization trick, the random matrix B of interest can be  embedded into a larger block matrix, which can be then separated as the sum of a deterministic matrix  and a random matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Instead of invoking the classic freeness over C, we are able to elevate them as the  operator-valued variables, which are shown to be asymptotically free in the operator-valued probability  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The limiting spectral distribution of their sum can be then obtained by using the operator-valued  free additive convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Linearization Trick and Operator-Valued Probability Space  Let n denote 2L + R + T and M = Mn(C) denote the algebra of n × n complex random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Although the original formulation of B is in the form of product of two random matrices that are not free  with respect to τR, we could instead construct a block matrix L ∈ M, whose operator-valued Cauchy  transform can be properly defined and is directly related to the conventional Cauchy transform of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  By using the Anderson’s linearization trick as described in [30, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='4], we can construct a block  matrix L ∈ M as follows  L =  �  ��  L(1,1)  L(1,2)  L(2,1)  L(2,2)  �  �� =  �  ���������  0R×R  0R×L  0R×T  G  0L×R  0L×L  F  −IL  0T×R  F†  −IT  0T×L  G†  −IL  0L×T  0L×L  �  ���������  ,  (22)  DRAFT  12  where the matrix blocks  �  L(i,j)�  correspond to the partitions shown on the right-hand-side (RHS) of (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  In addition, let us consider the sub-algebra D ⊂ M as the n×n block diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' For each K ∈ D,  it is defined as  K = blkdiag  �  �C, D, �D, C  �  ,  (23)  where �C is a R × R sub-matrix and �D is a T × T sub-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The L × L block diagonal matrices C  and D are defined as C = blkdiag {C0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , CK} and D = blkdiag {D0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , DK}, respectively, where  Ck and Dk are Lk × Lk sub-matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' In (23), all the involved sub-matrices �C, �D, {Ck}0≤k≤K, and  {Dk}0≤k≤K are Hermitian matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  For X ∈ M, we define X �C, X �D, {XCk}0≤k≤K, and {XDk}0≤k≤K as the sub-blocks of X, corre-  sponding to the same diagonal sub-blocks �C, �D, {Ck}0≤k≤K, and {Dk}0≤k≤K in the matrix K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Then,  we define the linear functional τD : M → D as  τD(X) = id ◦ ED [X] ,  (24)  where id denotes the identity operator on a Hilbert space and the expectation ED [X] is defined as  ED [X] =  �  ���������  E[X �C]  E[XD]  E[X �D]  E[XC]  �  ���������  ,  (25)  and E[XC] = blkdiag {E[XC0], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , E[XCK]}, E[XD] = blkdiag {E[XD0], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , E[XDK]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Then, we  can define an operator-valued probability space (M, τD, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' For the M-valued random variable L ∈  (M, τD, D), its D-valued Cauchy transform is defined as  GD  L (Λ(z)) = id ◦ ED  �  (Λ(z) − L)−1�  = τD  �  (Λ(z) − L)−1�  ,  (26)  where Λ(z) ∈ M denotes the n × n diagonal matrix as  Λ(z) =  �  ��  zIR  0R×(2L+T)  0(2L+T)×R  0(2L+T)×(2L+T)  �  �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (27)  DRAFT  13  By substituting (22) and (27) into (26) and invoking Lemma 2 in the Appendix A, we obtain  GD  L (Λ(z)) = id ◦ ED  �  ��  �  zIR + L(1,2) �  L(2,2)�−1 L(2,1)�−1  −L(1,2) �  zL(2,2) + L(2,1)L(1,2)�−1  −  �  zL(2,2) + L(2,1)L(1,2)�−1 L(2,1)  −  �  L(2,2) + z−1L(2,1)L(1,2)�−1  �  �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' (28)  In particular, the upper-left block of (28) can be explicitly written as  �  zIR + L(1,2) �  L(2,2)�−1  L(2,1)  �−1  =  �  zIR − GFF†G†�−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (29)  Therefore, the Cauchy transform of B over C is related to the D-valued Cauchy transform of L as  GB(z) = 1  RTr  ��  GD  L (Λ(z))  �(1,1)�  ,  (30)  where {·}(1,1) denotes the upper-left R × R matrix block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Operator-Valued Free Additive Convolution  Let us introduce the following notations:  G =  �  IR  √ρ1G1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  √ρKGK  �  ,  (31)  �G =  �  0R  √ρ1 �G1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  √ρK �GK  �  ,  (32)  F =  �  F  T  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  F  T  K  �T  ,  (33)  �F =  �  �FT  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  �FT  K  �T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (34)  The linearization matrix L can be further expressed as  L = L + �L,  (35)  where L and �L contain the deterministic and the random parts of L, respectively, and are given as follows:  L =  �  ���������  G  F  −IL  F  †  −IT  G  †  −IL  �  ���������  ,  (36)  DRAFT  14  �L =  �  ���������  �G  �F  �F†  �G†  �  ���������  ,  (37)  where we omit the all-zero matrix blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  The advantage of working with L as well as its D-valued Cauchy transform GD  L is that the elements  of �L are monomials of �G, �G†, �F, and �F†, which are decoupled from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' This is in contrast  to the Cauchy transform of B over C, where the random variables are mixed together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Then, following  similar steps as in [32], �L is shown to be an operator-valued semicircular variable and is free from the  deterministic matrix L over D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Thus, the limiting spectral distribution of L can be determined by the  operator-valued free additive convolution of L and �L, over the sub-algebra D, which are summarized in  the following propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The random variable �L is semicircular and is free from L over D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Proof: The proof of Proposition 1 is given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Due to Proposition 1, the operator-valued Cauchy transform of L in (30) can be calculated as the free  additive convolution between �L and L, by using a subordination formula [30] as follows:  GD  L (Λ(z)) = GD  L  �  Λ(z) − RD  �L  �  GD  L (Λ(z))  ��  = ED  ��  Λ(z) − RD  �L  �  GD  L (Λ(z))  �  − L  �−1�  ,  (38)  where RD  �L (·) denotes the operator-valued R-transform of L over D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Then, GB(z) can be determined by  the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The Cauchy transform of B, with z ∈ C+, is given by  GB(z) = 1  RTr  ��  �Ψ(z) − GΞ(z)−1G  †�−1�  ,  (39)  where  Ξ(z) = Ψ(z) −  �  �Φ(z) − FΦ(z)−1F  †�−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (40)  DRAFT  15  The matrix-valued function �Ψ(z), Ψ(z), �Φ(z), Φ(z) satisfy the following fixed-point equations  �Ψ(z) = zIR −  K  �  k=1  �ηk(GCk(z)),  (41)  Ψ(z) = blkdiag  �  0R, −η1(G �C(z)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , −ηK(G �C(z))  �  ,  (42)  �Φ(z) = blkdiag  �  −�ζ0(G �D(z)), −�ζ1(G �D(z)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , −�ζK(G �D(z))  �  ,  (43)  Φ(z) = IT −  K  �  k=0  ζk(GDk(z)),  (44)  where blkdiag {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , An} constructs a block diagonal matrix with square matrices A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , An being  the diagonal blocks, and G �C(z), GCk(z), G �D(z), GDk(z) are given by  G �C(z) =  �  �Ψ(z) − GΞ(z)−1G  †�−1  ,  (45)  GCk(z) =  ��  Ψ(z) − G  † �Ψ(z)−1G −  �  �Φ(z) − FΦ(z)−1F  †�−1�−1�  k+1  ,  1 ≤ k ≤ K,  (46)  G �D(z) =  �  Φ(z) − F  †  �  �Φ(z) −  �  Ψ(z) − G  † �Ψ(z)−1G  �−1�−1  F  �−1  ,  (47)  GDk(z) =  ��  �Φ(z) − FΦ(z)−1F  † −  �  Ψ(z) − G  † �Ψ(z)−1G  �−1�−1�  k+1  ,  0 ≤ k ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (48)  The notation {A}k+1 with n × n matrix A denotes the (k + 1)-th diagonal matrix block containing  entries from �k−1  i=0 Li + 1 to �k  i=0 Li rows and columns of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Proof: The proof of Proposition 2 is given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  As indicated by Proposition 2, the Cauchy transform GB(z) as well as the matrix-valued functions  �Ψ(z), Ψ(z), �Φ(z), Φ(z) can be determined by solving the fixed-point equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The numerical value  of GB(z) can be obtained by iterating the set of equations (41)-(44) and (45)-(48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' NUMERICAL RESULTS  In this section, numerical simulations are conducted to study the spectral distribution of the RIS-assisted  MIMO channel as well as its mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' In particular, we examine the impacts of the number of  RIS panels, the number of antennas at the transceivers, and the Rician factor of the propagation channels  DRAFT  16  on the mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The mutual information I(γ) and the eigenvalue PDF fB(t) are calculated  by (17) and (20), respectively, where the involved Cauchy transform GB(z) is given in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' In  each simulation case, the MIMO system without RIS deployment is included for comparison, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=', K = 0,  where the eigenvalue PDF and the Cauchy transform can be calculated by using existing result from [32,  Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Each simulation curve is obtained by averaging over 106 independent channel realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  In the simulations, the antenna elements of the transceivers and the reflecting elements of the RIS  panels are arranged as the uniform planar arrays (UPAs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Denote T = T (H) × T (V ), R = R(H) × R(V ),  and Lk = L(H)  k  × L(V )  k  , where the numbers with the superscripts H and V represent the numbers of  elements aligned in the horizontal and vertical dimensions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The specular component of each  channel is the line-of-sight propagation component between two uniform planar arrays (UPAs), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=',  Fk = a  �  ϕ(F)  k  , ν(F)  k  , L(H)  k  , L(V )  k  �  a† �  θ(F)  k  , φ(F)  k  , T (H), T (V )�  ,  0 ≤ k ≤ K,  (49)  Gk = a  �  ϕ(G)  k  , ν(G)  k  , R(H), R(V )�  a† �  θ(G)  k  , φ(G)  k  , L(H)  k  , L(V )  k  �  ,  1 ≤ k ≤ K,  (50)  where θ(i)  k  and φ(i)  k  are the azimuth and elevation angles of the k-th departing UPA, while ϕ(i)  k  and ν(i)  k  are the azimuth and elevation angles of the k-th arriving UPA, i ∈ {F, G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The function a(·) denotes  the steering vector of an M × N UPA and is defined as  a(α, β, M, N) =  �  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , eiπ(n sin(α) sin(β)+m cos(β)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , eiπ((N−1) sin(α) sin(β)+(M−1) cos(β))�T  ,  (51)  where 0 ≤ m ≤ M − 1 and 0 ≤ n ≤ N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2 shows the empirical and asymptotic eigenvalue PDF of the RIS-assisted MIMO channels HH†,  assuming the number of RIS panels is K = 0, 1, 2, and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' In all the cases, the numbers of  transmit and receive antennas are set to T = R = 64, and the number of reflecting elements in each RIS  panel is set to 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The channel statistics, such as Fk, Uk, Vk, Mk in (3), and Gk, Wk, Sk, Nk in  (4) are randomly generated but fix for the Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The numerical results show that the  asymptotic PDF calculated by (20) provides an excellent approximation to the simulated PDF for all the  considered parameter configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' By increasing the number of deployed RIS panels, it is possible to  increase the maximum eigenvalue, therefore, improve amplitude of the eigen-channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  DRAFT  17  0  1  2  3  4  5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='5  PDF  (a) K = 0  0  1  2  3  4  5  6  7  8  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='9  1  PDF  (b) K = 1  0  1  2  3  4  5  6  7  8  9  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='9  1  PDF  (c) K = 2  0  2  4  6  8  10  12  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='7  PDF  (d) K = 4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Comparisons of empirical and asymptotic eigenvalue PDFs of the RIS-assisted MIMO channels HH† with different  numbers of RIS panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The numbers of transmit and receive antennas are set to T = R = 64, and the number of reflecting  elements of each RIS panel is set to 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 3, we investigate the impacts of the SNR, the number of antennas, and the Rician factors  on the mutual information of the RIS-assisted MIMO channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Equal number of antennas is set at the  transmitter and the receiver, where T = R = 4 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 3 (a) and T = R = 8 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 3 (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  The MIMO communication is assisted by K = 6 RIS panels, and each RIS panel is composed of  16 reflecting elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Compared to the direct link F0, the relative channel gains [ρ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , ρ6] in (15)  corresponding to the reflected links are configured as [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' All the Rician factors  are set equal as κ = κ(F)  k  = κ(G)  k  , where κ is set to 1, 10, or 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' In presence of non-degenerate random  DRAFT  18  scattering components �Fk in (3) and �Gk in (4), the RIS-assisted MIMO channels are full-rank, and the  mutual information at large SNR linearly increases as min{T, R}/10 log10(e) nats/s/Hz for every 1 dB  SNR improvement, depicted as the dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' However, as the Rician factor becomes large,  although the mutual information has the same scaling law, it requires larger SNR levels to exhibit the  linear improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' This is illustrated in the insets of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' When the Rician factors are κ = 1, 10,  and 100, the asymptotic mutual information has at least 5% deviation from the high-SNR scaling law at  SNRs 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='8 dB, 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='5 dB, and 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='2 dB when T = R = 4, and at SNRs 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='7 dB, 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='8 dB, and 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='1 dB  when T = R = 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' This is due to the fact that as the Rician factor increases, the random  scattering components to maintain the rank of the channel have less contributions to the overall MIMO  channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  To further investigate the impacts of the Rician factor on the mutual information of the MIMO channels,  we plot Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 4 to show the mutual information as a function of κ, with the numbers of RIS panels K  set to 0, 1, 2, and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The number of transmit and receive antennas are set to T = 16 and  R = 8, while the performance of the MIMO system is evaluated at SNR γ = 10 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' It is observed that  when κ is less than 1, the mutual information can be improved as κ increases, while it monotonically  decreases for κ > 1 in all the considered cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' When the number of RIS panels is larger, the mutual  information degradation is less prominent as each RIS provides independent reflected link, which increases  the richness of the MIMO channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 5, the impact of the numbers of RIS panels is investigated in more details, when the mutual  information is evaluated for different transmit antennas T = 8, 16, 32, and 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The number of receive  antennas is fixed to R = 10, and each RIS panel has 8 reflecting elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' In this simulation setting, we  consider the urban canyon communication scenario as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 1, where the specular components  of {Fk} channels and of {Gk} channels have relatively small angular variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' That is, in (49)  and (50), we assume that the departing angles  �  θ(F)  k  , φ(F)  k  �  0≤k≤K of the transmitter UPA and the  arriving angles  �  ϕ(G)  k  , ν(G)  k  �  1≤k≤K of the receiver UPA are uniformly and randomly generated in some  fixed intervals having length 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='05π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The departing angles  �  θ(G)  k  , φ(G)  k  �  1≤k≤K and the arriving angles  DRAFT  19  0  10  20  30  40  5  10  15  20  25  30  35  40  Mutual information (nats/s/Hz)  Asymptotic MI  High-SNR MI  Simulation  20  25  30  35  15  16  17  18  = 1, 10, 100  (a) T = R = 4  0  10  20  30  40  50  0  10  20  30  40  50  60  70  80  Mutual information (nats/s/Hz)  Asymptotic MI  High-SNR MI  Simulation  22  24  26  28  30  32  34  36  38  26  28  30  32  34  = 1, 10, 100  (b) T = R = 8  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Mutual information of RIS-assisted MIMO channels at varying SNR γ, when the number of antennas at the transceivers  is T = R = 4 in (a) and T = R = 8 in (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' In each case, the Rician factor of the component channels is set equal  to κ = 1, 10, or 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' There are K = 6 deployed RIS panels, each of which has 16 reflecting elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Insets show the 5%  deviations of the asymptotic mutual information from the high-SNR scaling law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  DRAFT  20  0  2  4  6  8  10  12  14  4  6  8  10  12  14  16  Mutual information (nat/s/Hz)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Mutual information of RIS-assisted MIMO channel for varying Rician factor κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The number of antennas at the transmitter  and the receiver are T = 16 and R = 8, respectively, and each RIS panel has 8 reflecting elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The SNR of the end-to-end  channel is set to γ = 10 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  �  ϕ(F)  k  , ν(F)  k  �  1≤k≤K of the RIS panels are randomly generated in some fixed intervals having length  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='1π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' As K increases, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 5 shows that the mutual information first improves at a larger rate between  0 ≤ K ≤ 5, and then becomes slower thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' This is due to the fact that the richness of the channels  can be improved more efficiently when the number of reflected links is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Since the angular ranges  are restricted, the added RIS panels have similar reflected links that cannot provide additional richness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Therefore, it is less effective to deploy more RIS panels to improve the mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Finally, as  shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 3-5, the mutual information calculated by (17) via the Cauchy transform (39) achieves a  good agreement with the simulation in all the considered simulation cases, and thus, can be applied to  evaluate the performance of the RIS-assisted MIMO channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' CONCLUSIONS  This paper studies the information-theoretic data rate of the RIS-assisted MIMO systems, where  multiple RIS panels are deployed to improve the scattering-limited MIMO channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' By using the  DRAFT  21  0  5  10  15  6  7  8  9  10  11  12  13  Mutual information (nat/s/Hz)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Mutual information of RIS-assisted MIMO channel for varying numbers of RIS panels K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The number of receive  antennas is R = 8, the number of elements in each RIS panel is 8, and the SNR of the channel is γ = 10 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  operator-valued free probability theory, the Cauchy transform of the MIMO matrix is obtained using  the general Rician MIMO model with Weichselberger’s correlation structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Based on this result, the  asymptotic eigenvalue distribution of the channel matrix as well as the mutual information of the MIMO  channel are calculated, which closely match the corresponding simulation results for practical system  configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Numerical results show that the additional reflected links created by the RIS panels can  increase the range of eigenvalues of the channel matrix, which can be leveraged to improve the amplitude  of the eigen-channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' In the MIMO communications, the negative impact of a large Rician factor on the  mutual information can be partly alleviated by deploying more RIS panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' However, the performance  improvement of the multi-RIS deployment slows down when the added reflected links have similar  arriving and departing angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  DRAFT  22  APPENDIX A  SOME USEFUL MATRIX INVERSION IDENTITIES  For the sake of completeness, the following matrix inversion identities are summarized in Lemmas 1-3,  which are repeatedly applied throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' For notational simplicity, in this appendix, we use  italic bold symbols to define matrices, which are different from those used in the main sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' (Woodbury matrix inversion identity [31, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=') Let A denote a m × m invertible  matrix, D denote a k × k matrix, B and C denote m × k and k × m matrices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Then the  following identity holds  (A + BDC)−1 = A−1 − A−1B  �  D−1 + CA−1B  �−1 CA−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (52)  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' (2 × 2 block matrix inversion identity [31, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=') Let A, B, C, and D be defined  as in Lemma 1, the inversion identity of the following 2 × 2 block matrix holds  �  ��  A  B  C  D  �  ��  −1  =  �  ��  A−1 + A−1B(D − CA−1B)−1CA−1  −A−1B(D − CA−1B)−1  −(D − CA−1B)−1CA−1  (D − CA−1B)−1  �  ��  =  �  ��  (A − BD−1C)−1  −A−1B(D − CA−1B)−1  −(D − CA−1B)−1CA−1  (D − CA−1B)−1  �  �� ,  (53)  where the second equality holds when D is also invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' (3 × 3 block matrix inversion identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=') Let the matrices E, F , G, H, J, K, L, M, and N  be the conformable partitions of the following 3 × 3 block matrix X  X =  �  ������  E  F  G  H  J  K  L  M  N  �  ������  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  DRAFT  23  When E is invertible, the inversion of X is given by  X−1 =  �  ������  E−1 + E−1(F A−1H + US−1V )E−1  −E−1(F − US−1C)A−1  −E−1US−1  −A−1(H − BS−1V )E−1  A−1 + A−1BS−1CA−1  −A−1BS−1  −S−1V E−1  −S−1CA−1  S−1  �  ������  ,  where  A = J − HE−1F ,  B = K − HE−1G,  C = M − LE−1F ,  D = N − LE−1G,  (54)  U = G − F A−1B,  V = L − CA−1H,  (55)  S = D − CA−1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (56)  Proof: Apply Lemma 2 to the inversion of X that is partitioned into a 2 × 2 block matrix as  X−1 =  �  ������  E  F  G  H  J  K  L  M  N  �  ������  −1  =  �  ��  P  Q  R  Z−1  �  �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (57)  where the matrix blocks P ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' and R are given by  P = E−1 + E−1  �  F  G  �  Z−1  �  ��  H  L  �  �� E−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (58)  Q = −E−1  �  F  G  �  Z−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (59)  R = −Z−1  �  ��  H  L  �  �� E−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (60)  and the matrix Z is a 2 × 2 block matrix such that  Z =  �  ��  J  K  M  N  �  �� −  �  ��  H  L  �  �� E−1  �  F  G  �  =  �  ��  A  B  C  D  �  �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (61)  where A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' and D are given in (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Applying again Lemma 2 to the inversion of Z, we obtain  Z−1 =  �  ��  A−1 + A−1BS−1CA−1  −A−1BS−1  −S−1CA−1  S−1  �  �� ,  (62)  DRAFT  24  where S is given in (56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Substituting (62) into (58), P can be rewritten as  P = E−1 + E−1  �  F A−1 − US−1CA−1  US−1  �  �  ��  H  L  �  �� E−1  = E−1 + E−1 �  F A−1H + US−1V  �  E−1,  (63)  where U and V are defined in (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Similarly, Q and R can be obtained as  Q =  �  −E−1 �  F − (G − F A−1B)S−1C  �  A−1  −E−1 �  G − F A−1B  �  S−1  �  =  �  −E−1 �  F − US−1C  �  A−1  −E−1US−1  �  ,  (64)  R =  �  ��  −A−1HE−1 + A−1BS−1(L − CA−1H)E−1  −S−1(L − CA−1H)E−1  �  �� =  �  ��  −A−1(H − BS−1V )E−1  −S−1V E−1  �  �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (65)  Finally, substituting (62)-(65) into (57) completes the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  APPENDIX B  PROOF OF PROPOSITION 1  A random variable �L ∈ M is said to be D-valued semicircular if the free cumulant  κD  m(�Lb1, �Lb2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , �Lbm−1, �L) = 0,  (66)  for all n ̸= 2, and all b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , bn−1 ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The free cumulant κD  m is a mapping from Mm to D and we  refer the reader to [29] for detailed explanations on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The proof is followed by expanding �L  into a sum of n × n matrices, such that  �L =  K  �  k=0  �L(F)  k  +  K  �  k=1  �L(G)  k  ,  (67)  DRAFT  25  where the matrices �L(F)  k  and �L(G)  k  are given by  �L(F)  k  =  �  ���������  0R×L  �Fk  �F†  k  0L×R  �  ���������  ,  (68)  �L(G)  k  =  �  ���������  �Gk  0L×T  0T×L  �G†  k  �  ���������  ,  (69)  where �Fk and �Gk are L × T and R × L matrices, respectively, and are given by  �Fk =  �  0T×L0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  �F†  k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  0T×LK  �†  ,  0 ≤ k ≤ K,  (70)  �Gk =  �  0R×R  0R×L1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  √ρk �Gk  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  0R×LK  �  ,  1 ≤ k ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (71)  Recalling the definitions of �Fk and �Gk in (3) and (4), we have  �L(F)  k  = A(F)  k  �X kA(F)†  k  ,  (72)  �L(G)  k  = A(G)  k  �YkA(G)†  k  ,  (73)  where the matrix �X k has the same structure as the block matrix �L(F)  k  in (68) while replacing �Fk in (70)  with �Xk = Mk ⊙ Xk, and �Yk has the same structure as the block matrix �L(G)  k  in (69) while replacing  �Gk in (71) with �Yk =  1  √rk Nk ⊙ Yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' The n × n matrices A(F)  k  and A(G)  k  are given by  A(F)  k  =  �  ��  �Uk  0(R+L)×(T+L)  0(T+L)×(R+L)  �Vk  �  �� ,  (74)  A(G)  k  =  �  ��  �  Wk  0(R+L)×(T+L)  0(T+L)×(R+L)  �Sk  �  �� ,  (75)  DRAFT  26  where �Uk, �Vk, �  Wk, �Sk are deterministic diagonal block matrices and are given by  �Uk = blkdiag(0R, 0L0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , Uk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , 0LK),  (76)  �Vk = blkdiag(Vk, 0L),  (77)  �  Wk = blkdiag(Wk, 0L),  (78)  �Sk = blkdiag(0T , 0L0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , Sk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , 0LK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (79)  Since { �X k}0≤k≤K, {�Yk}1≤k≤K are Wigner matrices and independent from each other, they are semi-  circular and free over the sub-algebra Dn ⊂ M of n × n diagonal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Then, following the same  arguments as in [32, Appendix B], {�L(F)  k  }0≤k≤K and {�L(G)  k  }1≤k≤K are semicircular and free over sub-  algebra of block diagonal matrices D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Therefore, the sum of �L(F)  k  and �L(G)  k  is also semicircular over D  and is free from any deterministic matrix from M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  APPENDIX C  PROOF OF PROPOSITION 2  Since �L is an operator-valued semicircular variable over D and �L are free from L over D, the  limiting spectral distribution of L is a free additive convolution of the limiting spectral distributions of  �L and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Specifically, the operator-valued Cauchy transform GD  L can be calculated via the subordination  formula (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Recall that the R-transform RD  �L (·) is the free cumulant generating function of �L with the  following formal power series expansion:  RD  �L (K) = κD  1 ( �K) + κD  2 (�LK, �L) + κD  3 (�LK, �LK, �L) + · · · ,  (80)  where κD  i denotes the i-th free cumulant of �L over D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' In addition, since �L is semicircular over D, all  its cumulants in (80) except κD  2 are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Therefore, the R-transform RD  �L (K) reduces to the covariance  DRAFT  27  function of �L over D parameterized by K, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=',  RD  �L(K) = ED  �  �LK�L  �  =  �  ���������  �K  k=1 �ηk(Ck)  �ζ( �D)  �K  k=0 ζk(Dk)  η(�C)  �  ���������  ,  (81)  where �ζ( �D) = blkdiag  �  �ζ0( �D), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , �ζK( �D)  �  and η(�C) = blkdiag  �  0R, η1(�C), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , ηK(�C)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Since GD  L (Λ(z)) ∈ D, by same matrix partitioning as in (23), GD  L (Λ(z)) is partitioned into  GD  L (Λ(z)) = blkdiag  �  G �C(z), GD(z), G �D(z), GC(z)  �  ,  (82)  where GD(z) = blkdiag {GD0(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , GDK(z)} and GC(z) = blkdiag {0R, GC1(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' , GCK(z)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Note  that the upper-left block  �  GD  L (Λ(z))  �(1,1) = G �C(z), which is then used to compute GB(z) = 1  RTr(G �C(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  By replacing K in (81) with GD  L (Λ(z)) in (82), and substituting L and RD  �L with (36) and (81),  respectively, we obtain GD  L (Λ(z)) as  GD  L (Λ(z)) =  �  ���������  G �C(z)  GD(z)  G �D(z)  GC(z)  �  ���������  = ED  �  �  �  �  �  �  �  �  �  �  �Ψ(z)  0  0  −G  0  �Φ(z)  −F  IL  0  −F  †  Φ(z)  0  −G  †  IL  0  Ψ(z)  �  �  �  �  �  �  �  �  �  �  −1  ,  (83)  where �Ψ(z), Ψ(z), �Φ(z), and Φ(z) are given in (41)-(44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' By invoking Lemma 2 to the RHS of (83)  and taking expectation over D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' the matrix-valued function G �C(z) = A−1  1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' and GD(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' G �D(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' GC(z) are  DRAFT  28  the diagonal blocks of the matrix A−1  2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' where A1 and A2 are given by  A1 = �Ψ(z) −  �  0  0  G  �  �  ������  �Φ(z)  −F  IL  −F  †  Φ(z)  0  IL  0  Ψ(z)  �  ������  −1 �  ������  0  0  G  †  �  ������  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (84)  A2 =  �  ������  �Φ(z)  −F  IL  −F  †  Φ(z)  0  IL  0  Ψ(z)  �  ������  −  �  ������  0  0  G  †  �  ������  �Ψ(z)−1  �  0  0  G  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (85)  Applying Lemma 3, the RHS of (84) can be further derived as  A1 = �Ψ(z) − GS−1G  †,  (86)  where S = Ξ(z) and is calculated in (56) as  S = Ξ(z) = Ψ(z) − �Φ(z)−1 − �Φ(z)−1F  �  Φ(z) − F  † �Φ(z)−1F  �−1  F  † �Φ(z)−1  = Ψ(z) −  �  �Φ(z) − FΦ(z)−1F  †�−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (87)  The second equality of (87) is obtained by applying Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Then, (45) is established by combining  (86) and (87).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  The inverse of A2 can be explicitly calculated via Lemma 3, where E = �Φ(z), F = H† = −F,  G = L = IL, J = Φ(z), K = M = 0, and N = Ψ(z) − G  † �Ψ(z)−1G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' We further let T =  Φ(z)−F  † �Φ(z)−1F and �T = �Φ(z)−FΦ(z)−1F  †.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Then, the matrix-valued functions GD(z), G �D(z), and  GC(z), being the diagonal blocks of A−1  2 , are given by  GD(z) = �Φ(z)−1 + �Φ(z)−1FT −1F  † �Φ(z)−1 + �T −1 �  N − �T −1�−1 �T −1,  (88)  G �D(z) = T −1 + T −1F  † �Φ(z)−1 �  N − �T −1�−1 �Φ(z)−1F T −1,  (89)  GC(z) =  �  N −  �  �Φ(z)−1 + �Φ(z)−1FT −1F  † �Φ(z)−1��−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  (90)  Finally, applying Lemma 1 to (88)-(90), we obtain GD(z), G �D(z), and GC(z) as in (46)-(48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  DRAFT  29  REFERENCES  [1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Saad, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Bennis, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Chen, “A vision of 6G wireless systems: Applications, trends, technologies, and open research  problems,” IEEE Netw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 34, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 134–142, May 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [2] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Telatar, “Capacity of multi-antenna Gaussian channels,” Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Telecommun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 10, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 585–595, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [3] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Shin and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Lee, “Capacity of multiple-antenna fading channels: Spatial fading correlation, double scattering, and  keyhole,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 49, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2636–2647, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [4] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Pan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Ren, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Kolb, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Elkashlan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Chen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Di Renzo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Hao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Wang, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Swindlehurst,  “Reconfigurable intelligent surfaces for 6G systems: Principles, applications, and research directions,” IEEE Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 59, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 14–20, June 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [5] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Wu and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Zhang, “Towards smart and reconfigurable environment: Intelligent reflecting surface aided wireless network,”  IEEE Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 58, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 106–112, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [6] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Di Renzo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Xiao, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Ai, “Performance analysis of RIS-aided systems with practical phase  shift and amplitude response,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Veh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 70, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 4501–4511, May 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [7] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Yang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Meng, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Wu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Da Costa, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Alouini, “Accurate closed-form approximations to channel distributions  of RIS-aided wireless systems,” IEEE Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 9, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 1985–1989, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Salhab and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Samuh, “Accurate performance analysis of reconfigurable intelligent surfaces over Rician fading  channels,” IEEE Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 10, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 1051–1055, May 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [9] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Dash, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Joshi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Aissa, “Envelope distribution of two correlated complex Gaussian random variables and  application to the performance evaluation of RIS-assisted communications,” IEEE Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 26, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  2018-2022, Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [10] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Xie, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Yi, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Liu, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Nallanathan, “Downlink multi-RIS aided transmission in backhaul limited networks,”  IEEE Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
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+page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 1458–1462, July 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [11] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Hashemi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Ali, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Mahmood, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Latva-aho, “Average rate and error probability analysis in short packet  communications over RIS-aided URLLC systems,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Veh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
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+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 70, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 10320–10334, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [12] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Yildirim, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Uyrus, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Basar, “Modeling and analysis of reconfigurable intelligent surfaces for indoor and outdoor  applications in future wireless networks,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 69, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 1290–1301, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [13] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Yang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Da Costa, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Trigui, “Outage probability and capacity scaling law of multiple RIS-aided  networks,” IEEE Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 10, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 256–260, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [14] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Do, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Kaddoum, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Nguyen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Da Costa, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Haas, “Multi-RIS-aided wireless systems: Statistical  characterization and performance analysis,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 69, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 8641–8658, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [15] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Shi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Fu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Ma, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Gao, “Outage analysis of reconfigurable intelligent surface aided MIMO  communications with statistical CSI,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 823–839, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  DRAFT  30  [16] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Carter and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Springer, “The distribution of products, quotients and powers of independent H-function variates,”  SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Applied Maths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 33, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 542–558, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [17] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Ayach, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Rajagopal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Abu-Surra, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Pi, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Heath, “Spatially sparse precoding in millimeter wave MIMO  systems,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 13, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 1499–1513, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [18] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Sun, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Han, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Tao, “Ergodic achievable rate analysis and optimization of RIS-assisted millimeter-  wave MIMO communication systems,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=', in press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [19] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Ma, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Yang, “On the sum-rate of RIS-assisted MIMO multiple-access channels over  spatially correlated Rician fading,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 69, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 8228–8241, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [20] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Yu, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Song, “Outage probability and finite-SNR DMT analysis for IRS-aided MIMO systems: How  large IRSs need to be?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=',” IEEE J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Sel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 16, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 1070–1085, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Ma, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Wen, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Jin, “Large system achievable rate analysis of RIS-assisted MIMO wireless  communication with statistical CSIT,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 20, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 5572–5585, Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [22] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Weichselberger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Herdin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' ¨Ozcelik, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Bonek, “A stochastic MIMO channel model with joint correlation of  both link ends,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 5, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 90-100, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [23] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Zheng, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Wei, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Speicher, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' M¨uller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' H¨am¨al¨ainen, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Corander, “Asymptotic analysis of Rayleigh product  channels: A free probability approach,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 63, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 1731–1745, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [24] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' M¨uller, “A random matrix model of communication via antenna arrays,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 48, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  2495–2506, Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [25] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Couillet and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Debbah, Random Matrix Methods for Wireless Communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Cambridge: Cambridge University  Press, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [26] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' M¨uller, “On the asymptotic eigenvalue distribution of concatenated vector-valued fading channels,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 48, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2086-2091, July 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [27] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Benaych-Georges, “Rectangular random matrices, related free entropy and free Fischer’s information,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Operator Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=',  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 62, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 371-419, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [28] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Capitaine and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Donati-Martin, “Strong asymptotic freeness for Wigner and Wishart matrices,” Indiana Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 56, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 767-803, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [29] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Mingo and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Speicher, Free Probability and Random Matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' New York: Springer, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [30] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Belinschi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Mai, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Speicher, “Analytic subordination theory of operator-valued free additive convolution and  the solution of a general random matrix problem,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Reine Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' (Crelles J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' ), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2017, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 732, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 21–53, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [31] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Horn and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Johnson, Matrix Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' New York: Cambridge University Press, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  [32] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Lu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Gao, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Xiao, “Free deterministic equivalents for the analysis of MIMO multiple access channel,” IEEE  Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 62, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 4604-4629, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
+page_content='  DRAFT' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFLT4oBgHgl3EQfri9A/content/2301.12144v1.pdf'}
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+Resonant triad interactions of gravity waves
+in cylindrical basins
+Matthew Durey1 and Paul A. Milewski2
+1School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8QQ, UK
+2Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK
+Abstract
+We present the results of a theoretical investigation into the existence, evolution and ex-
+citation of resonant triads of nonlinear free-surface gravity waves confined to a cylinder of
+finite depth. It is well known that resonant triads are impossible for gravity waves in laterally
+unbounded domains; we demonstrate, however, that horizontal confinement of the fluid may
+induce resonant triads for particular fluid depths. For any three correlated wave modes arising
+in a cylinder of arbitrary cross-section, we prove necessary and sufficient conditions for the
+existence of a depth at which nonlinear resonance may arise, and show that the resultant crit-
+ical depth is unique. We enumerate the low-frequency triads for circular cylinders, including
+a new class of resonances between standing and counter-propagating waves, and also briefly
+discuss annular and rectangular cylinders. Upon deriving the triad amplitude equations for a
+finite-depth cylinder of arbitrary cross-section, we deduce that the triad evolution is always
+periodic, and determine parameters controlling the efficiency of energy exchange. In order to
+excite a particular triad, we explore the influence of external forcing; in this case, the triad
+evolution may be periodic, quasi-periodic, or chaotic. Finally, our results have potential im-
+plications on resonant water waves in man-made and natural basins, such as industrial-scale
+fluid tanks, harbours and bays.
+1
+Introduction
+Nonlinear resonance is a mechanism by which energy is continuously transferred between a small
+number of linear wave modes. This phenomenon, first observed in Wilton’s analysis of gravity-
+capillary wave trains [63], has been the subject of frequent investigation over the past century
+[36, 37, 38, 54, 52, 19, 15]; indeed, nonlinear resonance has since observed for wave trains in a
+growing number of dispersive wave systems, including gravity waves [48, 21, 33, 2], acoustic-gravity
+waves [27, 26], flexural-gravity waves [58], two-layer flows [1, 25, 53], and atmospheric flows [51, 50].
+Whilst the aforementioned studies typically consider nonlinear resonance for laterally unbounded
+domains, the purpose of this study is to demonstrate that energy exchange between free-surface
+gravity waves may be induced and accentuated by horizontal confinement.
+We focus our study on the collective resonance of three linear wave modes, henceforth referred to
+as a triad [5]. In laterally unbounded domains, the monotonic and concave form of the dispersion
+curve precludes the existence of resonant triads for gravity wave trains at finite depth [48, 21],
+with resonant quartets instead being the smallest possible collective resonant interaction [2, 33, 4].
+However, confinement of the fluid to a vertical cylinder results in linear wave modes that differ in
+1
+arXiv:2301.02163v1  [physics.flu-dyn]  5 Jan 2023
+
+form to sinusoidal plane waves (except for a rectangular cylinder), so the preclusion of resonant
+triads no longer applies. Indeed, our study demonstrates that, under certain conditions, resonant
+triads may arise in cylinders of arbitrary cross-section for specific values of the fluid depth. As
+resonant triads evolve over a much faster time scale than that of resonant quartets, the exchange
+of energy in gravity waves is thus more efficient under the influence of lateral confinement [40],
+with potential implications on resonant sloshing in man-made and natural basins [8].
+Prior investigations of confined resonant free-surface gravity waves have predominantly focused
+on the so-called 1:2 resonance, which arises when two of the three linear wave modes comprising
+a triad coincide. For axisymmetric standing waves in a circular cylinder, Mack [34] determined a
+condition for the existence of critical depth-to-radius ratios at which a 1:2 resonance may arise, a
+result later generalised to cylinders of arbitrary cross-section [44]. Miles [42, 43] then characterised
+the weakly nonlinear evolution of such internal resonances, demonstrating that a 1:2 resonance is
+impossible in a rectangular cylinder [42]. Although Miles’ seminal results provide an informative
+view of the weakly nonlinear dynamics, the influence of fully nonlinear effects was later assessed
+by Bryant [8] and Yang et al. [64]. For the case of a circular cylinder of finite depth, Bryant [8]
+and Yang et al. [64] both characterised new steadily propagating nonlinear waves arising in the
+vicinity of a 1:2 resonance, and Yang et al. [64] also computed nonlinear near-resonant axisymmetric
+standing waves. Finally, broader mathematical properties of water waves exhibiting O(2) symmetry
+(of which a circular cylinder is one example) were analysed by Bridges & Dias [6] and Chossat &
+Dias [12].
+Given the restrictive set of critical depths at which a 1:2 resonance may arise [8, 64], it is
+natural to explore the possibility of nonlinear resonance in cylinders whose depth departs from the
+depths that trigger a 1:2 resonance. To the best of our knowledge, the first and only such study
+was the seminal experimental investigation performed by Michel [40], who focused on resonant
+triads arising for free-surface gravity waves confined to a finite-depth circular cylinder. Notably,
+the cylinder depth in Michel’s experiment was judiciously chosen so as to isolate a specific triad.
+Michel utilised bandlimited random horizontal vibration so as to excite two members of the triad,
+whose nonlinear interaction led to the growth of the third mode. Significantly, the energy of the
+third mode was, on average, the product of the energies of the remaining two modes, thereby
+satisfying the quadratic energy exchange typical of resonant triads.
+In order to exemplify the mechanism of nonlinear resonance, Michel [40] also calculated the
+response of a child mode due to the nonlinear interaction between two parent modes (where all
+three wave modes comprise the triad). Notably, Michel’s calculation is restricted to the early stages
+of growth and to particular relative phases of the wave modes. In addition Michel considered a
+fluid of infinite depth for all but the resonance conditions, for which finite-depth corrections were
+included. In contrast, we consider general resonances in arbitrary cylinders of finite depth and
+derive equations for the triad evolution over long time-scales. We also believe some nonlinear
+contributions to the interactions were omitted from Michel’s calculation, resulting in quantitative
+differences (see §4.3).
+The goal of our study is to unify the existence and evolution of 1:2 and triadic resonances
+into a single mathematical framework, effectively characterising all triad interactions of this type.
+Based on existing theory, it is unclear how the existence of resonant triads depends on the form
+of the cylinder cross-section, and which combinations of wave modes are permissible for judicious
+choice of the fluid depth. Furthermore, the range of depths that may excite a particular triad
+is uncertain, with 1:2 resonances only excited in a very narrow window about each critical depth
+[34, 44]. Once a particular triad is excited, one anticipates that the triad evolution will be governed
+by the canonical triad equations [5, 15]; however, quantifying the triad evolution and relative energy
+2
+
+exchange requires computation of the triad coupling coefficients. Finally, it is unclear how best to
+excite triads in arbitrary cylinders, both with and without external forcing.
+We here present a relatively comprehensive characterisation of the existence, evolution and
+excitation of resonant triads for gravity waves confined to a cylinder of arbitrary cross-section
+and finite depth. In order to reduce the problem to its key components, we first truncate the
+Euler equations, recasting the fluid evolution in terms of a finite-depth Benney-Luke equation
+(§2), incorporating only the nonlinear interactions necessary for resonant triads. In §3, we prove
+necessary and sufficient conditions for there to exist a finite depth at which three linear wave
+modes may form a resonant triad. In particular, we prove that resonant triads are impossible
+for rectangular cylinders, yet there is an abundance of resonant triads for circular cylinders. We
+then use multiple-scales analysis to determine the long-time evolution of a triad in a cylinder of
+arbitrary cross-section (§4), from which we characterise the relative coupling of different triads.
+Finally, we explore the excitation of resonant triads (§5), and discuss the potential extension of
+our theoretical developments to the cases of applied forcing and two-layer flows (§6).
+2
+Formulation
+We consider the irrotational flow of an inviscid, incompressible liquid that is bounded above by
+a free surface, confined laterally by the vertical walls of a cylinder whose horizontal cross-section,
+D, is enclosed by the curve ∂D, and bounded below by a rigid horizontal plane lying a distance H
+below the undisturbed free surface; see figure 1. We consider the fluid evolution in dimensionless
+variables, taking the cylinder’s typical horizontal extent, a, as the unit of length, and
+�
+ag−1 as
+the unit of time, where g is the acceleration due to gravity. It follows that the dimensionless
+free-surface elevation, η(x, t), and velocity potential, φ(x, z, t), evolve according to the equations
+∆φ + φzz = 0
+for x ∈ D,
+−h < z < ϵη,
+(1a)
+φt + η + ϵ
+2
+�
+|∇φ|2 + φ2
+z
+�
+= 0
+for x ∈ D,
+z = ϵη,
+(1b)
+ηt + ϵ∇φ · ∇η = φz
+for x ∈ D,
+z = ϵη,
+(1c)
+n · ∇φ = 0
+for x ∈ ∂D,
+−h < z < ϵη,
+(1d)
+φz = 0
+for x ∈ D,
+z = −h,
+(1e)
+corresponding to the continuity equation, dynamic and kinematic boundary conditions, and no-
+flux through the vertical walls and horizontal base, respectively. In equation (1), the dimensionless
+parameter ϵ is proportional to the typical wave slope, h = H/a is the ratio of the fluid depth to
+the typical horizontal extent, n is a unit vector normal to the boundary ∂D, and the operators ∇
+and ∆ denote the horizontal gradient and Laplacian, respectively. Moreover, conservation of mass
+implies that the free surface satisfies
+��
+D η dA = 0 for all time. Finally, in dimensional variables,
+ax is the two-dimensional horizontal coordinate, az is the upward-pointing vertical coordinate,
+�
+ag−1t denotes time, ϵaη is the free-surface displacement, and ϵa√agφ is the velocity potential.
+We aim to develop a broad framework for understanding resonant triads in a cylinder of finite
+depth; however, care must be taken when modelling fluid-boundary interactions and determining
+the class of permissible cylinder cross-sections.
+From a modelling perspective, we employ an
+assumption generally implicit to the water-wave problem in bounded domains; specifically, we
+neglect the meniscus and dissipation arising near the vertical walls [41], thus determining that
+the free surface intersects the boundary normally, i.e. n · ∇η = 0 for x ∈ ∂D [46]. In order to
+3
+
+Figure 1: Schematic diagram of the cylindrical tank (with cross-section D and boundary ∂D)
+partially filled with liquid. The undisturbed free surface (dashed lines) lies on z = 0, a distance H
+above the rigid bottom plane (grey). The disturbed free surface is sketched in dash-dotted lines.
+maximise the generality of our investigation, we allow the cylinder cross-section, D, to be fairly
+arbitrary; however, the mathematical developments presented herein require D to be bounded
+with a piecewise-smooth boundary, thereby allowing us to utilise the spectral theorem for compact
+self-adjoint operators [29] and the divergence theorem. As most cylinders of practical interest
+consist of a piecewise-smooth boundary, this mathematical restriction fails to limit the breadth of
+our study.
+2.1
+Derivation of the Benney-Luke equation
+As our study is focused on the weakly nonlinear evolution of small-amplitude waves, we proceed
+to simplify (1) in the case 0 < ϵ ≪ 1 and h = O(1). We begin by expanding the dynamic and
+kinematic boundary conditions (equations (1b)–(1c)) about z = 0 in powers of ϵ, which, upon
+eliminating η, gives rise to the equation [2, 47]
+φtt + φz = ϵ
+�
+∂t(φtzφt) + φzzφt − ∂t(|∇φ|2) − φzφzt
+�
++ O(ϵ2)
+for
+x ∈ D,
+z = 0.
+(2)
+To reduce the fluid evolution to the dynamics arising on the linearised free surface, z = 0, we
+define the Dirichlet-to-Neumann operator, L , so that L φ|z=0 = φz|z=0. Here φ satisfies Laplace’s
+equation (1a) over the linearised domain −h < z < 0, with ∂zφ = 0 on z = −h (see equation (1e))
+and n·∇φ = 0 for x ∈ ∂D (see equation (1d)). Notably, the Dirichlet-to-Neumann operator may be
+defined in terms of its spectral representation, as detailed in §2.2. By denoting u(x, t) = φ(x, 0, t),
+we finally obtain the finite-depth Benney-Luke equation [2, 3, 47]
+utt + L u + ϵ
+�
+ut
+�
+L 2 + ∆
+�
+u + ∂
+∂t
+�
+(L u)2 + |∇u|2��
+= O(ϵ2)
+for
+x ∈ D,
+(3)
+where we have simplified the nonlinear terms in equation (2) using φzz = −∆φ and utt = −L u +
+O(ϵ).
+4
+
+H
+DThe remainder of our investigation will be focused on the evolution of resonant triads governed
+by the Benney-Luke equation (3). As resonant triads arising in confined geometries are governed
+primarily by quadratic nonlinearities, it is sufficient to neglect terms of size O(ϵ2) in equation
+(3); however, higher-order corrections to the Benney-Luke equation may be derived by following
+a similar expansion procedure [2, 47, 4].
+Although our investigation is mainly focused on the
+evolution of the velocity potential, u, one may recover the leading-order free-surface elevation from
+the dynamic boundary condition (1b), namely η = −ut + O(ϵ).
+2.2
+Spectral representation of the Dirichlet-to-Neumann operator
+The Dirichlet-to-Neumann operator, L , may be understood in terms of the discrete set of orthog-
+onal eigenfunctions of the horizontal Laplacian operator [29]. Specifically, we consider the set of
+real-valued eigenfunctions, Φn(x), satisfying
+−∆Φn = k2
+nΦn for n = 0, 1, . . . ,
+where the corresponding eigenvalues, k2
+n, are ordered so that 0 = k0 < k1 ≤ k2 ≤ . . .. Moreover,
+each eigenfunction satisfies the boundary condition n · ∇Φn = 0 on ∂D, as motivated by the no-
+flux condition (1d). Finally, the orthogonal eigenfunctions are normalised so that ⟨Φm, Φn⟩ = δmn,
+where
+⟨f, g⟩ = 1
+S
+��
+D
+fg dA
+defines an inner product for real functions f and g, S is the area of D, and δmn is the Kronecker
+delta. Notably, Φ0(x) = 1 is the constant eigenfunction, with corresponding eigenvalue k0 = 0.
+To determine the Dirichlet-to-Neumann operator for sufficiently smooth φ, we first substitute
+the series expansion φ(x, z) = �∞
+n=0 φn(z)Φn(x) into Laplace’s equation (1a), where we have
+temporally omitted the time dependence. We then solve the resulting equation for φn(z) over the
+linearised domain −h < z < 0, in conjunction with the no-flux condition on z = −h (see equation
+(1e)). It follows that ∂zφn(0) =
+ˆ
+Lnφn(0), where
+ˆ
+Ln = kn tanh(knh)
+(4)
+is the spectral multiplier of the Dirichlet-to-Neumann operator, L .
+By expressing the time-
+dependent free-surface velocity potential, u = φ|z=0, in terms of the basis expansion u(x, t) =
+�∞
+n=0 un(t)Φn(x), it follows that the Dirichlet-to-Neumann map has the spectral representation
+L u = �∞
+n=0
+ˆ
+LnunΦn.
+3
+The existence of resonant triads
+Resonant triads arise due to the exchange of energy between linear wave modes, an effect induced
+by nonlinear wave interactions. In order to define resonant triads mathematically, it is necessary
+to first determine the angular frequency associated with each linear wave mode. In the limit ϵ → 0,
+the Benney-Luke equation (3) reduces to the linear equation utt + L u = 0. By seeking a solution
+to the linearised Benney-Luke equation of the form u(x, t) = Φn(x)e−iωnt, we conclude that the
+angular frequency, ωn, satisfies ω2
+n =
+ˆ
+Ln, or the more familiar [32]
+ω2
+n = kn tanh(knh).
+(5)
+5
+
+As we will see, a crucial aspect of the following analysis is that the angular frequency depends on
+the fluid depth, i.e. ωn(h). Finally, we note that the angular frequency is larger for more oscillatory
+eigenfunctions (i.e. for larger values of kn); by analogy to the evolution of plane gravity waves, we
+refer to kn as a ‘wavenumber’ henceforth.
+We proceed by considering three linear wave modes, enumerated n1, n2 and n3, where we denote
+Ωj = ωnj,
+Kj = knj,
+and
+Ψj(x) = Φnj(x)
+for j = 1, 2, 3.
+Notably, we exclude the wavenumber k0 = 0 from consideration as the corresponding eigenmode,
+Φ0, simply reflects the invariance of the Benney-Luke equation (3) under the mapping u �→ u +
+constant; henceforth, we consider only wavenumbers Kj > 0. The three linear wave modes form a
+resonant triad if there is a critical fluid depth, hc, satisfying
+Ω1(hc) ± Ω2(hc) ± Ω3(hc) = 0,
+(6)
+where all four sign combinations are permissible (we consider Ωj > 0 without loss of generality).
+To simplify notation in the following arguments, we restrict our attention to the particular case
+Ω1(hc) + Ω2(hc) = Ω3(hc),
+(7)
+where the other three sign combinations in equation (6) may be recovered by suitable re-indexing
+of the Ωj terms. However, as we will see in §4, an additional constraint necessary for triads to
+exist is the eigenmode correlation condition,
+��
+D
+Ψ1Ψ2Ψ3 dA ̸= 0,
+(8)
+which implies that the product of any two eigenmodes is non-orthogonal to the remaining eigen-
+mode.
+3.1
+The existence of a critical depth
+We proceed to determine necessary and sufficient conditions on the wavenumbers, Kj, for there
+to exist a depth, hc, at which a resonant triad forms, where such a critical depth is unique. We
+summarise our results in terms of the following theorem.
+Theorem 1. There exists a positive and finite value of h such that Ω1 + Ω2 = Ω3 if and only if
+K1 + K2 < K3 <
+��
+K1 +
+�
+K2
+�2.
+(9)
+When this pair of inequalities is satisfied, the corresponding value of h is unique.
+We briefly sketch the proof of Theorem 1, with full details presented in appendix A.
+We
+first demonstrate that no solutions to Ω1 + Ω2 = Ω3 are possible when the bounds in equation
+(9) are violated, i.e. when K1 + K2 ≥ K3 or when √K1 + √K2 ≤ √K3. We then consider the
+case where the inequalities (9) are satisfied and determine the existence of positive roots to the
+function F(h) = (Ω1(h) + Ω2(h))/Ω3(h) − 1. In this case, we demonstrate that limh→0 F(h) < 0
+and limh→∞ F(h) > 0, from which we conclude that F(h) has at least one root (by continuity of
+F). Finally, we deduce that this root is unique by proving that F(h) is a strictly monotonically
+increasing function of h when the inequalities (9) are satisfied.
+6
+
+Two important conclusions may be deduced from Theorem 1. First, it follows from equation (9)
+that the wavenumber, K3, corresponding to the largest angular frequency, Ω3, is larger than both
+the other two wavenumbers (K1 and K2), but it cannot be arbitrarily large (as supplied by the
+upper bound). For a given pair of eigenmodes (say Ψ1 and Ψ2), we conclude that there are likely
+to be only finitely many eigenmodes that can resonate with this pair (indeed, that number might
+fairly small, or even zero). Second, when modes 1 and 2 coincide (a 1:2 resonance), one deduces
+that Ω1 = Ω2 and K1 = K2; as such, the existence bounds (9) simplify to 2K1 < K3 < 4K1, or
+2 < K3/K1 < 4 [34, 44].
+3.2
+Determining the critical depth
+Although Theorem 1 determines necessary and sufficient conditions on the wavenumbers, Kj, for
+there to be a critical depth, hc, at which a resonant triad exists, the critical depth remains to be
+determined. In general, the critical depth must be computed numerically (being the unique root
+of the nonlinear function F(h)); however, we demonstrate that useful quantitative and qualitative
+information may be obtained via asymptotic analysis. For the remainder of this section, we consider
+the rescaled wavenumbers, ξ1 = K1/K3 and ξ2 = K2/K3, and the rescaled depth, ζ = K3h; it
+remains to determine the root, ζc, of
+F(ζ) =
+�
+ξ1 tanh(ξ1ζ)
+tanh(ζ)
++
+�
+ξ2 tanh(ξ2ζ)
+tanh(ζ)
+− 1
+(10)
+when ξ1, ξ2 > 0 satisfy
+ξ1 + ξ2 < 1 <
+�
+ξ1 +
+�
+ξ2.
+(11)
+In figure 2(a), we present contours of the critical rescaled depth, ζc, in the (ξ1, ξ2)-plane, restricted
+to the region demarcated by equation (11). Consistent with the limits limζ→0 F(ζ) = ξ1 + ξ2 − 1
+and limζ→∞ F(ζ) = √ξ1+√ξ2−1, we observe that the root, ζc, tends to zero at the line ξ1+ξ2 = 1,
+and approaches infinity at the curve √ξ1 + √ξ2 = 1. Furthermore, the uniqueness of the root of
+F for given (ξ1, ξ2) is reflected in the observation that the contours of ζc do not cross. Finally, we
+note that the contours are symmetric about the line ξ1 = ξ2, which is a direct consequence of the
+invariance of F(ζ) under the mapping ξ1 ↔ ξ2 (see equation (10)).
+Although we are primarily interested in the physically relevant case for which the cylinder’s
+depth-to-width ratio, h, is of size O(1), an informative analytic result may be obtained by consid-
+ering F(ζ) in the limit ζ ≪ 1 (or K3h ≪ 1). By utilising the Taylor expansion
+�
+tanh(x) ∼ √x
+�
+1 − x2
+6 + 19
+360x4 + O
+�
+x6��
+,
+we obtain
+�
+ξ1 tanh(ξ1ζ)+
+�
+ξ2 tanh(ξ2ζ)−
+�
+tanh(ζ) ∼
+�
+ζ
+��
+ξ1 +ξ2 −1
+�
+− ζ2
+6
+�
+ξ3
+1 +ξ3
+2 −1
+�
++O(ζ4)
+�
+(12)
+for 0 < ζ ≪ 1. Whilst deriving equation (12), we have utilised the bound ξ1, ξ2 < 1 (see equation
+(11)), which additionally ensures that 0 < ξjζ ≪ 1 for j = 1, 2. We note that the left-hand side of
+equation (12) is equal to F(ζ) tanh(ζ), so ζc satisfies
+ξ1 + ξ2 − 1 − ζ2
+c
+6
+�
+ξ3
+1 + ξ3
+2 − 1
+�
+= O(ζ4
+c ),
+(13)
+7
+
+Figure 2: Contours of the rescaled critical depth, ζc = hcK3, as a function of the rescaled wavenum-
+bers, ξ1 = K1/K3 and ξ2 = K2/K3. (a) The contours computed numerically from equation (10).
+The black lines indicate the limiting cases of ζc → 0 (at ξ1+ξ2 = 1) and ζc → ∞ (at √ξ1+√ξ2 = 1).
+(b) The contours are overlaid by the leading-order approximation (equation (17); circles) and the
+higher-order correction (equation (18); diamonds) for ζc equal to 0.5, 1, 1.5 and 2.
+provided that 0 < ζc ≪ 1. By neglecting terms of size O(ζ4
+c ) in equation (13), one may then easily
+solve for ζc in terms of ξ1 and ξ2.
+Alternatively, a more succinct expression for ζc may be found by first noting that
+ξ3
+1 + ξ3
+2 = (ξ1 + ξ2)3 − 3ξ1ξ2(ξ1 + ξ2) = 1 − 3ξ1ξ2 + O(ζ2
+c ),
+(14)
+where we have utilised the leading-order approximation ξ1 + ξ2 = 1 + O(ζ2
+c ) (see equation (13)) to
+determine the second equality. Upon substituting equation (14) into equation (13), we find that
+ξ1, ξ2 and ζc are now related by the notably simpler expression
+ξ1 + ξ2 − 1 + ζ2
+c
+2 ξ1ξ2 = O(ζ4
+c ).
+(15)
+By neglecting terms of O(ζ4
+c ), the leading-order approximation for the rescaled critical depth, ζc,
+is given by
+ζc ∼
+�
+2(1 − ξ1 − ξ2)
+ξ1ξ2
+,
+(16)
+an expression valid when 0 < ζc ≪ 1 and ξ1 + ξ2 < 1 (see equation (11)). Alternatively, one may
+deduce from equation (15) that the contours of ζc satisfy the approximate form
+ξ2 ∼
+1 − ξ1
+1 + 1
+2ζ2
+c ξ1
+,
+(17)
+where the term in the denominator is responsible for the increased ‘bending’ of the contours as
+ζc becomes progressively larger (see figure 2). We note that the additional simplification afforded
+8
+
+(a)
+(b)
+hcK3 = 0.5
+ Leading-order approx.
+hcK3 = 1
+Higher-order correction
+hcK3 = 1.5
+0.8
+0.8
+hcK3 = 2
+hcK3 = 3
+hcK3 = 5
+0.6
+0.6
+K2
+hc
+hcK3 = 10
+个
+K3
+0.4
+0.4
+3
+0.2
+0.2
+8
+0
+0
+0.2
+0.4
+0.6
+0.8
+0.2
+0.4
+0.6
+0.8
+0
+1
+0
+1
+Ki/K3
+Ki/K3by equation (14) allows for a far more tractable representation of the contours relative to solving
+equation (13) directly for ξ2 given ξ1 and ζc.
+Despite being derived under the assumption 0 < ζc ≪ 1, we see in figure 2(b) that the contours
+given by equation (17) agree favorably with the numerical solution even up to ζc ≈ 1. However, it
+is readily verified from equation (16) that the asymptotic approximation of each contour crosses
+the boundary curve √ξ1 + √ξ2 = 1 at ζc = 4 (for which ξ1 = ξ2 = 1
+4), thereby demonstrating
+that the reduced asymptotic form has limited applicability (even in a qualitative sense) for slightly
+larger values of ζc. One may further improve the quantitative (and, to an extent, qualitative)
+agreement between the asymptotic analysis and numerical computation by including terms of size
+O(ζ4) in equation (12); indeed, an analogous calculation gives rise to the following higher-order
+correction to equation (15):
+ξ1 + ξ2 − 1 + ζ2
+c
+2 ξ1ξ2 + ζ4
+c
+72ξ1ξ2
+�
+ξ1ξ2 − 1
+�
+= O(ζ6
+c ).
+(18)
+Although one may then solve for ζc given ξ1 and ξ2 (or, alternatively, determine the contours of
+ζc) by truncating terms of O(ζ6
+c ) in equation (18), the resulting algebraic expressions yield little
+qualitative information. However, one may, in principle, use this reduced form as a reasonable
+initial guess for a numerical root-finding algorithm for determining the root of F(ζ), provided that
+ζc is not too large.
+3.3
+Example cavities
+Our investigation into the emergence of resonant triads has been focused, thus far, on finite-depth
+cylinders with arbitrary horizontal cross-section. However, it is convenient to understand how
+the results of Theorem 1 influence the formation (or not) of resonant triads for some specific
+cross-sections, namely rectangular, circular, and annular cylinders.
+3.3.1
+Rectangular cylinder
+It is well known that resonant triads are impossible for plane gravity waves evolving across an
+unbounded horizontal domain of finite depth [48, 21]. 1 We now utilise Theorem 1 to demonstrate
+a similar result: resonant triads are impossible for gravity waves evolving within a rectangular
+cylinder of finite depth. Our result generalises the special case of a 1:2 resonance, for which the
+impossibility of internal resonance in a rectangular cylinder was demonstrated by Miles [42].
+To proceed, we consider a rectangular cylinder with side lengths Lx and Ly. By orientating the
+Cartesian coordinate system, x = (x, y), so that the cylinder cross-section is defined by the region
+0 < x < Lx and 0 < y < Ly, the eigenmodes are of the form
+Φmn(x, y) =
+1
+Nmn
+cos(pmx) cos(qny),
+where Nmn > 0 is a normalisation constant.
+Notably, the wavenumbers pm = mπ/Lx and
+qn = nπ/Ly are chosen so that the no-flux condition is satisfied (see equation (1d)). For a triad de-
+termined by the non-negative integers mj and nj (for j = 1, 2, 3), the corresponding wavenumbers,
+Pj = pmj and Qj = qnj, must satisfy P1 + P2 = P3 and Q1 + Q2 = Q3 (under suitable reordering
+1Weak interactions are possible, however, in the shallow-water limit, Kjh → 0, for which tanh(Kjh) in the
+dispersion relation (5) is replaced by its leading-order approximation, Kjh [48, 7, 42].
+9
+
+of the subscripts) in order for the eigenmode correlation condition (8) to be satisfied. By defining
+the wave vector kj = (Pj, Qj), the conditions on Pj and Qj simplify to the single requirement
+k1 + k2 = k3, where the triangle inequality supplies that |k3| ≤ |k1| + |k2|. As the eigenvalues,
+K2
+j , of the negative Laplacian operator are related to the wave vectors via Kj = |kj|, we deduce
+that K3 ≤ K1 + K2. Owing to the violation of the left-hand bound in equation (9), we conclude
+that resonant triads cannot exist in a rectangular cylinder of finite depth.
+3.3.2
+Circular cylinder
+We consider a circular cylinder of unit radius in dimensionless variables (i.e. the dimensional radius
+is equal to a; see §2). For polar coordinates x = (r, θ), it is well known that the corresponding
+(complex-valued) eigenmodes may be expressed in the form
+Φmn(r, θ) =
+1
+Nmn
+Jm(kmnr)eimθ,
+where
+Nmn =
+��Jm(kmn)
+��
+�
+1 − m2
+k2
+mn
+(19)
+is the normalisation factor and m is the azimuthal wavenumber (an integer). Furthermore, the
+no-flux condition (1d) determines that the radial wavenumbers, denoted kmn, satisfy J′
+m(kmn) =
+0, where 0 < km1 < km2 < . . . (we exclude k00 = 0 from consideration; see §3).
+Notably,
+the eigenvalues of the negative Laplacian operator are precisely the squared wavenumbers, k2
+mn;
+consequently, the antinodes of each Bessel function play a pivotal role in determining the existence
+of resonant triads.
+Akin to the rectangular cylinder, we find that the eigenmode correlation condition imparts
+an important restriction on the combination of eigenmodes that may resonate.
+For given mj
+and nj (for j = 1, 2, 3), we denote Kj = kmjnj, Ψj = Φmjnj and Nj = Nmjnj. Although the
+correlation condition given in equation (8) is defined for real eigenmodes, a similar condition holds
+for complex-valued eigenmodes, namely
+��
+D Ψ1Ψ2Ψ∗
+3 dA ̸= 0. By considering the quantity
+��
+D
+Ψ1Ψ2Ψ∗
+3 dA =
+1
+N1N2N3
+� � 1
+0
+rJm1(K1r)Jm2(K2r)Jm3(K3r) dr
+�� � 2π
+0
+ei(m1+m2−m3)θ dθ
+�
+,
+we deduce from the azimuthal integral that a necessary condition for the correlation integral to
+be nonzero is m1 + m2 = m3 [40]. This condition thus restricts the permissible combinations of
+angular wavenumbers in a manner similar to the restriction on the permissible planar wavenumbers
+for the case of a rectangular cylinder. Unlike rectangular cylinders, however, we demonstrate that
+resonant triads are possible in a circular cylinder.
+Despite the apparent restriction of the Bessel antinodes, Kj, and summation condition on the
+azimuthal wavenumbers, mj, Theorem 1 determines that a vast array of resonant triads may be
+excited for judicious choices of the fluid depth. In table 1, we list a small number of resonant
+triads and their corresponding critical depth, hc, subject to the restrictions |mj| ≤ 3 and nj ≤ 3;
+for larger values of |mj| and nj, the corresponding wave field becomes increasingly oscillatory, to
+the extent that the effects of surface tension and dissipation may become appreciable. Moreover,
+even marginally relaxing the upper bounds on |mj| and nj vastly increases the number of resonant
+triads; indeed, the restriction |mj| ≤ 4 and nj ≤ 4 introduces 70 additional resonant triads relative
+to table 1. As the upper bounds for |mj| and nj are further increased, the typical difference between
+the various critical depths decreases. Finally, the correlation condition,
+��
+D Ψ1Ψ2Ψ∗
+3 dA ̸= 0, is
+satisfied for each triad; however, the integral is very close to zero in some cases (e.g. triad 18),
+corresponding to an elongation of the triad evolution time-scale (see §4.1).
+10
+
+No.
+m1
+m2
+m3
+n1
+n2
+n3
+K1
+K2
+K3
+hc
+��
+D Ψ1Ψ2Ψ∗
+3 dA
+1
+-3
+3
+0
+1
+1
+3
+4.201
+4.201
+10.173
+0.14591
+0.19061
+2
+-2
+2
+0
+1
+1
+2
+3.054
+3.054
+7.016
+0.17030
+0.46429
+3
+-2
+2
+0
+1
+1
+3
+3.054
+3.054
+10.173
+0.39129
+-0.03050
+4
+-2
+2
+0
+1
+2
+3
+3.054
+6.706
+10.173
+0.06331
+0.68257
+5
+-1
+1
+0
+1
+1
+1
+1.841
+1.841
+3.832
+0.15227
+1.28795
+6
+-1
+1
+0
+1
+1
+2
+1.841
+1.841
+7.016
+1.00970
+-0.02032
+7
+-1
+1
+0
+1
+2
+3
+1.841
+5.331
+10.173
+0.30197
+-0.00603
+8
+0
+0
+0
+1
+1
+3
+3.832
+3.832
+10.173
+0.19814
+0.03327
+9
+-2
+3
+1
+1
+1
+3
+3.054
+4.201
+8.536
+0.15767
+0.30704
+10
+-1
+2
+1
+1
+1
+2
+1.841
+3.054
+5.331
+0.17266
+0.85581
+11
+-1
+2
+1
+1
+1
+3
+1.841
+3.054
+8.536
+0.60375
+-0.02595
+12
+-1
+2
+1
+2
+1
+3
+5.331
+3.054
+8.536
+0.04664
+0.99088
+13
+0
+1
+1
+1
+1
+3
+3.832
+1.841
+8.536
+0.38516
+0.00542
+14
+-1
+3
+2
+1
+1
+2
+1.841
+4.201
+6.706
+0.16313
+0.64211
+15
+-1
+3
+2
+1
+1
+3
+1.841
+4.201
+9.969
+0.48152
+-0.02717
+16
+-1
+3
+2
+1
+2
+3
+1.841
+8.015
+9.969
+0.03928
+1.08903
+17
+-1
+3
+2
+2
+1
+3
+5.331
+4.201
+9.969
+0.06286
+0.66930
+18
+0
+2
+2
+1
+1
+3
+3.832
+3.054
+9.969
+0.26387
+-0.00087
+19
+1
+1
+2
+1
+1
+2
+1.841
+1.841
+6.706
+0.83138
+0.02801
+20
+1
+1
+2
+1
+2
+3
+1.841
+5.331
+9.969
+0.28691
+0.01818
+21
+0
+3
+3
+1
+1
+3
+3.832
+4.201
+11.346
+0.21395
+-0.00640
+22
+0
+3
+3
+2
+1
+3
+7.016
+4.201
+11.346
+0.02782
+1.00669
+23
+1
+2
+3
+1
+1
+2
+1.841
+3.054
+8.015
+0.50595
+0.03712
+24
+1
+2
+3
+1
+2
+3
+1.841
+6.706
+11.346
+0.23678
+0.02590
+25
+1
+2
+3
+2
+1
+3
+5.331
+3.054
+11.346
+0.19839
+0.01522
+Table 1: Combinations of the angular wavenumbers, mj, and radial mode indices, nj, that form
+a resonant triad (m1 + m2 = m3 and Ω1 + Ω2 = Ω3) at critical depth, hc, in a circular cylinder
+of unit radius. For each triad, the corresponding wavenumbers, Kj = kmjnj, satisfy (9), and the
+correlation condition,
+��
+D Ψ1Ψ2Ψ∗
+3 dA ̸= 0, is met. The list is restricted to resonant triads arising
+for |mj|, nj ≤ 3, and we consider m1 ≤ m2 and m3 ≥ 0 without loss of generality. We have omitted
+resonances that give rise to the same critical depth, but with the roles of modes 1 and 2 swapped.
+The triad numbers (left column) and shaded rows are referenced in the text.
+11
+
+At this juncture, it is informative to assess how the triads listed in table 1 relate to the
+resonances explored in prior investigations. First, triad 7 in table 1 (dark grey row) was explored
+by Michel [40] for a circular cylinder of radius 9.45 cm and an approximate fluid depth of 3 cm; it
+follows that the depth-to-radius ratio in Michel’s experiment was approximately 0.317, close to the
+value of 0.30197 reported in table 1. Furthermore, table 1 (grey rows) incorporates two well-known
+examples of a 1:2 resonance, for which modes 1 and 2 coincide: (i) the critical depth hc = 0.83138
+(triad 19) corresponds to the second-harmonic resonance with the fundamental mode [42, 43, 8, 64];
+(ii) the critical depth hc = 0.19814 (triad 8) corresponds to a standing wave composed of two
+resonant axisymmetric modes [34, 64]. Finally, triads 1, 2, 3, 5 and 6 (table 1, light grey rows)
+form an interesting class of resonant triad, for which an axisymmetric mode (m3 = 0) interacts
+with two identical counter-propagating non-axisymmetric modes (m1 = −m2 ̸= 0 and n1 = n2).
+In fact, our investigation in §5.1 demonstrates that the axisymmetric mode is the so-called pump
+mode, and may thus excite the non-axisymmetric modes, even when the initial energy in each
+non-axisymmetric mode is negligible. We draw an analogy between this novel class of resonant
+triad and the excitation of beach edge waves [18] in §6.
+We conclude our exploration of resonant triads arising in a circular cylinder by remarking
+that the fluid depth may, in some cases, be judiciously chosen so as to excite multiple triads. In
+general, the condition on the angular frequencies, Ω1 + Ω2 = Ω3 (see equation (7)), cannot be
+satisfied for two distinct triads at the same fluid depth; however, nonlinear resonance may persist
+for both triads provided that each condition on the angular frequencies is approximately satisfied
+[5, 39, 15], at the cost of weak detuning (see §4.3.1 for further details). Specifically, if triads 1 and
+2 have critical depths hc,1 and hc,2, respectively, then there is potential excitement of both triads
+when the fluid depth, h, satisfies |h−hc,j| = O(ϵ) for j = 1, 2 (where 0 < ϵ ≪ 1 is the typical wave
+slope; see §2), giving rise to the approximation Ω1 + Ω2 − Ω3 = O(ϵ) for each triad. For example,
+if 0 < hc,2 − hc,1 ≪ 1, then it may be sufficient to excite both triads at an intermediate depth,
+hc,1 ≤ h ≤ hc,2. We note, however, that the excitation of multiple triads at a single fluid depth is
+not possible when the depth discrepancy, |h− hc,j|, becomes too large (relative to the typical wave
+slope) for any of the triads under consideration.
+To demonstrate the potential for the simultaneous excitation of two triads within a circular
+cylinder of finite depth, we consider two scenarios: (i) the excitation of two triads that share a
+common wave mode; and (ii) the excitation of two triads that do not share any common wave
+modes. Heuristically, case (ii) is more common than case (i) owing to the number of similar fluid
+depths in table 1; however, case (i) will likely generate a far richer set of dynamics owing to
+the nonlinear interaction between the two triads [35, 15, 13, 11]. As an example of case (i), we
+consider triads 21 and 24 in table 1, with nearby critical depths hc,1 = 0.21395 and hc,2 = 0.23678,
+respectively. As mode (m3, n3) = (3, 3) is common to both triads, inter-triad resonance may arise
+at an intermediate depth, e.g. h = 0.225. Finally, an example of case (ii) arises for triads 8 and 25
+in table 1, with nearby critical depths hc,1 = 0.19814 and hc,2 = 0.19839. Neither of these triads
+share a common wave mode, so one would not expect the inter-triad energy exchange discussed
+in case (i). Nevertheless, one might anticipate a signature of these two triads to be visible in
+the surface evolution for an intermediate depth, e.g. h = 0.19825. The theoretical and numerical
+exploration of coupled triads in a circular cylinder will be the focus of future investigation.
+3.3.3
+Annular cylinder
+A natural variation upon a circular cylinder is an annulus of inner radius r0 ∈ (0, 1) and outer
+radius 1. By varying r0, the annulus approaches a circular cylinder as r0 → 0+, and a quasi-one-
+12
+
+0
+0.5
+1
+0
+0.5
+1
+0
+0.5
+1
+0
+2
+4
+6
+0
+0.5
+1
+0
+0.5
+1
+Figure 3: The existence and predominant characteristics of a triad in an annular cylinder with inner
+radius r0 and outer radius 1. The triad bifurcates from the critical depth hc = 0.17266 as r0 → 0
+(the limiting case of a circular cylinder), with corresponding wavenumbers presented in table 1
+(see triad 10). (a) The critical depth, hc (blue curve), with hc → ∞ as r0 → rc, where rc ≈ 0.57
+(black line). (b) The corresponding wavenumbers, Kj, all of which remain finite for r0 < rc (black
+line). (c) The normalised wavenumbers, K1/K3 and K2/K3, parametrised by increasing r0 (blue
+arrow), with the limiting case r0 → 0 denoted by the white dot. The wavenumbers leave the triad
+existence region (see Theorem 1) via the left-hand boundary (black curve) as r0 → r−
+c .
+dimensional periodic ring as r0 → 1−. Notably, resonant triads are impossible for a one-dimensional
+periodic ring, as can be shown by modifying the arguments presented for the case of a rectangular
+cylinder (see §3.3.1). Thus, one might anticipate that the existence of triads in an annular cylinder
+depends critically on the inner radius, r0. Rather than enumerating some possible triads for given
+values of r0, we instead track the corresponding critical depth, hc, for the triads identified for a
+circular cylinder (see table 1) as r0 is progressively increased from zero. Of particular interest is
+determining whether a given triad exists for all r0 < 1, or whether there is some critical inner
+radius, rc, beyond which the triad ceases to exist, with either hc → 0 or hc → ∞ as r0 → r−
+c .
+The (complex-valued) eigenmodes in an annular domain are cylinder functions of the form
+Φmn(r, θ) =
+1
+Nmn
+�
+Jm(kmnr) cos(γmnπ) + Ym(kmnr) sin(γmnπ)
+�
+eimθ,
+(20)
+where Nmn > 0 is a normalisation constant, Ym is the Bessel function of the second kind with
+order m (an integer), and γmn ∈ [0, 1] determines the weighting between the two Bessel functions.
+As shown in appendix B, the no-flux condition (see equation (1d)) on the inner and outer walls
+determines that the wavenumbers, kmn(r0), satisfy the equation
+J′
+m(kmnr0)Y′
+m(kmn) − J′
+m(kmn)Y′
+m(kmnr0) = 0.
+(21)
+A formula for the corresponding value of γmn is determined in appendix B.
+Once again, the
+wavenumbers, kmn, are ordered so that 0 < km1 < km2 < . . . (excluding k00 = 0) and satisfy
+−∆Φmn = k2
+mnΦmn. Three correlated wave modes may form a resonant triad (for a judicious
+choice of the fluid depth) provided that the corresponding wavenumbers, Kj, which depend on the
+channel width, 1 − r0, satisfy the bounds given in Theorem 1.
+Bifurcating from the limiting case of a circular cylinder, we track the critical depth (when such
+a depth exists) of different triads as r0 is progressively increased. The predominant behaviour is
+13
+
+characterised by the example presented in figure 3, for which we consider the triad whose critical
+depth is hc = 0.17266 as r0 → 0+ (see triad 10 in table 1). Given that hc is fairly small in this limit,
+one might anticipate that the triad ceases to exist with hc → 0; somewhat surprisingly, however,
+the opposite scenario arises, with hc → ∞ as r0 → r−
+c (rc ≈ 0.57 in this example). It follows,
+therefore, that the triad may persist for narrow channels only when the fluid is sufficiently deep.
+We note, however, that there exist (at least) two relatively rare transitions for increasing r0, which
+we briefly describe as follows: (i) the triad ceases to exist when hc → 0 as r0 → r−
+c , which may
+arise when bifurcating from a sufficiently shallow circular cylinder (e.g. triad 22 in table 1); and (ii)
+the triad continues to exist for all r0 < 1, with hc → 0 and Kj → ∞ as r0 → 1, yet the normalised
+depth, hcK3, remains finite, and the normalised wavenumbers, K1/K3 and K2/K3, remain within
+the triad existence region (e.g. triad 8 in table 1). Owing to the appreciable influence of viscous
+effects for relatively shallow fluids, the physical relevance of these latter two scenarios is somewhat
+nebulous, however.
+4
+The evolution of resonant triads
+Having established the existence of resonant triads, we now determine the long-time triad evolution,
+utilising the method of multiple scales. Ostensibly, the calculations necessary for determining the
+triad equations are a variation upon the pioneering work of McGoldrick [36, 37, 38] in the absence
+of surface tension. However, the confinement of the fluid to a cylinder imposes some additional
+considerations, the salient details of which we outline below. Finally, we note that an alternative
+approach to multiple scales is Whitham’s technique of averaging the system’s Lagrangian [59,
+60, 61, 62], which has the advantage of streamlining some algebraic calculations [54, 42, 43];
+nevertheless, multiple-scales analysis is sufficient for our purposes and allows for the possible
+inclusion of higher-order corrections in the asymptotic expansion [38].
+In a manner similar to §3, we consider three linear wave modes (with real-valued eigenfunctions),
+enumerated n1, n2 and n3, where we denote
+Ωj = ωnj,
+Kj = knj,
+Lj =
+ˆ
+Lnj,
+and
+Ψj(x) = Φnj(x)
+for j = 1, 2, 3.
+In contrast to §3, however, we now allow each (nonzero) angular frequency to be either negative
+or positive: the resonance condition on the angular frequencies is henceforth defined
+Ω1 + Ω2 + Ω3 = 0.
+(22)
+The modified requirement on the angular frequencies (equation (22)) is not restrictive on the
+possible triad combinations; one may recover equation (7) by mapping Ω3 �→ −Ω3, for example.
+The decision behind the summation condition on the angular frequencies is motivated by the
+cyclical symmetry of equation (22), a property that will be inherited by the resultant amplitude
+equations [54]. As a consequence, one need only derive the amplitude equation for one of the wave
+modes; the amplitude equations for the remaining two wave modes follow by cyclic permutation
+of the subscripts (1, 2, 3).
+Before embarking on the multiple-scales analysis presented in §4.1, we remark upon two caveats.
+First, we note that equation (22) corresponds to an exact resonance, for which the fluid depth, h,
+is chosen to be precisely equal to the critical depth, hc. In practice, however, there may be a small
+discrepancy between h and hc, resulting in a the sum of the angular frequencies being slightly
+offset from zero. When the frequency detuning is sufficiently weak, e.g. Ω1 + Ω2 + Ω3 = O(ϵ), one
+14
+
+may modify the following asymptotic analysis to derive a similar set of amplitude equations (see
+§4.3.1). Second, our analysis in §4.1 is not valid when two of the wave modes coincide. This case
+corresponds to a 1:2 resonance, for which the corresponding evolution equations were derived by
+Miles [42] using Whitham modulation theory (as summarised in §4.3.2).
+4.1
+Multiple-scales analysis
+In order to determine the evolution of each of the three dominant wave modes involved in an exact
+resonance, we utilise the method of multiple scales [28, 55]. Specifically, we seek a perturbation
+solution to the Benney-Luke equation (3) of the form u ∼ u0+ϵu1+O(ϵ2). The leading-order terms
+in equation (3) determine that u0 satisfies ∂ttu0 + L u0 = 0; we choose to consider a leading-order
+solution comprised only of the three triad modes (all other modes are assumed to be smaller in
+magnitude and appear at higher order), giving rise to the leading-order form
+u0(x, t, τ) =
+3
+�
+j=1
+�
+Aj(τ)Ψj(x)e−iΩjt + c.c.
+�
+.
+(23)
+In equation (23), we have introduced the slow time-scale τ = ϵt, which governs the evolution of
+each complex amplitude, Aj. As ϵ and t are both independent variables, we treat τ and t as
+independent time-scales, giving rise to the transformation of derivatives ∂t �→ ∂t + ϵ∂τ. Finally,
+c.c. denotes the complex conjugate of the preceding term, a contribution necessary for real u0.
+So as to determine coupled evolution equations for each complex amplitude, Aj, we consider
+terms of O(ϵ) in the Benney-Luke equation (3). By substituting the leading-order solution, u0,
+into the nonlinear terms and applying the triad condition for the angular frequencies (equation
+(22)), we obtain the following problem for u1:
+∂ttu1 + L u1 = −
+�
+3
+�
+j=1
+fj(x, τ)e−iΩjt + c.c.
+�
++ nonresonant terms.
+(24)
+As we will see below, each of the functions fj(x, τ) appearing on the right-hand side of equation
+(24) will play a fundamental role when determining the amplitude equations; specifically,
+f1 = −2iΩ1
+dA1
+dτ Ψ1 + iA∗
+2A∗
+3
+��
+Ω2
+�
+L2
+3 − K2
+3
+�
++ Ω3
+�
+L2
+2 − K2
+2
+�
+− 2Ω1L2L3
+�
+Ψ2Ψ3 − 2Ω1∇Ψ2 · ∇Ψ3
+�
+,
+where f2 and f3 follow upon cyclic permutation of the subscripts (1, 2, 3). Finally, we note that
+the ‘nonresonant terms’ in equation (24) are of the general form p(x, τ)eiςt, where we assume that
+the angular frequency, ς, is not equal (or close) to any of the angular frequencies, ±ωn, associated
+with linear wave modes (see §3).
+We proceed by projecting equation (24) onto each of the three wave modes, giving rise to
+differential equations of the form (for j = 1, 2, 3)
+∂ttˆu1,j + Ljˆu1,j = −
+�
+⟨Ψj, fj⟩e−iΩjt + c.c.
+�
++ nonresonant terms,
+(25)
+where ˆu1,j = ⟨Ψj, u1⟩ is the projection of u1 onto the mode Ψj. By recalling that Lj = Ω2
+j, we
+immediately see that the term in square brackets in equation (25) is itself a solution to the linear
+operator ∂tt+Ω2
+j. It follows that the solution of equation (25) comprises of particular solutions that
+15
+
+have temporal dependence te±iΩjt, leading to an ill-posed asymptotic expansion when ϵt = O(1).
+The resolution to this problem is achieved via the solubility condition ⟨Ψj, fj⟩ = 0, which suppresses
+the secular growth.
+By applying the solubility condition ⟨Ψj, fj⟩ = 0 for j = 1, 2, 3, we conclude that the com-
+plex amplitude, Aj(τ), of each wave mode, Ψj(x)e−iΩjt, evolves according to the triad system of
+canonical form [5, 15]
+dA1
+dτ = α1A∗
+2A∗
+3,
+dA2
+dτ = α2A∗
+1A∗
+3,
+dA3
+dτ = α3A∗
+1A∗
+2,
+(26)
+where
+α1 =
+1
+2Ω1
+��
+Ω2
+�
+L2
+3 − K2
+3
+�
++ Ω3
+�
+L2
+2 − K2
+2
+�
+− 2Ω1L2L3
+�
+C − 2Ω1
+�
+Ψ1, ∇Ψ2 · ∇Ψ3
+��
+,
+(27)
+while α2 and α3 follow by cyclic coefficient of the subscripts (1, 2, 3). Furthermore, the correlation
+integral, C , is defined
+C = 1
+S
+��
+D
+Ψ1Ψ2Ψ3 dA,
+(28)
+where we recall that S is the area of the cylinder cross-section (see §2.2). As the triad equations
+(26) are valid for τ = O(1) (or t = O(1/ϵ)), their dynamics yield an informative view of the
+long-time evolution of the resonant triad.
+In order to assess the influence of the triad coefficients on the triad evolution (see §4.2), we first
+simplify the algebraic form given in equation (27). As shown by Miles [42], one may simplify the
+inner product ⟨Ψ1, ∇Ψ2 · ∇Ψ3⟩ by repeated application of the divergence theorem and utilisation
+of the relationship −∆Ψj = K2
+j Ψj; it follows that
+�
+Ψ1, ∇Ψ2 · ∇Ψ3
+�
+= 1
+2
+�
+K2
+2 + K2
+3 − K2
+1
+�
+C ,
+(29)
+where C is the correlation integral defined in equation (28). We then substitute equation (29) into
+equation (27) and simplify using the relation Ω1 + Ω2 + Ω3 = 0. After some algebra, we derive the
+reduced expression
+α1 = C
+2Ω1
+�
+Ω2L2
+3 + Ω3L2
+2 − 2Ω1L2L3 +
+3
+�
+l=1
+ΩlK2
+l
+�
+,
+(30)
+where α2 and α3 follow similarly. Finally, we demonstrate in appendix C that the algebraic form
+of the triad coefficients may be further reduced to
+αj = C β
+2Ωj
+for
+j = 1, 2, 3,
+(31)
+where
+β =
+3
+�
+l=1
+ΩlK2
+l − 1
+2Ω1Ω2Ω3
+�
+Ω2
+1 + Ω2
+2 + Ω2
+3
+�
+.
+(32)
+Equations (26), (28), (31) and (32) constitute the triad equations for resonant gravity waves
+confined to a cylinder of finite depth. Although the triad equations are of canonical form [5],
+the novelty of our investigation is the computation of the coefficients, αj, whose algebraic form is
+specific to our system.
+16
+
+Figure 4: Contours of β/K2
+3 (see equation (32)) for the case Ω1, Ω2 > 0 and Ω3 < 0 (with
+Ω1 + Ω2 + Ω3 = 0), for which β < 0 (see equation (33)).
+4.2
+Properties of the triad coefficients
+The simplified form of the coefficients, αj (equation (31)), allows for some important theoretical
+observations that were obfuscated by the more complicated expressions for αj given in equations
+(27) and (30). In particular, as exactly two of the angular frequencies, Ωj, have the same sign, we
+deduce from equation (31) that the two corresponding coefficients, αj, also have the same sign,
+with the third coefficient having the opposite sign. By utilising the well-known results pertaining
+to the canonical triad equations, we conclude that all solutions to the triad equations (26) are
+periodic in time, with solutions expressible in terms of elliptic functions [1, 5, 54, 15]. Typically,
+these solutions result in an exchange of energy between the comprising modes, although there is a
+class of periodic solution that, perhaps counter-intuitively, results in zero energy exchange for all
+time [9, 10]. Moreover, it is readily verified that the leading-order energy density, E1 + E2 + E3, is
+conserved, where Ej = Ω2
+j|Aj|2, consistent with the Hamiltonian structure of the Euler equations
+[5, 15]. The reader is directed to the work of Craik [15] for a more detailed account of the various
+properties of the canonical triad equations.
+Of particular relevance to the evolution of the triad is the quantity β (see equation (32)), which,
+together with C , determines the time scale over which energy exchange arises. In particular, we
+present the form of β in figure 4 for the case Ω1, Ω2 > 0 and Ω3 < 0. As we will demonstrate
+below, β < 0 in this case; in general, the sign of β is the same as the sign of the largest (in
+magnitude) angular frequency, Ωj. Notably, |β| decreases sharply towards zero as K1 + K2 → K3,
+corresponding to the limit hc → 0. Similarly, |β| approaches zero in the limiting cases K1 ≪
+K3 or K2 ≪ K3, corresponding to one low-oscillatory wave mode interacting with two highly-
+oscillatory wave modes. Away from these limiting cases, however, |β| depends only weakly on
+the wavenumbers, Kj, suggesting that the correlation integral, C , predominantly controls the
+time-scale of the triad evolution. Finally, we observe that β is symmetric about the line K1 = K2,
+consistent with the invariance of equation (32) under the mapping K1 ↔ K2 (and hence, Ω1 ↔ Ω2).
+17
+
+0
+-0.1
+0.8
+-0.2
+0.6
+-0.3
+K2
+K3
+-0.4
+0.4
+-0.5
+0.2
+-0.6
+-0.7
+0
+0
+0.2
+0.4
+0.6
+0.8
+1
+Ki/K3We conclude this section by proving that β < 0 in the case Ω1, Ω2 > 0 and Ω3 < 0. By
+comparing the forms of equations (32) and (30), and then permuting the subscripts (1, 2, 3) �→
+(3, 1, 2), we first note that β may be equivalently expressed as
+β = Ω1L2
+2 + Ω2L2
+1 − 2Ω3L1L2 +
+3
+�
+l=1
+ΩlK2
+l ,
+or
+β = Ω1(L2
+2 + K2
+1) + Ω2(L2
+1 + K2
+2) + |Ω3|(2L1L2 − K2
+3).
+By bounding Lj = Kj tanh(Kjhc) < Kj for 0 < hc < ∞ and utilising the relation Ω1 + Ω2 = |Ω3|,
+we obtain
+β < |Ω3|
+�
+K2
+1 + K2
+2 + 2K1K2 − K2
+3
+�
+= |Ω3|
+�
+(K1 + K2)2 − K2
+3
+�
+.
+(33)
+As resonant triads exist only when K1 + K2 < K3 (see Theorem 1), we conclude that β < 0 in this
+case.
+4.3
+Summary
+To summarise our theoretical developments, the velocity potential, u, at the fluid rest level (z = 0)
+evolves according to
+u(x, t) ∼
+3
+�
+j=1
+�
+Aj(τ)Ψj(x)e−iΩjt + c.c.
+�
++ O(ϵ),
+(34)
+where the complex amplitudes, Aj(τ), evolve over the slow time-scale, τ = ϵt, according to the triad
+equations (26). In particular, the triad coefficients, αj (see equation (31)), are defined in terms
+of the correlation integral, C (equation (28)), and the coefficient β (equation (32)). Notably, we
+assume that C is nonzero; if this condition were violated then all three of the triad coefficients, αj,
+would be equal to zero, giving rise to non-interacting wave modes at leading order (contradicting
+the notion of a triad).
+Indeed, the condition C ̸= 0 is identical to the correlation condition
+detailed in equation (8), the origins of which we have now justified. Finally, the evolution of the
+free surface, η, may be recovered by recalling that η = −ut + O(ϵ): we conclude that η(x, t) has
+a similar leading-order form to u(x, t), but each complex amplitude, Aj(τ), in (34) is replaced by
+iΩjAj(τ) (see equation (37) below).
+We briefly contrast our investigation of triad interaction with the early-time calculation of
+Michel [40], who characterised the initial linear growth of a child mode induced by the nonlinear
+interaction of two parent modes (where all three modes comprise the triad).
+If modes 1 and
+2 are the parent modes and mode 3 is the child mode, then the initial linear growth may be
+deduced directly from triad equations (26) in the limit |A3| ≪ |A1| ∼ |A2|. Specifically, the initial
+variation of A1 and A2 is slow relative to that of A3, which has the approximate early-time form
+A3(τ) ≈ α3C∗
+1C∗
+2τ + C3, where Cj = Aj(0). Notably, the linear growth rate of the child mode
+depends on the corresponding triad coefficient, α3, and the product of the initial amplitudes of the
+two parent modes. However, our result for circular cylinders differs to that of Michel; we believe
+that the author neglected some important nonlinear contributions (compare Michel’s equation
+(A2) to equations (2.4) and (2.4a) of Longuet-Higgins [33]). As Michel’s experiment verified the
+scaling of the interaction only up to a proportionality constant, this discrepancy was not captured.
+18
+
+4.3.1
+The influence of weak detuning
+As discussed earlier in §4, the analysis in §§4.1 and 4.2 does not account for weak detuning of the
+angular frequencies, as might arise when the fluid depth, h, differs slightly from the critical depth,
+hc. We now briefly consider the case of weak detuning, for which equation (22) is replaced by the
+condition Ω1 +Ω2 +Ω3 = ϵσ (see §3.3.2); here ϵ is the small parameter representative of the typical
+wave slope (see §2) and σ = O(1) determines the extent of the detuning [5, 39]. By following
+a very similar multiple-scales procedure to the case σ = 0, we obtain amplitude equations that
+are now augmented by a time-dependent modulation. Specifically, each complex amplitude now
+evolves according to
+dA1
+dτ = α1A∗
+2A∗
+3eiστ,
+dA2
+dτ = α2A∗
+1A∗
+3eiστ,
+dA3
+dτ = α3A∗
+1A∗
+2eiστ,
+where each coefficient, αj, is defined in equation (31). Although detuning yields non-autonomous
+amplitude equations, autonomous equations may be derived by mapping Aj(τ) �→ Aj(τ)eiστ/3 for
+all j = 1, 2, 3 [15]. Finally, we note that the energy, E1 + E2 + E3, is not exactly conserved when
+considering the effects of detuning; instead, the energy slowly oscillates about a constant value
+[15].
+4.3.2
+The case of a 1:2 resonance
+A 1:2 resonance is a resonant triad for which two modes comprising the triad coincide. For this
+case, we define two angular frequencies, Ω1 and Ω2, so that Ω2 = 2Ω1 [42], where the connection
+to resonant triads is clear when writing Ω1 + Ω1 = Ω2. By following a very similar multiple-scales
+procedure to that outlined in §4.1, we obtain
+u(x, t) ∼
+2
+�
+j=1
+�
+Aj(τ)Ψj(x)e−iΩjt + c.c.
+�
++ O(ϵ),
+where
+dA1
+dτ = −γA∗
+1A2
+and
+dA2
+dτ = γ
+4A2
+1.
+(35)
+In particular, the evolution of the amplitude equations (35) depends on the coefficient γ = C
+�
+K2
+2 −
+K2
+1 − 3Ω4
+1
+�
+, where C = 1
+S
+��
+D Ψ2
+1Ψ2 dA is the correlation integral. Indeed, the amplitude equations
+(35) and coefficient, γ, are consistent with the results of Miles [42] when expressing the evolution
+of each complex amplitude, Aj, in polar form (with appropriate rescaling). Finally, we note that a
+weak detuning (see §4.3.1) may also be incorporated within the amplitude equations (35), thereby
+accounting for a slight mismatch between the fluid depth, h, and the corresponding critical depth,
+hc [42].
+Of particular interest is the evolution of weakly nonlinear waves steadily propagating around
+a circular cylinder of unit radius, focusing on the case where the fluid depth is precisely equal to
+the critical depth of a 1:2 resonance [64]. For the complex-valued eigenmodes defined in equation
+(19), the correlation condition,
+��
+D Ψ2
+1Ψ∗
+2 dA ̸= 0, determines that the angular wavenumbers satisfy
+m2 = 2m1 [12, 64]. By expressing the complex wave amplitudes in polar form, Aj(τ) = aj(τ)eiθj(τ)
+(for j = 1, 2), equation (35) may be recast as [42]
+da1
+dτ = −γa1a2 cos Θ,
+da2
+dτ = γ
+4a2
+1 cos Θ,
+dΘ
+dτ = 2γa2
+�
+1 − a2
+1
+8a2
+2
+�
+sin Θ,
+19
+
+where Θ(τ) = θ2(τ) − 2θ1(τ) is the time-dependent phase shift.
+Steadily propagating waves
+correspond to time-independent solutions for a1, a2 (both nonzero) and Θ, from which we deduce
+that cos Θ = 0 and a1/a2 = 2
+√
+2.
+Indeed, it is remarkable that the amplitude ratio of the
+two dominant (normalised) wave modes is independent of the angular wavenumbers, mj, the
+radial wavenumbers, Kj, and the corresponding angular frequencies, Ωj (see §3.3.2 for details).
+Furthermore, one may readily determine the relationship between the angular velocity of the
+steady wave rotation and the corresponding wave amplitude, which may then be compared to the
+numerical solution of the full Euler equations [64]. This comparison, as well as a comparison to
+steadily propagating waves computed from various truncations of the Euler equations, will be the
+subject of future investigation.
+5
+The excitation of resonant triads
+Having established the existence and evolution of resonant triads, we now focus on the excitation
+of a particular triad via external forcing. So as to motivate the method of excitation, we first
+recall (§5.1) the well-known result that one mode in the triad may, or may not, excite the other
+two modes [16, 22, 54]; in the case of excitation, the initial mode is referred to as the pump mode
+[15]. We will then utilise the criterion of the pump mode to excite all three modes in the triad via
+a pulsating pressure source (§5.2). Throughout this section, we continue with the convention that
+the triad angular frequencies satisfy Ω1 + Ω2 + Ω3 = 0, as set forth in §4.
+5.1
+Excitation via the triad pump mode
+To first identify the triad pump mode and then characterise the resultant excitation, we consider
+the case for which A3, say, is much larger in magnitude than the other two mode amplitudes, so
+|A1|, |A2| ≪ |A3| [16, 22, 54]. By linearising the triad equations (26), we obtain
+dA1
+dτ = α1A∗
+2A∗
+3,
+dA2
+dτ = α2A∗
+1A∗
+3,
+dA3
+dτ = 0,
+(36)
+from which we immediately conclude that A3 is constant (whilst the linearisation assumption
+holds); we denote A3(τ) = C for some given complex number C. By considering second derivatives
+of A1 and A2, we deduce the linearised evolution equations [15]
+d2A1
+dτ 2 = α1α2|C|2A1
+and
+d2A2
+dτ 2 = α1α2|C|2A2,
+where α1α2 = C 2β2/(4Ω1Ω2) (see equation (31)). We conclude that A1(τ) and A2(τ) grow ex-
+ponentially in time (whilst the linearisation approximation holds) when Ω1Ω2 > 0, and exhibit
+sinusoidal oscillations when Ω1Ω2 < 0 [16, 22, 15].
+Thus, mode 3 may excite modes 1 and 2
+when Ω1 and Ω2 have the same sign (and likewise for other mode permutations). As one angular
+frequency must have a different sign from the other two (so as to satisfy Ω1 + Ω2 + Ω3 = 0), we
+conclude that the mode whose angular frequency is largest in magnitude (i.e. differs in sign) is the
+triad pump mode [15]. Equivalently, the pump mode is the mode with largest wavenumber, Kj.
+To visualise the influence of the pump mode on the resultant free-surface pattern, we present the
+solution of the triad equations (26) and the corresponding pump-mode approximation (equation
+(36)) in figure 5. By recalling that the free surface satisfies η = −ut + O(ϵ), we first deduce that
+η(x, t) ∼
+3
+�
+j=1
+�
+iΩjAj(τ)Ψj(x)e−iΩjt + c.c.
+�
++ O(ϵ).
+(37)
+20
+
+Figure 5: Excitation of a triad via its pump mode for the case of a circular cylinder. We consider
+triad 24 in table 1, but with m3 �→ −m3. We choose Ω1, Ω2 > 0 and Ω3 < 0, so that mode 3
+is the pump mode. (a) Evolution of the free-surface, η ∼ −ut, over the slow time-scale, τ = ϵt,
+with ϵ = 10−3. (b) The evolution of the wave amplitudes, |Aj|, according to the triad equations
+(equation (26), solid curves) and the pump-mode approximation (equation (36), dashed-dotted
+curves). Insets: modes 1 (blue), 2 (red) and 3 (gold) at τ = 0; all three modes rotate counter-
+clockwise. The simulations were initialised from A1(0) = 0.01 and A2(0) = 0.01i, where A3(0) was
+chosen to be the positive real number satisfying E1 + E2 + E3 = 1, with Ej = Ω2
+j|Aj|2 (see §4.2).
+21
+
+T=O
+8
+T= 16
+24
+T川
+a
+(6)
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+0
+5
+10
+15
+20
+25
+30
+=For the case of a circular cylinder, we utilise the complex-valued eigenmodes defined in equation
+(19), corresponding to the superposition of steadily propagating waves for mj ̸= 0 (the rotation di-
+rection depends on the sign of Ωj/mj). Upon initialising the system so that the energy is primarily
+within the pump mode (mode 3), modes 1 and 2 are gradually excited due to nonlinear interaction,
+with exponential growth evident for τ ≲ 10. As time further increases, the dynamics depart from
+the pump-mode approximation: the energy in the pump mode appreciably decreases, whilst the
+energy in modes 1 and 2 saturates. The free surface varies qualitatively during this evolution, with
+an appreciable change in pattern structure visible by τ = 24 (primarily a superposition of modes 1
+and 2). Notably, the system evolution is periodic, which becomes apparent over longer time scales.
+5.2
+Excitation via an applied pressure source
+Based on the ideas of the previous section, we consider a methodology for exciting the pump
+mode of a triad, which will subsequently excite the remaining two modes (provided that the initial
+disturbance of each of the remaining modes is nonzero). Notably, several methods for exciting
+internal resonances have been considered in prior investigations, primarily focusing on imposed
+motion of the fluid vessel via horizontal [42, 45] or vertical vibration [42, 44, 46, 24]. Furthermore,
+one may, in principle, utilise sinusoidal paddles or plungers to excite a particular triad’s pump
+mode for a given geometry (similar wave makers are used in rectangular wave tanks [37, 23]).
+However, for large-scale fluid tanks, imposed motion of the vessel may be impractical (if the tank
+were set in a concrete base, for example), and it may be challenging to determine the correct
+paddle motion necessary to excite a chosen pump mode for geometrically complex cylinders. We
+choose, therefore, to consider a slightly different approach: we instead excite the pump mode via
+a pulsating pressure source located just above the free surface (e.g. an air blower).
+In order to incorporate a pressure source within our mathematical framework, we first refor-
+mulate the dimensionless dynamic boundary condition (equation (1b)) as
+φt + η + ϵ
+2
+�
+|∇φ|2 + φ2
+z
+�
++ ϵP(x, t) = 0
+for
+x ∈ D,
+z = ϵη,
+where the dimensional pressure is ϵ2aρgP for fluid density ρ (P = 0 corresponds to atmospheric
+pressure). The pressure source is chosen to be small in magnitude so that the resultant wave
+excitation arises over the slow time-scale, τ = ϵt, and may thus be saturated by weakly nonlinear
+effects. By modifying the developments outlined in §2.1, we derive the forced Benney-Luke equation
+utt + L u + ϵ
+�
+ut
+�
+L 2 + ∆
+�
+u + ∂
+∂t
+�
+(L u)2 + |∇u|2�
++ ∂tP
+�
+= O(ϵ2)
+for
+x ∈ D,
+(38)
+which will be the starting point for the asymptotic analysis.
+Before proceeding further, we first describe two forms of the pressure source relevant to our
+investigation. For a stationary pressure source oscillating periodically over the fast time-scale, t,
+we express P(x, t) = f(τ)s(x)e−iΩpt + c.c., where s(x) is a fixed spatial profile (generally spanning
+the cavity), f(τ) accounts for a slow modulation in the magnitude of the pressure, and Ωp is the
+pulsation angular frequency. We choose Ωp to be close to the angular frequency of the pump
+mode, which, without loss of generality, we assume to be mode 3 (i.e. Ω3 has the opposite sign
+from Ω1 and Ω2). We denote, therefore, Ωp = Ω3 + ϵµ, where µ = O(1) determines the extent
+of the frequency mismatch. For a pressure source orbiting the centre of a circular cylinder at a
+constant angular velocity, we instead posit that P has the form P(r, θ, t) = f(τ)s(r, θ−Ωpt), where
+22
+
+0
+5
+10
+1
+2
+3
+4
+0
+20
+40
+60
+80
+100
+0
+1
+2
+3
+4
+0
+50
+100
+150
+200
+250
+300
+1
+2
+3
+4
+5
+Figure 6: Evolution of the forced triad equations (39) for σ = µ = 0 and constant f. We consider
+the same triad as figure 5, with s3f = 0.1. In all three panels, A1(0) = 0.02i and A2(0) = 0.01.
+For A3(0) = 0.01, we observe (a) the initial excitation of the triad and (b) the resultant periodic
+dynamics (the initial growth is highlighted within the grey box). (c) For A3(0) = 0.01i, the triad
+evolution is chaotic.
+Ωp = (Ω3 + ϵµ)/m3 is the angular velocity of the pressure source (assuming that the pump mode
+is non-axisymmetric, i.e. m3 ̸= 0).
+For both standing and orbiting pressure sources, we now follow a similar multiple-scales pro-
+cedure to that outlined in §4.1, starting from the forced Benney-Luke equation (38). So as to
+discount the possibility that the pressure source excites more than one mode in the triad, we
+assume that neither |Ω1| or |Ω2| are close to |Ω3|. Furthermore, we incorporate a weak detuning
+in the triad angular frequencies, denoting Ω1 + Ω2 + Ω3 = ϵσ (see §4.3.1). It follows that each
+complex amplitude, Aj(τ), evolves according to
+dA1
+dτ = α1A∗
+2A∗
+3eiστ,
+dA2
+dτ = α2A∗
+1A∗
+3eiστ,
+dA3
+dτ = α3A∗
+1A∗
+2eiστ − Ω3s3f(τ)e−iµτ,
+(39)
+where the coefficients, αj, are defined in equation (31). Notably, the pump mode may only be
+excited provided that the corresponding eigenmode is non-orthogonal to the pressure source, cor-
+responding to a nonzero projection, i.e. s3 ̸= 0, where s3 = ⟨Ψ3, s⟩. Similar equations describing
+the evolution of forced resonant triads have been explored by McEwan et al. [35] (with the inclusion
+of linear damping) and Raupp & Silva Dias [50].
+In the special case of time-independent forcing (f constant) and no frequency detuning (σ =
+µ = 0), the dynamics of the forced triad equations has been analysed by Harris et al. [20], with
+both periodic and quasi-periodic dynamics reported. We also consider this case, leaving the effects
+of detuning and variable forcing for future investigation. In this setting, when |A1|, |A2| and |A3|
+23
+
+are initially small relative to the magnitude of the forcing, |Ω3s3f|, the initial growth in A3 is
+approximately linear (see figure 6(a)). As mode 3 is the pump mode, the growth in A3 excites A1
+and A2, thus activating the triad. The conservation laws of the forced triad equations [20] result
+in a temporary diminution of mode 3, which is later augmented by the external forcing; whence
+the process repeats. In some parameter regimes, the resulting evolution of the forced triad is
+periodic in time (see figure 6(b) and Raupp & Silva Dias [50]); in contrast to the findings of Harris
+et al. [20], however, we also identify initial conditions (with all other parameters unchanged) that
+result in hitherto unidentified chaotic dynamics (see figure 6(c)). The chaotic nature of this latter
+example may be verified via estimation of the maximal Lyapunov exponent [55], which is found
+to be positive (i.e. initially adjacent trajectories diverge exponentially in phase space); however, a
+more thorough investigation of the chaotic dynamics of the forced triad equations, and the subtle
+dependence on initial conditions, will be presented elsewhere.
+6
+Discussion
+We have performed a systematic investigation into nonlinear resonant triads of free-surface gravity
+waves confined to a cylinder of finite depth; previously studied 1:2 resonances are obtained as
+special cases. A key result of our study is Theorem 1, which determines whether there exists
+a fluid depth at which three given wave modes resonate due to the nonlinear evolution of the
+fluid. Equipped with this result, we determined the long-time fluid evolution using multiple-scales
+analysis, from which we deduced that all solutions to the triad equations are periodic in time.
+Finally, we determined that a given triad may be excited via external forcing of the triad’s pump
+mode, thereby providing a mechanism for exciting a given triad in a wave tank. All our results
+are derived for cylinders of arbitrary cross-section (barring some technical assumptions; see §2),
+thus forming a broad framework for characterising nonlinear resonance of confined free-surface
+gravity waves. In particular, our theoretical developments buttress experimental observations [40]
+and demonstrate the potential generality of confinement as a mechanism for promoting nonlinear
+resonance.
+A second fundamental component of our study is the influence of the cylinder cross-section
+on the existence of resonant triads; for example, resonant triads are impossible in rectangular
+cylinders, yet abundant within circular and annular cylinders (for particular fluid depths). Of the
+vast array of resonances arising in a circular cylinder, triads consisting of an axisymmetric pump
+mode and two identical counter-propagating waves are of notable interest. This combination of
+axisymmetric and non-axisymmetric modes possesses an interesting analogy to the excitation of
+counter-propagating subharmonic beach edge waves due to a normally incident standing wave
+[18]. Specifically, the wave crests of the standing axisymmetric mode are always parallel to the
+bounding wall of the circular cylinder, and may excite steadily propagating waves that are periodic
+in the azimuthal direction. For the special case for which the amplitudes of the two counter-
+propagating modes coincide, one observes the resonant interaction of standing axisymmetric and
+non-axisymmetric waves.
+So as to gain a deeper insight into the influence of nonlinearity on resonant triads, a primary
+focus for future investigations will be the simulation of the Euler equations within a cylindrical
+domain, with consideration of various truncated systems [14, 47, 4, 57]. From a computational
+perspective, the most natural geometry to consider is a circular cylinder [49]; this geometry has
+been previously explored in the context of steadily propagating nonlinear waves in the vicinity of
+a 1:2 resonance [8, 64], but it remains to assess the efficacy of the amplitude equations (26) for
+24
+
+predicting the evolution of nonlinear triads. Indeed, exploration of the nonlinear dynamics may
+reveal additional resonant triads arising beyond the small-wave-amplitude limit explored herein.
+Of similar interest is the fluid evolution when multiple triads are excited at a single depth, with
+the potential for energy exchange via triad-triad interactions [35, 15, 13, 11]. The simulation of
+free-surface gravity waves in non-circular cylinders presents a more formidable challenge, however,
+except for cylinder cross-sections that possess a tractable eigenmode decomposition.
+A second natural avenue for future investigation is to characterise the influence of applied
+forcing on resonant triads. For example, when the fluid bath is subjected to sufficiently vigorous
+vertical vibration, Faraday waves [17, 30] may appear on the free surface; although this scenario
+has been studied in the case of a 1:2 internal resonance [44, 46, 24], resonant triads may give rise to
+the formation of more exotic free-surface patterns, particularly at fluid depths that differ from that
+of a 1:2 resonance. In a similar vein, horizontal vibration [42, 45] or a pulsating pressure source
+at the frequency of the triad’s pump mode may lead to a wealth of periodic and quasi-periodic
+dynamics, as predicted by the forced triad equations [20].
+Our study has indicated, however,
+that chaotic dynamics are also possible in some parameter regimes, and might thus be excited in
+numerical simulation or experiments. Lastly, our study has focused on flat-bottomed cylinders; it
+seems plausible, however, that submerged topography may enhance or mitigate certain resonances,
+which may be an important consideration in the design of industrial-scale fluid tanks.
+Finally, our study has focused on the special case of a liquid-air interface, for which the dynamics
+of the air are neglected within the Euler equations. It is natural, however, to extend our formulation
+to the case of two-layer flows (in the absence of surface tension), with two immiscible fluids (e.g.
+air and water) confined within a cylinder whose lid and base are both rigid. In this setting, the
+density difference across the fluid-fluid interface has a strong influence of the system dynamics; it
+seems plausible, therefore, that additional resonances may be excited in this configuration, relative
+to the liquid-air interface considered herein. Notably, the anticipated resonances would arise across
+a single interface, in contrast to the cross-interface resonances explored in previous investigations
+[1, 54, 25, 53, 56, 11].
+Finally, exploring the influence of parametric forcing [31] on resonant
+triads arising for two-layer flows opens up exciting new vistas in nonlinear resonance induced by
+confinement.
+A
+Proof of Theorem 1
+Proof. To prove Theorem 1, we first show that there are no values of h ∈ (0, ∞) satisfying Ω1+Ω2 =
+Ω3 when K1+K2 ≥ K3 or when √K1+√K2 ≤ √K3, where we recall that Ωj(h) =
+�
+Kj tanh(Kjh)
+and Kj > 0 for j = 1, 2, 3. We then prove that there exists a solution to Ω1 + Ω2 = Ω3 when
+K1 + K2 < K3 < (√K1 + √K2)2, and that this solution is unique.
+In the case K1 + K2 ≥ K3, we first define χ(K; h) =
+�
+K tanh(Kh). For fixed h > 0, we
+observe that
+χ(K3; h) ≤ χ(K1 + K2; h) < χ(K1; h) + χ(K2; h),
+where we have utilised that χ(K; h) is a positive, monotonically increasing, concave function of
+K > 0. We conclude that Ω3 < Ω1 + Ω2 for any h > 0, so there are no values of h for which
+Ω1 + Ω2 = Ω3.
+In the case √K1 + √K2 ≤ √K3, we first note that the lower bound Kj > 0 (for j = 1, 2, 3)
+implies that K1 < K3 and K2 < K3.
+Furthermore, as tanh(x) is a monotonically increasing
+function for x > 0, we conclude that tanh(Kjh) < tanh(K3h) for j = 1, 2 and all h > 0. We now
+25
+
+utilise this property to deduce that
+�
+K1 tanh(K1h) +
+�
+K2 tanh(K2h) <
+��
+K1 +
+�
+K2
+��
+tanh(K3h) ≤
+�
+K3 tanh(K3h).
+We conclude that Ω3 > Ω1 + Ω2 for any h > 0, so there are no values of h for which Ω1 + Ω2 = Ω3.
+For the remainder of the proof, we consider the case
+K1 + K2 < K3
+and
+�
+K3 <
+�
+K1 +
+�
+K2,
+(40)
+which is equivalent to the pair of inequalities given by equation (9). Indeed, we will show that there
+exists a unique value of h > 0 satisfying Ω1 + Ω2 = Ω3 in this case. Equivalently, we demonstrate
+that F(h) =
+�
+Ω1(h) + Ω2(h)
+�
+/Ω3(h) − 1 has a unique positive root, where we express
+F(h) =
+�
+ψ1(h) +
+�
+ψ2(h) − 1,
+with the positive functions ψ1 and ψ2 defined
+ψj(h) = Kj tanh(Kjh)
+K3 tanh(K3h)
+for j = 1, 2.
+In order to show the existence of a root of F(h), we first note that
+lim
+h→0 F(h) = K1 + K2
+K3
+− 1 < 0
+and
+lim
+h→∞ F(h) =
+√K1 + √K2
+√K3
+− 1 > 0,
+where we have used the limits limx→0(tanh(x)/x) = 1 and limx→∞ tanh(x) = 1, respectively,
+and implemented the inequalities given in equation (40). As F(h) is a continuous function, the
+intermediate-value theorem determines that F(h) has at least one positive root.
+To prove that such a root is unique, we demonstrate that F(h) is a strictly monotonically
+increasing function for h > 0. Specifically, we note that (for j = 1, 2)
+dψj
+dh = 2K3ψj(h)
+�Kj
+K3
+cosech(2Kjh) − cosech(2K3h)
+�
+> 0
+for 0 < Kj < K3,
+where the inequality follows from the convexity of cosech(x) for x > 0, i.e. b cosech(bx) > cosech(x)
+for 0 < b < 1 and all x > 0 (associating x = 2K3h and b = Kj/K3). As the bounds K1 < K3
+and K2 < K3 incorporate the region determined by equation (40), we deduce that F(h) is strictly
+monotonically increasing. We conclude, therefore, that the root of F(h) must be unique, thereby
+completing the proof.
+B
+Wavenumbers in an annulus
+The no-flux condition (equation (1d)) on the inner and outer radii of an annulus requires that
+∂rΦmn(r0, θ) = 0 and ∂rΦmn(1, θ) = 0 for all θ, where Φmn(r, θ) is the cylinder function defined
+in equation (20). It follows, therefore, that the corresponding wavenumber, kmn, and weighting
+factor, γmn, satisfy the equations
+J′(kmnr0) cos(γmnπ) + Y′
+m(kmnr0) sin(γmnπ) = 0,
+(41a)
+J′(kmn) cos(γmnπ) + Y′
+m(kmn) sin(γmnπ) = 0.
+(41b)
+26
+
+By rearranging equation (41), we determine the following expressions for tan(γmnπ):
+tan(γmnπ) = − J′
+m(kmnr0)
+Y′
+m(kmnr0)
+and
+tan(γmnπ) = − J′
+m(kmn)
+Y′
+m(kmn).
+(42)
+By eliminating tan(γmnπ) and rearranging, we find that kmn > 0 satisfies equation (21). Upon
+computing kmn, one may then determine γmn ∈ [0, 1] using either of the equivalent expressions for
+tan(γmnπ) given in equation (42).
+C
+Reduction of the triad coefficients
+As motivated by the form of α1 given in equation (30), we demonstrate that
+Ω2L2
+3 + Ω3L2
+2 − 2Ω1L2L3 = −1
+2Ω1Ω2Ω3
+�
+Ω2
+1 + Ω2
+2 + Ω2
+3
+�
+,
+(43)
+where we recall that Ω1 + Ω2 + Ω3 = 0 and Lj = Ω2
+j. In fact, the equality given in equation
+(43) holds under cyclic permutation of the indices (1, 2, 3) (as is necessary when defining α2 and
+α3), where we note that the right-hand side is unchanged under such permutations. We conclude
+that α2 and α3 may be simplified in a similar manner, with the right-hand side of equation (43)
+appearing as a constant term in all three coefficients (see §4.2).
+We now detail the algebraic manipulations necessary to transform the left-hand side of equation
+(43) into the right-hand side. By substituting Lj = Ω2
+j into the left-hand side of equation (43) and
+factorising, we obtain
+Ω2L2
+3 + Ω3L2
+2 − 2Ω1L2L3 = Ω4
+2Ω3 + Ω2Ω2
+3
+�
+Ω2
+3 − 2Ω1Ω2
+�
+.
+(44)
+Next, we substitute
+Ω2
+3 = (Ω2
+1 + Ω2
+2) = Ω2
+1 + 2Ω1Ω2 + Ω2
+2
+(45)
+into equation (44), yielding
+Ω2L2
+3 + Ω3L2
+2 − 2Ω1L2L3 = Ω2Ω3
+�
+Ω3
+2 + Ω3
+�
+Ω2
+1 + Ω2
+2
+��
+.
+(46)
+We proceed by substituting Ω3 = −(Ω1 + Ω2) within the square brackets in equation (46); by
+distributing and cancelling common terms, we obtain
+Ω2L2
+3 + Ω3L2
+2 − 2Ω1L2L3 = −Ω1Ω2Ω3
+�
+Ω2
+1 + Ω1Ω2 + Ω2
+2
+�
+.
+(47)
+Finally, we rearrange equation (45) to give
+Ω1Ω2 = 1
+2
+�
+Ω2
+3 − Ω2
+1 − Ω2
+2
+�
+,
+which, upon substitution into equation (47), supplies the required result (equation (43)).
+27
+
+References
+[1] F. K. Ball. Energy transfer between external and internal gravity waves. J. Fluid Mech.,
+19(3):465–478, 1964.
+[2] D. J. Benney. Non-linear gravity wave interactions. J. Fluid Mech., 14(4):577–584, 1962.
+[3] D. J. Benney and J. C. Luke. On the Interactions of Permanent Waves of Finite Amplitude.
+J. Math. Phys., 43:309–313, 1964.
+[4] K. M. Berger and P. A. Milewski. Simulation of Wave Interactions and Turbulence in One-
+Dimensional Water Waves. SIAM J. Appl. Math., 63(4):1121–1140, 2003.
+[5] F. Bretherton. Resonant interactions between waves. the case of discrete oscillations. J. Fluid
+Mech., 20(3):457–479, 1964.
+[6] T. J. Bridges and F. Dias. An analysis of two-dimensional water waves based on O(2) sym-
+metry. Nonlinear Anal. Theory Methods Appl., 14(9):733–764, 1990.
+[7] P. J. Bryant. Periodic waves in shallow water. J. Fluid Mech., 59(4):625–644, 1973.
+[8] P. J. Bryant. Nonlinear progressive free waves in a circular basin. J. Fluid Mech., 205:453–467,
+1989.
+[9] K. M. Case and S. C. Chiu. Three-wave resonant interactions of gravity-capillary waves. Phys.
+Fluids, 20:742, 1977.
+[10] M. Chabane and W. Choi. On resonant interactions of gravity-capillary waves without energy
+exchange. Stud. Appl. Math., 142:528–550, 2019.
+[11] W. Choi, M. Chabane, and T. M. A. Taklo. Two-dimensional resonant triad interactions in
+a two-layer system. J. Fluid Mech., 907:A5, 2021.
+[12] P. Chossat and F. Dias. The 1:2 Resonance with O(2) Symmetry and Its Applications in
+Hydrodynamics. J. Nonlinear Sci., 5:105–129, 1995.
+[13] C. Chow, D. Henderson, and H. Segur. A generalized stability criterion for resonant triad
+interactions. J. Fluid Mech., 319:67–76, 1996.
+[14] W. Craig and C. Sulem. Numerical Simulation of Gravity Waves. J. Comp. Phys., 108(1):73–
+83, 1993.
+[15] A. D. D. Craik. Wave interactions and fluid flows. Cambridge Monographs on Mechanics.
+Cambridge University Press, 1986.
+[16] R. E. Davis and A. Acrivos.
+The stability of oscillatory internal waves.
+J. Fluid Mech.,
+30(4):723–736, 1967.
+[17] M. Faraday. On a Peculiar Class of Acoustical Figures; and on Certain Forms Assumed by
+Groups of Particles upon Vibrating Elastic Surfaces. Philos. Trans. R. Soc., 121:299–340,
+1831.
+28
+
+[18] R. T. Guza and R. E. Davis. Excitation of Edge Waves by Waves Incident on a Beach. J.
+Geophys. Res., 79(9):1285– 1291, 1974.
+[19] J. L. Hammack and D. M. Henderson. Resonant Interactions Among Surface Water Waves.
+Annu. Rev. Fluid Mech., 25(1):55–97, 1993.
+[20] J. Harris, M. D. Bustamante, and C. Connaughton. Externally forced triads of resonantly
+interacting waves: Boundedness and integrability properties. Commun. Nonlinear Sci. Numer.
+Simulat., 17:4988–5006, 2012.
+[21] K. Hasselmann. On the non-linear energy transfer in a gravity-wave spectrum Part 1. General
+theory. J. Fluid Mech., 12(4):481–500, 1961.
+[22] K. Hasselmann. A criterion for nonlinear wave stability. J. Fluid Mech., 30(4):737–739, 1967.
+[23] D. M. Henderson and J. L. Hammack. Experiments on ripple instabilities. Part 1. Resonant
+triads. J. Fluid Mech., 184:15–41, 1987.
+[24] D. M. Henderson and J. W. Miles. Faraday waves in 2:1 internal resonance. J. Fluid Mech.,
+222:449–470, 1991.
+[25] T. M. Joyce. Nonlinear interactions among standing surface and internal gravity waves. J.
+Fluid Mech., 63(4):801–825, 1974.
+[26] U. Kadri and T. R. Akylas. On resonant triad interactions of acoustic-gravity waves. J. Fluid
+Mech., 788:R1, 2016.
+[27] U. Kadri and M. Stiassnie. Generation of an acoustic-gravity wave by two gravity waves, and
+their subsequent mutual interaction. J. Fluid Mech., 735:R6, 2013.
+[28] J. K. Kevorkian and J. D. Cole. Multiple Scale and Singular Perturbation Methods, volume
+114 of Appl. Math. Sci. Springer, New York, 1996.
+[29] E. Kreyszig. Introductory Functional Analysis with Applications. John Wiley & Sons, 1989.
+[30] K. Kumar. Linear theory of Faraday instability in viscous fluids. Proc. R. Soc. Lond. A,
+452:1113–1126, 1996.
+[31] K. Kumar and L. S. Tuckerman. Parametric instability of the interface between two fluids.
+J. Fluid Mech., 279:49–68, 1994.
+[32] H. Lamb. Hydrodynamics (6th ed.). Dover Publications, 1932.
+[33] M. S. Longuet-Higgins. Resonant interactions between two trains of gravity waves. J. Fluid
+Mech., 12(3):321–332, 1962.
+[34] L. R. Mack.
+Periodic, finite-amplitude, axisymmetric gravity waves.
+J. Geophys. Res.,
+67(2):829–843, 1962.
+[35] A. McEwan, D. Mander, and R. Smith. Forced resonant second-order interaction between
+damped internal waves. J. Fluid Mech., 55(4):589–608, 1972.
+29
+
+[36] L. F. McGoldrick.
+Resonant interactions among capillary-gravity waves.
+J. Fluid Mech.,
+21(2):305–331, 1965.
+[37] L. F. McGoldrick. An experiment on second-order capillary gravity resonant wave interactions.
+J. Fluid Mech., 40(2):251–271, 1970.
+[38] L. F. McGoldrick. On Wilton’s ripples: a special case of resonant interactions. J. Fluid Mech.,
+42(1):193–200, 1970.
+[39] L. F. McGoldrick. On the rippling of small waves: a harmonic nonlinear nearly resonant
+interaction. J. Fluid Mech., 52(4):725–751, 1972.
+[40] G. Michel. Three-wave interactions among surface gravity waves in a cylindrical container.
+Phys. Rev. Fluids, 4:012801(R), 2019.
+[41] J. W. Miles. Surface-wave damping in closed basins. Proc. Roy. Soc. A, 297(1451):459–475,
+1967.
+[42] J. W. Miles. Nonlinear surface waves in closed basins. J. Fluid Mech., 75(3):419–448, 1976.
+[43] J. W. Miles. Internally resonant surface waves in a circular cylinder. J. Fluid Mech., 149:1–14,
+1984.
+[44] J. W. Miles. Nonlinear Faraday resonance. J. Fluid Mech., 146:285–302, 1984.
+[45] J. W. Miles. Resonantly forced surface waves in a circular cylinder. J. Fluid Mech., 149:15–31,
+1984.
+[46] J. W. Miles and D. M. Henderson. Parametrically forced surface waves. Annu. Rev. Fluid
+Mech., 22:143–165, 1990.
+[47] P. A. Milewski and J. B. Keller. Three-Dimensional Water Waves. Stud. Appl. Math., 97:149–
+166, 1996.
+[48] O. M. Phillips. On the dynamics of unsteady gravity waves of finite amplitude Part 1. The
+elementary interactions. J. Fluid Mech., 9(2):193–217, 1960.
+[49] S. Qadeer and J. A. Wilkening. Computing the Dirichlet–Neumann Operator on a Cylinder.
+SIAM J. Numer. Anal., 57(3):1183–1204, 2019.
+[50] C. F. M. Raupp and P. L. Silva Dias.
+Resonant Wave Interactions in the Presence of a
+Diurnally Varying Heat Source. J. Atmos. Sci., 66(10):3165–3183, 2009.
+[51] C. F. M. Raupp, P. L. Silva Dias, E. G. Tabak, and P. A. Milewski. Resonant wave interactions
+in the equatorial waveguide. J. Atmos. Sci., 65(11):3398–3418, 2008.
+[52] L. Schwartz and J.-M. Vanden-Broeck. Numerical solution of the exact equations for capil-
+lary–gravity waves. J. Fluid Mech., 95(1):119–139, 1979.
+[53] H. Segur. Resonant interactions of surface and internal gravity waves. Phys. Fluids, 23:2556,
+1980.
+30
+
+[54] W. F. Simmons. A variational method for weak resonant wave interactions. Proc. Roy. Soc.
+Lond. A, 309(1499):551–577, 1969.
+[55] S. H. Strogatz.
+Nonlinear Dynamics and Chaos: With Applications to Physics, Biology,
+Chemistry and Engineering (2nd ed.). CRC Press, 2015.
+[56] T. M. A. Taklo and W. Choi. Group resonant interactions between surface and internal gravity
+waves in a two-layer system. J. Fluid Mech., 892:A14, 2020.
+[57] Z. Wang and P. A. Milewski. Dynamics of gravity-capillary solitary waves in deep water. J.
+Fluid Mech., 708:480–501, 2012.
+[58] Z. Wang, J.-M. Vanden-Broeck, and P. A. Milewski. Two-dimensional flexural–gravity waves
+of finite amplitude in deep water. IMA J. App. Math., 78(4):750–761, 2013.
+[59] G. B. Whitham. A general approach to linear and non-linear dispersive waves using a La-
+grangian. J. Fluid Mech., 22(2):273–283, 1965.
+[60] G. B. Whitham. Non-Linear Dispersive Waves. Proc. Roy. Soc. Lond. A, 283(1393):238–261,
+1965.
+[61] G. B. Whitham. Non-linear dispersion of water waves. J. Fluid Mech., 27(2):399–412, 1967.
+[62] G. B. Whitham. Variational methods and applications to water waves. Proc. Roy. Soc. Lond.
+A, 299:6–25, 1967.
+[63] J. R. Wilton. On ripples. Phil. Mag. Ser. 6, 29(173):688–700, 1915.
+[64] X. Yang, F. Dias, Z. Liu, and S. Liao. Finite-amplitude steady-state resonant waves in a
+circular basin. J. Fluid Mech., 915:A136, 2021.
+31
+
diff --git a/8NA0T4oBgHgl3EQfOf_i/content/tmp_files/load_file.txt b/8NA0T4oBgHgl3EQfOf_i/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf,len=1324
+page_content='Resonant triad interactions of gravity waves  in cylindrical basins  Matthew Durey1 and Paul A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Milewski2  1School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8QQ, UK  2Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK  Abstract  We present the results of a theoretical investigation into the existence, evolution and ex-  citation of resonant triads of nonlinear free-surface gravity waves confined to a cylinder of  finite depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' It is well known that resonant triads are impossible for gravity waves in laterally  unbounded domains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' we demonstrate, however, that horizontal confinement of the fluid may  induce resonant triads for particular fluid depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' For any three correlated wave modes arising  in a cylinder of arbitrary cross-section, we prove necessary and sufficient conditions for the  existence of a depth at which nonlinear resonance may arise, and show that the resultant crit-  ical depth is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We enumerate the low-frequency triads for circular cylinders, including  a new class of resonances between standing and counter-propagating waves, and also briefly  discuss annular and rectangular cylinders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Upon deriving the triad amplitude equations for a  finite-depth cylinder of arbitrary cross-section, we deduce that the triad evolution is always  periodic, and determine parameters controlling the efficiency of energy exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' In order to  excite a particular triad, we explore the influence of external forcing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' in this case, the triad  evolution may be periodic, quasi-periodic, or chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Finally, our results have potential im-  plications on resonant water waves in man-made and natural basins, such as industrial-scale  fluid tanks, harbours and bays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  1  Introduction  Nonlinear resonance is a mechanism by which energy is continuously transferred between a small  number of linear wave modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' This phenomenon, first observed in Wilton’s analysis of gravity-  capillary wave trains [63], has been the subject of frequent investigation over the past century  [36, 37, 38, 54, 52, 19, 15];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' indeed, nonlinear resonance has since observed for wave trains in a  growing number of dispersive wave systems, including gravity waves [48, 21, 33, 2], acoustic-gravity  waves [27, 26], flexural-gravity waves [58], two-layer flows [1, 25, 53], and atmospheric flows [51, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Whilst the aforementioned studies typically consider nonlinear resonance for laterally unbounded  domains, the purpose of this study is to demonstrate that energy exchange between free-surface  gravity waves may be induced and accentuated by horizontal confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  We focus our study on the collective resonance of three linear wave modes, henceforth referred to  as a triad [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' In laterally unbounded domains, the monotonic and concave form of the dispersion  curve precludes the existence of resonant triads for gravity wave trains at finite depth [48, 21],  with resonant quartets instead being the smallest possible collective resonant interaction [2, 33, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  However, confinement of the fluid to a vertical cylinder results in linear wave modes that differ in  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='02163v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='flu-dyn]  5 Jan 2023  form to sinusoidal plane waves (except for a rectangular cylinder), so the preclusion of resonant  triads no longer applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Indeed, our study demonstrates that, under certain conditions, resonant  triads may arise in cylinders of arbitrary cross-section for specific values of the fluid depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' As  resonant triads evolve over a much faster time scale than that of resonant quartets, the exchange  of energy in gravity waves is thus more efficient under the influence of lateral confinement [40],  with potential implications on resonant sloshing in man-made and natural basins [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Prior investigations of confined resonant free-surface gravity waves have predominantly focused  on the so-called 1:2 resonance, which arises when two of the three linear wave modes comprising  a triad coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' For axisymmetric standing waves in a circular cylinder, Mack [34] determined a  condition for the existence of critical depth-to-radius ratios at which a 1:2 resonance may arise, a  result later generalised to cylinders of arbitrary cross-section [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Miles [42, 43] then characterised  the weakly nonlinear evolution of such internal resonances, demonstrating that a 1:2 resonance is  impossible in a rectangular cylinder [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Although Miles’ seminal results provide an informative  view of the weakly nonlinear dynamics, the influence of fully nonlinear effects was later assessed  by Bryant [8] and Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' For the case of a circular cylinder of finite depth, Bryant [8]  and Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' [64] both characterised new steadily propagating nonlinear waves arising in the  vicinity of a 1:2 resonance, and Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' [64] also computed nonlinear near-resonant axisymmetric  standing waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Finally, broader mathematical properties of water waves exhibiting O(2) symmetry  (of which a circular cylinder is one example) were analysed by Bridges & Dias [6] and Chossat &  Dias [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Given the restrictive set of critical depths at which a 1:2 resonance may arise [8, 64], it is  natural to explore the possibility of nonlinear resonance in cylinders whose depth departs from the  depths that trigger a 1:2 resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' To the best of our knowledge, the first and only such study  was the seminal experimental investigation performed by Michel [40], who focused on resonant  triads arising for free-surface gravity waves confined to a finite-depth circular cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Notably,  the cylinder depth in Michel’s experiment was judiciously chosen so as to isolate a specific triad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Michel utilised bandlimited random horizontal vibration so as to excite two members of the triad,  whose nonlinear interaction led to the growth of the third mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Significantly, the energy of the  third mode was, on average, the product of the energies of the remaining two modes, thereby  satisfying the quadratic energy exchange typical of resonant triads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  In order to exemplify the mechanism of nonlinear resonance, Michel [40] also calculated the  response of a child mode due to the nonlinear interaction between two parent modes (where all  three wave modes comprise the triad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Notably, Michel’s calculation is restricted to the early stages  of growth and to particular relative phases of the wave modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' In addition Michel considered a  fluid of infinite depth for all but the resonance conditions, for which finite-depth corrections were  included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' In contrast, we consider general resonances in arbitrary cylinders of finite depth and  derive equations for the triad evolution over long time-scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We also believe some nonlinear  contributions to the interactions were omitted from Michel’s calculation, resulting in quantitative  differences (see §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  The goal of our study is to unify the existence and evolution of 1:2 and triadic resonances  into a single mathematical framework, effectively characterising all triad interactions of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Based on existing theory, it is unclear how the existence of resonant triads depends on the form  of the cylinder cross-section, and which combinations of wave modes are permissible for judicious  choice of the fluid depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Furthermore, the range of depths that may excite a particular triad  is uncertain, with 1:2 resonances only excited in a very narrow window about each critical depth  [34, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Once a particular triad is excited, one anticipates that the triad evolution will be governed  by the canonical triad equations [5, 15];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' however, quantifying the triad evolution and relative energy  2  exchange requires computation of the triad coupling coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Finally, it is unclear how best to  excite triads in arbitrary cylinders, both with and without external forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  We here present a relatively comprehensive characterisation of the existence, evolution and  excitation of resonant triads for gravity waves confined to a cylinder of arbitrary cross-section  and finite depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' In order to reduce the problem to its key components, we first truncate the  Euler equations, recasting the fluid evolution in terms of a finite-depth Benney-Luke equation  (§2), incorporating only the nonlinear interactions necessary for resonant triads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' In §3, we prove  necessary and sufficient conditions for there to exist a finite depth at which three linear wave  modes may form a resonant triad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' In particular, we prove that resonant triads are impossible  for rectangular cylinders, yet there is an abundance of resonant triads for circular cylinders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We  then use multiple-scales analysis to determine the long-time evolution of a triad in a cylinder of  arbitrary cross-section (§4), from which we characterise the relative coupling of different triads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Finally, we explore the excitation of resonant triads (§5), and discuss the potential extension of  our theoretical developments to the cases of applied forcing and two-layer flows (§6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  2  Formulation  We consider the irrotational flow of an inviscid, incompressible liquid that is bounded above by  a free surface, confined laterally by the vertical walls of a cylinder whose horizontal cross-section,  D, is enclosed by the curve ∂D, and bounded below by a rigid horizontal plane lying a distance H  below the undisturbed free surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' see figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We consider the fluid evolution in dimensionless  variables, taking the cylinder’s typical horizontal extent, a, as the unit of length, and  �  ag−1 as  the unit of time, where g is the acceleration due to gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' It follows that the dimensionless  free-surface elevation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' η(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' and velocity potential,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' φ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' evolve according to the equations  ∆φ + φzz = 0  for x ∈ D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  −h < z < ϵη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  (1a)  φt + η + ϵ  2  �  |∇φ|2 + φ2  z  �  = 0  for x ∈ D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  z = ϵη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  (1b)  ηt + ϵ∇φ · ∇η = φz  for x ∈ D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  z = ϵη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  (1c)  n · ∇φ = 0  for x ∈ ∂D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  −h < z < ϵη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  (1d)  φz = 0  for x ∈ D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  z = −h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  (1e)  corresponding to the continuity equation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' dynamic and kinematic boundary conditions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' and no-  flux through the vertical walls and horizontal base,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' In equation (1), the dimensionless  parameter ϵ is proportional to the typical wave slope, h = H/a is the ratio of the fluid depth to  the typical horizontal extent, n is a unit vector normal to the boundary ∂D, and the operators ∇  and ∆ denote the horizontal gradient and Laplacian, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Moreover, conservation of mass  implies that the free surface satisfies  ��  D η dA = 0 for all time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Finally, in dimensional variables,  ax is the two-dimensional horizontal coordinate, az is the upward-pointing vertical coordinate,  �  ag−1t denotes time, ϵaη is the free-surface displacement, and ϵa√agφ is the velocity potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  We aim to develop a broad framework for understanding resonant triads in a cylinder of finite  depth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' however, care must be taken when modelling fluid-boundary interactions and determining  the class of permissible cylinder cross-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  From a modelling perspective, we employ an  assumption generally implicit to the water-wave problem in bounded domains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' specifically, we  neglect the meniscus and dissipation arising near the vertical walls [41], thus determining that  the free surface intersects the boundary normally, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' n · ∇η = 0 for x ∈ ∂D [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' In order to  3  Figure 1: Schematic diagram of the cylindrical tank (with cross-section D and boundary ∂D)  partially filled with liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' The undisturbed free surface (dashed lines) lies on z = 0, a distance H  above the rigid bottom plane (grey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' The disturbed free surface is sketched in dash-dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  maximise the generality of our investigation, we allow the cylinder cross-section, D, to be fairly  arbitrary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' however, the mathematical developments presented herein require D to be bounded  with a piecewise-smooth boundary, thereby allowing us to utilise the spectral theorem for compact  self-adjoint operators [29] and the divergence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' As most cylinders of practical interest  consist of a piecewise-smooth boundary, this mathematical restriction fails to limit the breadth of  our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='1  Derivation of the Benney-Luke equation  As our study is focused on the weakly nonlinear evolution of small-amplitude waves, we proceed  to simplify (1) in the case 0 < ϵ ≪ 1 and h = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We begin by expanding the dynamic and  kinematic boundary conditions (equations (1b)–(1c)) about z = 0 in powers of ϵ, which, upon  eliminating η, gives rise to the equation [2, 47]  φtt + φz = ϵ  �  ∂t(φtzφt) + φzzφt − ∂t(|∇φ|2) − φzφzt  �  + O(ϵ2)  for  x ∈ D,  z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  (2)  To reduce the fluid evolution to the dynamics arising on the linearised free surface, z = 0, we  define the Dirichlet-to-Neumann operator, L , so that L φ|z=0 = φz|z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Here φ satisfies Laplace’s  equation (1a) over the linearised domain −h < z < 0, with ∂zφ = 0 on z = −h (see equation (1e))  and n·∇φ = 0 for x ∈ ∂D (see equation (1d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Notably, the Dirichlet-to-Neumann operator may be  defined in terms of its spectral representation, as detailed in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' By denoting u(x, t) = φ(x, 0, t),  we finally obtain the finite-depth Benney-Luke equation [2, 3, 47]  utt + L u + ϵ  �  ut  �  L 2 + ∆  �  u + ∂  ∂t  �  (L u)2 + |∇u|2��  = O(ϵ2)  for  x ∈ D,  (3)  where we have simplified the nonlinear terms in equation (2) using φzz = −∆φ and utt = −L u +  O(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  4  H  DThe remainder of our investigation will be focused on the evolution of resonant triads governed  by the Benney-Luke equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' As resonant triads arising in confined geometries are governed  primarily by quadratic nonlinearities, it is sufficient to neglect terms of size O(ϵ2) in equation  (3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' however, higher-order corrections to the Benney-Luke equation may be derived by following  a similar expansion procedure [2, 47, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Although our investigation is mainly focused on the  evolution of the velocity potential, u, one may recover the leading-order free-surface elevation from  the dynamic boundary condition (1b), namely η = −ut + O(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='2  Spectral representation of the Dirichlet-to-Neumann operator  The Dirichlet-to-Neumann operator, L , may be understood in terms of the discrete set of orthog-  onal eigenfunctions of the horizontal Laplacian operator [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Specifically, we consider the set of  real-valued eigenfunctions, Φn(x), satisfying  −∆Φn = k2  nΦn for n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' ,  where the corresponding eigenvalues, k2  n, are ordered so that 0 = k0 < k1 ≤ k2 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='. Moreover,  each eigenfunction satisfies the boundary condition n · ∇Φn = 0 on ∂D, as motivated by the no-  flux condition (1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Finally, the orthogonal eigenfunctions are normalised so that ⟨Φm, Φn⟩ = δmn,  where  ⟨f, g⟩ = 1  S  ��  D  fg dA  defines an inner product for real functions f and g, S is the area of D, and δmn is the Kronecker  delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Notably, Φ0(x) = 1 is the constant eigenfunction, with corresponding eigenvalue k0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  To determine the Dirichlet-to-Neumann operator for sufficiently smooth φ, we first substitute  the series expansion φ(x, z) = �∞  n=0 φn(z)Φn(x) into Laplace’s equation (1a), where we have  temporally omitted the time dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We then solve the resulting equation for φn(z) over the  linearised domain −h < z < 0, in conjunction with the no-flux condition on z = −h (see equation  (1e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' It follows that ∂zφn(0) =  ˆ  Lnφn(0), where  ˆ  Ln = kn tanh(knh)  (4)  is the spectral multiplier of the Dirichlet-to-Neumann operator, L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  By expressing the time-  dependent free-surface velocity potential, u = φ|z=0, in terms of the basis expansion u(x, t) =  �∞  n=0 un(t)Φn(x), it follows that the Dirichlet-to-Neumann map has the spectral representation  L u = �∞  n=0  ˆ  LnunΦn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  3  The existence of resonant triads  Resonant triads arise due to the exchange of energy between linear wave modes, an effect induced  by nonlinear wave interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' In order to define resonant triads mathematically, it is necessary  to first determine the angular frequency associated with each linear wave mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' In the limit ϵ → 0,  the Benney-Luke equation (3) reduces to the linear equation utt + L u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' By seeking a solution  to the linearised Benney-Luke equation of the form u(x, t) = Φn(x)e−iωnt, we conclude that the  angular frequency, ωn, satisfies ω2  n =  ˆ  Ln, or the more familiar [32]  ω2  n = kn tanh(knh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  (5)  5  As we will see, a crucial aspect of the following analysis is that the angular frequency depends on  the fluid depth, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' ωn(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Finally, we note that the angular frequency is larger for more oscillatory  eigenfunctions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' for larger values of kn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' by analogy to the evolution of plane gravity waves, we  refer to kn as a ‘wavenumber’ henceforth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  We proceed by considering three linear wave modes, enumerated n1, n2 and n3, where we denote  Ωj = ωnj,  Kj = knj,  and  Ψj(x) = Φnj(x)  for j = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Notably, we exclude the wavenumber k0 = 0 from consideration as the corresponding eigenmode,  Φ0, simply reflects the invariance of the Benney-Luke equation (3) under the mapping u �→ u +  constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' henceforth, we consider only wavenumbers Kj > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' The three linear wave modes form a  resonant triad if there is a critical fluid depth, hc, satisfying  Ω1(hc) ± Ω2(hc) ± Ω3(hc) = 0,  (6)  where all four sign combinations are permissible (we consider Ωj > 0 without loss of generality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  To simplify notation in the following arguments, we restrict our attention to the particular case  Ω1(hc) + Ω2(hc) = Ω3(hc),  (7)  where the other three sign combinations in equation (6) may be recovered by suitable re-indexing  of the Ωj terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' However, as we will see in §4, an additional constraint necessary for triads to  exist is the eigenmode correlation condition,  ��  D  Ψ1Ψ2Ψ3 dA ̸= 0,  (8)  which implies that the product of any two eigenmodes is non-orthogonal to the remaining eigen-  mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='1  The existence of a critical depth  We proceed to determine necessary and sufficient conditions on the wavenumbers, Kj, for there  to exist a depth, hc, at which a resonant triad forms, where such a critical depth is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We  summarise our results in terms of the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' There exists a positive and finite value of h such that Ω1 + Ω2 = Ω3 if and only if  K1 + K2 < K3 <  ��  K1 +  �  K2  �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  (9)  When this pair of inequalities is satisfied, the corresponding value of h is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  We briefly sketch the proof of Theorem 1, with full details presented in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  We  first demonstrate that no solutions to Ω1 + Ω2 = Ω3 are possible when the bounds in equation  (9) are violated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' when K1 + K2 ≥ K3 or when √K1 + √K2 ≤ √K3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We then consider the  case where the inequalities (9) are satisfied and determine the existence of positive roots to the  function F(h) = (Ω1(h) + Ω2(h))/Ω3(h) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' In this case, we demonstrate that limh→0 F(h) < 0  and limh→∞ F(h) > 0, from which we conclude that F(h) has at least one root (by continuity of  F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Finally, we deduce that this root is unique by proving that F(h) is a strictly monotonically  increasing function of h when the inequalities (9) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  6  Two important conclusions may be deduced from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' First, it follows from equation (9)  that the wavenumber, K3, corresponding to the largest angular frequency, Ω3, is larger than both  the other two wavenumbers (K1 and K2), but it cannot be arbitrarily large (as supplied by the  upper bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' For a given pair of eigenmodes (say Ψ1 and Ψ2), we conclude that there are likely  to be only finitely many eigenmodes that can resonate with this pair (indeed, that number might  fairly small, or even zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Second, when modes 1 and 2 coincide (a 1:2 resonance), one deduces  that Ω1 = Ω2 and K1 = K2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' as such, the existence bounds (9) simplify to 2K1 < K3 < 4K1, or  2 < K3/K1 < 4 [34, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='2  Determining the critical depth  Although Theorem 1 determines necessary and sufficient conditions on the wavenumbers, Kj, for  there to be a critical depth, hc, at which a resonant triad exists, the critical depth remains to be  determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' In general, the critical depth must be computed numerically (being the unique root  of the nonlinear function F(h));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' however, we demonstrate that useful quantitative and qualitative  information may be obtained via asymptotic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' For the remainder of this section, we consider  the rescaled wavenumbers, ξ1 = K1/K3 and ξ2 = K2/K3, and the rescaled depth, ζ = K3h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' it  remains to determine the root, ζc, of  F(ζ) =  �  ξ1 tanh(ξ1ζ)  tanh(ζ)  +  �  ξ2 tanh(ξ2ζ)  tanh(ζ)  − 1  (10)  when ξ1, ξ2 > 0 satisfy  ξ1 + ξ2 < 1 <  �  ξ1 +  �  ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  (11)  In figure 2(a), we present contours of the critical rescaled depth, ζc, in the (ξ1, ξ2)-plane, restricted  to the region demarcated by equation (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Consistent with the limits limζ→0 F(ζ) = ξ1 + ξ2 − 1  and limζ→∞ F(ζ) = √ξ1+√ξ2−1, we observe that the root, ζc, tends to zero at the line ξ1+ξ2 = 1,  and approaches infinity at the curve √ξ1 + √ξ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Furthermore, the uniqueness of the root of  F for given (ξ1, ξ2) is reflected in the observation that the contours of ζc do not cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Finally, we  note that the contours are symmetric about the line ξ1 = ξ2, which is a direct consequence of the  invariance of F(ζ) under the mapping ξ1 ↔ ξ2 (see equation (10)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Although we are primarily interested in the physically relevant case for which the cylinder’s  depth-to-width ratio, h, is of size O(1), an informative analytic result may be obtained by consid-  ering F(ζ) in the limit ζ ≪ 1 (or K3h ≪ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' By utilising the Taylor expansion  �  tanh(x) ∼ √x  �  1 − x2  6 + 19  360x4 + O  �  x6��  ,  we obtain  �  ξ1 tanh(ξ1ζ)+  �  ξ2 tanh(ξ2ζ)−  �  tanh(ζ) ∼  �  ζ  ��  ξ1 +ξ2 −1  �  − ζ2  6  �  ξ3  1 +ξ3  2 −1  �  +O(ζ4)  �  (12)  for 0 < ζ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Whilst deriving equation (12), we have utilised the bound ξ1, ξ2 < 1 (see equation  (11)), which additionally ensures that 0 < ξjζ ≪ 1 for j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We note that the left-hand side of  equation (12) is equal to F(ζ) tanh(ζ), so ζc satisfies  ξ1 + ξ2 − 1 − ζ2  c  6  �  ξ3  1 + ξ3  2 − 1  �  = O(ζ4  c ),  (13)  7  Figure 2: Contours of the rescaled critical depth, ζc = hcK3, as a function of the rescaled wavenum-  bers, ξ1 = K1/K3 and ξ2 = K2/K3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' (a) The contours computed numerically from equation (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  The black lines indicate the limiting cases of ζc → 0 (at ξ1+ξ2 = 1) and ζc → ∞ (at √ξ1+√ξ2 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  (b) The contours are overlaid by the leading-order approximation (equation (17);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' circles) and the  higher-order correction (equation (18);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' diamonds) for ζc equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='5, 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  provided that 0 < ζc ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' By neglecting terms of size O(ζ4  c ) in equation (13), one may then easily  solve for ζc in terms of ξ1 and ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Alternatively, a more succinct expression for ζc may be found by first noting that  ξ3  1 + ξ3  2 = (ξ1 + ξ2)3 − 3ξ1ξ2(ξ1 + ξ2) = 1 − 3ξ1ξ2 + O(ζ2  c ),  (14)  where we have utilised the leading-order approximation ξ1 + ξ2 = 1 + O(ζ2  c ) (see equation (13)) to  determine the second equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Upon substituting equation (14) into equation (13), we find that  ξ1, ξ2 and ζc are now related by the notably simpler expression  ξ1 + ξ2 − 1 + ζ2  c  2 ξ1ξ2 = O(ζ4  c ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  (15)  By neglecting terms of O(ζ4  c ), the leading-order approximation for the rescaled critical depth, ζc,  is given by  ζc ∼  �  2(1 − ξ1 − ξ2)  ξ1ξ2  ,  (16)  an expression valid when 0 < ζc ≪ 1 and ξ1 + ξ2 < 1 (see equation (11)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Alternatively, one may  deduce from equation (15) that the contours of ζc satisfy the approximate form  ξ2 ∼  1 − ξ1  1 + 1  2ζ2  c ξ1  ,  (17)  where the term in the denominator is responsible for the increased ‘bending’ of the contours as  ζc becomes progressively larger (see figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We note that the additional simplification afforded  8  (a)  (b)  hcK3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='5  Leading-order approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  hcK3 = 1  Higher-order correction  hcK3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='8  hcK3 = 2  hcK3 = 3  hcK3 = 5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='6  K2  hc  hcK3 = 10  个  K3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='4  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='2  8  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='8  0  1  0  1  Ki/K3  Ki/K3by equation (14) allows for a far more tractable representation of the contours relative to solving  equation (13) directly for ξ2 given ξ1 and ζc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Despite being derived under the assumption 0 < ζc ≪ 1, we see in figure 2(b) that the contours  given by equation (17) agree favorably with the numerical solution even up to ζc ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' However, it  is readily verified from equation (16) that the asymptotic approximation of each contour crosses  the boundary curve √ξ1 + √ξ2 = 1 at ζc = 4 (for which ξ1 = ξ2 = 1  4), thereby demonstrating  that the reduced asymptotic form has limited applicability (even in a qualitative sense) for slightly  larger values of ζc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' One may further improve the quantitative (and, to an extent, qualitative)  agreement between the asymptotic analysis and numerical computation by including terms of size  O(ζ4) in equation (12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' indeed, an analogous calculation gives rise to the following higher-order  correction to equation (15):  ξ1 + ξ2 − 1 + ζ2  c  2 ξ1ξ2 + ζ4  c  72ξ1ξ2  �  ξ1ξ2 − 1  �  = O(ζ6  c ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  (18)  Although one may then solve for ζc given ξ1 and ξ2 (or, alternatively, determine the contours of  ζc) by truncating terms of O(ζ6  c ) in equation (18), the resulting algebraic expressions yield little  qualitative information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' However, one may, in principle, use this reduced form as a reasonable  initial guess for a numerical root-finding algorithm for determining the root of F(ζ), provided that  ζc is not too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='3  Example cavities  Our investigation into the emergence of resonant triads has been focused, thus far, on finite-depth  cylinders with arbitrary horizontal cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' However, it is convenient to understand how  the results of Theorem 1 influence the formation (or not) of resonant triads for some specific  cross-sections, namely rectangular, circular, and annular cylinders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='1  Rectangular cylinder  It is well known that resonant triads are impossible for plane gravity waves evolving across an  unbounded horizontal domain of finite depth [48, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' 1 We now utilise Theorem 1 to demonstrate  a similar result: resonant triads are impossible for gravity waves evolving within a rectangular  cylinder of finite depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Our result generalises the special case of a 1:2 resonance, for which the  impossibility of internal resonance in a rectangular cylinder was demonstrated by Miles [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  To proceed, we consider a rectangular cylinder with side lengths Lx and Ly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' By orientating the  Cartesian coordinate system, x = (x, y), so that the cylinder cross-section is defined by the region  0 < x < Lx and 0 < y < Ly, the eigenmodes are of the form  Φmn(x, y) =  1  Nmn  cos(pmx) cos(qny),  where Nmn > 0 is a normalisation constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Notably, the wavenumbers pm = mπ/Lx and  qn = nπ/Ly are chosen so that the no-flux condition is satisfied (see equation (1d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' For a triad de-  termined by the non-negative integers mj and nj (for j = 1, 2, 3), the corresponding wavenumbers,  Pj = pmj and Qj = qnj, must satisfy P1 + P2 = P3 and Q1 + Q2 = Q3 (under suitable reordering  1Weak interactions are possible, however, in the shallow-water limit, Kjh → 0, for which tanh(Kjh) in the  dispersion relation (5) is replaced by its leading-order approximation, Kjh [48, 7, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  9  of the subscripts) in order for the eigenmode correlation condition (8) to be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' By defining  the wave vector kj = (Pj, Qj), the conditions on Pj and Qj simplify to the single requirement  k1 + k2 = k3, where the triangle inequality supplies that |k3| ≤ |k1| + |k2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' As the eigenvalues,  K2  j , of the negative Laplacian operator are related to the wave vectors via Kj = |kj|, we deduce  that K3 ≤ K1 + K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Owing to the violation of the left-hand bound in equation (9), we conclude  that resonant triads cannot exist in a rectangular cylinder of finite depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='2  Circular cylinder  We consider a circular cylinder of unit radius in dimensionless variables (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' the dimensional radius  is equal to a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' see §2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' For polar coordinates x = (r, θ), it is well known that the corresponding  (complex-valued) eigenmodes may be expressed in the form  Φmn(r, θ) =  1  Nmn  Jm(kmnr)eimθ,  where  Nmn =  ��Jm(kmn)  ��  �  1 − m2  k2  mn  (19)  is the normalisation factor and m is the azimuthal wavenumber (an integer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Furthermore, the  no-flux condition (1d) determines that the radial wavenumbers, denoted kmn, satisfy J′  m(kmn) =  0, where 0 < km1 < km2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' (we exclude k00 = 0 from consideration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' see §3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Notably,  the eigenvalues of the negative Laplacian operator are precisely the squared wavenumbers, k2  mn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  consequently, the antinodes of each Bessel function play a pivotal role in determining the existence  of resonant triads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Akin to the rectangular cylinder, we find that the eigenmode correlation condition imparts  an important restriction on the combination of eigenmodes that may resonate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  For given mj  and nj (for j = 1, 2, 3), we denote Kj = kmjnj, Ψj = Φmjnj and Nj = Nmjnj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Although the  correlation condition given in equation (8) is defined for real eigenmodes, a similar condition holds  for complex-valued eigenmodes, namely  ��  D Ψ1Ψ2Ψ∗  3 dA ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' By considering the quantity  ��  D  Ψ1Ψ2Ψ∗  3 dA =  1  N1N2N3  � � 1  0  rJm1(K1r)Jm2(K2r)Jm3(K3r) dr  �� � 2π  0  ei(m1+m2−m3)θ dθ  �  ,  we deduce from the azimuthal integral that a necessary condition for the correlation integral to  be nonzero is m1 + m2 = m3 [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' This condition thus restricts the permissible combinations of  angular wavenumbers in a manner similar to the restriction on the permissible planar wavenumbers  for the case of a rectangular cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Unlike rectangular cylinders, however, we demonstrate that  resonant triads are possible in a circular cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Despite the apparent restriction of the Bessel antinodes, Kj, and summation condition on the  azimuthal wavenumbers, mj, Theorem 1 determines that a vast array of resonant triads may be  excited for judicious choices of the fluid depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' In table 1, we list a small number of resonant  triads and their corresponding critical depth, hc, subject to the restrictions |mj| ≤ 3 and nj ≤ 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  for larger values of |mj| and nj, the corresponding wave field becomes increasingly oscillatory, to  the extent that the effects of surface tension and dissipation may become appreciable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Moreover,  even marginally relaxing the upper bounds on |mj| and nj vastly increases the number of resonant  triads;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' indeed, the restriction |mj| ≤ 4 and nj ≤ 4 introduces 70 additional resonant triads relative  to table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' As the upper bounds for |mj| and nj are further increased, the typical difference between  the various critical depths decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Finally, the correlation condition,  ��  D Ψ1Ψ2Ψ∗  3 dA ̸= 0, is  satisfied for each triad;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' however, the integral is very close to zero in some cases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' triad 18),  corresponding to an elongation of the triad evolution time-scale (see §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  10  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
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+page_content='00640  22  0  3  3  2  1  3  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
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+page_content='02782  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='00669  23  1  2  3  1  1  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
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+page_content='50595  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='03712  24  1  2  3  1  2  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='841  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='706  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
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+page_content='23678  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='02590  25  1  2  3  2  1  3  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='331  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='054  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='346  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='19839  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='01522  Table 1: Combinations of the angular wavenumbers, mj, and radial mode indices, nj, that form  a resonant triad (m1 + m2 = m3 and Ω1 + Ω2 = Ω3) at critical depth, hc, in a circular cylinder  of unit radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' For each triad, the corresponding wavenumbers, Kj = kmjnj, satisfy (9), and the  correlation condition,  ��  D Ψ1Ψ2Ψ∗  3 dA ̸= 0, is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' The list is restricted to resonant triads arising  for |mj|, nj ≤ 3, and we consider m1 ≤ m2 and m3 ≥ 0 without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We have omitted  resonances that give rise to the same critical depth, but with the roles of modes 1 and 2 swapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  The triad numbers (left column) and shaded rows are referenced in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  11  At this juncture, it is informative to assess how the triads listed in table 1 relate to the  resonances explored in prior investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' First, triad 7 in table 1 (dark grey row) was explored  by Michel [40] for a circular cylinder of radius 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='45 cm and an approximate fluid depth of 3 cm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' it  follows that the depth-to-radius ratio in Michel’s experiment was approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='317, close to the  value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='30197 reported in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Furthermore, table 1 (grey rows) incorporates two well-known  examples of a 1:2 resonance, for which modes 1 and 2 coincide: (i) the critical depth hc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='83138  (triad 19) corresponds to the second-harmonic resonance with the fundamental mode [42, 43, 8, 64];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  (ii) the critical depth hc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='19814 (triad 8) corresponds to a standing wave composed of two  resonant axisymmetric modes [34, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Finally, triads 1, 2, 3, 5 and 6 (table 1, light grey rows)  form an interesting class of resonant triad, for which an axisymmetric mode (m3 = 0) interacts  with two identical counter-propagating non-axisymmetric modes (m1 = −m2 ̸= 0 and n1 = n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  In fact, our investigation in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='1 demonstrates that the axisymmetric mode is the so-called pump  mode, and may thus excite the non-axisymmetric modes, even when the initial energy in each  non-axisymmetric mode is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We draw an analogy between this novel class of resonant  triad and the excitation of beach edge waves [18] in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  We conclude our exploration of resonant triads arising in a circular cylinder by remarking  that the fluid depth may, in some cases, be judiciously chosen so as to excite multiple triads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' In  general, the condition on the angular frequencies, Ω1 + Ω2 = Ω3 (see equation (7)), cannot be  satisfied for two distinct triads at the same fluid depth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' however, nonlinear resonance may persist  for both triads provided that each condition on the angular frequencies is approximately satisfied  [5, 39, 15], at the cost of weak detuning (see §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='1 for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Specifically, if triads 1 and  2 have critical depths hc,1 and hc,2, respectively, then there is potential excitement of both triads  when the fluid depth, h, satisfies |h−hc,j| = O(ϵ) for j = 1, 2 (where 0 < ϵ ≪ 1 is the typical wave  slope;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' see §2), giving rise to the approximation Ω1 + Ω2 − Ω3 = O(ϵ) for each triad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' For example,  if 0 < hc,2 − hc,1 ≪ 1, then it may be sufficient to excite both triads at an intermediate depth,  hc,1 ≤ h ≤ hc,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We note, however, that the excitation of multiple triads at a single fluid depth is  not possible when the depth discrepancy, |h− hc,j|, becomes too large (relative to the typical wave  slope) for any of the triads under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  To demonstrate the potential for the simultaneous excitation of two triads within a circular  cylinder of finite depth, we consider two scenarios: (i) the excitation of two triads that share a  common wave mode;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' and (ii) the excitation of two triads that do not share any common wave  modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Heuristically, case (ii) is more common than case (i) owing to the number of similar fluid  depths in table 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' however, case (i) will likely generate a far richer set of dynamics owing to  the nonlinear interaction between the two triads [35, 15, 13, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' As an example of case (i), we  consider triads 21 and 24 in table 1, with nearby critical depths hc,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='21395 and hc,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='23678,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' As mode (m3, n3) = (3, 3) is common to both triads, inter-triad resonance may arise  at an intermediate depth, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Finally, an example of case (ii) arises for triads 8 and 25  in table 1, with nearby critical depths hc,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='19814 and hc,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='19839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Neither of these triads  share a common wave mode, so one would not expect the inter-triad energy exchange discussed  in case (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Nevertheless, one might anticipate a signature of these two triads to be visible in  the surface evolution for an intermediate depth, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='19825.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' The theoretical and numerical  exploration of coupled triads in a circular cylinder will be the focus of future investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='3  Annular cylinder  A natural variation upon a circular cylinder is an annulus of inner radius r0 ∈ (0, 1) and outer  radius 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' By varying r0, the annulus approaches a circular cylinder as r0 → 0+, and a quasi-one-  12  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='5  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='5  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='5  1  0  2  4  6  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='5  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='5  1  Figure 3: The existence and predominant characteristics of a triad in an annular cylinder with inner  radius r0 and outer radius 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' The triad bifurcates from the critical depth hc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='17266 as r0 → 0  (the limiting case of a circular cylinder), with corresponding wavenumbers presented in table 1  (see triad 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' (a) The critical depth, hc (blue curve), with hc → ∞ as r0 → rc, where rc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='57  (black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' (b) The corresponding wavenumbers, Kj, all of which remain finite for r0 < rc (black  line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' (c) The normalised wavenumbers, K1/K3 and K2/K3, parametrised by increasing r0 (blue  arrow), with the limiting case r0 → 0 denoted by the white dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' The wavenumbers leave the triad  existence region (see Theorem 1) via the left-hand boundary (black curve) as r0 → r−  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  dimensional periodic ring as r0 → 1−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Notably, resonant triads are impossible for a one-dimensional  periodic ring, as can be shown by modifying the arguments presented for the case of a rectangular  cylinder (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Thus, one might anticipate that the existence of triads in an annular cylinder  depends critically on the inner radius, r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Rather than enumerating some possible triads for given  values of r0, we instead track the corresponding critical depth, hc, for the triads identified for a  circular cylinder (see table 1) as r0 is progressively increased from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Of particular interest is  determining whether a given triad exists for all r0 < 1, or whether there is some critical inner  radius, rc, beyond which the triad ceases to exist, with either hc → 0 or hc → ∞ as r0 → r−  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  The (complex-valued) eigenmodes in an annular domain are cylinder functions of the form  Φmn(r, θ) =  1  Nmn  �  Jm(kmnr) cos(γmnπ) + Ym(kmnr) sin(γmnπ)  �  eimθ,  (20)  where Nmn > 0 is a normalisation constant, Ym is the Bessel function of the second kind with  order m (an integer), and γmn ∈ [0, 1] determines the weighting between the two Bessel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  As shown in appendix B, the no-flux condition (see equation (1d)) on the inner and outer walls  determines that the wavenumbers, kmn(r0), satisfy the equation  J′  m(kmnr0)Y′  m(kmn) − J′  m(kmn)Y′  m(kmnr0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  (21)  A formula for the corresponding value of γmn is determined in appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Once again, the  wavenumbers, kmn, are ordered so that 0 < km1 < km2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' (excluding k00 = 0) and satisfy  −∆Φmn = k2  mnΦmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Three correlated wave modes may form a resonant triad (for a judicious  choice of the fluid depth) provided that the corresponding wavenumbers, Kj, which depend on the  channel width, 1 − r0, satisfy the bounds given in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Bifurcating from the limiting case of a circular cylinder, we track the critical depth (when such  a depth exists) of different triads as r0 is progressively increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' The predominant behaviour is  13  characterised by the example presented in figure 3, for which we consider the triad whose critical  depth is hc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='17266 as r0 → 0+ (see triad 10 in table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Given that hc is fairly small in this limit,  one might anticipate that the triad ceases to exist with hc → 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' somewhat surprisingly, however,  the opposite scenario arises, with hc → ∞ as r0 → r−  c (rc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='57 in this example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' It follows,  therefore, that the triad may persist for narrow channels only when the fluid is sufficiently deep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  We note, however, that there exist (at least) two relatively rare transitions for increasing r0, which  we briefly describe as follows: (i) the triad ceases to exist when hc → 0 as r0 → r−  c , which may  arise when bifurcating from a sufficiently shallow circular cylinder (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' triad 22 in table 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' and (ii)  the triad continues to exist for all r0 < 1, with hc → 0 and Kj → ∞ as r0 → 1, yet the normalised  depth, hcK3, remains finite, and the normalised wavenumbers, K1/K3 and K2/K3, remain within  the triad existence region (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' triad 8 in table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Owing to the appreciable influence of viscous  effects for relatively shallow fluids, the physical relevance of these latter two scenarios is somewhat  nebulous, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  4  The evolution of resonant triads  Having established the existence of resonant triads, we now determine the long-time triad evolution,  utilising the method of multiple scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Ostensibly, the calculations necessary for determining the  triad equations are a variation upon the pioneering work of McGoldrick [36, 37, 38] in the absence  of surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' However, the confinement of the fluid to a cylinder imposes some additional  considerations, the salient details of which we outline below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Finally, we note that an alternative  approach to multiple scales is Whitham’s technique of averaging the system’s Lagrangian [59,  60, 61, 62], which has the advantage of streamlining some algebraic calculations [54, 42, 43];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  nevertheless, multiple-scales analysis is sufficient for our purposes and allows for the possible  inclusion of higher-order corrections in the asymptotic expansion [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  In a manner similar to §3, we consider three linear wave modes (with real-valued eigenfunctions),  enumerated n1, n2 and n3, where we denote  Ωj = ωnj,  Kj = knj,  Lj =  ˆ  Lnj,  and  Ψj(x) = Φnj(x)  for j = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  In contrast to §3, however, we now allow each (nonzero) angular frequency to be either negative  or positive: the resonance condition on the angular frequencies is henceforth defined  Ω1 + Ω2 + Ω3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  (22)  The modified requirement on the angular frequencies (equation (22)) is not restrictive on the  possible triad combinations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' one may recover equation (7) by mapping Ω3 �→ −Ω3, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  The decision behind the summation condition on the angular frequencies is motivated by the  cyclical symmetry of equation (22), a property that will be inherited by the resultant amplitude  equations [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' As a consequence, one need only derive the amplitude equation for one of the wave  modes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' the amplitude equations for the remaining two wave modes follow by cyclic permutation  of the subscripts (1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Before embarking on the multiple-scales analysis presented in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='1, we remark upon two caveats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  First, we note that equation (22) corresponds to an exact resonance, for which the fluid depth, h,  is chosen to be precisely equal to the critical depth, hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' In practice, however, there may be a small  discrepancy between h and hc, resulting in a the sum of the angular frequencies being slightly  offset from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' When the frequency detuning is sufficiently weak, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Ω1 + Ω2 + Ω3 = O(ϵ), one  14  may modify the following asymptotic analysis to derive a similar set of amplitude equations (see  §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Second, our analysis in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='1 is not valid when two of the wave modes coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' This case  corresponds to a 1:2 resonance, for which the corresponding evolution equations were derived by  Miles [42] using Whitham modulation theory (as summarised in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='1  Multiple-scales analysis  In order to determine the evolution of each of the three dominant wave modes involved in an exact  resonance, we utilise the method of multiple scales [28, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Specifically, we seek a perturbation  solution to the Benney-Luke equation (3) of the form u ∼ u0+ϵu1+O(ϵ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' The leading-order terms  in equation (3) determine that u0 satisfies ∂ttu0 + L u0 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' we choose to consider a leading-order  solution comprised only of the three triad modes (all other modes are assumed to be smaller in  magnitude and appear at higher order), giving rise to the leading-order form  u0(x, t, τ) =  3  �  j=1  �  Aj(τ)Ψj(x)e−iΩjt + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  (23)  In equation (23), we have introduced the slow time-scale τ = ϵt, which governs the evolution of  each complex amplitude, Aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' As ϵ and t are both independent variables, we treat τ and t as  independent time-scales, giving rise to the transformation of derivatives ∂t �→ ∂t + ϵ∂τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Finally,  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' denotes the complex conjugate of the preceding term, a contribution necessary for real u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  So as to determine coupled evolution equations for each complex amplitude, Aj, we consider  terms of O(ϵ) in the Benney-Luke equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' By substituting the leading-order solution, u0,  into the nonlinear terms and applying the triad condition for the angular frequencies (equation  (22)), we obtain the following problem for u1:  ∂ttu1 + L u1 = −  �  3  �  j=1  fj(x, τ)e−iΩjt + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  �  + nonresonant terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  (24)  As we will see below, each of the functions fj(x, τ) appearing on the right-hand side of equation  (24) will play a fundamental role when determining the amplitude equations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' specifically,  f1 = −2iΩ1  dA1  dτ Ψ1 + iA∗  2A∗  3  ��  Ω2  �  L2  3 − K2  3  �  + Ω3  �  L2  2 − K2  2  �  − 2Ω1L2L3  �  Ψ2Ψ3 − 2Ω1∇Ψ2 · ∇Ψ3  �  ,  where f2 and f3 follow upon cyclic permutation of the subscripts (1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Finally, we note that  the ‘nonresonant terms’ in equation (24) are of the general form p(x, τ)eiςt, where we assume that  the angular frequency, ς, is not equal (or close) to any of the angular frequencies, ±ωn, associated  with linear wave modes (see §3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  We proceed by projecting equation (24) onto each of the three wave modes, giving rise to  differential equations of the form (for j = 1, 2, 3)  ∂ttˆu1,j + Ljˆu1,j = −  �  ⟨Ψj, fj⟩e−iΩjt + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  �  + nonresonant terms,  (25)  where ˆu1,j = ⟨Ψj, u1⟩ is the projection of u1 onto the mode Ψj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' By recalling that Lj = Ω2  j, we  immediately see that the term in square brackets in equation (25) is itself a solution to the linear  operator ∂tt+Ω2  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' It follows that the solution of equation (25) comprises of particular solutions that  15  have temporal dependence te±iΩjt, leading to an ill-posed asymptotic expansion when ϵt = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  The resolution to this problem is achieved via the solubility condition ⟨Ψj, fj⟩ = 0, which suppresses  the secular growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  By applying the solubility condition ⟨Ψj, fj⟩ = 0 for j = 1, 2, 3, we conclude that the com-  plex amplitude, Aj(τ), of each wave mode, Ψj(x)e−iΩjt, evolves according to the triad system of  canonical form [5, 15]  dA1  dτ = α1A∗  2A∗  3,  dA2  dτ = α2A∗  1A∗  3,  dA3  dτ = α3A∗  1A∗  2,  (26)  where  α1 =  1  2Ω1  ��  Ω2  �  L2  3 − K2  3  �  + Ω3  �  L2  2 − K2  2  �  − 2Ω1L2L3  �  C − 2Ω1  �  Ψ1, ∇Ψ2 · ∇Ψ3  ��  ,  (27)  while α2 and α3 follow by cyclic coefficient of the subscripts (1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Furthermore, the correlation  integral, C , is defined  C = 1  S  ��  D  Ψ1Ψ2Ψ3 dA,  (28)  where we recall that S is the area of the cylinder cross-section (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' As the triad equations  (26) are valid for τ = O(1) (or t = O(1/ϵ)), their dynamics yield an informative view of the  long-time evolution of the resonant triad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  In order to assess the influence of the triad coefficients on the triad evolution (see §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='2), we first  simplify the algebraic form given in equation (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' As shown by Miles [42], one may simplify the  inner product ⟨Ψ1, ∇Ψ2 · ∇Ψ3⟩ by repeated application of the divergence theorem and utilisation  of the relationship −∆Ψj = K2  j Ψj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' it follows that  �  Ψ1, ∇Ψ2 · ∇Ψ3  �  = 1  2  �  K2  2 + K2  3 − K2  1  �  C ,  (29)  where C is the correlation integral defined in equation (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We then substitute equation (29) into  equation (27) and simplify using the relation Ω1 + Ω2 + Ω3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' After some algebra, we derive the  reduced expression  α1 = C  2Ω1  �  Ω2L2  3 + Ω3L2  2 − 2Ω1L2L3 +  3  �  l=1  ΩlK2  l  �  ,  (30)  where α2 and α3 follow similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Finally, we demonstrate in appendix C that the algebraic form  of the triad coefficients may be further reduced to  αj = C β  2Ωj  for  j = 1, 2, 3,  (31)  where  β =  3  �  l=1  ΩlK2  l − 1  2Ω1Ω2Ω3  �  Ω2  1 + Ω2  2 + Ω2  3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  (32)  Equations (26), (28), (31) and (32) constitute the triad equations for resonant gravity waves  confined to a cylinder of finite depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Although the triad equations are of canonical form [5],  the novelty of our investigation is the computation of the coefficients, αj, whose algebraic form is  specific to our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  16  Figure 4: Contours of β/K2  3 (see equation (32)) for the case Ω1, Ω2 > 0 and Ω3 < 0 (with  Ω1 + Ω2 + Ω3 = 0), for which β < 0 (see equation (33)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='2  Properties of the triad coefficients  The simplified form of the coefficients, αj (equation (31)), allows for some important theoretical  observations that were obfuscated by the more complicated expressions for αj given in equations  (27) and (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' In particular, as exactly two of the angular frequencies, Ωj, have the same sign, we  deduce from equation (31) that the two corresponding coefficients, αj, also have the same sign,  with the third coefficient having the opposite sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' By utilising the well-known results pertaining  to the canonical triad equations, we conclude that all solutions to the triad equations (26) are  periodic in time, with solutions expressible in terms of elliptic functions [1, 5, 54, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Typically,  these solutions result in an exchange of energy between the comprising modes, although there is a  class of periodic solution that, perhaps counter-intuitively, results in zero energy exchange for all  time [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Moreover, it is readily verified that the leading-order energy density, E1 + E2 + E3, is  conserved, where Ej = Ω2  j|Aj|2, consistent with the Hamiltonian structure of the Euler equations  [5, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' The reader is directed to the work of Craik [15] for a more detailed account of the various  properties of the canonical triad equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Of particular relevance to the evolution of the triad is the quantity β (see equation (32)), which,  together with C , determines the time scale over which energy exchange arises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' In particular, we  present the form of β in figure 4 for the case Ω1, Ω2 > 0 and Ω3 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' As we will demonstrate  below, β < 0 in this case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' in general, the sign of β is the same as the sign of the largest (in  magnitude) angular frequency, Ωj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Notably, |β| decreases sharply towards zero as K1 + K2 → K3,  corresponding to the limit hc → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Similarly, |β| approaches zero in the limiting cases K1 ≪  K3 or K2 ≪ K3, corresponding to one low-oscillatory wave mode interacting with two highly-  oscillatory wave modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Away from these limiting cases, however, |β| depends only weakly on  the wavenumbers, Kj, suggesting that the correlation integral, C , predominantly controls the  time-scale of the triad evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Finally, we observe that β is symmetric about the line K1 = K2,  consistent with the invariance of equation (32) under the mapping K1 ↔ K2 (and hence, Ω1 ↔ Ω2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  17  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='3  K2  K3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='7  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='8  1  Ki/K3We conclude this section by proving that β < 0 in the case Ω1, Ω2 > 0 and Ω3 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' By  comparing the forms of equations (32) and (30), and then permuting the subscripts (1, 2, 3) �→  (3, 1, 2), we first note that β may be equivalently expressed as  β = Ω1L2  2 + Ω2L2  1 − 2Ω3L1L2 +  3  �  l=1  ΩlK2  l ,  or  β = Ω1(L2  2 + K2  1) + Ω2(L2  1 + K2  2) + |Ω3|(2L1L2 − K2  3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  By bounding Lj = Kj tanh(Kjhc) < Kj for 0 < hc < ∞ and utilising the relation Ω1 + Ω2 = |Ω3|,  we obtain  β < |Ω3|  �  K2  1 + K2  2 + 2K1K2 − K2  3  �  = |Ω3|  �  (K1 + K2)2 − K2  3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  (33)  As resonant triads exist only when K1 + K2 < K3 (see Theorem 1), we conclude that β < 0 in this  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='3  Summary  To summarise our theoretical developments, the velocity potential, u, at the fluid rest level (z = 0)  evolves according to  u(x, t) ∼  3  �  j=1  �  Aj(τ)Ψj(x)e−iΩjt + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  �  + O(ϵ),  (34)  where the complex amplitudes, Aj(τ), evolve over the slow time-scale, τ = ϵt, according to the triad  equations (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' In particular, the triad coefficients, αj (see equation (31)), are defined in terms  of the correlation integral, C (equation (28)), and the coefficient β (equation (32)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Notably, we  assume that C is nonzero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' if this condition were violated then all three of the triad coefficients, αj,  would be equal to zero, giving rise to non-interacting wave modes at leading order (contradicting  the notion of a triad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Indeed, the condition C ̸= 0 is identical to the correlation condition  detailed in equation (8), the origins of which we have now justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Finally, the evolution of the  free surface, η, may be recovered by recalling that η = −ut + O(ϵ): we conclude that η(x, t) has  a similar leading-order form to u(x, t), but each complex amplitude, Aj(τ), in (34) is replaced by  iΩjAj(τ) (see equation (37) below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  We briefly contrast our investigation of triad interaction with the early-time calculation of  Michel [40], who characterised the initial linear growth of a child mode induced by the nonlinear  interaction of two parent modes (where all three modes comprise the triad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  If modes 1 and  2 are the parent modes and mode 3 is the child mode, then the initial linear growth may be  deduced directly from triad equations (26) in the limit |A3| ≪ |A1| ∼ |A2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Specifically, the initial  variation of A1 and A2 is slow relative to that of A3, which has the approximate early-time form  A3(τ) ≈ α3C∗  1C∗  2τ + C3, where Cj = Aj(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Notably, the linear growth rate of the child mode  depends on the corresponding triad coefficient, α3, and the product of the initial amplitudes of the  two parent modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' However, our result for circular cylinders differs to that of Michel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' we believe  that the author neglected some important nonlinear contributions (compare Michel’s equation  (A2) to equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='4a) of Longuet-Higgins [33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' As Michel’s experiment verified the  scaling of the interaction only up to a proportionality constant, this discrepancy was not captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  18  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='1  The influence of weak detuning  As discussed earlier in §4, the analysis in §§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='2 does not account for weak detuning of the  angular frequencies, as might arise when the fluid depth, h, differs slightly from the critical depth,  hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We now briefly consider the case of weak detuning, for which equation (22) is replaced by the  condition Ω1 +Ω2 +Ω3 = ϵσ (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' here ϵ is the small parameter representative of the typical  wave slope (see §2) and σ = O(1) determines the extent of the detuning [5, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' By following  a very similar multiple-scales procedure to the case σ = 0, we obtain amplitude equations that  are now augmented by a time-dependent modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Specifically, each complex amplitude now  evolves according to  dA1  dτ = α1A∗  2A∗  3eiστ,  dA2  dτ = α2A∗  1A∗  3eiστ,  dA3  dτ = α3A∗  1A∗  2eiστ,  where each coefficient, αj, is defined in equation (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Although detuning yields non-autonomous  amplitude equations, autonomous equations may be derived by mapping Aj(τ) �→ Aj(τ)eiστ/3 for  all j = 1, 2, 3 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Finally, we note that the energy, E1 + E2 + E3, is not exactly conserved when  considering the effects of detuning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' instead, the energy slowly oscillates about a constant value  [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='2  The case of a 1:2 resonance  A 1:2 resonance is a resonant triad for which two modes comprising the triad coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' For this  case, we define two angular frequencies, Ω1 and Ω2, so that Ω2 = 2Ω1 [42], where the connection  to resonant triads is clear when writing Ω1 + Ω1 = Ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' By following a very similar multiple-scales  procedure to that outlined in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='1, we obtain  u(x, t) ∼  2  �  j=1  �  Aj(τ)Ψj(x)e−iΩjt + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  �  + O(ϵ),  where  dA1  dτ = −γA∗  1A2  and  dA2  dτ = γ  4A2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  (35)  In particular, the evolution of the amplitude equations (35) depends on the coefficient γ = C  �  K2  2 −  K2  1 − 3Ω4  1  �  , where C = 1  S  ��  D Ψ2  1Ψ2 dA is the correlation integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Indeed, the amplitude equations  (35) and coefficient, γ, are consistent with the results of Miles [42] when expressing the evolution  of each complex amplitude, Aj, in polar form (with appropriate rescaling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Finally, we note that a  weak detuning (see §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='1) may also be incorporated within the amplitude equations (35), thereby  accounting for a slight mismatch between the fluid depth, h, and the corresponding critical depth,  hc [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Of particular interest is the evolution of weakly nonlinear waves steadily propagating around  a circular cylinder of unit radius, focusing on the case where the fluid depth is precisely equal to  the critical depth of a 1:2 resonance [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' For the complex-valued eigenmodes defined in equation  (19), the correlation condition,  ��  D Ψ2  1Ψ∗  2 dA ̸= 0, determines that the angular wavenumbers satisfy  m2 = 2m1 [12, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' By expressing the complex wave amplitudes in polar form, Aj(τ) = aj(τ)eiθj(τ)  (for j = 1, 2), equation (35) may be recast as [42]  da1  dτ = −γa1a2 cos Θ,  da2  dτ = γ  4a2  1 cos Θ,  dΘ  dτ = 2γa2  �  1 − a2  1  8a2  2  �  sin Θ,  19  where Θ(τ) = θ2(τ) − 2θ1(τ) is the time-dependent phase shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Steadily propagating waves  correspond to time-independent solutions for a1, a2 (both nonzero) and Θ, from which we deduce  that cos Θ = 0 and a1/a2 = 2  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Indeed, it is remarkable that the amplitude ratio of the  two dominant (normalised) wave modes is independent of the angular wavenumbers, mj, the  radial wavenumbers, Kj, and the corresponding angular frequencies, Ωj (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Furthermore, one may readily determine the relationship between the angular velocity of the  steady wave rotation and the corresponding wave amplitude, which may then be compared to the  numerical solution of the full Euler equations [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' This comparison, as well as a comparison to  steadily propagating waves computed from various truncations of the Euler equations, will be the  subject of future investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  5  The excitation of resonant triads  Having established the existence and evolution of resonant triads, we now focus on the excitation  of a particular triad via external forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' So as to motivate the method of excitation, we first  recall (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='1) the well-known result that one mode in the triad may, or may not, excite the other  two modes [16, 22, 54];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' in the case of excitation, the initial mode is referred to as the pump mode  [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We will then utilise the criterion of the pump mode to excite all three modes in the triad via  a pulsating pressure source (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Throughout this section, we continue with the convention that  the triad angular frequencies satisfy Ω1 + Ω2 + Ω3 = 0, as set forth in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='1  Excitation via the triad pump mode  To first identify the triad pump mode and then characterise the resultant excitation, we consider  the case for which A3, say, is much larger in magnitude than the other two mode amplitudes, so  |A1|, |A2| ≪ |A3| [16, 22, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' By linearising the triad equations (26), we obtain  dA1  dτ = α1A∗  2A∗  3,  dA2  dτ = α2A∗  1A∗  3,  dA3  dτ = 0,  (36)  from which we immediately conclude that A3 is constant (whilst the linearisation assumption  holds);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' we denote A3(τ) = C for some given complex number C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' By considering second derivatives  of A1 and A2, we deduce the linearised evolution equations [15]  d2A1  dτ 2 = α1α2|C|2A1  and  d2A2  dτ 2 = α1α2|C|2A2,  where α1α2 = C 2β2/(4Ω1Ω2) (see equation (31)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We conclude that A1(τ) and A2(τ) grow ex-  ponentially in time (whilst the linearisation approximation holds) when Ω1Ω2 > 0, and exhibit  sinusoidal oscillations when Ω1Ω2 < 0 [16, 22, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Thus, mode 3 may excite modes 1 and 2  when Ω1 and Ω2 have the same sign (and likewise for other mode permutations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' As one angular  frequency must have a different sign from the other two (so as to satisfy Ω1 + Ω2 + Ω3 = 0), we  conclude that the mode whose angular frequency is largest in magnitude (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' differs in sign) is the  triad pump mode [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Equivalently, the pump mode is the mode with largest wavenumber, Kj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  To visualise the influence of the pump mode on the resultant free-surface pattern, we present the  solution of the triad equations (26) and the corresponding pump-mode approximation (equation  (36)) in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' By recalling that the free surface satisfies η = −ut + O(ϵ), we first deduce that  η(x, t) ∼  3  �  j=1  �  iΩjAj(τ)Ψj(x)e−iΩjt + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  �  + O(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  (37)  20  Figure 5: Excitation of a triad via its pump mode for the case of a circular cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We consider  triad 24 in table 1, but with m3 �→ −m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We choose Ω1, Ω2 > 0 and Ω3 < 0, so that mode 3  is the pump mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' (a) Evolution of the free-surface, η ∼ −ut, over the slow time-scale, τ = ϵt,  with ϵ = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' (b) The evolution of the wave amplitudes, |Aj|, according to the triad equations  (equation (26), solid curves) and the pump-mode approximation (equation (36), dashed-dotted  curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Insets: modes 1 (blue), 2 (red) and 3 (gold) at τ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' all three modes rotate counter-  clockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' The simulations were initialised from A1(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='01 and A2(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='01i, where A3(0) was  chosen to be the positive real number satisfying E1 + E2 + E3 = 1, with Ej = Ω2  j|Aj|2 (see §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  21  T=O  8  T= 16  24  T川  a  (6)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='1  0  0  5  10  15  20  25  30  =For the case of a circular cylinder, we utilise the complex-valued eigenmodes defined in equation  (19), corresponding to the superposition of steadily propagating waves for mj ̸= 0 (the rotation di-  rection depends on the sign of Ωj/mj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Upon initialising the system so that the energy is primarily  within the pump mode (mode 3), modes 1 and 2 are gradually excited due to nonlinear interaction,  with exponential growth evident for τ ≲ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' As time further increases, the dynamics depart from  the pump-mode approximation: the energy in the pump mode appreciably decreases, whilst the  energy in modes 1 and 2 saturates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' The free surface varies qualitatively during this evolution, with  an appreciable change in pattern structure visible by τ = 24 (primarily a superposition of modes 1  and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Notably, the system evolution is periodic, which becomes apparent over longer time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='2  Excitation via an applied pressure source  Based on the ideas of the previous section, we consider a methodology for exciting the pump  mode of a triad, which will subsequently excite the remaining two modes (provided that the initial  disturbance of each of the remaining modes is nonzero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Notably, several methods for exciting  internal resonances have been considered in prior investigations, primarily focusing on imposed  motion of the fluid vessel via horizontal [42, 45] or vertical vibration [42, 44, 46, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Furthermore,  one may, in principle, utilise sinusoidal paddles or plungers to excite a particular triad’s pump  mode for a given geometry (similar wave makers are used in rectangular wave tanks [37, 23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  However, for large-scale fluid tanks, imposed motion of the vessel may be impractical (if the tank  were set in a concrete base, for example), and it may be challenging to determine the correct  paddle motion necessary to excite a chosen pump mode for geometrically complex cylinders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We  choose, therefore, to consider a slightly different approach: we instead excite the pump mode via  a pulsating pressure source located just above the free surface (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' an air blower).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  In order to incorporate a pressure source within our mathematical framework, we first refor-  mulate the dimensionless dynamic boundary condition (equation (1b)) as  φt + η + ϵ  2  �  |∇φ|2 + φ2  z  �  + ϵP(x, t) = 0  for  x ∈ D,  z = ϵη,  where the dimensional pressure is ϵ2aρgP for fluid density ρ (P = 0 corresponds to atmospheric  pressure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' The pressure source is chosen to be small in magnitude so that the resultant wave  excitation arises over the slow time-scale, τ = ϵt, and may thus be saturated by weakly nonlinear  effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' By modifying the developments outlined in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='1, we derive the forced Benney-Luke equation  utt + L u + ϵ  �  ut  �  L 2 + ∆  �  u + ∂  ∂t  �  (L u)2 + |∇u|2�  + ∂tP  �  = O(ϵ2)  for  x ∈ D,  (38)  which will be the starting point for the asymptotic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Before proceeding further, we first describe two forms of the pressure source relevant to our  investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' For a stationary pressure source oscillating periodically over the fast time-scale, t,  we express P(x, t) = f(τ)s(x)e−iΩpt + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', where s(x) is a fixed spatial profile (generally spanning  the cavity), f(τ) accounts for a slow modulation in the magnitude of the pressure, and Ωp is the  pulsation angular frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We choose Ωp to be close to the angular frequency of the pump  mode, which, without loss of generality, we assume to be mode 3 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Ω3 has the opposite sign  from Ω1 and Ω2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We denote, therefore, Ωp = Ω3 + ϵµ, where µ = O(1) determines the extent  of the frequency mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' For a pressure source orbiting the centre of a circular cylinder at a  constant angular velocity, we instead posit that P has the form P(r, θ, t) = f(τ)s(r, θ−Ωpt), where  22  0  5  10  1  2  3  4  0  20  40  60  80  100  0  1  2  3  4  0  50  100  150  200  250  300  1  2  3  4  5  Figure 6: Evolution of the forced triad equations (39) for σ = µ = 0 and constant f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We consider  the same triad as figure 5, with s3f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' In all three panels, A1(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='02i and A2(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  For A3(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='01, we observe (a) the initial excitation of the triad and (b) the resultant periodic  dynamics (the initial growth is highlighted within the grey box).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' (c) For A3(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='01i, the triad  evolution is chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Ωp = (Ω3 + ϵµ)/m3 is the angular velocity of the pressure source (assuming that the pump mode  is non-axisymmetric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' m3 ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  For both standing and orbiting pressure sources, we now follow a similar multiple-scales pro-  cedure to that outlined in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='1, starting from the forced Benney-Luke equation (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' So as to  discount the possibility that the pressure source excites more than one mode in the triad, we  assume that neither |Ω1| or |Ω2| are close to |Ω3|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Furthermore, we incorporate a weak detuning  in the triad angular frequencies, denoting Ω1 + Ω2 + Ω3 = ϵσ (see §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' It follows that each  complex amplitude, Aj(τ), evolves according to  dA1  dτ = α1A∗  2A∗  3eiστ,  dA2  dτ = α2A∗  1A∗  3eiστ,  dA3  dτ = α3A∗  1A∗  2eiστ − Ω3s3f(τ)e−iµτ,  (39)  where the coefficients, αj, are defined in equation (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Notably, the pump mode may only be  excited provided that the corresponding eigenmode is non-orthogonal to the pressure source, cor-  responding to a nonzero projection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' s3 ̸= 0, where s3 = ⟨Ψ3, s⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Similar equations describing  the evolution of forced resonant triads have been explored by McEwan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' [35] (with the inclusion  of linear damping) and Raupp & Silva Dias [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  In the special case of time-independent forcing (f constant) and no frequency detuning (σ =  µ = 0), the dynamics of the forced triad equations has been analysed by Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' [20], with  both periodic and quasi-periodic dynamics reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We also consider this case, leaving the effects  of detuning and variable forcing for future investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' In this setting, when |A1|, |A2| and |A3|  23  are initially small relative to the magnitude of the forcing, |Ω3s3f|, the initial growth in A3 is  approximately linear (see figure 6(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' As mode 3 is the pump mode, the growth in A3 excites A1  and A2, thus activating the triad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' The conservation laws of the forced triad equations [20] result  in a temporary diminution of mode 3, which is later augmented by the external forcing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' whence  the process repeats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' In some parameter regimes, the resulting evolution of the forced triad is  periodic in time (see figure 6(b) and Raupp & Silva Dias [50]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' in contrast to the findings of Harris  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' [20], however, we also identify initial conditions (with all other parameters unchanged) that  result in hitherto unidentified chaotic dynamics (see figure 6(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' The chaotic nature of this latter  example may be verified via estimation of the maximal Lyapunov exponent [55], which is found  to be positive (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' initially adjacent trajectories diverge exponentially in phase space);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' however, a  more thorough investigation of the chaotic dynamics of the forced triad equations, and the subtle  dependence on initial conditions, will be presented elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  6  Discussion  We have performed a systematic investigation into nonlinear resonant triads of free-surface gravity  waves confined to a cylinder of finite depth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' previously studied 1:2 resonances are obtained as  special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' A key result of our study is Theorem 1, which determines whether there exists  a fluid depth at which three given wave modes resonate due to the nonlinear evolution of the  fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Equipped with this result, we determined the long-time fluid evolution using multiple-scales  analysis, from which we deduced that all solutions to the triad equations are periodic in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Finally, we determined that a given triad may be excited via external forcing of the triad’s pump  mode, thereby providing a mechanism for exciting a given triad in a wave tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' All our results  are derived for cylinders of arbitrary cross-section (barring some technical assumptions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' see §2),  thus forming a broad framework for characterising nonlinear resonance of confined free-surface  gravity waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' In particular, our theoretical developments buttress experimental observations [40]  and demonstrate the potential generality of confinement as a mechanism for promoting nonlinear  resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  A second fundamental component of our study is the influence of the cylinder cross-section  on the existence of resonant triads;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' for example, resonant triads are impossible in rectangular  cylinders, yet abundant within circular and annular cylinders (for particular fluid depths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Of the  vast array of resonances arising in a circular cylinder, triads consisting of an axisymmetric pump  mode and two identical counter-propagating waves are of notable interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' This combination of  axisymmetric and non-axisymmetric modes possesses an interesting analogy to the excitation of  counter-propagating subharmonic beach edge waves due to a normally incident standing wave  [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Specifically, the wave crests of the standing axisymmetric mode are always parallel to the  bounding wall of the circular cylinder, and may excite steadily propagating waves that are periodic  in the azimuthal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' For the special case for which the amplitudes of the two counter-  propagating modes coincide, one observes the resonant interaction of standing axisymmetric and  non-axisymmetric waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  So as to gain a deeper insight into the influence of nonlinearity on resonant triads, a primary  focus for future investigations will be the simulation of the Euler equations within a cylindrical  domain, with consideration of various truncated systems [14, 47, 4, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' From a computational  perspective, the most natural geometry to consider is a circular cylinder [49];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' this geometry has  been previously explored in the context of steadily propagating nonlinear waves in the vicinity of  a 1:2 resonance [8, 64], but it remains to assess the efficacy of the amplitude equations (26) for  24  predicting the evolution of nonlinear triads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Indeed, exploration of the nonlinear dynamics may  reveal additional resonant triads arising beyond the small-wave-amplitude limit explored herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Of similar interest is the fluid evolution when multiple triads are excited at a single depth, with  the potential for energy exchange via triad-triad interactions [35, 15, 13, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' The simulation of  free-surface gravity waves in non-circular cylinders presents a more formidable challenge, however,  except for cylinder cross-sections that possess a tractable eigenmode decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  A second natural avenue for future investigation is to characterise the influence of applied  forcing on resonant triads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' For example, when the fluid bath is subjected to sufficiently vigorous  vertical vibration, Faraday waves [17, 30] may appear on the free surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' although this scenario  has been studied in the case of a 1:2 internal resonance [44, 46, 24], resonant triads may give rise to  the formation of more exotic free-surface patterns, particularly at fluid depths that differ from that  of a 1:2 resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' In a similar vein, horizontal vibration [42, 45] or a pulsating pressure source  at the frequency of the triad’s pump mode may lead to a wealth of periodic and quasi-periodic  dynamics, as predicted by the forced triad equations [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Our study has indicated, however,  that chaotic dynamics are also possible in some parameter regimes, and might thus be excited in  numerical simulation or experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Lastly, our study has focused on flat-bottomed cylinders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' it  seems plausible, however, that submerged topography may enhance or mitigate certain resonances,  which may be an important consideration in the design of industrial-scale fluid tanks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Finally, our study has focused on the special case of a liquid-air interface, for which the dynamics  of the air are neglected within the Euler equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' It is natural, however, to extend our formulation  to the case of two-layer flows (in the absence of surface tension), with two immiscible fluids (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  air and water) confined within a cylinder whose lid and base are both rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' In this setting, the  density difference across the fluid-fluid interface has a strong influence of the system dynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' it  seems plausible, therefore, that additional resonances may be excited in this configuration, relative  to the liquid-air interface considered herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Notably, the anticipated resonances would arise across  a single interface, in contrast to the cross-interface resonances explored in previous investigations  [1, 54, 25, 53, 56, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Finally, exploring the influence of parametric forcing [31] on resonant  triads arising for two-layer flows opens up exciting new vistas in nonlinear resonance induced by  confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  A  Proof of Theorem 1  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' To prove Theorem 1, we first show that there are no values of h ∈ (0, ∞) satisfying Ω1+Ω2 =  Ω3 when K1+K2 ≥ K3 or when √K1+√K2 ≤ √K3, where we recall that Ωj(h) =  �  Kj tanh(Kjh)  and Kj > 0 for j = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We then prove that there exists a solution to Ω1 + Ω2 = Ω3 when  K1 + K2 < K3 < (√K1 + √K2)2, and that this solution is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  In the case K1 + K2 ≥ K3, we first define χ(K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' h) =  �  K tanh(Kh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' For fixed h > 0, we  observe that  χ(K3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' h) ≤ χ(K1 + K2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' h) < χ(K1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' h) + χ(K2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' h),  where we have utilised that χ(K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' h) is a positive, monotonically increasing, concave function of  K > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We conclude that Ω3 < Ω1 + Ω2 for any h > 0, so there are no values of h for which  Ω1 + Ω2 = Ω3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  In the case √K1 + √K2 ≤ √K3, we first note that the lower bound Kj > 0 (for j = 1, 2, 3)  implies that K1 < K3 and K2 < K3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Furthermore, as tanh(x) is a monotonically increasing  function for x > 0, we conclude that tanh(Kjh) < tanh(K3h) for j = 1, 2 and all h > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We now  25  utilise this property to deduce that  �  K1 tanh(K1h) +  �  K2 tanh(K2h) <  ��  K1 +  �  K2  ��  tanh(K3h) ≤  �  K3 tanh(K3h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  We conclude that Ω3 > Ω1 + Ω2 for any h > 0, so there are no values of h for which Ω1 + Ω2 = Ω3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  For the remainder of the proof, we consider the case  K1 + K2 < K3  and  �  K3 <  �  K1 +  �  K2,  (40)  which is equivalent to the pair of inequalities given by equation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Indeed, we will show that there  exists a unique value of h > 0 satisfying Ω1 + Ω2 = Ω3 in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Equivalently, we demonstrate  that F(h) =  �  Ω1(h) + Ω2(h)  �  /Ω3(h) − 1 has a unique positive root, where we express  F(h) =  �  ψ1(h) +  �  ψ2(h) − 1,  with the positive functions ψ1 and ψ2 defined  ψj(h) = Kj tanh(Kjh)  K3 tanh(K3h)  for j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  In order to show the existence of a root of F(h), we first note that  lim  h→0 F(h) = K1 + K2  K3  − 1 < 0  and  lim  h→∞ F(h) =  √K1 + √K2  √K3  − 1 > 0,  where we have used the limits limx→0(tanh(x)/x) = 1 and limx→∞ tanh(x) = 1, respectively,  and implemented the inequalities given in equation (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' As F(h) is a continuous function, the  intermediate-value theorem determines that F(h) has at least one positive root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  To prove that such a root is unique, we demonstrate that F(h) is a strictly monotonically  increasing function for h > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Specifically, we note that (for j = 1, 2)  dψj  dh = 2K3ψj(h)  �Kj  K3  cosech(2Kjh) − cosech(2K3h)  �  > 0  for 0 < Kj < K3,  where the inequality follows from the convexity of cosech(x) for x > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' b cosech(bx) > cosech(x)  for 0 < b < 1 and all x > 0 (associating x = 2K3h and b = Kj/K3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' As the bounds K1 < K3  and K2 < K3 incorporate the region determined by equation (40), we deduce that F(h) is strictly  monotonically increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We conclude, therefore, that the root of F(h) must be unique, thereby  completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  B  Wavenumbers in an annulus  The no-flux condition (equation (1d)) on the inner and outer radii of an annulus requires that  ∂rΦmn(r0, θ) = 0 and ∂rΦmn(1, θ) = 0 for all θ, where Φmn(r, θ) is the cylinder function defined  in equation (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' It follows, therefore, that the corresponding wavenumber, kmn, and weighting  factor, γmn, satisfy the equations  J′(kmnr0) cos(γmnπ) + Y′  m(kmnr0) sin(γmnπ) = 0,  (41a)  J′(kmn) cos(γmnπ) + Y′  m(kmn) sin(γmnπ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  (41b)  26  By rearranging equation (41), we determine the following expressions for tan(γmnπ):  tan(γmnπ) = − J′  m(kmnr0)  Y′  m(kmnr0)  and  tan(γmnπ) = − J′  m(kmn)  Y′  m(kmn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  (42)  By eliminating tan(γmnπ) and rearranging, we find that kmn > 0 satisfies equation (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Upon  computing kmn, one may then determine γmn ∈ [0, 1] using either of the equivalent expressions for  tan(γmnπ) given in equation (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  C  Reduction of the triad coefficients  As motivated by the form of α1 given in equation (30), we demonstrate that  Ω2L2  3 + Ω3L2  2 − 2Ω1L2L3 = −1  2Ω1Ω2Ω3  �  Ω2  1 + Ω2  2 + Ω2  3  �  ,  (43)  where we recall that Ω1 + Ω2 + Ω3 = 0 and Lj = Ω2  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' In fact, the equality given in equation  (43) holds under cyclic permutation of the indices (1, 2, 3) (as is necessary when defining α2 and  α3), where we note that the right-hand side is unchanged under such permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' We conclude  that α2 and α3 may be simplified in a similar manner, with the right-hand side of equation (43)  appearing as a constant term in all three coefficients (see §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  We now detail the algebraic manipulations necessary to transform the left-hand side of equation  (43) into the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' By substituting Lj = Ω2  j into the left-hand side of equation (43) and  factorising, we obtain  Ω2L2  3 + Ω3L2  2 − 2Ω1L2L3 = Ω4  2Ω3 + Ω2Ω2  3  �  Ω2  3 − 2Ω1Ω2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  (44)  Next, we substitute  Ω2  3 = (Ω2  1 + Ω2  2) = Ω2  1 + 2Ω1Ω2 + Ω2  2  (45)  into equation (44), yielding  Ω2L2  3 + Ω3L2  2 − 2Ω1L2L3 = Ω2Ω3  �  Ω3  2 + Ω3  �  Ω2  1 + Ω2  2  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  (46)  We proceed by substituting Ω3 = −(Ω1 + Ω2) within the square brackets in equation (46);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' by  distributing and cancelling common terms, we obtain  Ω2L2  3 + Ω3L2  2 − 2Ω1L2L3 = −Ω1Ω2Ω3  �  Ω2  1 + Ω1Ω2 + Ω2  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  (47)  Finally, we rearrange equation (45) to give  Ω1Ω2 = 1  2  �  Ω2  3 − Ω2  1 − Ω2  2  �  ,  which, upon substitution into equation (47), supplies the required result (equation (43)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  27  References  [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Energy transfer between external and internal gravity waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=',  19(3):465–478, 1964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Benney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Non-linear gravity wave interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 14(4):577–584, 1962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Benney and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Luke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' On the Interactions of Permanent Waves of Finite Amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 43:309–313, 1964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [4] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Berger and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Milewski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Simulation of Wave Interactions and Turbulence in One-  Dimensional Water Waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 63(4):1121–1140, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [5] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Bretherton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Resonant interactions between waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' the case of discrete oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid  Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 20(3):457–479, 1964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [6] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Bridges and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Dias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' An analysis of two-dimensional water waves based on O(2) sym-  metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Nonlinear Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Theory Methods Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 14(9):733–764, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [7] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Bryant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Periodic waves in shallow water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 59(4):625–644, 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [8] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Bryant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Nonlinear progressive free waves in a circular basin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 205:453–467,  1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [9] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Case and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Chiu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Three-wave resonant interactions of gravity-capillary waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Fluids, 20:742, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Chabane and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Choi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' On resonant interactions of gravity-capillary waves without energy  exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Stud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 142:528–550, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [11] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Choi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Chabane, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Taklo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Two-dimensional resonant triad interactions in  a two-layer system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 907:A5, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [12] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Chossat and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Dias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' The 1:2 Resonance with O(2) Symmetry and Its Applications in  Hydrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Nonlinear Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 5:105–129, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [13] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Chow, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Henderson, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Segur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' A generalized stability criterion for resonant triad  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 319:67–76, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [14] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Craig and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Sulem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Numerical Simulation of Gravity Waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 108(1):73–  83, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Craik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Wave interactions and fluid flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Cambridge Monographs on Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Cambridge University Press, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [16] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Davis and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Acrivos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  The stability of oscillatory internal waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=',  30(4):723–736, 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Faraday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' On a Peculiar Class of Acoustical Figures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' and on Certain Forms Assumed by  Groups of Particles upon Vibrating Elastic Surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Philos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 121:299–340,  1831.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  28  [18] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Guza and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Davis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Excitation of Edge Waves by Waves Incident on a Beach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 79(9):1285– 1291, 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Hammack and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Henderson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Resonant Interactions Among Surface Water Waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 25(1):55–97, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Harris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Bustamante, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Connaughton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Externally forced triads of resonantly  interacting waves: Boundedness and integrability properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Nonlinear Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Simulat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 17:4988–5006, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [21] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Hasselmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' On the non-linear energy transfer in a gravity-wave spectrum Part 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' General  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 12(4):481–500, 1961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [22] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Hasselmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' A criterion for nonlinear wave stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 30(4):737–739, 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [23] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Henderson and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Hammack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Experiments on ripple instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Part 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Resonant  triads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 184:15–41, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [24] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Henderson and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Miles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Faraday waves in 2:1 internal resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=',  222:449–470, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [25] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Joyce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Nonlinear interactions among standing surface and internal gravity waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 63(4):801–825, 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [26] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Kadri and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Akylas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' On resonant triad interactions of acoustic-gravity waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid  Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 788:R1, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [27] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Kadri and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Stiassnie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Generation of an acoustic-gravity wave by two gravity waves, and  their subsequent mutual interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 735:R6, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [28] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Kevorkian and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Cole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Multiple Scale and Singular Perturbation Methods, volume  114 of Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Springer, New York, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [29] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Kreyszig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Introductory Functional Analysis with Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' John Wiley & Sons, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [30] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Kumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Linear theory of Faraday instability in viscous fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' A,  452:1113–1126, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [31] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Kumar and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Tuckerman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Parametric instability of the interface between two fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 279:49–68, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [32] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Lamb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Hydrodynamics (6th ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Dover Publications, 1932.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [33] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Longuet-Higgins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Resonant interactions between two trains of gravity waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid  Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 12(3):321–332, 1962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [34] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Mack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Periodic, finite-amplitude, axisymmetric gravity waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=',  67(2):829–843, 1962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [35] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' McEwan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Mander, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Smith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Forced resonant second-order interaction between  damped internal waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 55(4):589–608, 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  29  [36] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' McGoldrick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Resonant interactions among capillary-gravity waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=',  21(2):305–331, 1965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [37] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' McGoldrick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' An experiment on second-order capillary gravity resonant wave interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 40(2):251–271, 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [38] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' McGoldrick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' On Wilton’s ripples: a special case of resonant interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=',  42(1):193–200, 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [39] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' McGoldrick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' On the rippling of small waves: a harmonic nonlinear nearly resonant  interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 52(4):725–751, 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [40] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Michel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Three-wave interactions among surface gravity waves in a cylindrical container.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluids, 4:012801(R), 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [41] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Miles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Surface-wave damping in closed basins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' A, 297(1451):459–475,  1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [42] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Miles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Nonlinear surface waves in closed basins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 75(3):419–448, 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [43] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Miles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Internally resonant surface waves in a circular cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 149:1–14,  1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [44] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Miles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Nonlinear Faraday resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 146:285–302, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [45] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Miles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Resonantly forced surface waves in a circular cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 149:15–31,  1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [46] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Miles and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Henderson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Parametrically forced surface waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid  Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 22:143–165, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [47] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Milewski and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Keller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Three-Dimensional Water Waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Stud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 97:149–  166, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [48] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Phillips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' On the dynamics of unsteady gravity waves of finite amplitude Part 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' The  elementary interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 9(2):193–217, 1960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [49] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Qadeer and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Wilkening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Computing the Dirichlet–Neumann Operator on a Cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 57(3):1183–1204, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [50] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Raupp and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Silva Dias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Resonant Wave Interactions in the Presence of a  Diurnally Varying Heat Source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Atmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 66(10):3165–3183, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [51] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Raupp, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Silva Dias, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Tabak, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Milewski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Resonant wave interactions  in the equatorial waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Atmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 65(11):3398–3418, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [52] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Schwartz and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Vanden-Broeck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Numerical solution of the exact equations for capil-  lary–gravity waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 95(1):119–139, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [53] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Segur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Resonant interactions of surface and internal gravity waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluids, 23:2556,  1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  30  [54] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Simmons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' A variational method for weak resonant wave interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' A, 309(1499):551–577, 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [55] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Strogatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Nonlinear Dynamics and Chaos: With Applications to Physics, Biology,  Chemistry and Engineering (2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' CRC Press, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [56] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Taklo and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Choi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Group resonant interactions between surface and internal gravity  waves in a two-layer system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 892:A14, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [57] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Wang and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Milewski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Dynamics of gravity-capillary solitary waves in deep water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 708:480–501, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [58] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Vanden-Broeck, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Milewski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Two-dimensional flexural–gravity waves  of finite amplitude in deep water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' IMA J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 78(4):750–761, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [59] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Whitham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' A general approach to linear and non-linear dispersive waves using a La-  grangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 22(2):273–283, 1965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [60] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Whitham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Non-Linear Dispersive Waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' A, 283(1393):238–261,  1965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [61] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Whitham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Non-linear dispersion of water waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 27(2):399–412, 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [62] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Whitham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Variational methods and applications to water waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  A, 299:6–25, 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [63] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Wilton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' On ripples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Phil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' 6, 29(173):688–700, 1915.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  [64] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Yang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Dias, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Liu, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Liao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Finite-amplitude steady-state resonant waves in a  circular basin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content=', 915:A136, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
+page_content='  31' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NA0T4oBgHgl3EQfOf_i/content/2301.02163v1.pdf'}
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+Bohm - de Broglie Cycles
+To my beloved Izabel
+Olivier Piguet∗
+January 30, 2023
+Abstract
+In the de Broglie-Bohm quantum theory, particles describe trajectories determined
+by the flux associated with their wave function. These trajectories are studied here for
+relativistic spin-one-half particles. Based in explicit numerical calculations for the case of
+a massless particle in dimension three space-time, it is shown that if the wave function
+is an eigenfunction of the total angular momentum, the trajectories begin as circles of
+slowly increasing radius until a transition time at which they tend to follow straight lines.
+Arrival times at some detector, as well as their probability distribution are calculated,
+too. The chosen energy and momentum parameters are of the orders of magnitude met
+in graphene’s physics.
+Keywords: Bohm - de Broglie, Quantum Mechanics, Transport properties, Graphene.
+1
+Introduction
+Since its beginning in the first decades of XXth century [3]-[9] Quantum Mechanics and its
+extension in the form of Quantum Field Theory led to an accurate description of atomic and
+subatomic phenomena, confirmed in an extraordinarily precise way by countless experiments.
+However, there is no such a broad consensus about its interpretation. Various ones are present
+in the literature, such as the Copenhagen [10], the Many-World [11], the Relational [12] or the
+de Broglie-Bohm (dBB) one. We will deal here with the latter interpretation, first proposed by
+Louis de Broglie [6] as the ”Pilot Wave Theory”, later on formulated by David Bohm [13, 14]
+as the ”Ontological Interpretation of Quantum Theory” and finally, critically defended by John
+∗Pra¸ca Graccho Cardoso, 76/504, 45015-180 Aracaju, SE, Brazil, E-mail: opiguet@yahoo.com
+1
+arXiv:2301.13251v1  [quant-ph]  30 Jan 2023
+
+Bell in a series of papers reproduced in [15]. This interpretation of Quantum Mechanics differs
+essentially from the largely more widespread Copenhagen interpretation by taking particle
+trajectories as elements of reality, i.e., a particle really follows a trajectory, the latter being
+defined by “guying conditions” first proposed by de Broglie. A probabilistic interpretation is
+maintained, but now in the sense of classical statistical mechanics. The trajectory followed by
+a particle is fully defined by giving boundary conditions such as, e.g., the coordinates of its
+initial position. The probability distribution of the particle following a particular trajectory
+is then given by the value of the squared modulus of the wave function taken at the initial
+time and position.
+Since this statistical distribution is equal to the “usual” (Copenhagen)
+quantum probability distribution and in particular satisfies the same conservation conditions,
+mean values of beables1 evolve identically in both interpretations of Quantum Mechanics. On
+the other side, the possibility of observable consequences of the existence of trajectories remains
+an open question. Let us however mention an experimental proposal [17, 18] on the measure
+of the times of arrival at a detector for a particle prepared in some initial state.
+The first aim of the present paper is to introduce the reader to the dBB theory by treat-
+ing simple physical examples. Since the main peculiarity of this interpretation is the factual
+existence of trajectories, some effort will be made in the study of their trajectories, looking for
+situations in which dBB could make a difference.
+The dBB trajectories we will calculate are those of a relativistic2, massive or massless, spin
+one half particle in dimension 3 space-time. Its quantum state will be supposed to be described
+by an eigenfunction of the total angular momentum J defined relatively to some space point3.
+Stationary as well as packets of stationary wave functions will be considered. The main result
+for the latter is that the dBB trajectories consist of an initial phase of quasi circles whose
+radius increases till a critical time at which the trajectory begins tending to a straight line –
+the straight line expected for a classical particle with definite angular momentum.
+The paper begins in Section 2 with an introduction to the dBB formalism and in Section
+3 for its application to the relativistic spin one half particle described by the Dirac equation.
+Numerical computations of dBB trajectories and arrival times for electrons in the context of
+graphene’s transport properties are presented in Section 3.3. Final considerations are given in
+the Conclusion Section.
+Most analytic and numerical calculations are made with the help of the software Mathe-
+matica [22].
+1Following John Bell [16], I use here the term ”beable“ , instead of the usual term “observable” which is
+somewhat related to the Copenhagen interpretation and to its postulate of the “wave function reduction”.
+2Only the theory of a single particle is considered here. See [2, 19, 20] for a discussion of the N-particle
+relativistic case.
+3See [21] for the calculation of such states in the case of the non-relativistic free particle.
+2
+
+2
+Summary of the de Broglie-Bohm theory
+In the ”usual” (i.e., Copenhagen) interpretation of Quantum Mechanics [10], the state of a
+physical system constituted of a single particle is described, in the Schr¨odinger picture, by a
+wave equation
+iℏ∂ψ(x, t)
+∂t
+= ˆHψ(x, t),
+(2.1)
+where ˆH is a self-adjoint partial derivative operator acting on a N-components wave function
+ψ(x, t) =
+�
+�
+ψ1(x, t)
+· · ·
+ψN(x, t)
+�
+� ,
+(2.2)
+belonging to a Hilbert space, with scalar product and norm defined by4
+⟨ψ|Φ⟩ =
+�
+Rdddx ψ†(x, t)φ(x, t) =
+�
+Rdddx
+N
+�
+α=1
+ψ∗
+αφα(x, t),
+||ψ|| = ⟨ψ|ψ⟩
+1
+2 .
+(2.3)
+x = (xi, i = 1, · · · , d) are the space coordinates, d the space dimension and †, ∗ denote the
+hermitian and complex conjugate, respectively. The number of components, N, depends on the
+particle’s spin and on the space dimension d. E.g., N = 1 for a scalar (spin 0) particle, N = 2
+for a non-relativistic particle of spin 1/2 in any dimension, N = 2 for a relativistic particle of
+spin 1/2 in 2 dimensions, N = 4 for the same in 3 dimensions, etc [23].
+The wave equation (2.1) implies the existence of a non-negative density
+ρ(x, t) = ψ†(x, t)ψ(x, t),
+(2.4)
+and an associate d-current density5 j(x, t) obeying a continuity equation
+∂tρ + ∇j = 0,
+(2.5)
+which assures the constancy of the norm defined in (2.3). Normalizing the norm to 1, one
+interprets ρ as a probability density and j as a probability flux.
+In the Copenhagen theory, the state of the system is completely characterized by the wave
+function ψ, solution of the wave equation (2.1), with an arbitrarily given initial wave function
+ψ(x, 0) = ψ0(x). The de Broglie-Bohm (dBB) theory completes the characterization of the
+state of the system by postulating the existence of a trajectory of the system (the particle,
+here), determined by the de Broglie “guidance conditions” [6] for the particle’s velocity
+v(x, t) = j(x, t)/ρ(x, t),
+(2.6)
+4This holds e.g., for a non-relativistic particle in general, or a spin 1/2 relativistic one. We do not consider
+here cases such as the relativistic spin zero particle described by the Klein-Gordon equation.
+5Its explicit form will be given below for the cases studied there.
+3
+
+the dBB trajectory xB(x0, t) being then a solution of the set of differential of equations
+∂xB(x0, t)
+∂t
+= v(xB(x0, t), t),
+(2.7)
+with an arbitrarily given initial position xB(x0, 0) = x0. In other word, the possible trajectories
+are the integral lines of the dBB vector field v(x, t), labelled by their initial position x0. Since
+the flux j and the density ρ turn out to be bilinear in ψ and ψ† (see later on), the dBB
+trajectories do not depend on the “intensity” of the wave function, but only on its “form”.
+Recall that, in the quantum probabilistic interpretation of Copenhagen, the density ρ repre-
+sents the probability of experimentally finding the particle at a given place at a given time. On
+the other hand, the dBB theory treats this density in a more “classical statistical mechanics”
+way: Trajectories “really happen”, and their probability distribution, which amounts to the
+probability distribution of their initial positions x0, is given [24] by the density ρ(x0, 0). The
+latter represents our lack of knowledge of the precise initial position. Thanks to the continuity
+equation (2.5), the probability for the particle being inside a co-moving volume6 V (t) is con-
+stant in time, which sustains this statistical interpretation, called “thermal equilibrium” in the
+literature [24].
+Such a formulation is called a theory with “hidden variables”, the hidden variables of the
+present one being, e.g., the components of the initial position x0.
+A heuristic justification for the guiding condition (2.6) may be found in analogy with fluid
+mechanics, considering ρ and v as the fluid mass density and velocity field, respectively. With
+j = ρv, Eq. (2.5) has the form of the continuity equation expressing the conservation of the
+fluid mass.
+Now, granted the existence of a trajectory, one can define properties of the particle along
+it, such as energy, spin, etc. , in the following way. First, given an “observable” A represented
+by a self-adjoint operator ˆA, one defines the associated beable field (See footnote 1)
+A(x, t) = ψ†(x, t) ˆA ψ(x, t)/ρ(x, t),
+(2.8)
+where ρ is the probability density (2.4). One then defines the instantaneous value of A, i.e., its
+value when the particle is at point xB(x0, t) at time t:
+AB(x0, t) = A(xB(x0, t), t).
+(2.9)
+Note that these definitions are independent of the normalization of the wave function. Their
+justification is obvious in the case of ˆA being the multiplication by a function fA(x, t), since
+in this case A(x, t) = fA(x, t). For the general case, including that of a matricidal and/or
+differential operator, one derives from (2.8) the identity
+�
+ddx ρ(x, t)A(x, t) =
+�
+ddx ψ†(x, t) ˆA ψ(x, t) = ⟨A⟩ (t).
+(2.10)
+6I.e., a volume whose boundary points move along the dBB trajectories.
+4
+
+The first integral is an expectation value in the dBB theory sense, i.e., in the “classical statisti-
+cal mechanical” sense, whereas the second one gives it in the conventional quantum mechanical
+sense. Their equality indeed shows the equivalence of both theories for what concerns expecta-
+tion values of observables.
+To summarize, the dBB theory proposes the existence of:
+1. A wave function or ”guidance field” (2.2), solution of the Schr¨odinger-like wave equation
+(2.1).
+2. A statistical ensemble of particle’s trajectories xB(x0, t) as the integral lines of the dBB
+vector field (2.6), solutions of the differential equations (2.7) parametrized by the value
+of the initial position.
+3. Beable fields and beable instantaneous values, defined by (2.8) and (2.9). Such beables
+may be energy, momentum, angular momentum, spin, etc.
+All considerations made in this subsection generalize easily to systems of N particles, x
+denoting a point of the d × N dimensional configuration space7.
+3
+Free relativistic spin one half particle in 3-dimensional
+space-time
+3.1
+Dirac equation
+A free, spin 1/2 relativistic particle of mass m in 3-dimensional space-time is described in the
+usual theory by a 2-components spinor wave function
+ψ(x) =
+� ψ1(x)
+ψ2(x)
+�
+,
+(3.1)
+solution of the free Dirac equation
+iℏcγµ∂µψ(x) − mc2ψ(x) = 0,
+(3.2)
+where x = (xµ, µ = 0, 1, 2) are the space-time coordinates space-time, whose metric is ηµν =
+diag(1, −1, −1). The Dirac matrices obey the anticommutation rules {γµ, γν} = 2ηµν. Our
+choice for them is given in Appendix A. c, ℏ are the speed of light and the reduced Planck
+constant, respectively. The Dirac equation may be cast in the form8 of (2.1):
+iℏ∂ψ(x, t)
+∂t
+= ˆHψ(x, t),
+(3.3)
+7At least in the non-relativistic case. See footnote 2.
+8In fact Dirac’s original form, reduced to 3 space-time dimensions.
+5
+
+with
+ˆH = −iℏc αi∂i + mc2 β,
+(3.4)
+with αi = γ0γi and β = γ0 (see Appendix A). The density and the flux obeying the continuity
+equation (2.5) are given by
+ρ = ψ†ψ,
+ji = c ψ†αiψ.
+(3.5)
+The theory is relativistic: cρ and ji are the time and space components of the space-time
+3-vector jµ = ¯ψγµψ, and the continuity equation reads ∂µjµ = 0.
+Another relativistic object is the scalar density
+σ(x, t) = 1
+2
+¯ψψ = 1
+2ψ†βψ.
+(3.6)
+jµ and σ fulfil the identity
+jµjµ = 4σ2,
+(3.7)
+consequence of the Pauli matrices identity
+3
+�
+i=1
+σi
+αβσi
+γδ = 2δαδδβγ − δαβδγδ.
+(3.8)
+Let us go now to the dBB theory. The identity (3.7) allows one to define the time-like 3-velocity
+field’
+uµ = jµ
+2|σ|,
+uµuµ = 1,
+u0 > 0.
+(3.9)
+The relativistic form of the dBB guidance equation (2.7) then reads9
+dxµ(λ)
+dλ
+= uµ(x(λ)),
+(3.10)
+with λ as curve’s parameter. This equation is equivalent to (2.7) and defines the space-time
+trajectories of the particle.
+In the same way as one defines the dBB velocity field (2.6) or (3.9), one can define a dBB
+spin field
+S(x, t) = ℏ σ(x, t)/ρ(x, t),
+(3.11)
+which takes values between −ℏ/2 and ℏ/2. The beable S(x, t) can thus be interpreted as the
+field associated to the spin operator
+ˆS = ℏ
+2σz,
+(3.12)
+according to the definition (2.8). Note that, in the instantaneous rest frame of the particle
+guided by the wave, where vi = ji = 0, the identity (3.7) implies S = ±ℏ/2, in agreement with
+the interpretation of S as the intrinsic angular momentum of a spin one half particle. One then
+gets the instantaneous spin of the particle according to (2.9):
+SB(x0, t) = S(xB(x0, t), t).
+(3.13)
+9The present discussion is the reduction to 3-dimensional space-time of the one made by [1, 2] in 4 dimensions.
+6
+
+3.2
+Eigenstates of the angular momentum and of the energy
+We will look for the solutions of the Dirac equation which are eigenfunctions of the total angular
+momentum and Hamiltonian operators. It will be useful to work in polar coordinates r, φ:
+x = r cos φ,
+y = r sin φ.
+In these coordinates, the Hamiltonian operator (3.4) reads
+ˆH = −iℏc
+��
+α1 cos φ + α2 sin φ
+�
+∂r + 1
+r
+�
+α2 cos φ − α1 sin φ
+�
+∂φ
+�
++ mc2β.
+(3.14)
+One easily checks, using the algebra of the Pauli matrices, that the total angular momentum
+operator with respect to the origin, which has a single component in the two-dimensional space’s
+case,
+ˆJ = −iℏ∂φ + ℏ
+2β,
+(3.15)
+commutes with the Hamiltonian operator. Spinor eigenfunctions of ˆJ with eigenvalue j are
+readily found to be of the form
+ψ(r, φ, t) =
+� ei(j− 1
+2 )φf1(r, t),
+ei(j+ 1
+2 )φf2(r, t)
+�
+.
+(3.16)
+One directly sees that the uniformity of the wave function requires j to be half-integer. With
+the result (3.16), solving the Dirac equation (3.2) amounts to solving the two radial equations
+for the functions f1(r, t) and f2(r, t):
+iℏ
+�
+r (∂tf1 + c ∂rf2) + c(j + 1
+2) f2
+�
+− mc2r f1 = 0,
+iℏ
+�
+r (∂tf2 + c ∂rf1) − c(j − 1
+2) f1
+�
++ mc2r f2 = 0.
+(3.17)
+In terms of the radial functions fα, the probability density ρ, the flux j and the scalar density
+σ defined by (3.5), (3.6) read:
+ρ(r, t) = |f1(r, t)|2 + |f2(r, t)|2,
+jx(r, φ, t) = c
+�
+eiφf ∗
+1(r, t)f2(r, t) + e−iφf ∗
+2(r, t)f1(r, t)
+�
+,
+jy(r, φ, t) = ic
+�
+−eiφf ∗
+1f2(r, t) + e−iφf ∗
+2(r, t)f1(r, t)
+�
+,
+σ(r, t) = 1
+2 (|f1(r, t)|2 − |f2(r, t)|2) .
+(3.18)
+ρ and σ, as well as the radial and azimuthal components of the flux j,
+jr(r, t) = cos φ jx(r, φ, t) + sin φ jy(r, φ, t) = c (f ∗
+1(r, t)f2(r, t) + f ∗
+2(r, t)f1(r, t)) ,
+jφ(r, t) = − sin φ jx(r, φ, t) + cos φ jy(r, φ, t) = ic (−f ∗
+1(r, t)f2(r, t) + f ∗
+2(r, t)f1(r, t)) ,
+(3.19)
+turn out to be independent of the angular coordinate.
+7
+
+3.2.1
+Instantaneous values of energy, spin and orbital angular momentum
+Before going to our main task, i.e., the concrete study of the dBB trajectories, let us look at
+the expressions of the instant values of energy, spin and orbital angular momentum, namely
+EB(x0, t), SB(x0, t) and LB(x0, t) along a dBB trajectory xB(x0, t) fixed by the initial position
+x0. They are generated by the corresponding fields E(x, t), S(x, t) and L(x, t) according to
+(2.9). One obtains
+E(x, t) = ψ†(x, t) ˆHψ(x, t)/ρ(x, t) = iℏ(f ∗
+1(r, t)∂tf1(r, t) + f ∗
+2(r, t)∂tf2(r, t))/ρ(r, t),
+EB(x0, , t) = E(xB(x0, t), t),
+(3.20)
+where f1, f2 are the radial components defined by (3.16)
+and ˆH is the Hamilton operator, the validity of the Schr¨odinger-like equation (2.1) being
+assumed;
+S(x, t) = ψ†(x, t) ˆSψ(x, t)/ρ(x, t) = 1
+2(f ∗
+1(r, t)f1(r, t) − f ∗
+2(r, t)f2(r, t))/ρ(r, t),
+SB(x0, t) = S(xB(x0, t), t),
+(3.21)
+where ˆS is the spin operator (3.12);
+L(x, t) = ψ†(x, t)ˆLψ(x, t)/ρ(x, t) = ℏ(j − S(x, t)),
+LB(x0, t) = L(xB(x0, t), t) = ℏ(j − S(x0, t)),
+(3.22)
+where ˆL is the orbital angular momentum operator (in Cartesian and polar coordinates)
+ˆL = −i (x∂y − y∂x) = −iℏ ∂φ.
+(3.23)
+Use has been made in (3.22) of the fact that ˆL = ˆJ − ˆS and of the wave function being an
+eigenfunction of the total angular momentum ˆJ with eigenvalue j.
+3.2.2
+Stationary solutions:
+Since the Hamiltonian (3.14) and the total angular momentum (3.15) commute, we can impose
+the stationarity condition
+ˆHψ = ℏωp ψ.
+(3.24)
+Thus the radial wave functions fα(r, t) take the form
+f p, stat
+α
+(r, t) = e−iωpt hp, stat
+α
+(r),
+α = 1, 2,
+(3.25)
+ℏωp being the energy of the stationary state. This leads to a pair of equations for the function
+hp, stat
+α
+(r), derived from (3.17) by substituting i∂t by ωp. The general solution of these equa-
+tions is a superposition of the Bessel functions of the first and second kind, Jj±1/2(pr/ℏ) and
+Yj±1/2(pr/ℏ), with p a function of ωp defined as the positive solution for p of
+ωp = 1
+ℏ
+�
+m2c4 + p2c2.
+(3.26)
+8
+
+Square integrability of the wave function at r = 0 leads us to discard the solutions involving
+Yj±1/2 because of the latter’s singularity at the origin (see (B.3) in Appendix B). The regular
+solution thus is
+hp, stat
+1
+(r) = icp Jj−1/2(pr/ℏ),
+hp, stat
+2
+(r) = −(ℏωp − mc2)Jj+1/2(pr/ℏ).
+(3.27)
+The general stationary solution of the Dirac equation for angular momentum eigenstates, non-
+singular at the origin, then reads
+ψp, stat(r, φ, t) =
+�
+� ei(j− 1
+2 )φf p, stat
+1
+(r, t),
+ei(j+ 1
+2 )φf p, stat
+2
+(r, t)
+�
+�
+= e−iωpt
+�
+icp ei(j−1/2)φJj−1/2(pr/ℏ),
+−(ℏωp − mc2)ei(j+1/2)φJj+1/2(pr/ℏ)
+�
+,
+(3.28)
+with ωp given by (3.26). These spinors form a basis for the solutions of the Dirac equation,
+however an improper one since they are not square integrable due to the asymptotic behaviour
+of the Bessel functions shown in (B.4) of Appendix B.
+We can nevertheless apply the dBB guidance principle, expressed in Eqs, (2.6) and (2.7),
+to such a basis element. From the result (3.28) we can compute explicitly the density ρ given
+in (3.18):
+ρp, stat(r) = c2p2J2
+j−1/2(pr/ℏ) + (ℏωp − mc2)2J2
+j+1/2(pr/ℏ),
+(3.29)
+as well as the flux components (3.19) which, divided through ρ according to the dBB condition,
+yields the radial and azimuthal components of the velocity vector field:
+vp, stat
+r
+(r) = 0
+vp, stat
+φ
+(r) = 2c
+cp(ℏωp − mc2)Jj−1/2(pr/ℏ)Jj+1/2(pr/ℏ)
+(cp)2J2
+j−1/2(pr/ℏ) + (ℏωp − mc2)2J2
+j+1/2(pr/ℏ).
+(3.30)
+Obviously all these expressions are time independent due to the stationarity condition. One sees
+that the radial component of the velocity field is vanishing and that its azimuthal component
+does not depend on the polar angle. Thus the dBB trajectories of the particle, defined as the
+integral curves of the velocity vector field, are circles of radius r centred at the origin travelled
+at a constant velocity vφ and whose radius dependent value is bounded by c, in the massive
+as well as in the massless case. In all cases the bound c is effectively reached, for a discrete
+set of values of the radius. Fig. 1 in Subsection 3.3 shows a typical behaviour of the azimuthal
+velocity vp, stat
+φ
+(r) in the massless particle’s case.
+One may observe that, in the massless case, in which ℏωp = pc, changing the sign of the
+total angular momentum implies a change of sign of the velocity field: vp, stat
+φ
+(r) → −vp, stat
+φ
+(r),
+hence of the orientation of the trajectories. However this does not hold for the massive case,
+and also not for the more general wave packets examined in Subsection (3.2.3).
+9
+
+3.2.3
+Gaussian wave packet
+The general solution of the Dirac equation for angular momentum eigenstates, non-singular at
+the origin, reads
+ψ(x, t) =
+� ∞
+0
+dp a(p)ψp, stat(x, t),
+(3.31)
+where ψp, stat is the stationary solution (3.28) of energy10 ℏωp given by (3.26), and the amplitude
+a(p) is an arbitrary complex function, but constrained by the requirement of ψ(x, t) to be a
+square integrable function of x.
+The angular momentum eigenstates we will consider in the following are described by the
+spinor wave functions of the form (3.31), with a(p) the Gaussian amplitude
+a(p) = √p e
+−(p − p0)2
+2Σ2
+.
+(3.32)
+Here and in the rest of this paper we consider only positive energy solutions of the Dirac
+equations.
+3.2.4
+Mean values
+We recall that mean values of observables obtained from the dBB or from the Copenhagen
+theory coincide.
+The wave functions (3.31) are normalizable and we can calculate the square norm ||ψ||2 in
+the following way:
+∥ψ∥2 =
+� 2π
+0
+dφ
+� ∞
+0
+dr r ρ(r, t),
+with ρ(r, t) the density (3.5). From (3.31) we get
+∥ψ∥2 = 2π
+� ∞
+0
+dr r
+� ∞
+0
+dp
+� ∞
+0
+dp′ a(p)a(p′)
+2
+�
+α=1
+f p, stat
+α
+(r, t)∗f p′, stat
+α
+(r, t)
+= 2π
+� ∞
+0
+dp
+� ∞
+0
+dp′ a(p)a(p′)ei(ωp−ωp′)
+� ∞
+0
+dr r
+�
+(mc2 + ℏωp)(mc2 + ℏωp′)Jj−1/2(pr/ℏ)Jj−1/2(pr′/ℏ) + ℏ2pp′Jj+1/2(pr/ℏ)Jj+1/2(pr′/ℏ)
+�
+.
+From (3.32) and the completeness identity [25] for the Bessel functions:
+� ∞
+0
+dr r Jl(kr)Jl(k′r) = 1
+kδ(k − k′),
+one gets
+∥ψ∥2 = 2πℏ2
+� ∞
+0
+dp 1
+pa2(p)
+�
+c2p2 + (ℏωp − mc2)2�
+.
+(3.33)
+10We restrict to positive energy contributions.
+10
+
+In the same way one establishes expressions for the mean energy:
+⟨E⟩ =
+� 2π
+0
+dφ
+� ∞
+0
+dr r ψ†(r, φ, t) ˆH ψ(r, φ, t) / ∥ψ∥2,
+where ˆH is the Hamiltonian operator (3.4), and for the standard energy deviation
+∆E =
+�
+⟨E2⟩ − ⟨E⟩2.
+The result is
+⟨E⟩ = 2πℏ2
+∥ψ∥2
+� ∞
+0
+dp 1
+pa2(p)
+�
+c2p2 + (ℏωp − mc2)2�
+ℏωp,
+∆E =
+�2πℏ2
+∥ψ∥2
+� ∞
+0
+dp 1
+pa2(p)
+�
+c2p2 + (ℏωp − mc2)2�
+(ℏωp)2 − ⟨E⟩2
+� 1
+2
+.
+(3.34)
+Finally, a computation of the mean value of the spin operator (3.12) yields
+⟨S⟩ = πℏ3
+∥ψ∥2
+� ∞
+0
+dp 1
+pa2(p)
+�
+−c2p2 + (ℏωp − mc2)2�
+.
+(3.35)
+Substituting ∥ψ∥2 in the denominator by its expression (3.33), one sees that this mean value
+obeys the bounds −ℏ/2 < ⟨S⟩ < ℏ/2.
+All these integrals can be computed analytically in the massless case m = 0 for the amplitude
+given by (3.32). One gets
+∥ψ∥2 = πℏ2c2Σ3 �√π(1 + 2z2)(1 + erf(z)) + 2ze−z2�
+,
+⟨E⟩ = cp0
+√πz(3 + 2z2) (1 + erf(z)) + 2(1 + z2)e−z2
+z(√π(1 + 2z2)(1 + erf(z)) + 2ze−z2)
+,
+(3.36)
+∆E = cp0
+�
+π(3 + 4z4)(1 + erf(z))2 + 8√πz(−1 + z2)(1 + erf(z))e−z2 + 4(−2 + z2)e−2z2
+2z2(π(1 + 2z2)2(1 + erf(z))2 + 4√πz(1 + 2z2)(1 + erf(z))e−z2 + 4z2e−2z2)
+(3.37)
+where we have set
+z = p0
+Σ ,
+(3.38)
+and erf(z) is the error function [26].
+Finally, the mean spin is null in this massless case:
+⟨S⟩ = 0,
+(3.39)
+11
+
+which implies ⟨L⟩ = ℏj for the orbital angular momentum since the states considered are
+eigenstates of the total angular momentum with eigenvalue ℏj. Associated to this mean orbital
+angular momentum, we can define an L-radius
+rL = ⟨L⟩
+⟨p⟩ = c ℏ j
+⟨E⟩ ,
+(3.40)
+where ⟨p⟩ is the mean value of the momentum p, equal to ⟨E⟩ /c in the present massless
+case. This definition mimics the classical relation between angular momentum and radius for
+a uniform circular motion.
+As one may expect, in the case of a very narrow width of the amplitude (3.32), i.e., z ≫ 1,
+the energy and the energy uncertainty approximate to the values
+⟨E⟩ ≃ cp0,
+∆E ≃
+c
+√
+2Σ.
+(3.41)
+The results (3.34) and (3.35) allow us to identify the expression
+˜ρ(p) = a2(p)c2p2 + (ℏωp − mc2)2
+p2
+(3.42)
+up to a due normalization, as the probability density in p-space, conjugate to the x-space
+probability defined in (3.5).
+3.3
+j - electrons in graphene
+Let us denote free electrons in eigenstates of the total angular momentum with eigenvalue j as
+“j-electrons”. Free electrons in monolayer graphene [27] with energy less than Ecrit ≈ 160 meV
+obey approximatively a relativistic-like massless dispersion law E ≈ p c, where p is the linear
+momentum and the “velocity of light”11 c ≈ 106 m s−1. The dynamics of these electron is that
+of free massless particles in dimension three space-time with pseudo-spin 1/2 obeying the Dirac
+equation (3.2) with m = 0 [28]. Pseudo-spin is a chirality effect due to the peculiar crystalline
+structure of graphene and should not be confused with the usual spin. Nevertheless, since the
+wave function obeys the Dirac equation (3.2), the pseudo-spin adds itself to the orbital angular
+momentum yielding the conserved total angular momentum j (3.15) as explained in Subsection
+3.2.
+We will consider first stationary states and then states defined by Gaussian-like wave pack-
+ets. Conventions and units used in the following are described in Appendix A.
+3.3.1
+Stationary states in graphene
+As already shown in Subsection 3.2.2, the dBB trajectories associated to the stationary wave
+functions (3.28) are circles centred at the origin, travelled at a constant dBB velocity vp, stat
+φ
+11This is the so-called critical velocity, which we will denote by c.
+12
+
+20
+40
+60
+80
+100
+r
+1000
+2000
+3000
+ρp,stat (r)
+20
+40
+60
+80
+100
+r
+-1.0
+-0.5
+0.5
+1.0
+vϕ
+p,stat (r)/c
+Figure 1: Stationary case: density ρp, stat(r) and azimuthal velocity vp, stat
+φ
+(r)/c for j = 5/2,
+p = 10−4 meV/c (E = 100 meV).
+j
+1/2
+3/2
+5/2
+7/2
+9/2
+11/2
+13/2
+15/2
+17/2
+αj
+0
+2.19
+3.45
+4.61
+5.74
+6.84
+7.93
+9.01
+10.08
+Table 1: Coefficients αj of Eq. (3.45) in function of the angular momentum j.
+which depends on the radius r according to (3.30). For m = 0, this velocity reads
+vp, stat
+φ
+(r) = 2c Jj−1/2(pr/ℏ)Jj+1/2(pr/ℏ)
+J2
+j−1/2(pr/ℏ) + J2
+j+1/2(pr/ℏ).
+(3.43)
+The dBB velocity value oscillates between −c and c. A typical behaviour, in function of the
+radius, of the probability density and of the dBB velocity, which depend on the energy E = c p
+and on the angular momentum j, is shown in Fig. 1 for j = 5/2 and E = 100 meV. The most
+probable radius ˆrj, is given by the position of the first maximum of the probability density
+ρp, stat (3.29) (See Fig. 1), which for m = 0 reads
+ρp, stat(r) = c2p2 �
+J2
+j−1/2(pr/ℏ) + J2
+j+1/2(pr/ℏ)
+�
+.
+(3.44)
+Since r appears in the combination pr/ℏ, the most probable radius ˆrj may be written as a
+function of p:
+ˆrj(p) = αj
+ℏ
+p,
+(3.45)
+the j-dependent coefficients αj being shown in Table 1 for some values of j. These results can
+be taken as an approximation for the more realistic case of a wave packet with a very small
+energy width.
+13
+
+0.002
+0.004
+0.006
+0.008
+0.010
+t
+40
+60
+80
+100
+120
+rB
+(a)
+0.002
+0.004
+0.006
+0.008
+0.010
+t
+-250
+-200
+-150
+-100
+-50
+ϕB
+(b)
+-50
+50
+xB
+-50
+50
+100
+yB
+(c)
+Figure 2: The most probable dBB tractory xB(ˆr0, t) for Gaussian wave packet parameters
+j = 5/2, p0 = 10−4 meV/c (Peak energy E0 = 100 meV), and Σ = 10−7 meV/c.
+This
+trajectory is fixed by the initial condition parameter = ˆr0 = 22.7. ˆr0 is the position of the peak
+of the probability density ρ at t = 0 (the blue point in Fig. 4a.)
+(a) Radial coordinate rB(ˆr0, t).
+(b) Azimuthal coordinate φB(ˆr0t).
+(c) dBB trajectory in the (x, y) plane for 0 ≤ t ≤ 0.010 ns. The trajectory performs 38 loops
+before the critical time (decay time) τ ∼ 0.006 ns.
+14
+
+0.002 0.004 0.006 0.008 0.010 0.012 0.014
+t
+500
+1000
+1500
+rB
+(a)
+0.002 0.004 0.006 0.008 0.010 0.012 0.014
+t
+-250
+-200
+-150
+-100
+-50
+ϕB
+(b)
+500
+1000
+1500
+xB
+-100
+100
+200
+300
+400
+500
+yB
+(c)
+Figure 3: dBB tractories for Gaussian wave packet: same parametrization as in Fig. 2, but
+with the larger time scale 0 ≤ t ≤ 0.015 ns.
+3.3.2
+Gaussian wave packets in graphene
+We turn now to the Gaussian wave functions defined by (3.31) and (3.32), which are normal-
+izable. We shall denote by
+xB(r0, t) = (rB(r0, t), φB(r0, t)) ,
+20
+40
+60
+80
+100
+r0
+5.×10-7
+1.×10-6
+1.5×10-6
+2.×10-6
+ρ(r0,0)
+(a)
+0.002
+0.004
+0.006
+0.008
+0.010
+0.012
+t
+200
+400
+600
+800
+rB
+(b)
+0.002
+0.004
+0.006
+0.008
+0.010
+0.012
+t
+-600
+-500
+-400
+-300
+-200
+-100
+ϕB
+(c)
+200
+400
+600
+800
+xB
+-300
+-200
+-100
+100
+yB
+(d)
+Figure 4: dBB tractories for Gaussian wave packet: same parametrization as in Figs. 2 and
+3, but with five trajectories corresponding to the initial radial positions r0 = 10, 17, 22.7,
+30 and 35 nm. The respective numbers of trajectory’s closed loops are 93, 66, 38, 10 and 1.
+Their respective relative probabilities are proportional to the heights of the coloured dots in
+the subfigure (a) showing the initial probability density ρ(r0, 0) in function of the initial radial
+position r0.
+15
+
+the (polar) coordinates of the dBB trajectory solution of the trajectory equation (2.7), fixed
+by the initial position
+xB(r0, 0) = (rB(r0, 0), φB(r0, 0) = (r0, 0).
+(3.46)
+We have made explicit, in our notation, the dependence on the trajectory parameter r0.
+Figs. 2 and 3 show the most probable dBB trajectory xB(ˆr0, t) for a particular wave func-
+tion’s parametrization in which the energy dispersion is very small, i.e., Σ ≪ p0.
+“Most
+probable” means that the initial particle’s position parameter ˆr0 is the value of the radial co-
+ordinate r which maximizes the initial probability density ρ(r, 0) This value, equal to 22.7 nm
+in the present example12 corresponds to the blue dot in Fig. 4a. Thus, the behaviour of this
+trajectory, shown by Figs. 2c or 3c in the (x, y)-plane, is very similar to the circular one shown
+in the corresponding stationary solution, but only up to a certain critical “decay time” τobs,
+approximately equal to 0.006 ns in this example. For later times the trajectory tends to a
+straight line, reproducing the expected classical behaviour. This is best observed in Figs. 2b or
+3b which show the time behaviour of the azimuthal angle φ: the angular velocity is almost con-
+stant until the time τobs, and almost vanishes thereafter. This decay time marks the transition
+from the almost circular regime to an almost straight-way, classical-like, regime.
+A theoretical lower bound for the decay time τ may be computed from the quantum ”time-
+energy uncertainty principle” [29]:
+τ ≥ τmin =
+ℏ
+2 ∆E ,
+(3.47)
+where ∆E is the quantum energy uncertainty given by (3.34) taken with m = 0.
+In our
+example, τmin = 0.00465 ns: this is the order of magnitude of the observed decay time for the
+most probable trajectory, τobs ∼ 0.006 nm, and the uncertainty inequality (3.47) is obeyed.
+Table 2 displays, for certain values of the wave parameters p0, Σ and j, quantities of interest
+such as mean energy ⟨E⟩ (3.36), standard energy deviation ∆E (3.37), τmin (3.47), (which are
+usual quantum theory quantities), and also, specifically concerning the most probable dBB
+trajectory, its approximate observed decay time τobs, its L-radius rL (3.40), its initial radial
+coordinate ˆr0 (see (3.46)) which fixes it and the number of closed loops, Nloops,’ it performs
+before passing to the straight-way regime.
+One can make the following remarks about the items of Table 2.
+1. The mean values ⟨E⟩, ∆E, hence τmin, do not depend on the quantum number j, which is
+obvious from the explicit expressions (3.36) and (3.37). ⟨E⟩ and ∆E tend towards their
+limit values (3.41) as the width Σ becomes narrower, as can be seen in the Table.
+2. The observed decay time τobs seen in the behaviour of the azimuthal angle φB shown, e.g.,
+in Figs. 4c or 4d, depends on the specific trajectory: it diminishes when the value of the
+12This value is very near of the one corresponding to the stationary wave function with same j = 5/2 and p
+equal to the momentum parameter p0 = 10−4. Indeed, Eq. (3.45) together with Table 1 yield ˆr= 22.6.
+16
+
+p0( meV
+c
+)
+Σ( meV
+c
+)
+⟨E⟩ (meV)
+∆E(meV)
+τmin(ns)
+τobs(ns)
+rL(nm)
+ˆr0(nm)
+Nloops
+j
+0.3
+32.91
+0
+757
+1/2
+10−8
+10.0000
+0.00707
+0.0465
+0.06
+164.6
+227
+39
+5/2
+0.05
+362.0
+450
+20
+11/2
+10−5
+0.05
+3258.
+3450
+3
+99/2
+0.001
+32.59
+0
+3
+1/2
+10−6
+10.0995
+0.704
+0.000468
+–
+162.9
+220
+< 1
+5/2
+–
+358.4
+435
+< 1
+11/2
+–
+3226.
+3250
+< 1
+99/2
+0.027
+3.291
+0
+676
+1/2
+10−7
+100.000
+0.0707
+0.00465
+0.006
+16.46
+22.7
+39
+5/2
+0.006
+36.20
+45.0
+20
+11/2
+10−4
+0.005
+325.8
+345
+3
+99/2
+0.0001
+3.259
+0
+2
+1/2
+10−5
+100.995
+7.04
+0.0000468
+–
+16.29
+22.0
+< 1
+5/2
+–
+35.84
+43.5
+< 1
+11/2
+–
+322.6
+325
+< 1
+99/2
+Table 2: Mean energy ⟨E⟩, standard energy deviation ∆E, decay time lower bound τmin,
+observed decay time τobs, L-radius rL, initial radial coordinate ˆr0 and number of trajectory
+loops Nloops for some values of the wave packet parameters p0, Σ and j. Trajectories concerned
+in columns 6 to 9 are the most probable ones.
+initial radial coordinate rB(r0, 0) = r0 augments. On the other hand, its value does not
+depend sensibly on the quantum number j, as can be seen in the table.
+3. Except for j = 1/2, the L-radius rL (3.40) is near of the value of the initial radial
+coordinate ˆr0 of the most probable trajectory.
+This is what can be expected for the
+nearly circular motion which takes place at times t < τobs.
+4. The number of revolutions also tends to decrease with increasing initial position r0, as
+shown in the example of Fig.
+4, which shows five trajectories corresponding to five
+different initial radial positions.
+5. The behaviours observed in these examples are generic, this being confirmed by all other
+cases we have numerically studied.
+Concluding this subsection, an important observation can be made. Although the minimum
+value for the decay-time τ was inferred from the usual quantum theoretical uncertainty principle
+for time-energy (3.47), it appears difficult to interpret τ in this framework. But it looks quite
+17
+
+natural in the dBB scheme, namely as a property of the dBB trajectories.
+May one even
+imagine an experimental way of discriminating the dBB trajectories by measuring it?
+3.3.3
+Instantaneous beables
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+t
+99.9999
+100.0000
+100.0000
+100.0000
+100.0000
+E
+(a)
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+t
+-0.4
+-0.2
+0.2
+0.4
+spin
+(b)
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+t
+200000
+400000
+600000
+800000
+1×106
+|v|
+(c)
+Figure 5: Instaneous energy (subfigure (a)), spin (subfigure (b)) and absolute velocity (subfigure
+(c)) in function of t for the solution shown in Figs. 2, 3 and 4. Colors correspond to the five
+different trajectories exhibited in 4
+Fig. 5 shows the time evolution of the instantaneous energy E(r0, t) (3.20), spin S(r0, t)
+(3.21) and absolute velocity |v|(r0, t) for the state already exhibited in the figures of the former
+subsection. The quantities are shown for five dBB trajectories, caracterized by their initial
+position parameter r0, and graphically by colours as in Fig. 4.
+Note the striking similarity between the time behaviours of both energy and spin.
+One further observes that, for a given trajectory, all three quantities tend to constant values
+above a certain threshold. E.g., for the blue one, which corresponds to the initial radial position
+r0 = 22.7, the threshold is at t ∼ 0.019 ns for the energy and spin, and at at t ∼ 0.017 ns for
+the velocity. This threshold is substantially higher than the decay time τ (∼ 0.006 ns here).
+The energy tends to its mean value ⟨E⟩ (∼ 100 meV here), the spin to its mean value 0 and the
+absolute velocity to the ”velocity of light” c = 106 ms−1. Note in particular that the energy is
+not conserved13 and that the velocity stays inferior to c before the time threshold, after which
+it goes rapidly to its asymptotic value c.
+3.3.4
+Times of flight
+The dBB theory offers a very natural way to define the time of flight of a particle which has
+followed a dBB trajectory xB(x0, t) from its initial position x0 to some target, e.g., consisting
+13This is a general feature of the dBB theory.
+Think of the obvious time dependence of the ”quantum
+potential” which defines the quantum contribution to the particle’s motion of a non-relativistic particle (see Eq.
+(3.6) of Ref. [14]).
+18
+
+of a detector. In our case, one can think of a detector occupying a circle centred at the origin
+and of radius R. This time of flight is then the solution tflight(R, r0) of the equation
+rB(r0, tflight) − R = 0,
+(3.48)
+where rB(r0, t) is the radial coordinate of the considered trajectory, characterized by its initial
+radial coordinate r0. In case the solution is not unique, one has to take the lowest one, corre-
+sponding to the first hit of the particle to the target [17, 18].However, this precaution is not
+needed in all cases we have investigated, where rB(r0, t) is a monotonically increasing function
+of t.
+The outcome of such an experiment is a probability distribution Π(τ), in terms of the time
+of flight tflight = τ, given by Eq. (9) of [17] and taking the form, in our context:
+Π(τ) = N2π
+� ∞
+0 dr0 r0 ρ(r0, 0) δ (tflight(R, r0) − τ)
+= N2πr0(R, τ) ρ(r0(R, τ), 0) |∂r0tflight(R, r0(R, τ))| ,
+(3.49)
+where r0(R, τ) is the inverse of the time of flight function tflight(R, r0), i.e., the solution (unique,
+here) of (3.48) for r0 in terms of R and tflight = τ. Recall that ρ(r0, 0) represents the probability
+distribution for the trajectory defined by its initial radial coordinate r0. N is a normalization
+factor ensuring the normaliztion condition
+� τmax
+0
+dτ Π(τ) = 1.
+(3.50)
+If the probability flux through the detector’s entry is always positive, which is the case in our
+examples, an alternative expression for the probability distribution is given by [30]
+ΠFlux(τ) = N
+�
+Σ
+ds · j(x, τ)
+=
+(here) N2πRjr(R, τ),
+(3.51)
+where j is the probability flux and Σ the detector entry’s surface. This result was proved in a
+scattering context by [30], and more generally, but in the one-dimensional case, by [31], and
+by [32] in the case of a spinless non-relativistic particle. We have checked numerically the
+equivalence of both formulae (3.49) and (3.51) in our specific situation for various parametriza-
+tions of the wave function.
+Figs. 6 and 7 show the time of flight in function of the trajectory
+parameter r0 and the corresponding probability distribution (3.49) at a circular target of radius
+30 and 500 nm, respectively.
+4
+Conclusion
+The trajectories predicted by the de Broglie-Bohm (dBB) quantum theory were calculated for
+the case of a guiding wave function being solution of the two-dimensional free Dirac equation,
+a solution constrained to be an eigenfunction of the total angular momentum operator relative
+19
+
+5
+10
+15
+20
+25
+30
+r0
+0.005
+0.010
+0.015
+0.020
+0.025
+tflight
+(a)
+0.005
+0.010
+0.015
+0.020
+0.025
+tflight
+20
+40
+60
+80
+100
+120
+Π
+(b)
+Figure 6: Times of flight tflight solutions of (3.48) and values of their probability density Π(tflight)
+(3.49) for the dBB trajectories shown in Fig. 4. The wave function parameters are the same
+as those in Figs. 2 to 5. The target is a circle centred at the origin, with radius R = 30 nm.
+The initial radial coordinate r0 varies between 2 and 30 nm.
+(a) Values of the time of flight for each dBB trajectory. The dots represent the numerically
+calculated values, and the continuous line an interpolation used for the calculation of the
+probability distribution.
+(b) Values of the corresponding probability density. Use of Eq. (3.51) has been made.
+100
+200
+300
+400
+500
+r0
+0.005
+0.010
+0.015
+tflight
+(a)
+0.005
+0.010
+0.015
+tflight
+20
+40
+60
+80
+100
+120
+Π
+(b)
+Figure 7: Same as Fig. 6, but with target’s radius R = 500 nm and initial radial coordinate r0
+in the interval 10 to 500 nm.
+to a given origin point of space. Numerical results have being provided for the case of massless
+particles with momentum-energy specifications corresponding to those of free electrons in mono-
+layer graphene.
+The trajectories corresponding to stationary wave functions turn out to be circles travelled
+at a constant speed. For Gaussian-like wave packets, the trajectories begin as quasi circles of
+20
+
+slowly increasing radius till a critical time at which they tend to straight lines approximating
+the behaviour expected for a classical free particle. This transition time decreases when the
+value of the initial radial coordinate which labels a particular trajectory increases, but appears
+to be insensible to the chosen value of the total angular momentum. It is worth noting that
+the transition time obtained in each example is of the order of magnitude of, but greater than,
+the lower bound given by the ”time-energy uncertainty principle”. Although the nature of this
+lower bound is of course purely quantum mechanical, a theory such as the dBB one appears
+necessary in order to interpret it. More, it is the use of the dBB theory which has allowed us
+to evidenciate this phenomenon.
+Given a wave function, the possible times of arrival of the particle at some region have
+also been calculated in function of its initial position for the same examples, taking profit
+of the objective reality of the trajectories in the dBB theory. The corresponding probability
+distribution of these arrival times has been calculated using the Das-D¨urr formula based on the
+dBB theory and also using the conventional quantum theory formula involving the probability
+flux. Both calculation’s results coincide, as can be expected from the equivalence’s proof given
+in [31] for the spin one-half particle in one-dimensional space and by [32] for the non-relativistic
+spinless particle. Note that this equivalence holds if the flux on any target is always positive
+– which is true in our examples. The importance of this probability distribution is that the
+latter may in principle be measured in a suitable physical context such as, e.g., the monolayer
+graphene.
+Acknowledgements
+I would like to thank Siddhant Das for the indication of interesting references and for his
+valuable comments.
+Appendices
+A
+Notations and conventions
+Units used in this paper are adapted to the physics of graphene. Length, time and energy are
+given in nm, ns and meV, respectively. The critical velocity and the Planck constant take the
+values
+c = 106 nm ns−1,
+ℏ = 6.5821 × 10−4 meV ns.
+(A.1)
+Space-time coordinate are denoted by xµ, µ = 0, 1, 2, space coordinates by x = (x, y), or (r, φ).
+Space-time metric is ηµν = diag(1, −1, −1)
+Dirac matrices are chosen in terms of the Pauli matrices as
+γ0 = σz,
+γ1 = γ0σx,
+γ2 = γ0σy.
+(A.2)
+21
+
+The Dirac matrices used in the non-relativistic formulation are
+α1 = σx,
+α2 = σy,
+β = σz.
+(A.3)
+B
+Some useful properties of the Bessel functions
+The general solution of the Bessel equation [33]
+z2f ′′(z) + zf ′(z) + (z2 − n2)f(z) = 0,
+(B.1)
+has the form
+f(z) = C1Jn(z) + C2Yn(z),
+(B.2)
+where Jn and Yn are the Bessel functions of the first [33], resp. second [33] kind, and C1, C2
+are two arbitrary complex constants. We shall restrict ourselves to an integer index n.
+The asymptotic behaviors of the Bessel functions at the origin are given by
+Jn(x) ∼ 1
+n!
+�x
+2
+�n
+(0 < x ≪ 1, n ≥ 0),
+Yn(x) ∼ −(n − 1)!
+π
+�2
+x
+�n
+(0 < x ≪ 1, n ≥ 1),
+Y0(x) ∼ 2
+π log
+�x
+2
+�
+(0 < x ≪ 1),
+(B.3)
+and at infinity by
+Jn(x) ∼
+�
+2
+πx cos
+�
+x − (n + 1
+2)π
+2
+�
+(x ≫ 1, n ≥ 0),
+Yn(x) ∼
+�
+2
+πx sin
+�
+x − (n + 1
+2)π
+2
+�
+(x ≫ 1, n ≥ 0).
+(B.4)
+Functions with a negative index are related to those with a positive one by the identities
+J−n(z) = (−1)nJn(z),
+Y−n(z) = (−1)nYn(z).
+(B.5)
+Under parity z → −z, the function Jn transforms as
+Jn(−z) = (−1)nJn(z).
+(B.6)
+An interesting orthogonality property is given by [33]
+� R
+0
+dr r Jn
+�zn,α r
+R
+�
+Jn
+�zn,β r
+R
+�
+= R2
+2 (Jn+1(zn,α)2 δαβ,
+(B.7)
+for n ≥ 0, where zn,α is the αth positive zero of the Bessel function Jn(z) [34]. Moreover, any
+function f(r) defined in the interval 0 ≤ r ≤ R with bounded variation and vanishing at the
+end point r = R can be represented as a “Fourier Bessel series” [35] as
+f(r) =
+∞
+�
+α=1
+cαJn
+�zn,α r
+R
+�
+,
+(B.8)
+for any n ≥ 0. The coefficients cα can be calculated using the orthogonality formula (B.7).
+22
+
+References
+[1] P.R. Holland, “The Dirac equation in the de Broglie-Bohm theory of motion”, Found.
+Phys. 22 (1992) 1287.
+[2] Peter R. Holland, “The quantum theory of motion”, Revised ed., Cambridge University
+Press (1995).
+[3] Max Planck, “Ueber das Gesetz der Energieverteilung im Normalspectrum” (English
+translation), Annalen der Physik 4 (1901) 553.
+[4] Niels Bohr, “On the Constitution of Atoms and Molecules”, Philos. Mag. 26 (1913) 1 and
+476.
+[5] Albert Einstein, “Concerning an Heuristic Point of View Toward the Emission and Trans-
+formation of Light”, Annalen der Physik 17 (1905) 132.
+[6] Louis de Broglie, “Recherches sur la th´eorie des quanta”, Thesis (Paris), 1924;
+Louis de Broglie, Ann. Phys. (Paris) 3, 22 (1925). Reprint in Ann. Found. Louis de
+Broglie 17 (1992) p. 22;
+Louis De Broglie, “La m´ecanique ondulatoire et la structure atomique de la mati`ere et du
+rayonnement”, J. Phys. Radium 8 (1927) 225, DOI 10.1051/jphysrad:0192700805022500.
+[7] Erwin Schr¨odinger, “Quantisierung als Eigenwertproblem”, Annalen der Physik 79 (1926),
+361, Annalen der Physik 79 (1926) 489, Annalen der Physik 80 (1926) 437, Annalen der
+Physik 81 (1926) 109.
+[8] Werner Heisenberg, ҬUber quantentheoretische Umdeutung kinematischer und mechanis-
+cher Beziehungen”, Z. Phys. 33 (1925) 879.
+[9] Paul A.M. Dirac, “The quantum theory of the electron”, Proc. R. Soc. A 117 (1928) 610
+and 118 (1928) 351.
+[10] Niels Bohr, “The Quantum Postulate and the Recent Development of Atomic Theory”,
+Supplement to ”Nature April 14 (1928) 580;
+Werner Heisenberg, “Physics and Philosophy”, Harper, New York (1958),
+[11] Hugh Everett, “Relative State Formulation of Quantum Mechanics”,
+Rev. Mod. Phys. 29 (1957) 454.
+[12] Carlo Rovelli, “Relational quantum mechanics”,
+Int. J. Theor. Phys. 35 (1996) 1637 e-Print: quant-ph/9609002 [quant-ph];
+Andrea Di Biagio and Carlo Rovelli, “Stable Facts, Relative Facts”,
+Found. Phys. (2021) 51:30.
+[13] David Bohm, “A Suggested interpretation of the quantum theory in terms of hidden
+variables 1, 2.”, Phys. Rev. 85 (1952) 166, 180.
+23
+
+[14] D. Bohm and B.J. Hiley, ”The Undivided Universe”, Routledge, London and New York
+(1995).
+[15] John S. Bell, “Speakable and Unspeakable in Quantum Mechanics”, Cambridge University
+Press, New York (2010).
+[16] John S. Bell, “Beables for quantum field theory”, preprint CERN TH-4035/84 (1984),
+reprinted in Cap. 19 of [15].
+[17] Siddhant Das and Detlef D¨urr, ”Arrival time distributions of spin-1/2 particles, Nature
+Scient. Rep. 9 (2019) 2242.
+[18] Siddhant Das, Markus N¨oth and Detlef D¨urr, “Exotic arrival times of spin-1/2 particles
+I - An analytical treatment”, Phys. Rev. A99 (2019) 052124.
+[19] Detlef D¨urr, Sheldon Goldstein, Travis Norsen, Ward Struyve and Nino Zangh`ı, “Can
+Bohmian mechanics be made relativistic?” Proc. R. Soc. A 470 (2013) 20130699
+[20] Roderich Tumulka, “On Bohmian Mechanics, Particle Creation, and Relativistic Space-
+Time: Happy 100th Birthday, David Bohm!”, Entropy 20 (2018) 462.
+[21] Dario Bressanini and Alessandro Ponti, “Angular Momentum and the Two-Dimensional
+Free Particle´´, J. Chem. Educ. 75 (1998) 916.
+[22] Wolfram Research, Inc., Mathematica, Champaign, IL.
+[23] Abraham Pais, “On Spinors in n Dimensions”, J. Math. Phys. 3 (1962) 1135; doi:
+10.1063/1.1703856.
+[24] D. D¨urr D., S. Goldstein and N. Zanghi, “Quantum equilibrium and the origin of absolute
+uncertainty”, J. Stat. Phys. 67 (1992) 843.
+[25] The Wolfram Functions Site,
+https://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/21/02/02/
+[26] WolframMathWorld,
+https://functions.wolfram.com/GammaBetaErf/Erf/
+[27] Mikhail I. Katsnelson, “The Physics of Graphene” 2nd Edition, Cambridge University
+Press, Cambridge (2020).
+[28] S. Das Sarma, Shaffique Adam, E. H. Hwang and Enrico Rossi, “Electronic transport
+in two-dimensional graphene”, Rev. Mod. Phys. 83 (2011) 407, e-Print: arXiv:1003.4731
+(cond-mat.mtrl-sci).
+[29] Albert Messiah, “Quantum Mechanics”, Vol. 1, Section VIII-13, Dover Publications, New
+York (2014) (English translation of “M´ecanique Quantique”, Dunod, Paris, (1962)).
+[30] M. Daumer, D. D¨urr, S. Goldstein and N. Zanghi, “On the Quantum Probability Flux
+Through Surfaces”, J. Stat. Phys. 88 (1997) 967.
+24
+
+[31] C. Richard Leavens, “Bohm Trajectory Approach to Timing Electrons”, p. 129 of “Time
+in Quantum Mechanics - Vol.1”, 2d Ed., J.G. Muga, R. Sala Mayato, ´I.L. Egusquiza
+(Eds.), Lecture Notes in Physics 734, Springer, Heidelberg, 2008.
+[32] Siddhant Das and Markus N¨oth, Times of arrival and gauge invariance, Proc. R. Soc. A
+477 (2021) 20210101, https://doi.org/10.1098/rspa.2021.0101.
+[33] WolframMathWorld, “Bessel function”,
+https://mathworld.wolfram.com/topics/BesselFunctions.html.
+[34] WolframMathWorld,
+https://mathworld.wolfram.com/BesselFunctionZeros.html
+[35] Eric W. Weisstein, “Fourier-Bessel Series.” From MathWorld–A Wolfram Web Resource.
+https://mathworld.wolfram.com/Fourier-BesselSeries.html
+25
+
diff --git a/8dFQT4oBgHgl3EQfIDXU/content/tmp_files/load_file.txt b/8dFQT4oBgHgl3EQfIDXU/content/tmp_files/load_file.txt
new file mode 100644
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--- /dev/null
+++ b/8dFQT4oBgHgl3EQfIDXU/content/tmp_files/load_file.txt
@@ -0,0 +1,823 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf,len=822
+page_content='Bohm - de Broglie Cycles  To my beloved Izabel  Olivier Piguet∗  January 30, 2023  Abstract  In the de Broglie-Bohm quantum theory, particles describe trajectories determined  by the flux associated with their wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' These trajectories are studied here for  relativistic spin-one-half particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Based in explicit numerical calculations for the case of  a massless particle in dimension three space-time, it is shown that if the wave function  is an eigenfunction of the total angular momentum, the trajectories begin as circles of  slowly increasing radius until a transition time at which they tend to follow straight lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Arrival times at some detector, as well as their probability distribution are calculated,  too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The chosen energy and momentum parameters are of the orders of magnitude met  in graphene’s physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Keywords: Bohm - de Broglie, Quantum Mechanics, Transport properties, Graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  1  Introduction  Since its beginning in the first decades of XXth century [3]-[9] Quantum Mechanics and its  extension in the form of Quantum Field Theory led to an accurate description of atomic and  subatomic phenomena, confirmed in an extraordinarily precise way by countless experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  However, there is no such a broad consensus about its interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Various ones are present  in the literature, such as the Copenhagen [10], the Many-World [11], the Relational [12] or the  de Broglie-Bohm (dBB) one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' We will deal here with the latter interpretation, first proposed by  Louis de Broglie [6] as the ”Pilot Wave Theory”, later on formulated by David Bohm [13, 14]  as the ”Ontological Interpretation of Quantum Theory” and finally, critically defended by John  ∗Pra¸ca Graccho Cardoso, 76/504, 45015-180 Aracaju, SE, Brazil, E-mail: opiguet@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='com  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='13251v1  [quant-ph]  30 Jan 2023  Bell in a series of papers reproduced in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' This interpretation of Quantum Mechanics differs  essentially from the largely more widespread Copenhagen interpretation by taking particle  trajectories as elements of reality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=', a particle really follows a trajectory, the latter being  defined by “guying conditions” first proposed by de Broglie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' A probabilistic interpretation is  maintained, but now in the sense of classical statistical mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The trajectory followed by  a particle is fully defined by giving boundary conditions such as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=', the coordinates of its  initial position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The probability distribution of the particle following a particular trajectory  is then given by the value of the squared modulus of the wave function taken at the initial  time and position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Since this statistical distribution is equal to the “usual” (Copenhagen)  quantum probability distribution and in particular satisfies the same conservation conditions,  mean values of beables1 evolve identically in both interpretations of Quantum Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' On  the other side, the possibility of observable consequences of the existence of trajectories remains  an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Let us however mention an experimental proposal [17, 18] on the measure  of the times of arrival at a detector for a particle prepared in some initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  The first aim of the present paper is to introduce the reader to the dBB theory by treat-  ing simple physical examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Since the main peculiarity of this interpretation is the factual  existence of trajectories, some effort will be made in the study of their trajectories, looking for  situations in which dBB could make a difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  The dBB trajectories we will calculate are those of a relativistic2, massive or massless, spin  one half particle in dimension 3 space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Its quantum state will be supposed to be described  by an eigenfunction of the total angular momentum J defined relatively to some space point3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Stationary as well as packets of stationary wave functions will be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The main result  for the latter is that the dBB trajectories consist of an initial phase of quasi circles whose  radius increases till a critical time at which the trajectory begins tending to a straight line –  the straight line expected for a classical particle with definite angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  The paper begins in Section 2 with an introduction to the dBB formalism and in Section  3 for its application to the relativistic spin one half particle described by the Dirac equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Numerical computations of dBB trajectories and arrival times for electrons in the context of  graphene’s transport properties are presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Final considerations are given in  the Conclusion Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Most analytic and numerical calculations are made with the help of the software Mathe-  matica [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  1Following John Bell [16], I use here the term ”beable“ , instead of the usual term “observable” which is  somewhat related to the Copenhagen interpretation and to its postulate of the “wave function reduction”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  2Only the theory of a single particle is considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' See [2, 19, 20] for a discussion of the N-particle  relativistic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  3See [21] for the calculation of such states in the case of the non-relativistic free particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  2  2  Summary of the de Broglie-Bohm theory  In the ”usual” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=', Copenhagen) interpretation of Quantum Mechanics [10], the state of a  physical system constituted of a single particle is described, in the Schr¨odinger picture, by a  wave equation  iℏ∂ψ(x, t)  ∂t  = ˆHψ(x, t),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='1)  where ˆH is a self-adjoint partial derivative operator acting on a N-components wave function  ψ(x, t) =  �  �  ψ1(x, t)  · ·  ψN(x, t)  �  � ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='2)  belonging to a Hilbert space, with scalar product and norm defined by4  ⟨ψ|Φ⟩ =  �  Rdddx ψ†(x, t)φ(x, t) =  �  Rdddx  N  �  α=1  ψ∗  αφα(x, t),  ||ψ|| = ⟨ψ|ψ⟩  1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='3)  x = (xi, i = 1, · · · , d) are the space coordinates, d the space dimension and †, ∗ denote the  hermitian and complex conjugate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The number of components, N, depends on the  particle’s spin and on the space dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=', N = 1 for a scalar (spin 0) particle, N = 2  for a non-relativistic particle of spin 1/2 in any dimension, N = 2 for a relativistic particle of  spin 1/2 in 2 dimensions, N = 4 for the same in 3 dimensions, etc [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  The wave equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='1) implies the existence of a non-negative density  ρ(x, t) = ψ†(x, t)ψ(x, t),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='4)  and an associate d-current density5 j(x, t) obeying a continuity equation  ∂tρ + ∇j = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='5)  which assures the constancy of the norm defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Normalizing the norm to 1, one  interprets ρ as a probability density and j as a probability flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  In the Copenhagen theory, the state of the system is completely characterized by the wave  function ψ, solution of the wave equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='1), with an arbitrarily given initial wave function  ψ(x, 0) = ψ0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The de Broglie-Bohm (dBB) theory completes the characterization of the  state of the system by postulating the existence of a trajectory of the system (the particle,  here), determined by the de Broglie “guidance conditions” [6] for the particle’s velocity  v(x, t) = j(x, t)/ρ(x, t),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='6)  4This holds e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=', for a non-relativistic particle in general, or a spin 1/2 relativistic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' We do not consider  here cases such as the relativistic spin zero particle described by the Klein-Gordon equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  5Its explicit form will be given below for the cases studied there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  3  the dBB trajectory xB(x0, t) being then a solution of the set of differential of equations  ∂xB(x0, t)  ∂t  = v(xB(x0, t), t),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='7)  with an arbitrarily given initial position xB(x0, 0) = x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' In other word, the possible trajectories  are the integral lines of the dBB vector field v(x, t), labelled by their initial position x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Since  the flux j and the density ρ turn out to be bilinear in ψ and ψ† (see later on), the dBB  trajectories do not depend on the “intensity” of the wave function, but only on its “form”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Recall that, in the quantum probabilistic interpretation of Copenhagen, the density ρ repre-  sents the probability of experimentally finding the particle at a given place at a given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' On  the other hand, the dBB theory treats this density in a more “classical statistical mechanics”  way: Trajectories “really happen”, and their probability distribution, which amounts to the  probability distribution of their initial positions x0, is given [24] by the density ρ(x0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The  latter represents our lack of knowledge of the precise initial position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Thanks to the continuity  equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='5), the probability for the particle being inside a co-moving volume6 V (t) is con-  stant in time, which sustains this statistical interpretation, called “thermal equilibrium” in the  literature [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Such a formulation is called a theory with “hidden variables”, the hidden variables of the  present one being, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=', the components of the initial position x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  A heuristic justification for the guiding condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='6) may be found in analogy with fluid  mechanics, considering ρ and v as the fluid mass density and velocity field, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' With  j = ρv, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='5) has the form of the continuity equation expressing the conservation of the  fluid mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Now, granted the existence of a trajectory, one can define properties of the particle along  it, such as energy, spin, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' , in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' First, given an “observable” A represented  by a self-adjoint operator ˆA, one defines the associated beable field (See footnote 1)  A(x, t) = ψ†(x, t) ˆA ψ(x, t)/ρ(x, t),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='8)  where ρ is the probability density (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' One then defines the instantaneous value of A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=', its  value when the particle is at point xB(x0, t) at time t:  AB(x0, t) = A(xB(x0, t), t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='9)  Note that these definitions are independent of the normalization of the wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Their  justification is obvious in the case of ˆA being the multiplication by a function fA(x, t), since  in this case A(x, t) = fA(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' For the general case, including that of a matricidal and/or  differential operator, one derives from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='8) the identity  �  ddx ρ(x, t)A(x, t) =  �  ddx ψ†(x, t) ˆA ψ(x, t) = ⟨A⟩ (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='10)  6I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=', a volume whose boundary points move along the dBB trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  4  The first integral is an expectation value in the dBB theory sense, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=', in the “classical statisti-  cal mechanical” sense, whereas the second one gives it in the conventional quantum mechanical  sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Their equality indeed shows the equivalence of both theories for what concerns expecta-  tion values of observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  To summarize, the dBB theory proposes the existence of:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' A wave function or ”guidance field” (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='2), solution of the Schr¨odinger-like wave equation  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' A statistical ensemble of particle’s trajectories xB(x0, t) as the integral lines of the dBB  vector field (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='6), solutions of the differential equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='7) parametrized by the value  of the initial position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Beable fields and beable instantaneous values, defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Such beables  may be energy, momentum, angular momentum, spin, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  All considerations made in this subsection generalize easily to systems of N particles, x  denoting a point of the d × N dimensional configuration space7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  3  Free relativistic spin one half particle in 3-dimensional  space-time  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='1  Dirac equation  A free, spin 1/2 relativistic particle of mass m in 3-dimensional space-time is described in the  usual theory by a 2-components spinor wave function  ψ(x) =  � ψ1(x)  ψ2(x)  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='1)  solution of the free Dirac equation  iℏcγµ∂µψ(x) − mc2ψ(x) = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='2)  where x = (xµ, µ = 0, 1, 2) are the space-time coordinates space-time, whose metric is ηµν =  diag(1, −1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The Dirac matrices obey the anticommutation rules {γµ, γν} = 2ηµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Our  choice for them is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' c, ℏ are the speed of light and the reduced Planck  constant, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The Dirac equation may be cast in the form8 of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='1):  iℏ∂ψ(x, t)  ∂t  = ˆHψ(x, t),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='3)  7At least in the non-relativistic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' See footnote 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  8In fact Dirac’s original form, reduced to 3 space-time dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  5  with  ˆH = −iℏc αi∂i + mc2 β,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='4)  with αi = γ0γi and β = γ0 (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The density and the flux obeying the continuity  equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='5) are given by  ρ = ψ†ψ,  ji = c ψ†αiψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='5)  The theory is relativistic: cρ and ji are the time and space components of the space-time  3-vector jµ = ¯ψγµψ, and the continuity equation reads ∂µjµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Another relativistic object is the scalar density  σ(x, t) = 1  2  ¯ψψ = 1  2ψ†βψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='6)  jµ and σ fulfil the identity  jµjµ = 4σ2,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='7)  consequence of the Pauli matrices identity  3  �  i=1  σi  αβσi  γδ = 2δαδδβγ − δαβδγδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='8)  Let us go now to the dBB theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The identity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='7) allows one to define the time-like 3-velocity  field’  uµ = jµ  2|σ|,  uµuµ = 1,  u0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='9)  The relativistic form of the dBB guidance equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='7) then reads9  dxµ(λ)  dλ  = uµ(x(λ)),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='10)  with λ as curve’s parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' This equation is equivalent to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='7) and defines the space-time  trajectories of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  In the same way as one defines the dBB velocity field (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='6) or (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='9), one can define a dBB  spin field  S(x, t) = ℏ σ(x, t)/ρ(x, t),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='11)  which takes values between −ℏ/2 and ℏ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The beable S(x, t) can thus be interpreted as the  field associated to the spin operator  ˆS = ℏ  2σz,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='12)  according to the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Note that, in the instantaneous rest frame of the particle  guided by the wave, where vi = ji = 0, the identity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='7) implies S = ±ℏ/2, in agreement with  the interpretation of S as the intrinsic angular momentum of a spin one half particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' One then  gets the instantaneous spin of the particle according to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='9):  SB(x0, t) = S(xB(x0, t), t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='13)  9The present discussion is the reduction to 3-dimensional space-time of the one made by [1, 2] in 4 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='2  Eigenstates of the angular momentum and of the energy  We will look for the solutions of the Dirac equation which are eigenfunctions of the total angular  momentum and Hamiltonian operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' It will be useful to work in polar coordinates r, φ:  x = r cos φ,  y = r sin φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  In these coordinates, the Hamiltonian operator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='4) reads  ˆH = −iℏc  ��  α1 cos φ + α2 sin φ  �  ∂r + 1  r  �  α2 cos φ − α1 sin φ  �  ∂φ  �  + mc2β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='14)  One easily checks, using the algebra of the Pauli matrices, that the total angular momentum  operator with respect to the origin, which has a single component in the two-dimensional space’s  case,  ˆJ = −iℏ∂φ + ℏ  2β,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='15)  commutes with the Hamiltonian operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Spinor eigenfunctions of ˆJ with eigenvalue j are  readily found to be of the form  ψ(r, φ, t) =  � ei(j− 1  2 )φf1(r, t),  ei(j+ 1  2 )φf2(r, t)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='16)  One directly sees that the uniformity of the wave function requires j to be half-integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' With  the result (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='16), solving the Dirac equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='2) amounts to solving the two radial equations  for the functions f1(r, t) and f2(r, t):  iℏ  �  r (∂tf1 + c ∂rf2) + c(j + 1  2) f2  �  − mc2r f1 = 0,  iℏ  �  r (∂tf2 + c ∂rf1) − c(j − 1  2) f1  �  + mc2r f2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='17)  In terms of the radial functions fα, the probability density ρ, the flux j and the scalar density  σ defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='5), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='6) read:  ρ(r, t) = |f1(r, t)|2 + |f2(r, t)|2,  jx(r, φ, t) = c  �  eiφf ∗  1(r, t)f2(r, t) + e−iφf ∗  2(r, t)f1(r, t)  �  ,  jy(r, φ, t) = ic  �  −eiφf ∗  1f2(r, t) + e−iφf ∗  2(r, t)f1(r, t)  �  ,  σ(r, t) = 1  2 (|f1(r, t)|2 − |f2(r, t)|2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='18)  ρ and σ, as well as the radial and azimuthal components of the flux j,  jr(r, t) = cos φ jx(r, φ, t) + sin φ jy(r, φ, t) = c (f ∗  1(r, t)f2(r, t) + f ∗  2(r, t)f1(r, t)) ,  jφ(r, t) = − sin φ jx(r, φ, t) + cos φ jy(r, φ, t) = ic (−f ∗  1(r, t)f2(r, t) + f ∗  2(r, t)f1(r, t)) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='19)  turn out to be independent of the angular coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='1  Instantaneous values of energy, spin and orbital angular momentum  Before going to our main task, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=', the concrete study of the dBB trajectories, let us look at  the expressions of the instant values of energy, spin and orbital angular momentum, namely  EB(x0, t), SB(x0, t) and LB(x0, t) along a dBB trajectory xB(x0, t) fixed by the initial position  x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' They are generated by the corresponding fields E(x, t), S(x, t) and L(x, t) according to  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' One obtains  E(x, t) = ψ†(x, t) ˆHψ(x, t)/ρ(x, t) = iℏ(f ∗  1(r, t)∂tf1(r, t) + f ∗  2(r, t)∂tf2(r, t))/ρ(r, t),  EB(x0, , t) = E(xB(x0, t), t),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='20)  where f1, f2 are the radial components defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='16)  and ˆH is the Hamilton operator, the validity of the Schr¨odinger-like equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='1) being  assumed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  S(x, t) = ψ†(x, t) ˆSψ(x, t)/ρ(x, t) = 1  2(f ∗  1(r, t)f1(r, t) − f ∗  2(r, t)f2(r, t))/ρ(r, t),  SB(x0, t) = S(xB(x0, t), t),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='21)  where ˆS is the spin operator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  L(x, t) = ψ†(x, t)ˆLψ(x, t)/ρ(x, t) = ℏ(j − S(x, t)),  LB(x0, t) = L(xB(x0, t), t) = ℏ(j − S(x0, t)),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='22)  where ˆL is the orbital angular momentum operator (in Cartesian and polar coordinates)  ˆL = −i (x∂y − y∂x) = −iℏ ∂φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='23)  Use has been made in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='22) of the fact that ˆL = ˆJ − ˆS and of the wave function being an  eigenfunction of the total angular momentum ˆJ with eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='2  Stationary solutions:  Since the Hamiltonian (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='14) and the total angular momentum (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='15) commute, we can impose  the stationarity condition  ˆHψ = ℏωp ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='24)  Thus the radial wave functions fα(r, t) take the form  f p, stat  α  (r, t) = e−iωpt hp, stat  α  (r),  α = 1, 2,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='25)  ℏωp being the energy of the stationary state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' This leads to a pair of equations for the function  hp, stat  α  (r), derived from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='17) by substituting i∂t by ωp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The general solution of these equa-  tions is a superposition of the Bessel functions of the first and second kind, Jj±1/2(pr/ℏ) and  Yj±1/2(pr/ℏ), with p a function of ωp defined as the positive solution for p of  ωp = 1  ℏ  �  m2c4 + p2c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='26)  8  Square integrability of the wave function at r = 0 leads us to discard the solutions involving  Yj±1/2 because of the latter’s singularity at the origin (see (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='3) in Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The regular  solution thus is  hp, stat  1  (r) = icp Jj−1/2(pr/ℏ),  hp, stat  2  (r) = −(ℏωp − mc2)Jj+1/2(pr/ℏ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='27)  The general stationary solution of the Dirac equation for angular momentum eigenstates, non-  singular at the origin, then reads  ψp, stat(r, φ, t) =  �  � ei(j− 1  2 )φf p, stat  1  (r, t),  ei(j+ 1  2 )φf p, stat  2  (r, t)  �  �  = e−iωpt  �  icp ei(j−1/2)φJj−1/2(pr/ℏ),  −(ℏωp − mc2)ei(j+1/2)φJj+1/2(pr/ℏ)  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='28)  with ωp given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' These spinors form a basis for the solutions of the Dirac equation,  however an improper one since they are not square integrable due to the asymptotic behaviour  of the Bessel functions shown in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='4) of Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  We can nevertheless apply the dBB guidance principle, expressed in Eqs, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='7),  to such a basis element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' From the result (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='28) we can compute explicitly the density ρ given  in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='18):  ρp, stat(r) = c2p2J2  j−1/2(pr/ℏ) + (ℏωp − mc2)2J2  j+1/2(pr/ℏ),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='29)  as well as the flux components (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='19) which, divided through ρ according to the dBB condition,  yields the radial and azimuthal components of the velocity vector field:  vp, stat  r  (r) = 0  vp, stat  φ  (r) = 2c  cp(ℏωp − mc2)Jj−1/2(pr/ℏ)Jj+1/2(pr/ℏ)  (cp)2J2  j−1/2(pr/ℏ) + (ℏωp − mc2)2J2  j+1/2(pr/ℏ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='30)  Obviously all these expressions are time independent due to the stationarity condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' One sees  that the radial component of the velocity field is vanishing and that its azimuthal component  does not depend on the polar angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Thus the dBB trajectories of the particle, defined as the  integral curves of the velocity vector field, are circles of radius r centred at the origin travelled  at a constant velocity vφ and whose radius dependent value is bounded by c, in the massive  as well as in the massless case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' In all cases the bound c is effectively reached, for a discrete  set of values of the radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 1 in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='3 shows a typical behaviour of the azimuthal  velocity vp, stat  φ  (r) in the massless particle’s case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  One may observe that, in the massless case, in which ℏωp = pc, changing the sign of the  total angular momentum implies a change of sign of the velocity field: vp, stat  φ  (r) → −vp, stat  φ  (r),  hence of the orientation of the trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' However this does not hold for the massive case,  and also not for the more general wave packets examined in Subsection (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='3  Gaussian wave packet  The general solution of the Dirac equation for angular momentum eigenstates, non-singular at  the origin, reads  ψ(x, t) =  � ∞  0  dp a(p)ψp, stat(x, t),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='31)  where ψp, stat is the stationary solution (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='28) of energy10 ℏωp given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='26), and the amplitude  a(p) is an arbitrary complex function, but constrained by the requirement of ψ(x, t) to be a  square integrable function of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  The angular momentum eigenstates we will consider in the following are described by the  spinor wave functions of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='31), with a(p) the Gaussian amplitude  a(p) = √p e  −(p − p0)2  2Σ2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='32)  Here and in the rest of this paper we consider only positive energy solutions of the Dirac  equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='4  Mean values  We recall that mean values of observables obtained from the dBB or from the Copenhagen  theory coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  The wave functions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='31) are normalizable and we can calculate the square norm ||ψ||2 in  the following way:  ∥ψ∥2 =  � 2π  0  dφ  � ∞  0  dr r ρ(r, t),  with ρ(r, t) the density (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='31) we get  ∥ψ∥2 = 2π  � ∞  0  dr r  � ∞  0  dp  � ∞  0  dp′ a(p)a(p′)  2  �  α=1  f p, stat  α  (r, t)∗f p′, stat  α  (r, t)  = 2π  � ∞  0  dp  � ∞  0  dp′ a(p)a(p′)ei(ωp−ωp′)  � ∞  0  dr r  �  (mc2 + ℏωp)(mc2 + ℏωp′)Jj−1/2(pr/ℏ)Jj−1/2(pr′/ℏ) + ℏ2pp′Jj+1/2(pr/ℏ)Jj+1/2(pr′/ℏ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='32) and the completeness identity [25] for the Bessel functions:  � ∞  0  dr r Jl(kr)Jl(k′r) = 1  kδ(k − k′),  one gets  ∥ψ∥2 = 2πℏ2  � ∞  0  dp 1  pa2(p)  �  c2p2 + (ℏωp − mc2)2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='33)  10We restrict to positive energy contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  10  In the same way one establishes expressions for the mean energy:  ⟨E⟩ =  � 2π  0  dφ  � ∞  0  dr r ψ†(r, φ, t) ˆH ψ(r, φ, t) / ∥ψ∥2,  where ˆH is the Hamiltonian operator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='4), and for the standard energy deviation  ∆E =  �  ⟨E2⟩ − ⟨E⟩2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  The result is  ⟨E⟩ = 2πℏ2  ∥ψ∥2  � ∞  0  dp 1  pa2(p)  �  c2p2 + (ℏωp − mc2)2�  ℏωp,  ∆E =  �2πℏ2  ∥ψ∥2  � ∞  0  dp 1  pa2(p)  �  c2p2 + (ℏωp − mc2)2�  (ℏωp)2 − ⟨E⟩2  � 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='34)  Finally, a computation of the mean value of the spin operator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='12) yields  ⟨S⟩ = πℏ3  ∥ψ∥2  � ∞  0  dp 1  pa2(p)  �  −c2p2 + (ℏωp − mc2)2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='35)  Substituting ∥ψ∥2 in the denominator by its expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='33), one sees that this mean value  obeys the bounds −ℏ/2 < ⟨S⟩ < ℏ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  All these integrals can be computed analytically in the massless case m = 0 for the amplitude  given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' One gets  ∥ψ∥2 = πℏ2c2Σ3 �√π(1 + 2z2)(1 + erf(z)) + 2ze−z2�  ,  ⟨E⟩ = cp0  √πz(3 + 2z2) (1 + erf(z)) + 2(1 + z2)e−z2  z(√π(1 + 2z2)(1 + erf(z)) + 2ze−z2)  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='36)  ∆E = cp0  �  π(3 + 4z4)(1 + erf(z))2 + 8√πz(−1 + z2)(1 + erf(z))e−z2 + 4(−2 + z2)e−2z2  2z2(π(1 + 2z2)2(1 + erf(z))2 + 4√πz(1 + 2z2)(1 + erf(z))e−z2 + 4z2e−2z2)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='37)  where we have set  z = p0  Σ ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='38)  and erf(z) is the error function [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Finally, the mean spin is null in this massless case:  ⟨S⟩ = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='39)  11  which implies ⟨L⟩ = ℏj for the orbital angular momentum since the states considered are  eigenstates of the total angular momentum with eigenvalue ℏj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Associated to this mean orbital  angular momentum, we can define an L-radius  rL = ⟨L⟩  ⟨p⟩ = c ℏ j  ⟨E⟩ ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='40)  where ⟨p⟩ is the mean value of the momentum p, equal to ⟨E⟩ /c in the present massless  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' This definition mimics the classical relation between angular momentum and radius for  a uniform circular motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  As one may expect, in the case of a very narrow width of the amplitude (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='32), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=', z ≫ 1,  the energy and the energy uncertainty approximate to the values  ⟨E⟩ ≃ cp0,  ∆E ≃  c  √  2Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='41)  The results (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='34) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='35) allow us to identify the expression  ˜ρ(p) = a2(p)c2p2 + (ℏωp − mc2)2  p2  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='42)  up to a due normalization, as the probability density in p-space, conjugate to the x-space  probability defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='3  j - electrons in graphene  Let us denote free electrons in eigenstates of the total angular momentum with eigenvalue j as  “j-electrons”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Free electrons in monolayer graphene [27] with energy less than Ecrit ≈ 160 meV  obey approximatively a relativistic-like massless dispersion law E ≈ p c, where p is the linear  momentum and the “velocity of light”11 c ≈ 106 m s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The dynamics of these electron is that  of free massless particles in dimension three space-time with pseudo-spin 1/2 obeying the Dirac  equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='2) with m = 0 [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Pseudo-spin is a chirality effect due to the peculiar crystalline  structure of graphene and should not be confused with the usual spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Nevertheless, since the  wave function obeys the Dirac equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='2), the pseudo-spin adds itself to the orbital angular  momentum yielding the conserved total angular momentum j (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='15) as explained in Subsection  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  We will consider first stationary states and then states defined by Gaussian-like wave pack-  ets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Conventions and units used in the following are described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='1  Stationary states in graphene  As already shown in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='2, the dBB trajectories associated to the stationary wave  functions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='28) are circles centred at the origin, travelled at a constant dBB velocity vp, stat  φ  11This is the so-called critical velocity, which we will denote by c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  12  20  40  60  80  100  r  1000  2000  3000  ρp,stat (r)  20  40  60  80  100  r  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='0  vϕ  p,stat (r)/c  Figure 1: Stationary case: density ρp, stat(r) and azimuthal velocity vp, stat  φ  (r)/c for j = 5/2,  p = 10−4 meV/c (E = 100 meV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  j  1/2  3/2  5/2  7/2  9/2  11/2  13/2  15/2  17/2  αj  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='19  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='45  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='61  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='74  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='84  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='93  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='01  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='08  Table 1: Coefficients αj of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='45) in function of the angular momentum j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  which depends on the radius r according to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' For m = 0, this velocity reads  vp, stat  φ  (r) = 2c Jj−1/2(pr/ℏ)Jj+1/2(pr/ℏ)  J2  j−1/2(pr/ℏ) + J2  j+1/2(pr/ℏ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='43)  The dBB velocity value oscillates between −c and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' A typical behaviour, in function of the  radius, of the probability density and of the dBB velocity, which depend on the energy E = c p  and on the angular momentum j, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 1 for j = 5/2 and E = 100 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The most  probable radius ˆrj, is given by the position of the first maximum of the probability density  ρp, stat (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='29) (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 1), which for m = 0 reads  ρp, stat(r) = c2p2 �  J2  j−1/2(pr/ℏ) + J2  j+1/2(pr/ℏ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='44)  Since r appears in the combination pr/ℏ, the most probable radius ˆrj may be written as a  function of p:  ˆrj(p) = αj  ℏ  p,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='45)  the j-dependent coefficients αj being shown in Table 1 for some values of j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' These results can  be taken as an approximation for the more realistic case of a wave packet with a very small  energy width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='010  t  40  60  80  100  120  rB  (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='010  t  250  200  150  100  50  ϕB  (b)  50  50  xB  50  50  100  yB  (c)  Figure 2: The most probable dBB tractory xB(ˆr0, t) for Gaussian wave packet parameters  j = 5/2, p0 = 10−4 meV/c (Peak energy E0 = 100 meV), and Σ = 10−7 meV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  This  trajectory is fixed by the initial condition parameter = ˆr0 = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' ˆr0 is the position of the peak  of the probability density ρ at t = 0 (the blue point in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=')  (a) Radial coordinate rB(ˆr0, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (b) Azimuthal coordinate φB(ˆr0t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (c) dBB trajectory in the (x, y) plane for 0 ≤ t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='010 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The trajectory performs 38 loops  before the critical time (decay time) τ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='006 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='014  t  500  1000  1500  rB  (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='014  t  250  200  150  100  50  ϕB  (b)  500  1000  1500  xB  100  100  200  300  400  500  yB  (c)  Figure 3: dBB tractories for Gaussian wave packet: same parametrization as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 2, but  with the larger time scale 0 ≤ t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='015 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='2  Gaussian wave packets in graphene  We turn now to the Gaussian wave functions defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='31) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='32), which are normal-  izable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' We shall denote by  xB(r0, t) = (rB(r0, t), φB(r0, t)) ,  20  40  60  80  100  r0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='×10-7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='×10-6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='5×10-6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='×10-6  ρ(r0,0)  (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='012  t  200  400  600  800  rB  (b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='012  t  600  500  400  300  200  100  ϕB  (c)  200  400  600  800  xB  300  200  100  100  yB  (d)  Figure 4: dBB tractories for Gaussian wave packet: same parametrization as in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 2 and  3, but with five trajectories corresponding to the initial radial positions r0 = 10, 17, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='7,  30 and 35 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The respective numbers of trajectory’s closed loops are 93, 66, 38, 10 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Their respective relative probabilities are proportional to the heights of the coloured dots in  the subfigure (a) showing the initial probability density ρ(r0, 0) in function of the initial radial  position r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  15  the (polar) coordinates of the dBB trajectory solution of the trajectory equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='7), fixed  by the initial position  xB(r0, 0) = (rB(r0, 0), φB(r0, 0) = (r0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='46)  We have made explicit, in our notation, the dependence on the trajectory parameter r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 2 and 3 show the most probable dBB trajectory xB(ˆr0, t) for a particular wave func-  tion’s parametrization in which the energy dispersion is very small, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=', Σ ≪ p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  “Most  probable” means that the initial particle’s position parameter ˆr0 is the value of the radial co-  ordinate r which maximizes the initial probability density ρ(r, 0) This value, equal to 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='7 nm  in the present example12 corresponds to the blue dot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Thus, the behaviour of this  trajectory, shown by Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 2c or 3c in the (x, y)-plane, is very similar to the circular one shown  in the corresponding stationary solution, but only up to a certain critical “decay time” τobs,  approximately equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='006 ns in this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' For later times the trajectory tends to a  straight line, reproducing the expected classical behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' This is best observed in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 2b or  3b which show the time behaviour of the azimuthal angle φ: the angular velocity is almost con-  stant until the time τobs, and almost vanishes thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' This decay time marks the transition  from the almost circular regime to an almost straight-way, classical-like, regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  A theoretical lower bound for the decay time τ may be computed from the quantum ”time-  energy uncertainty principle” [29]:  τ ≥ τmin =  ℏ  2 ∆E ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='47)  where ∆E is the quantum energy uncertainty given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='34) taken with m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  In our  example, τmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='00465 ns: this is the order of magnitude of the observed decay time for the  most probable trajectory, τobs ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='006 nm, and the uncertainty inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='47) is obeyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Table 2 displays, for certain values of the wave parameters p0, Σ and j, quantities of interest  such as mean energy ⟨E⟩ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='36), standard energy deviation ∆E (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='37), τmin (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='47), (which are  usual quantum theory quantities), and also, specifically concerning the most probable dBB  trajectory, its approximate observed decay time τobs, its L-radius rL (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='40), its initial radial  coordinate ˆr0 (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='46)) which fixes it and the number of closed loops, Nloops,’ it performs  before passing to the straight-way regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  One can make the following remarks about the items of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The mean values ⟨E⟩, ∆E, hence τmin, do not depend on the quantum number j, which is  obvious from the explicit expressions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='36) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' ⟨E⟩ and ∆E tend towards their  limit values (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='41) as the width Σ becomes narrower, as can be seen in the Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The observed decay time τobs seen in the behaviour of the azimuthal angle φB shown, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=',  in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 4c or 4d, depends on the specific trajectory: it diminishes when the value of the  12This value is very near of the one corresponding to the stationary wave function with same j = 5/2 and p  equal to the momentum parameter p0 = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Indeed, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='45) together with Table 1 yield ˆr= 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  16  p0( meV  c  )  Σ( meV  c  )  ⟨E⟩ (meV)  ∆E(meV)  τmin(ns)  τobs(ns)  rL(nm)  ˆr0(nm)  Nloops  j  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='3  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='91  0  757  1/2  10−8  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
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+page_content='0465  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='06  164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='6  227  39  5/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='05  362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
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+page_content='9  220  < 1  5/2  –  358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
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+page_content='  3250  < 1  99/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='027  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='291  0  676  1/2  10−7  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
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+page_content='005  325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
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+page_content='995  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
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+page_content='0000468  –  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='29  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='0  < 1  5/2  –  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='84  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='5  < 1  11/2  –  322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='6  325  < 1  99/2  Table 2: Mean energy ⟨E⟩, standard energy deviation ∆E, decay time lower bound τmin,  observed decay time τobs, L-radius rL, initial radial coordinate ˆr0 and number of trajectory  loops Nloops for some values of the wave packet parameters p0, Σ and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Trajectories concerned  in columns 6 to 9 are the most probable ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  initial radial coordinate rB(r0, 0) = r0 augments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' On the other hand, its value does not  depend sensibly on the quantum number j, as can be seen in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Except for j = 1/2, the L-radius rL (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='40) is near of the value of the initial radial  coordinate ˆr0 of the most probable trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  This is what can be expected for the  nearly circular motion which takes place at times t < τobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The number of revolutions also tends to decrease with increasing initial position r0, as  shown in the example of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  4, which shows five trajectories corresponding to five  different initial radial positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The behaviours observed in these examples are generic, this being confirmed by all other  cases we have numerically studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Concluding this subsection, an important observation can be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Although the minimum  value for the decay-time τ was inferred from the usual quantum theoretical uncertainty principle  for time-energy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='47), it appears difficult to interpret τ in this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' But it looks quite  17  natural in the dBB scheme, namely as a property of the dBB trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  May one even  imagine an experimental way of discriminating the dBB trajectories by measuring it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='3  Instantaneous beables  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='030  t  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='9999  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='0000  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='0000  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='0000  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='0000  E  (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='030  t  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='4  spin  (b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='030  t  200000  400000  600000  800000  1×106  |v|  (c)  Figure 5: Instaneous energy (subfigure (a)), spin (subfigure (b)) and absolute velocity (subfigure  (c)) in function of t for the solution shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 2, 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Colors correspond to the five  different trajectories exhibited in 4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 5 shows the time evolution of the instantaneous energy E(r0, t) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='20), spin S(r0, t)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='21) and absolute velocity |v|(r0, t) for the state already exhibited in the figures of the former  subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The quantities are shown for five dBB trajectories, caracterized by their initial  position parameter r0, and graphically by colours as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Note the striking similarity between the time behaviours of both energy and spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  One further observes that, for a given trajectory, all three quantities tend to constant values  above a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=', for the blue one, which corresponds to the initial radial position  r0 = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='7, the threshold is at t ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='019 ns for the energy and spin, and at at t ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='017 ns for  the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' This threshold is substantially higher than the decay time τ (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='006 ns here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  The energy tends to its mean value ⟨E⟩ (∼ 100 meV here), the spin to its mean value 0 and the  absolute velocity to the ”velocity of light” c = 106 ms−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Note in particular that the energy is  not conserved13 and that the velocity stays inferior to c before the time threshold, after which  it goes rapidly to its asymptotic value c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='4  Times of flight  The dBB theory offers a very natural way to define the time of flight of a particle which has  followed a dBB trajectory xB(x0, t) from its initial position x0 to some target, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=', consisting  13This is a general feature of the dBB theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Think of the obvious time dependence of the ”quantum  potential” which defines the quantum contribution to the particle’s motion of a non-relativistic particle (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='6) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  18  of a detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' In our case, one can think of a detector occupying a circle centred at the origin  and of radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' This time of flight is then the solution tflight(R, r0) of the equation  rB(r0, tflight) − R = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='48)  where rB(r0, t) is the radial coordinate of the considered trajectory, characterized by its initial  radial coordinate r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' In case the solution is not unique, one has to take the lowest one, corre-  sponding to the first hit of the particle to the target [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='However, this precaution is not  needed in all cases we have investigated, where rB(r0, t) is a monotonically increasing function  of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  The outcome of such an experiment is a probability distribution Π(τ), in terms of the time  of flight tflight = τ, given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' (9) of [17] and taking the form, in our context:  Π(τ) = N2π  � ∞  0 dr0 r0 ρ(r0, 0) δ (tflight(R, r0) − τ)  = N2πr0(R, τ) ρ(r0(R, τ), 0) |∂r0tflight(R, r0(R, τ))| ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='49)  where r0(R, τ) is the inverse of the time of flight function tflight(R, r0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=', the solution (unique,  here) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='48) for r0 in terms of R and tflight = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Recall that ρ(r0, 0) represents the probability  distribution for the trajectory defined by its initial radial coordinate r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' N is a normalization  factor ensuring the normaliztion condition  � τmax  0  dτ Π(τ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='50)  If the probability flux through the detector’s entry is always positive, which is the case in our  examples, an alternative expression for the probability distribution is given by [30]  ΠFlux(τ) = N  �  Σ  ds · j(x, τ)  =  (here) N2πRjr(R, τ),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='51)  where j is the probability flux and Σ the detector entry’s surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' This result was proved in a  scattering context by [30], and more generally, but in the one-dimensional case, by [31], and  by [32] in the case of a spinless non-relativistic particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' We have checked numerically the  equivalence of both formulae (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='49) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='51) in our specific situation for various parametriza-  tions of the wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 6 and 7 show the time of flight in function of the trajectory  parameter r0 and the corresponding probability distribution (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='49) at a circular target of radius  30 and 500 nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  4  Conclusion  The trajectories predicted by the de Broglie-Bohm (dBB) quantum theory were calculated for  the case of a guiding wave function being solution of the two-dimensional free Dirac equation,  a solution constrained to be an eigenfunction of the total angular momentum operator relative  19  5  10  15  20  25  30  r0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='025  tflight  (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='025  tflight  20  40  60  80  100  120  Π  (b)  Figure 6: Times of flight tflight solutions of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='48) and values of their probability density Π(tflight)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='49) for the dBB trajectories shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The wave function parameters are the same  as those in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 2 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The target is a circle centred at the origin, with radius R = 30 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  The initial radial coordinate r0 varies between 2 and 30 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (a) Values of the time of flight for each dBB trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The dots represent the numerically  calculated values, and the continuous line an interpolation used for the calculation of the  probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (b) Values of the corresponding probability density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Use of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='51) has been made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  100  200  300  400  500  r0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='015  tflight  (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='015  tflight  20  40  60  80  100  120  Π  (b)  Figure 7: Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 6, but with target’s radius R = 500 nm and initial radial coordinate r0  in the interval 10 to 500 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  to a given origin point of space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Numerical results have being provided for the case of massless  particles with momentum-energy specifications corresponding to those of free electrons in mono-  layer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  The trajectories corresponding to stationary wave functions turn out to be circles travelled  at a constant speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' For Gaussian-like wave packets, the trajectories begin as quasi circles of  20  slowly increasing radius till a critical time at which they tend to straight lines approximating  the behaviour expected for a classical free particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' This transition time decreases when the  value of the initial radial coordinate which labels a particular trajectory increases, but appears  to be insensible to the chosen value of the total angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' It is worth noting that  the transition time obtained in each example is of the order of magnitude of, but greater than,  the lower bound given by the ”time-energy uncertainty principle”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Although the nature of this  lower bound is of course purely quantum mechanical, a theory such as the dBB one appears  necessary in order to interpret it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' More, it is the use of the dBB theory which has allowed us  to evidenciate this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Given a wave function, the possible times of arrival of the particle at some region have  also been calculated in function of its initial position for the same examples, taking profit  of the objective reality of the trajectories in the dBB theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The corresponding probability  distribution of these arrival times has been calculated using the Das-D¨urr formula based on the  dBB theory and also using the conventional quantum theory formula involving the probability  flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Both calculation’s results coincide, as can be expected from the equivalence’s proof given  in [31] for the spin one-half particle in one-dimensional space and by [32] for the non-relativistic  spinless particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Note that this equivalence holds if the flux on any target is always positive  – which is true in our examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The importance of this probability distribution is that the  latter may in principle be measured in a suitable physical context such as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=', the monolayer  graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Acknowledgements  I would like to thank Siddhant Das for the indication of interesting references and for his  valuable comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Appendices  A  Notations and conventions  Units used in this paper are adapted to the physics of graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Length, time and energy are  given in nm, ns and meV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The critical velocity and the Planck constant take the  values  c = 106 nm ns−1,  ℏ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='5821 × 10−4 meV ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='1)  Space-time coordinate are denoted by xµ, µ = 0, 1, 2, space coordinates by x = (x, y), or (r, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Space-time metric is ηµν = diag(1, −1, −1)  Dirac matrices are chosen in terms of the Pauli matrices as  γ0 = σz,  γ1 = γ0σx,  γ2 = γ0σy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='2)  21  The Dirac matrices used in the non-relativistic formulation are  α1 = σx,  α2 = σy,  β = σz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='3)  B  Some useful properties of the Bessel functions  The general solution of the Bessel equation [33]  z2f ′′(z) + zf ′(z) + (z2 − n2)f(z) = 0,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='1)  has the form  f(z) = C1Jn(z) + C2Yn(z),  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='2)  where Jn and Yn are the Bessel functions of the first [33], resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' second [33] kind, and C1, C2  are two arbitrary complex constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' We shall restrict ourselves to an integer index n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  The asymptotic behaviors of the Bessel functions at the origin are given by  Jn(x) ∼ 1  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  �x  2  �n  (0 < x ≪ 1, n ≥ 0),  Yn(x) ∼ −(n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  π  �2  x  �n  (0 < x ≪ 1, n ≥ 1),  Y0(x) ∼ 2  π log  �x  2  �  (0 < x ≪ 1),  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='3)  and at infinity by  Jn(x) ∼  �  2  πx cos  �  x − (n + 1  2)π  2  �  (x ≫ 1, n ≥ 0),  Yn(x) ∼  �  2  πx sin  �  x − (n + 1  2)π  2  �  (x ≫ 1, n ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='4)  Functions with a negative index are related to those with a positive one by the identities  J−n(z) = (−1)nJn(z),  Y−n(z) = (−1)nYn(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='5)  Under parity z → −z, the function Jn transforms as  Jn(−z) = (−1)nJn(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='6)  An interesting orthogonality property is given by [33]  � R  0  dr r Jn  �zn,α r  R  �  Jn  �zn,β r  R  �  = R2  2 (Jn+1(zn,α)2 δαβ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='7)  for n ≥ 0, where zn,α is the αth positive zero of the Bessel function Jn(z) [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Moreover, any  function f(r) defined in the interval 0 ≤ r ≤ R with bounded variation and vanishing at the  end point r = R can be represented as a “Fourier Bessel series” [35] as  f(r) =  ∞  �  α=1  cαJn  �zn,α r  R  �  ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='8)  for any n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' The coefficients cα can be calculated using the orthogonality formula (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  22  References  [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Holland, “The Dirac equation in the de Broglie-Bohm theory of motion”, Found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 22 (1992) 1287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  [2] Peter R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Holland, “The quantum theory of motion”, Revised ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=', Cambridge University  Press (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  [3] Max Planck, “Ueber das Gesetz der Energieverteilung im Normalspectrum” (English  translation), Annalen der Physik 4 (1901) 553.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  [4] Niels Bohr, “On the Constitution of Atoms and Molecules”, Philos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 26 (1913) 1 and  476.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  [5] Albert Einstein, “Concerning an Heuristic Point of View Toward the Emission and Trans-  formation of Light”, Annalen der Physik 17 (1905) 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  [6] Louis de Broglie, “Recherches sur la th´eorie des quanta”, Thesis (Paris), 1924;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Louis de Broglie, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' (Paris) 3, 22 (1925).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Reprint in Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Louis de  Broglie 17 (1992) p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Louis De Broglie, “La m´ecanique ondulatoire et la structure atomique de la mati`ere et du  rayonnement”, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Radium 8 (1927) 225, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='1051/jphysrad:0192700805022500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  [7] Erwin Schr¨odinger, “Quantisierung als Eigenwertproblem”, Annalen der Physik 79 (1926),  361, Annalen der Physik 79 (1926) 489, Annalen der Physik 80 (1926) 437, Annalen der  Physik 81 (1926) 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  [8] Werner Heisenberg, “¨Uber quantentheoretische Umdeutung kinematischer und mechanis-  cher Beziehungen”, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 33 (1925) 879.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  [9] Paul A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Dirac, “The quantum theory of the electron”, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' A 117 (1928) 610  and 118 (1928) 351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  [10] Niels Bohr, “The Quantum Postulate and the Recent Development of Atomic Theory”,  Supplement to ”Nature April 14 (1928) 580;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Werner Heisenberg, “Physics and Philosophy”, Harper, New York (1958),  [11] Hugh Everett, “Relative State Formulation of Quantum Mechanics”,  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 29 (1957) 454.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  [12] Carlo Rovelli, “Relational quantum mechanics”,  Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 35 (1996) 1637 e-Print: quant-ph/9609002 [quant-ph];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  Andrea Di Biagio and Carlo Rovelli, “Stable Facts, Relative Facts”,  Found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' (2021) 51:30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  [13] David Bohm, “A Suggested interpretation of the quantum theory in terms of hidden  variables 1, 2.”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 85 (1952) 166, 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  23  [14] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Bohm and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Hiley, ”The Undivided Universe”, Routledge, London and New York  (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  [15] John S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Bell, “Speakable and Unspeakable in Quantum Mechanics”, Cambridge University  Press, New York (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  [16] John S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Bell, “Beables for quantum field theory”, preprint CERN TH-4035/84 (1984),  reprinted in Cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 19 of [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  [17] Siddhant Das and Detlef D¨urr, ”Arrival time distributions of spin-1/2 particles, Nature  Scient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 9 (2019) 2242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  [18] Siddhant Das, Markus N¨oth and Detlef D¨urr, “Exotic arrival times of spin-1/2 particles  I - An analytical treatment”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' A99 (2019) 052124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  [19] Detlef D¨urr, Sheldon Goldstein, Travis Norsen, Ward Struyve and Nino Zangh`ı, “Can  Bohmian mechanics be made relativistic?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' A 470 (2013) 20130699  [20] Roderich Tumulka, “On Bohmian Mechanics, Particle Creation, and Relativistic Space-  Time: Happy 100th Birthday, David Bohm!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=', Entropy 20 (2018) 462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  [21] Dario Bressanini and Alessandro Ponti, “Angular Momentum and the Two-Dimensional  Free Particle´´, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Educ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 75 (1998) 916.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  [22] Wolfram Research, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=', Mathematica, Champaign, IL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  [23] Abraham Pais, “On Spinors in n Dimensions”, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 3 (1962) 1135;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='1703856.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  [24] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' D¨urr D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Goldstein and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Zanghi, “Quantum equilibrium and the origin of absolute  uncertainty”, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 67 (1992) 843.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  [25] The Wolfram Functions Site,  https://functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='wolfram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='com/Bessel-TypeFunctions/BesselJ/21/02/02/  [26] WolframMathWorld,  https://functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='wolfram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='com/GammaBetaErf/Erf/  [27] Mikhail I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Katsnelson, “The Physics of Graphene” 2nd Edition, Cambridge University  Press, Cambridge (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  [28] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Das Sarma, Shaffique Adam, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Hwang and Enrico Rossi, “Electronic transport  in two-dimensional graphene”, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 83 (2011) 407, e-Print: arXiv:1003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='4731  (cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='mtrl-sci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  [29] Albert Messiah, “Quantum Mechanics”, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 1, Section VIII-13, Dover Publications, New  York (2014) (English translation of “M´ecanique Quantique”, Dunod, Paris, (1962)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  [30] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Daumer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' D¨urr, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Goldstein and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Zanghi, “On the Quantum Probability Flux  Through Surfaces”, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 88 (1997) 967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  24  [31] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Richard Leavens, “Bohm Trajectory Approach to Timing Electrons”, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' 129 of “Time  in Quantum Mechanics - Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='1”, 2d Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Muga, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Sala Mayato, ´I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Egusquiza  (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' ), Lecture Notes in Physics 734, Springer, Heidelberg, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='  [32] Siddhant Das and Markus N¨oth, Times of arrival and gauge invariance, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content=' A  477 (2021) 20210101, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
+page_content='1098/rspa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQfIDXU/content/2301.13251v1.pdf'}
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+Examination of saturation coverage of short polymers using 
+random sequential adsorption algorithm  
+Aref Abbasi Moud1* 
+1 Department of Chemical and Petroleum Engineering, University of Calgary, 2500 University Dr NW, 
+Calgary, AB, T2N 1N4, Canada 
+*Author to whom correspondence should be addressed; electronic mail:  aabbasim@ucalgary.ca 
+Abstract: We filled a void with a regular or asymmetric pattern without overlap using a time-dependent 
+packing method termed random sequential adsorption (RSA). In the infinite-time limit, the density of 
+coverage frequently hits a limit. This study focused on the saturation packing of squares and their dimers, 
+trimers, tetramers, pentamers, and hexamers, all of which were orientated in two randomly chosen 
+orientations (vertical and horizontal). Our results concurred with those of previous extrapolation-based 
+research1. We used the "separating axis theorem" to detect if freshly added polygons and previously put 
+ones overlapped. When RSA insertion became disproportionately sluggish, we concluded that saturation 
+had been attained. We also discovered that the system's capacity to fill the area decreased as squares were 
+stretched into dimers and trimmers. The microstructure of the resultant saturation was also thoroughly 
+investigated, including block function and the structural arrangement of dimers and trimers.  
+Keywords: RSA, short polymers, separating axis theorem, trimers, and dimers 
+1. Introduction 
+ 
+One of the most popular nonequilibrium packing models is the random sequential addition (RSA) packing 
+method, which is a time-dependent process. Unpredictable sphere packings are created using a method 
+similar to earlier established techniques; see References2-3. The RSA packing process in the three spatial 
+dimensions4 has been used to represent a range of scenarios, including protein adsorption5, polymer 
+oxidation, particles in cell membranes, and ion implantation in semiconductors2.  
+Euclidean space (d-dimensional) particles with certain shapes are randomly and progressively introduced 
+into the volume subject with the restriction that they do not overlap to carry out the simulation starting from 
+a big, vacant zone. The freshly created particles are only maintained if they do not overlap any already 
+existing particles; otherwise, they are deleted. After the simulation has begun, this procedure can be stopped 
+at any time, making the density that has been acquired time dependent. Density reaches a "saturation" or 
+"jamming" limit as time goes on6.  
+In its simplest form, an RSA sphere packing may be obtained by randomly, irreversibly, and sequentially 
+adding nonoverlapping objects into a huge volume that is initially empty of spheres. A further attempt is 
+made until the sphere can be added without doing so if an attempt to add a sphere (or any other polygon) at 
+time t overlaps with a sphere that is already present in the packing. An RSA configuration with a time-
+dependent packing fraction can be created by selecting any moment in finite time t as the process' endpoint. 
+The maximum saturation packing fraction, which occurs at the infinite-time and thermodynamic limits, 
+prevents this figure from becoming higher.  
+RSA implementations fall into two groups. The two fundamental categories into which the RSA models 
+may be split are continuum and lattice models. Based on object kinds, they are further split into two groups: 
+RSA of finite (nonzero) area objects and RSA of zero area objects. Things having finite area in this sense 
+are those that enclose a certain amount of space, whereas those with zero area lack this geometric feature 
+
+of enclosure. Therefore, upon adsorption on the substrate, items with finite area take up some space, 
+whereas those with zero area take up no space. Lattice models are defined by the realisation of the jammed 
+state7, regardless of the item types.  
+In general, in case of RSA of finite area objects, the approach of instantaneous coverage 𝜃(𝑡) to the jammed 
+state coverage 𝜃𝑚𝑎𝑥 is found to follow a power law 𝜃𝑚𝑎𝑥 − 𝜃(𝑡)~𝑡−𝑝. Researchers have proposed certain 
+laws about the value of the exponents p by researching the RSA of objects with a variety of geometries, 
+including circular, elliptical, rectangular, and sphero-cylindrical. Feder 8 is credited with being the first to 
+link the observed value of the exponent p in the RSA of circular objects in two dimensions to the object 
+dimensionality and to propose the general rule that p = 1/D for d-dimensional circles on a two-dimensional 
+continuum platform. Swendsen 9 subsequently demonstrated that, if the items are placed with random 
+orientations, the same ought to apply for RSA of items of any arbitrary form.  
+Since its invention by Feder 8 , Random Sequential Adsorption (RSA) has gained widespread acceptance 
+as a technique for simulating adsorption characteristics, particularly for spherical molecules. However, 
+employing RSA to replicate the adsorption of more complex particles like polymers or proteins raises 
+concerns about how RSA's inherent features alter when non-spherical molecules are involved. For simple 
+forms like spheroids, spherocylinders, rectangles, needles, and others, the subject has already been 
+addressed10-12. Recent research, however, indicates that these geometries are insufficient for simulating the 
+adsorption of common proteins, such as fibrinogen, for instance13. As a result, researchers' focus has 
+recently turned to coarse-grained modelling of complex biomolecules and polymers14-16.  
+In 1-D case, the saturation packing fraction can be obtained analytically as 0.747597920 17 however for 2-
+D and 3-D the saturation packing fraction for discs and spheres has been estimated only through numerical 
+simulations; the most precise ones are 0.5470735 ± 0.0000028 for 2-d and 0.3841307 ± 0.000002 for 3-D 
+cases2. 
+Other 
+figures 
+reported 
+for 
+2-D 
+simulations 
+of 
+discs 
+are 
+0.54707352, 0.54706718, 0.54707019, 0.547069020, 0.5470021, 
+0.5471122, 0.547223, 
+0.5478, 
+0.547924. 
+Similarly for 2-D aligned squares saturation coverage reported in the literature is 0.56200925, 0.562324, 
+0.5628, 0.556526, 0.562527, 0.544428, 0.562929, 0.56230. 
+In this study, we used the RSA technique to determine the saturation packing limit for squares and its dimer 
+and trimers. As the length of square increases, the results indicated that samples eventually generated 
+structures with less and less packing.  In this study, the "separating axis theorem" approach was used to 
+detect whether there was a collision between two polygons; more information on the procedure is provided 
+in the following sections. Polygons here refer to squares that encompasses its polymers as well as a 
+constructing monomer. Our preliminary findings on RSA packing with respect to polygons, which we just 
+published31, are the basis for this study, which extends that work by extending the polygon (square) into its 
+polymers.  
+2. Model and simulation procedure  
+ 
+When colloidal particles or molecules are being adsorbed, they frequently diffuse close to the surface. This 
+process might lead to the formation of a film consisting of molecules that were randomly adsorbed because 
+of adhesion. Here, we focus on adsorbate monolayers formed by irreversible adsorption. The most 
+straightforward technique for quantitatively modelling these processes is molecular dynamics (MD). The 
+advantages of MD include accurate forecasting and management of most environmental factors, such as 
+temperature and the diffusion constant. The main issue is the performance deficiency. As a result, we 
+decided to utilise a new method, continuum Random Sequential Adsorption (RSA), which has been 
+successfully employed to investigate colloidal and other systems32.  
+
+To simulate, a virtual particle was created (square, its polymers), and its location on an area was chosen at 
+random based on a uniform probability distribution. 
+- The overlap with the previously adsorbed nearest neighbours of a virtual particle was tested (the topic of 
+the next section). The result of this test tells us whether the surface-to-surface distance of a particle is greater 
+than zero. 
+- If there was no overlap, the virtual particle was adsorbed and added to an existing covering layer. 
+- If there was overlap, the virtual particle was dropped and abandoned.  
+2.1 Proposed algorithm 
+ 
+Numerous methods may be used to determine if two polygons intersect or not. One method for determining 
+if two polygons are overlapping uses mathematical equations and is known as the "separating axis 
+theorem"33. 
+The separating axis theorem states that if a line divides two convex polygons, they cannot intersect. The 
+separation axis, which is a line, may be thought of as the normal to one of the edges of each polygon. 
+Using the separating axis theorem, the following procedures can be used to determine if two polygons cross:  
+• 
+Determine each polygon A edge's edge normal, then project both polygons onto that value. 
+ 
+• 
+Establish the minimum and maximum projections of each polygon onto the normal. 
+ 
+• 
+If the maximum projection of polygon A is less than the minimum projection of polygon B, or the 
+other way around, the polygons do not overlap. 
+ 
+• 
+If the projections overlap, repeat the procedure for each edge in polygon B. 
+ 
+• 
+If the projections of the polygons onto the separating axes do not overlap, the polygons are 
+connected.  
+It is likely that non-convex polygons or polygons with holes will not be covered by this theorem, even 
+though the separating axis theory may be used to determine if two convex polygons overlap. In some 
+situations, it could be necessary to verify for intersection using alternative techniques.  
+3. Results and discussion 
+ 
+We construct saturated RSA configurations of polymers (dimer to hexamer) and compare saturation 
+packing with other findings reported in the literature, particularly in ref2, where authors employed a 
+different approach and orientation was random, to show the precision and utility of our algorithm. We 
+generate 1000 variations for each particle form using the system size that results in the fastest and densest 
+packing.  
+
+ 
+Figure 1. Typical monolayer samples (from trimer’s sample) for three different coverages: θ = 0.1, θ = 0.3, 
+θ = 0.4 and θ = 0.5 for trimers. The collector side length was equal to 50[-]. Fixed boundary conditions 
+were used. Figure shows truncated images of the distribution for a better visibility (20 by 20). 
+Using the greatest area (50 by 50) with fixed bounds, most of the results presented later in the study were 
+achieved. We made sure that the adoption of periodic boundary conditions had no discernible impact on 
+the results that were made (Equivalent of periodic and fixed boundary condition). Figure 1 shows the 
+outcomes of one of the simulations, in which trimers are placed in an area measuring 20 by 20 with a 
+monomer having a side length of 0.5 units. Particles are gradually added to the surface as the simulation 
+progresses.  
+ 
+
+0=10%
+0=30%
+0=40%
+0=50% 
+Figure 2. Hexamer units put irrevocably inside a 50 by 50 space are the focus of the RSA algorithm. (a) 
+Asymptotic observation of coverage as a function of total simulation duration indicated by t. (b) The 
+instantaneous time τ determined by based on coverage. The line represents an exponential fit.  
+To look at the development in more details, surface coverage was depicted as a function of simulation time 
+to the power of 𝑡−1/3 and results are shown in Figure 2a. Clearly at long simulation times of ~104 coverage 
+very slowly reaches its asymptotic limit that is 0.4968±0.0011.  Similarly using same schemes, we arrived 
+at 0.5631±0.0002 surface coverage for squares. This surface coverage corresponds very well with the results 
+reported in the literature for squares 8, 24-30.  
+The number of attempts needed to add a new particle to the grid (or collector both terms in congruency with 
+literature has been used here interchangeably) can be known, and this information can be used to model the 
+blocking of further adsorption through monitoring time. Clearly, as more of the surface is covered, adding 
+new particles should be more difficult, which can be represented by a lower probability (See Figure 2b) 
+that is it takes considerably more amount of time for a particle to be added. Adsorption kinetics in a real 
+experiment typically depends on two variables: the effectiveness of the transport process (primarily 
+diffusion or convection depending on the experimental setup) that moves the adsorbate from the bulk to the 
+surface, and the likelihood of catching particles that are nearby 34-39. Authors in other reports40 have 
+concentrated on the second aspect in this case, which is defined by the blocking function, also known as 
+the available surface function (ASF). The simulation makes it simple to obtain it as a ratio of successful 
+attempts to all RSA attempts. Equivalent to available surface function that is represented in shape of time 
+simulation is presented in Figure 2b. Figure 2b shows simulation time as a function of coverage; 
+statistically, it is evident that more trials are required to attain adsorption because the surface is already 
+rather packed. The exponential fit is, τ = 0.0006 exp (24.03 𝜃), thus describing increasing time required to 
+place an additional point onto the grid. Discussion on ASF is subject of next section.  
+The main objectives of this work were to determine the maximum random adsorption ratio for squares and 
+their polymers and compare it to the results for hard circles (spheres). That ratio ought to be provided for 
+an infinite grid area and adsorption duration. Although one must deal with constrained simulation durations, 
+one must also manage the accuracy issue brought on by the finite grid size. Because it is unclear if there 
+would be any possibility of adsorption after the simulation time, particularly in the case of large grids, the 
+determination of maximal coverage depends on the RSA kinetics model. As a result of prior research in the 
+region 9, 41-42, there have been a number of works in the area, and asymptotically:  
+𝜃𝑚𝑎𝑥 − 𝜃(𝑡)~𝑡−1
+𝐷     eq.1 
+
+0.5
+4000
+0.45
+3000
+0.4
+2000
+0.35
+1000
+0.3
+0.01
+0.02
+0.03Regarding the irreversible deposition of discs or squares that are not orientated (formerly known as p = -
+1/D). Despite controversy, D here specifies the grid's dimension9. When adsorbed particles are organised, 
+the situation is altered9, 42. Figure 2a previously in this post showed an example of the results of fitting 
+Equation 1. Asymptotic observations of coverage for squares seem to neatly match Equation 1. Although 
+Equation 1 hasn't been definitively proved, its validity has been vigorously defended12 by analytical and 
+numerical grounds. It should be noted that Equation 1 simplifies to the standard Feder's law8 for isotropic 
+objects since n equals the number of dimensions.  
+For instance, RSA of discs on a two-dimensional plane has D = 2, whereas RSA of rectangles, ellipses, and 
+other rigid but noticeably anisotropic structures has D = 3 37, 43. It appears that parameter D generally 
+correlates to the degrees of freedom of a number of packed objects, which has been validated for the random 
+packing of hyperspheres in higher dimensions 2, 21, not only the integral ones44-45. The power law (Equation 
+1) is satisfied for the RSA of polymers examined here, however the exponent -1/D strongly relies on a 
+polymer length. The parameter D is about equivalent to 3, which is the value recognised for anisotropic 
+molecules, for a small number of vertexes such as pentagon and squares. However, as number of vertices 
+increases parameter D converges to 2. This finding is consistent with those made for the RSA of spherical 
+beads examined in Ref.46. However, unlike what was shown in the cases of spherical beads46  or generalized 
+dimers 40, there is no abrupt transition between these two limitations.  
+The results are averaged across 10 simulation runs with time t in the order of 5 × 108 for each run in order 
+to determine parameter D for different polymers. These runs' data are not displayed here, and we will go 
+into more depth about the outcomes in our upcoming paper.  
+ 
+3.1 RSA for polymers 
+ 
+In the last part, we laid the foundation for using the RSA approach to create oriented squares and trimmers. 
+Results showing the behaviour of adsorption at asymptotic limits, the kinetics of the adsorption index (p), 
+and the relationship between simulation time and coverage were given. Additionally, results and discs were 
+compared. Utilizing the extrapolation method shown in Figure 2a previously, Table 1 generates 
+saturation densities for various polymer lengths. Figure 3 displays a sample of saturation densities for 
+various forms.  
+To arrive at the values reported in Table 1 following equation has been used: 
+𝜃(𝑡) = 𝜃𝑚𝑎𝑥 + 𝑏/𝑡𝑝     eq.2 
+When arriving at the values shown in table 1, we gave the data from longer simulation times more weight. 
+The approach's possible downside is that each data point is given the same weighting factor, assuming all 
+values are given the same weight. Because there are a lot more of these points in the higher part of the 
+asymptotic area, it is sort of underweighted. Therefore, we investigated a novel strategy that introduces a 
+bias favouring the longer durations. These changes are in line with the accounts in ref12.  
+Table 1. Saturation density, index, for square-based polymers with a monomer to simulation box length 
+ratio of 0.01 and their respective 95% confidence intervals.  
+Shape (oriented) 
+𝜃𝑚𝑎𝑥 [-] (95% confidence 
+bounds)  
+p [-] 
+(95% 
+confidence 
+bounds) 
+b [-] 
+(95% 
+confidence 
+bounds) 
+Square  
+0.5631(0.5629, 0.5633) 
+0.5138 (0.5125, 0.5151) 
+-4.226 (-4.475, -4.561) 
+
+Dimer 
+0.57 (0.5697, 0.5704) 
+0.4599 (0.4595, 0.4603) 
+-4.805 (-4.819, -4.792) 
+Trimers 
+0.5621 (0.5612, 0.5629) 
+0.466 (0.4652, 0.4668) 
+-8.003 (-8.066, -7.941) 
+Tetramer  
+0.5558 (0.5539, 0.5577)  
+0.4998 (0.499, 0.5007) 
+-6.046 (-6.15, -5.942) 
+Pentamer 
+0.5504 (0.5481, 0.5527) 
+0.46 (0.4576, 0.4624) 
+-5.007 (-5.122, -4.892) 
+Hexamer 
+0.4968 (0.4949, 0.4987) 
+0.5 (0.499, 0.501) 
+-6.264 (-6.458, -6.069) 
+Discs 
+0.5470732 
+- 
+- 
+ 
+As outlined in introduction section, similarly for 2-D aligned squares saturation coverage reported in the 
+literature is 0.56200925, 0.562324, 0.5628, 0.556526, 0.562527, 0.544428, 0.562929, 0.56230. Our values for 
+square are very well within range of values reported elsewhere. However, as particles get longer and become 
+Trimers, saturation has dropped since longer particles require more accessible area for deposition. For dimer 
+and trimer, the greater aspect ratio of the dimer is projected to result in somewhat higher saturation for 
+dimers. These findings are crucial because they suggest that it gets progressively harder for molecules to 
+adhere to surfaces as they become longer; an example of superiority of simulation over experiment in giving 
+researcher a tool to examine parameters hard to measure through experiments.  
+Results from this study can also be extrapolated to higher dimensions. For instance, the efficiency of a 
+sequential adsorption process with hard materials decreases with increasing size. It is noteworthy to note 
+that, as a general rule, the saturation coverage in D dimensions is very well estimated by that in one 
+dimension raised to power D (for the RSA of spherical particles, 𝜃𝑚𝑎𝑥≃ 0.75 for D = 1, 𝜃𝑚𝑎𝑥≃ 0.55 for D 
+= 2, 𝜃𝑚𝑎𝑥 ≃ 0.38 for D = 3, etc)47. Therefore, results obtained here can be extended to 1-D and 3-D cases 
+with good approximation, for instance for squares for cubes is predicted to lie around 0.38 and in 1-D case 
+around 0.73.  
+
+ 
+Figure 3. Square, dimers, and trimers near saturation points for samples with monomer’s side length of 0.5 
+and distributed within area of 50 by 50. Fixed boundary condition has been applied. Figure shows truncated 
+images of the distribution for a better visibility (20 by 20). For improved visibility, the horizontally oriented 
+polymers have been coloured blue, while the vertically oriented ones have been painted red.  
+Clearly visually samples experience a bit higher coverage for dimers and less coverage for trimers.   
+ 
+
+Square
+Dimers
+Trimers 
+Figure 4. RSA saturation density for polymers, with a line created to guide the viewer's eyes. Each data 
+point has an almost imperceptible error bar. For discs, RSA saturation density (red dotted line). All the 
+polymers in Table 1's saturation density changes as a function of simulation duration (a) The dependence 
+of 𝜃𝑚𝑎𝑥 on length of the polymer (c) Kinetic index as a function of the monomer length.  
+Figure 4 shows the RSA saturation density as a function of polymer length together with an eye-guiding 
+line. The saturation density coverage first increases somewhat as the polymer's length rises, but as it 
+continues to grow, it starts to decline. This outcome is in line with the outcomes for rectangles with various 
+aspect ratios and ellipses that have been reported in the literature5, 48. Additionally, we discovered that 
+saturation is somewhat lower for polymers with aspect ratios longer than 2, and we hypothesise that this is 
+because, as was already noted, longer particles are more difficult to pack efficiently.  
+The random coverage ratio dropped exponentially with polymer size in earlier investigations refs43, 49. On 
+a continuous surface, at least two competing variables can affect the maximum random coverage ratio. 
+First, there is less chance of finding a large enough uncovered section to place on a collector, making it 
+harder to separate larger particles than in the lattice case. The second point is that a polymer globule is more 
+likely to form a cluster when necessary because it has a greater monomer packing ratio than a group of 
+individual monomers. For continuous collectors as opposed to lattice ones, this second element is more 
+important.  
+
+0.6
+0.65
+0.5
+0.6
+0.4
+0.55
+Sguare
+0.3
+0.5
+Dimer
+Trimer
+0.2
+0.45
+Tetramer
+A
+Hexame
+0.4
+a
+b
+0.35
+3
+4
+5
+2
+X10 4
+0.65
+Monomerlength
+0.6
+DiscS
+0.55
+0.5
+0.45
+0.4
+0.35 
+3.2 Block function 
+ 
+Knowing how many tries are necessary to add a new particle to the grid allows us to simulate how further 
+adsorption is blocked over time. It is obvious that when more of the surface is covered, adding more 
+particles should become more challenging, which may be represented by a decreased probability (as shown 
+in Figure 2b, which depicts the same trend with time as the dependent variable)  
+ 
+ 
+ 
+Figure 5. The ratio of successful attempts to blocking attempts versus coverage. The green line represents 
+simulation data in b and the dotted line represent polynomial fit.; details of the fits are given in table 2.   
+ 
+Equation 3 describe the function with a simple second order polynomial perfectly describing the decreasing 
+trend in probability of a successful adsorption.  
+ 
+𝐴𝑆𝐹(𝜑) = 𝐶1𝜃2 + 𝐶2𝜃 + 𝐶3 eq.3 
+ 
+In which C1, C2 and C3 are pre-factors. Details of the fit of equation 3 is given in the Table 2. Figure 5 
+shows that the likelihood of adsorption for trimers drops more quickly than for dimers and squares, and the 
+dimer with respect to the square exhibits the same pattern. This is because consecutive adsorption is much 
+less likely to occur quickly in trimers and dimers than in squares due to their greater aspect ratio and unusual 
+orientation.  
+ 
+Table 2. Represents polynomial fit to the data in Figure 5 along with corresponding 95% confidence 
+bounds. 
+ 
+Shape (oriented) 
+C1 [-] (95% confidence 
+bounds)  
+C2 [-] 
+(95% 
+confidence 
+bounds) 
+C3 [-] 
+(95% 
+confidence 
+bounds) 
+Square  
+0.5439 (0.5296, 0.5582) 
+-2.244 (-2.252, -2.237) 
+1.008 (1.007, 1.009) 
+Dimer 
+2.508 (2.484, 2.532) 
+-3.18 (-3.194, -3.167) 
+0.9796 (0.978, 0.9811) 
+Trimers 
+5.483 (5.382, 5.585) 
+-4.463 (-4.511, -4.414) 
+0.9623 (0.9574, 0.9671) 
+Tetramer  
+5.359 (5.176, 5.543) 
+-4.338 (-4.437, -4.238) 
+0.8921 (0.8809, 0.9034) 
+Pentamer  
+6.352 (6.063, 6.641) 
+-4.828 (-4.94, -4.715) 
+0.97 (0.9621, 0.9779) 
+
+1.5
+b
+a
+Square
+Square
+ • Dimer
+Fit
+Trimer
+0.5
+0.5
+0.2
+0.4
+0.6
+0.2
+0.4
+0.6Hexamer 
+6.572 (6.359, 6.785) 
+-5.038 (-5.116, -4.961) 
+0.9933 (0.9881, 0.9986) 
+ 
+In the case of square, dimers, trimers simulations show that C1 = 0.5439-6.572 and C2 =-2.244-5.038, 
+whereas those parameters for hard circles adsorption are analytically derived as C2 =-4 and C1 = 3.3150 (we 
+obtained coefficients of C1=2.426  (1.071, 3.782) and C2=-2.907  (-3.644, -2.169) with aid of our 
+simulation) . Contrary to discs, the coefficient for squares suggests that they have an easier time adhering 
+to the surface. Higher saturation coverage for squares is another manifestation of this phenomenon. 
+Therefore, dimers values are very close to the values reported for hard discs.  
+ 
+ 
+According to Figure 5, as coverage levels increase, we get closer to the asymptotic phase, where dynamics 
+are well understood, and finally the jamming limit. Thus, the RSA procedure has now been completely 
+explained. Since it is challenging to conduct adsorption investigations near to the jamming limit51, 
+measuring the terms of the RSA process is the most effective technique to show that adsorption follows an 
+RSA process. It's crucial to remember that words up to 𝜃2 don't reveal anything about the properties of the 
+adsorption process (i.e., the degree of irreversibility). This implies that any experiment that involves the 
+adsorption of particles that resemble hard discs is susceptible to such an extension (to second order). Our 
+strategy is also applicable to combinations and non-circular particles.  
+ 
+3.3 Ordering and orientation 
+ 
+ 
+We study the presence of any orientational order in a monolayer using the amorphous form of a polymer. 
+Although most of these studies have used a collector surface lattice structure, as in ref. 52, such ordering has 
+been well investigated. It could also have an impact on the RSA kinetics mentioned before. Based on the 
+polymer structure, we offer the following function to measure orientational order in our continuous system:  
+ 
+𝑆(𝜑) =
+1
+𝑁 ∑
+(𝑥𝑖 cos(𝜑) + 𝑦𝑖 sin(𝜑))
+𝑁
+𝑖=1
+ eq.4 
+ 
+where (xi, yi) are positions along the i-th molecule in a layer for a unit vector. It is clear that 𝑆(𝜑) is an 
+average scalar product between both the orientation of molecules and the direction determined by an angle. 
+As a result, for a perfectly aligned layer, 𝑆(𝜑) will swing between 0 and 1, with highest values for angles 
+parallel to molecules and minimum values for angles perpendicular to the alignment direction. 𝑆(𝜑) will 
+always be a constant and equal to 0.5 for pure random alignment. 
+ 
+For trimers as coverage increases across simulation time, ordering hovered 0.49, 0.53,0.50,0.51 as surface 
+coverage increased from 10 to 30, 40 and 50%. Clearly ordering in trimer population is very close to random 
+due to simulation being designed to give equal chance to parallel or vertical orientation of trimers as shown 
+in Figure 3. Situation is very similar for dimers as well.  
+ 
+Figure 4 illustrates how ordering in trimers may be further examined as a function of the radius of the 
+particle neighbours. Figure 4 was made using a similar idea to the pair correlation function by treating the 
+trimer centre as a circle with a radius of 0.25. 𝑆(𝜑)  fluctuates in small regions because dense clusters of 
+horizontally or vertically oriented trimers are more likely to form, but as the sweeping radius grows, this 
+fluctuation decreases to a value that is very similar to a randomly oriented arrangement. Dimers also face a 
+similar set of circumstances. As a result, at r5, local order in each system vanishes. Small amplitude 
+fluctuations continue after r=5, although their amplitudes and frequency are higher for trimers.  
+ 
+
+ 
+Figure 4. Local ordering as a function of radius for two RSA packings made with dimers and trimers near 
+their saturation coverage.  (a) dimers (b) trimers. 
+ 
+ 
+Similar trends are expected for tetramers due to similarity of behavior we have refrained from exploring 
+them further here.  As an example, liquid crystals are one type of orientationally organised structure that an 
+elongated particle (such as polymers here with aspect ratio>2) can produce. When particle orientations are 
+chosen at random using a uniform probability distribution for RSA on an infinite collector, the global 
+orientational order is not expected to exist. However, since parallelly aligned particles take up less space, 
+the formation of local ordered domains is feasible53-54.  
+ 
+3.4 Radial distribution function  
+ 
+The radial distribution function (G(r)), also known as the pair correlation function, in a system of particles 
+(such as atoms, molecules, colloids, etc.) explains how density changes in response to distance from a 
+reference particle. G(r) is the radial distribution function 55 obtained from following equation:  
+ 
+𝐺(𝑟) =
+1
+𝜌 〈∑
+𝛿(𝑟 − 𝑟𝑖)
+𝑖≠0
+〉 eq.5 
+ 
+ 
+The monolayer's first crucial characteristic is the particle autocorrelation. Squares are assumed to have a 
+radius of 0.25 and to be treated equally regardless of whether they are made of the same polymer or a 
+different one in order to compute G(r). Figure 5 displays the average structures seen by various RSA 
+packings. We consistently saw a peak at a distance of r=0.5 (right on the edge of the particles). In other 
+words, the function reaches its maximum for the closest neighbour, r = 0.5, and then begins to degrade 
+because of the volume that is lacking. The similar trend is seen in the trimer and hexamer, but there are 
+more peaks. Hexamer contains additional peaks, for instance, at r=1, 1.5, and 2, while the trimer exhibits 
+an additional peak at 1.  
+ 
+As the radius gets bigger, these peaks get weaker. Due to the coverage's randomness, these oscillations 
+superexponentially vanish21, and after normalisation, the function stabilises at a value of 1. In addition, 
+when the number of monomers inside the polymer rises, the first peak corresponding to the nearest 
+neighbour grows progressively sharper. According to this behaviour, particles that may be seen as a chain 
+of squares pack more well even if the saturation coverage is smaller for monomers (squares) and short 
+oligomers (dimers).  
+
+.5
+1.5
+b
+a
+S( Φ),Dimers
+.- S( Φ),Trimers
+0.5
++
+0.5
+:
+.
+L
+10
+15
+20
+25
+5
+10
+15
+20
+25 
+Figure 5. Functions of autocorrelation. Behavior autocorrelation function is depicted as a function of radius 
+for square, dimer, trimer, tetramer, pentamer and hexamer.  
+ 
+The average structure seen by a generic particle of the system described by G(r) displayed in Figure 5, 
+shows a full agreement with the predicted theoretical regimes found in literature 56-57. In all cases, we 
+observe a pronounced peak at a distance r~0.5, with the sphere diameter that corresponds to the distance of 
+the nearest neighbors in contact. For r larger than the diameter, the probability to find neighbors decreases. 
+In fluid-like systems, theoretically for 𝜑 ≲ 0.55, 56-58 the G(r) is known to oscillate with decreasing 
+amplitude. 
+ 
+Conclusions  
+ 
+For a range of stiff polymers produced using squared monomers, we show the maximum random coverage 
+or saturation coverage in this paper and contrast our findings with those reported in the literature. In order 
+to do this, we enhanced an algorithm that was described in Ref 32. We prove the validity of our method by 
+calculating the RSA saturation densities of polymers (dimer, trimer, and tetramer) and showing their 
+consistency with prior findings in the literature. 
+The RSA model shown here may be extended to squares that may change into rectangles with larger aspect 
+ratios to incorporate anisotropic particles in future research. Moreover, like ref43 it can also include branched 
+or more flexible polymers. Biological molecules are usually non-spherical, as seen by the previous example, 
+and when their surface area in contact with the substrate is greatest, they firmly cling. According to 
+experimental results, Schaaf et al. 59 discovered that the maximum substrate coverage they were able to 
+achieve during the adsorption of fibrinogen—a non-spherical protein with an aspect ratio of roughly 7.5—
+was only about 40%, which was lower than the absorption coverage predicted by the RSA of hard discs—
+which is around 55%—and seen in experiments involving reasonably spherical globular proteins8 (A similar 
+impact was noted for albumin adsorption 60 ).  
+Here are some pertinent queries: 
+How does increasing the aspect ratio affect the saturation coverage of the substrate? 
+How does the particle shape impact the kinetics over both short and long time periods? 
+
+Square
+5
+. Trimer
+  .Hexamer
+6
+8What are the similarities and differences between equilibrium configurations produced by RSA and 
+configurations with equivalent surface coverage? 
+These questions will get their solutions in upcoming publications. This study's findings are pertinent since 
+they considered a variety of particle morphologies, including those of asphaltene, graphene, cellulose 
+nanocrystals, and kaolinite, among others 61-64.  
+The findings are important because they might help to understand how polymers behave when they are 
+close to surfaces. For instance, numerous biological processes depend on proteins adhering to different 
+surfaces. Understanding and having control over how protein molecules attach to surfaces and interact with 
+them is essential when creating biomaterials. For instance, among other things, the production of 
+biocompatible materials requires decreasing the adsorption of blood proteins to the material's surface. It is 
+generally known that platelet adhesion followed by blood protein adsorption can result in surface-induced 
+thrombosis. When protein adsorption is prevented or diminished, there is very little platelet adhesion to the 
+surface. Eliminating lysozyme buildup from the surface of contact lenses is another illustration. In other 
+circumstances, we would like to promote the adsorption of some proteins while inhibiting the adsorption 
+of others.  
+Conflict of interest statement: Author declares no conflict of interest 
+ 
+Data availability statement: The datasets generated during and/or analysed during the current study are 
+not publicly available but are available from the corresponding author on reasonable request. 
+ 
+References 
+ 
+1. 
+Zhang, G., Precise algorithm to generate random sequential adsorption of hard polygons at 
+saturation. Physical Review E 2018, 97 (4), 043311. 
+2. 
+Zhang, G.; Torquato, S., Precise algorithm to generate random sequential addition of hard 
+hyperspheres at saturation. Physical Review E 2013, 88 (5), 053312. 
+3. 
+Zhang, G.; Stillinger, F. H.; Torquato, S., Transport, geometrical, and topological properties of 
+stealthy disordered hyperuniform two-phase systems. the J. Chem. Phys. 2016, 145 (24), 244109. 
+4. 
+Torquato, S., Perspective: Basic understanding of condensed phases of matter via packing models. 
+the J. Chem. Phys. 2018, 149 (2), 020901. 
+5. 
+Petrone, L.; Cieśla, M., Random sequential adsorption of oriented rectangles with random aspect 
+ratio. Physical Review E 2021, 104 (3), 034903. 
+6. 
+Abbasi Moud, A., Colloidal and Sedimentation Behavior of Kaolinite Suspension in Presence of 
+Non-Ionic Polyacrylamide (PAM). Gels 2022, 8 (12), 807. 
+7. 
+Shelke, P. B., Random sequential adsorption of n-star objects. Surface Science 2016, 644, 34-40. 
+8. 
+Feder, J.; Giaever, I., Adsorption of ferritin. journal colloid interface sci. 1980, 78 (1), 144-154. 
+9. 
+Swendsen, R. H., Dynamics of random sequential adsorption. Physical Review A 1981, 24 (1), 504. 
+10. 
+Talbot, J.; Tarjus, G.; Schaaf, P., Unexpected asymptotic behavior in random sequential adsorption 
+of nonspherical particles. Physical Review A 1989, 40 (8), 4808. 
+11. 
+Vigil, R. D.; Ziff, R. M., Random sequential adsorption of unoriented rectangles onto a plane. the 
+J. Chem. Phys. 1989, 91 (4), 2599-2602. 
+12. 
+Viot, P.; Tarjus, G., Random sequential addition of unoriented squares: breakdown of Swendsen's 
+conjecture. EPL (Europhysics Letters) 1990, 13 (4), 295. 
+13. 
+Adamczyk, Z.; Barbasz, J.; Cieśla, M., Kinetics of fibrinogen adsorption on hydrophilic substrates. 
+Langmuir 2010, 26 (14), 11934-11945. 
+
+14. 
+Adamczyk, Z.; Barbasz, J.; Cieśla, M., Mechanisms of fibrinogen adsorption at solid substrates. 
+Langmuir 2011, 27 (11), 6868-6878. 
+15. 
+Rabe, M.; Verdes, D.; Seeger, S., Understanding protein adsorption phenomena at solid surfaces. 
+advances Colloid Interface Sci. 2011, 162 (1-2), 87-106. 
+16. 
+Finch, C. A.; Hickman, J. J. In A continuum hard-sphere model of protein adsorption, Proceedings 
+of the ACM Conference on Bioinformatics, Computational Biology and Biomedicine, 2012; pp 573-575. 
+17. 
+Rényi, A., On a one-dimensional problem concerning space-filling. Publ. Math. Inst. Hungar. 
+Acad. Sci. 1958, 3, 109-127. 
+18. 
+Cieśla, M.; Ziff, R. M., Boundary conditions in random sequential adsorption. Journal of Statistical 
+Mechanics: Theory and Experiment 2018, 2018 (4), 043302. 
+19. 
+Cieśla, M.; Nowak, A., Managing numerical errors in random sequential adsorption. Surface 
+Science 2016, 651, 182-186. 
+20. 
+Wang, J.-S., A fast algorithm for random sequential adsorption of discs. International Journal of 
+Modern Physics C 1994, 5 (04), 707-715. 
+21. 
+Torquato, S.; Uche, O.; Stillinger, F., Random sequential addition of hard spheres in high Euclidean 
+dimensions. Physical Review E 2006, 74 (6), 061308. 
+22. 
+Chen, E. R.; Holmes-Cerfon, M., Random sequential adsorption of discs on constant-curvature 
+surfaces: plane, sphere, hyperboloid, and projective plane. arXiv preprint arXiv:1709.05029 2017. 
+23. 
+Hinrichsen, E. L.; Feder, J.; Jøssang, T., Random packing of disks in two dimensions. Physical 
+Review A 1990, 41 (8), 4199. 
+24. 
+Wang, J.-S., Series expansion and computer simulation studies of random sequential adsorption. 
+colloids Surf. Physicochem. Eng. Asp. 2000, 165 (1-3), 325-343. 
+25. 
+Brosilow, B. J.; Ziff, R. M.; Vigil, R. D., Random sequential adsorption of parallel squares. 
+Physical Review A 1991, 43 (2), 631. 
+26. 
+Blaisdell, B. E.; Solomon, H., On random sequential packing in the plane and a conjecture of 
+Palasti. Journal of Applied Probability 1970, 7 (3), 667-698. 
+27. 
+Dickman, R.; Wang, J. S.; Jensen, I., Random sequential adsorption: Series and virial expansions. 
+the J. Chem. Phys. 1991, 94 (12), 8252-8257. 
+28. 
+Tory, E. M.; Jodrey, W.; Pickard, D., Simulation of random sequential adsorption: Efficient 
+methods and resolution of conflicting results. Journal of Theoretical Biology 1983, 102 (3), 439-445. 
+29. 
+Akeda, Y.; Hori, M., On random sequential packing in two and three dimensions. Biometrika 1976, 
+63 (2), 361-366. 
+30. 
+Jodrey, W.; Tory, E., Random sequential packing in Rn. Journal of statistical computation and 
+simulation 1980, 10 (2), 87-93. 
+31. 
+Moud, A. A., Examination of saturation coverage of polygons using random sequential adsorption 
+algorithm. arXiv preprint arXiv:2212.13849 2022. 
+32. 
+Adamczyk, Z.; Siwek, B.; Zembala, M.; Weroński, P., Influence of polydispersity on random 
+sequential adsorption of spherical particles. journal colloid interface sci. 1997, 185 (1), 236-244. 
+33. 
+Gottschalk, S.; Lin, M. C.; Manocha, D. In OBBTree: A hierarchical structure for rapid 
+interference detection, Proceedings of the 23rd annual conference on Computer graphics and interactive 
+techniques, 1996; pp 171-180. 
+34. 
+Adamczyk, Z.; Weroński, P., Random sequential adsorption of spheroidal particles: kinetics and 
+jamming limit. the J. Chem. Phys. 1996, 105 (13), 5562-5573. 
+35. 
+Schaaf, P.; Talbot, J., Surface exclusion effects in adsorption processes. the J. Chem. Phys. 1989, 
+91 (7), 4401-4409. 
+36. 
+Adamczyk, Z.; Siwek, B.; Zembala, M., Kinetics of localized adsorption of particles on 
+homogeneous surfaces. journal colloid interface sci. 1992, 151 (2), 351-369. 
+37. 
+Viot, P.; Tarjus, G.; Ricci, S.; Talbot, J., Random sequential adsorption of anisotropic particles. I. 
+Jamming limit and asymptotic behavior. the J. Chem. Phys. 1992, 97 (7), 5212-5218. 
+38. 
+Evans, J. W., Random and cooperative sequential adsorption. Reviews of modern physics 1993, 65 
+(4), 1281. 
+
+39. 
+Talbot, J.; Tarjus, G.; Van Tassel, P.; Viot, P., From car parking to protein adsorption: an overview 
+of sequential adsorption processes. colloids Surf. Physicochem. Eng. Asp. 2000, 165 (1-3), 287-324. 
+40. 
+Cieśla, M., Properties of random sequential adsorption of generalized dimers. Physical Review E 
+2014, 89 (4), 042404. 
+41. 
+Hinrichsen, E. L.; Feder, J.; Jøssang, T., Geometry of random sequential adsorption. Journal of 
+statistical physics 1986, 44 (5), 793-827. 
+42. 
+Privman, V.; Wang, J.-S.; Nielaba, P., Continuum limit in random sequential adsorption. Physical 
+Review B 1991, 43 (4), 3366. 
+43. 
+Cieśla, M., Continuum random sequential adsorption of polymer on a flat and homogeneous 
+surface. Physical Review E 2013, 87 (5), 052401. 
+44. 
+Ciesla, M.; Barbasz, J., Random sequential adsorption on fractals. the J. Chem. Phys. 2012, 137 
+(4), 044706. 
+45. 
+Cieśla, M.; Barbasz, J., Random packing of spheres in Menger sponge. the J. Chem. Phys. 2013, 
+138 (21), 214704. 
+46. 
+Cieśla, M.; Barbasz, J., Kinetics of random sequential adsorption of nearly spherically symmetric 
+particles. Physical Review E 2014, 89 (2), 022401. 
+47. 
+Palásti, I., On some random space filling problems. Publ. Math. Inst. Hung. Acad. Sci 1960, 5, 353-
+359. 
+48. 
+Shelke, P. B.; Limaye, A., Dynamics of Random Sequential Adsorption (RSA) of linear chains 
+consisting of n circular discs—Role of aspect ratio and departure from convexity. Surface Science 2015, 
+637, 1-4. 
+49. 
+Budinski-Petković, L.; Kozmidis-Luburić, U., Jamming configurations for irreversible deposition 
+on a square lattice. Physica A: Statistical Mechanics and its Applications 1997, 236 (3-4), 211-219. 
+50. 
+Schaaf, P.; Talbot, J., Kinetics of random sequential adsorption. physical rev. lett. 1989, 62 (2), 
+175. 
+51. 
+Speedy, R. J., Accurate theory of the hard sphere fluid. Journal of the Chemical Society, Faraday 
+Transactions 2: Molecular and Chemical Physics 1977, 73 (5), 714-721. 
+52. 
+Lebovka, N. I.; Karmazina, N. N.; Tarasevich, Y. Y.; Laptev, V. V., Random sequential adsorption 
+of partially oriented linear k-mers on a square lattice. Physical Review E 2011, 84 (6), 061603. 
+53. 
+Cieśla, M.; Barbasz, J., Ordering in fibrinogen layers: A numerical study. colloids Surf. B 
+Biointerfaces 2013, 110, 178-182. 
+54. 
+Barbasz, J.; Cieśla, M., Domain structure created by irreversible adsorption of dimers. arXiv 
+preprint arXiv:1301.4697 2013. 
+55. 
+Hansen, J.-P.; McDonald, I. R., Theory of simple liquids: with applications to soft matter. 
+Academic press: 2013. 
+56. 
+Richard, P.; Oger, L.; Troadec, J.-P.; Gervois, A., Geometrical characterization of hard-sphere 
+systems. Physical Review E 1999, 60 (4), 4551. 
+57. 
+Donev, A.; Torquato, S.; Stillinger, F. H., Pair correlation function characteristics of nearly jammed 
+disordered and ordered hard-sphere packings. Physical Review E 2005, 71 (1), 011105. 
+58. 
+Aste, T.; Di Matteo, T., Structural transitions in granular packs: statistical mechanics and statistical 
+geometry investigations. The European Physical Journal B 2008, 64 (3), 511-517. 
+59. 
+Schaaf, P.; Déjardin, P.; Johner, A.; Schmitt, A., Characteristic time scales for the adsorption 
+process for fibrinogen on silica. Langmuir 1992, 8 (2), 514-517. 
+60. 
+Mura-Galelli, M.; Voegel, J.; Behr, S.; Bres, E.; Schaaf, P., Adsorption/desorption of human serum 
+albumin on hydroxyapatite: a critical analysis of the Langmuir model. Proceedings of the National Academy 
+of Sciences 1991, 88 (13), 5557-5561. 
+61. 
+Moud, A. A., Asphaltene induced changes in rheological properties: A review. Fuel 2022, 316, 
+123372. 
+62. 
+Abbasi Moud, A.; Hatzikiriakos, S. G., Kaolinite colloidal suspensions under the influence of 
+sodium dodecyl sulfate. physics Fluid 2022, 34 (1), 013107. 
+
+63. 
+Moud, A. A., Recent advances in utility of artificial intelligence towards multiscale colloidal based 
+materials design. colloid Interface Sci. Commun. 2022, 47, 100595. 
+64. 
+Piette, J.; Moud, A. A.; Poisson, J.; Derakhshandeh, B.; Hudson, Z. M.; Hatzikiriakos, S. G., 
+Rheology of mature fine tailings. physics Fluid 2022, 34 (6), 063104. 
+ 
+ 
+ 
+
diff --git a/9dAyT4oBgHgl3EQf3Pmp/content/tmp_files/load_file.txt b/9dAyT4oBgHgl3EQf3Pmp/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..23e90c30f9a6c338b55b061382c8197172455626
--- /dev/null
+++ b/9dAyT4oBgHgl3EQf3Pmp/content/tmp_files/load_file.txt
@@ -0,0 +1,1013 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf,len=1012
+page_content='Examination of saturation coverage of short polymers using  random sequential adsorption algorithm  Aref Abbasi Moud1*  1 Department of Chemical and Petroleum Engineering, University of Calgary, 2500 University Dr NW,  Calgary, AB, T2N 1N4, Canada  Author to whom correspondence should be addressed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' electronic mail:  aabbasim@ucalgary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='ca  Abstract: We filled a void with a regular or asymmetric pattern without overlap using a time-dependent  packing method termed random sequential adsorption (RSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' In the infinite-time limit, the density of  coverage frequently hits a limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' This study focused on the saturation packing of squares and their dimers,  trimers, tetramers, pentamers, and hexamers, all of which were orientated in two randomly chosen  orientations (vertical and horizontal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Our results concurred with those of previous extrapolation-based  research1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' We used the "separating axis theorem" to detect if freshly added polygons and previously put  ones overlapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' When RSA insertion became disproportionately sluggish, we concluded that saturation  had been attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=" We also discovered that the system's capacity to fill the area decreased as squares were  stretched into dimers and trimmers." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' The microstructure of the resultant saturation was also thoroughly  investigated, including block function and the structural arrangement of dimers and trimers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Keywords: RSA, short polymers, separating axis theorem, trimers, and dimers  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Introduction  One of the most popular nonequilibrium packing models is the random sequential addition (RSA) packing  method, which is a time-dependent process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Unpredictable sphere packings are created using a method  similar to earlier established techniques;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' see References2-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' The RSA packing process in the three spatial  dimensions4 has been used to represent a range of scenarios, including protein adsorption5, polymer  oxidation, particles in cell membranes, and ion implantation in semiconductors2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Euclidean space (d-dimensional) particles with certain shapes are randomly and progressively introduced  into the volume subject with the restriction that they do not overlap to carry out the simulation starting from  a big, vacant zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' The freshly created particles are only maintained if they do not overlap any already  existing particles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' otherwise, they are deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' After the simulation has begun, this procedure can be stopped  at any time, making the density that has been acquired time dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Density reaches a "saturation" or  "jamming" limit as time goes on6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  In its simplest form, an RSA sphere packing may be obtained by randomly, irreversibly, and sequentially  adding nonoverlapping objects into a huge volume that is initially empty of spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' A further attempt is  made until the sphere can be added without doing so if an attempt to add a sphere (or any other polygon) at  time t overlaps with a sphere that is already present in the packing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=" An RSA configuration with a time-  dependent packing fraction can be created by selecting any moment in finite time t as the process' endpoint." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  The maximum saturation packing fraction, which occurs at the infinite-time and thermodynamic limits,  prevents this figure from becoming higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  RSA implementations fall into two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' The two fundamental categories into which the RSA models  may be split are continuum and lattice models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Based on object kinds, they are further split into two groups:  RSA of finite (nonzero) area objects and RSA of zero area objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Things having finite area in this sense  are those that enclose a certain amount of space, whereas those with zero area lack this geometric feature  of enclosure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Therefore, upon adsorption on the substrate, items with finite area take up some space,  whereas those with zero area take up no space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Lattice models are defined by the realisation of the jammed  state7, regardless of the item types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  In general, in case of RSA of finite area objects, the approach of instantaneous coverage 𝜃(𝑡) to the jammed  state coverage 𝜃𝑚𝑎𝑥 is found to follow a power law 𝜃𝑚𝑎𝑥 − 𝜃(𝑡)~𝑡−𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Researchers have proposed certain  laws about the value of the exponents p by researching the RSA of objects with a variety of geometries,  including circular, elliptical, rectangular, and sphero-cylindrical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Feder 8 is credited with being the first to  link the observed value of the exponent p in the RSA of circular objects in two dimensions to the object  dimensionality and to propose the general rule that p = 1/D for d-dimensional circles on a two-dimensional  continuum platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Swendsen 9 subsequently demonstrated that, if the items are placed with random  orientations, the same ought to apply for RSA of items of any arbitrary form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Since its invention by Feder 8 , Random Sequential Adsorption (RSA) has gained widespread acceptance  as a technique for simulating adsorption characteristics, particularly for spherical molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=" However,  employing RSA to replicate the adsorption of more complex particles like polymers or proteins raises  concerns about how RSA's inherent features alter when non-spherical molecules are involved." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' For simple  forms like spheroids, spherocylinders, rectangles, needles, and others, the subject has already been  addressed10-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Recent research, however, indicates that these geometries are insufficient for simulating the  adsorption of common proteins, such as fibrinogen, for instance13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=" As a result, researchers' focus has  recently turned to coarse-grained modelling of complex biomolecules and polymers14-16." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  In 1-D case, the saturation packing fraction can be obtained analytically as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='747597920 17 however for 2-  D and 3-D the saturation packing fraction for discs and spheres has been estimated only through numerical  simulations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' the most precise ones are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5470735 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='0000028 for 2-d and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='3841307 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='000002 for 3-D  cases2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Other  figures  reported  for  2-D  simulations  of  discs  are  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='54707352, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='54706718, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='54707019, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='547069020, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5470021,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5471122, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='547223,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5478,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='547924.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Similarly for 2-D aligned squares saturation coverage reported in the literature is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='56200925, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='562324,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5628, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='556526, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='562527, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='544428, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='562929, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='56230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  In this study, we used the RSA technique to determine the saturation packing limit for squares and its dimer  and trimers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' As the length of square increases, the results indicated that samples eventually generated  structures with less and less packing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' In this study, the "separating axis theorem" approach was used to  detect whether there was a collision between two polygons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' more information on the procedure is provided  in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Polygons here refer to squares that encompasses its polymers as well as a  constructing monomer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Our preliminary findings on RSA packing with respect to polygons, which we just  published31, are the basis for this study, which extends that work by extending the polygon (square) into its  polymers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Model and simulation procedure  When colloidal particles or molecules are being adsorbed, they frequently diffuse close to the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' This  process might lead to the formation of a film consisting of molecules that were randomly adsorbed because  of adhesion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Here, we focus on adsorbate monolayers formed by irreversible adsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' The most  straightforward technique for quantitatively modelling these processes is molecular dynamics (MD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' The  advantages of MD include accurate forecasting and management of most environmental factors, such as  temperature and the diffusion constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' The main issue is the performance deficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' As a result, we  decided to utilise a new method, continuum Random Sequential Adsorption (RSA), which has been  successfully employed to investigate colloidal and other systems32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  To simulate, a virtual particle was created (square, its polymers), and its location on an area was chosen at  random based on a uniform probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  The overlap with the previously adsorbed nearest neighbours of a virtual particle was tested (the topic of  the next section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' The result of this test tells us whether the surface-to-surface distance of a particle is greater  than zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  If there was no overlap, the virtual particle was adsorbed and added to an existing covering layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  If there was overlap, the virtual particle was dropped and abandoned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='1 Proposed algorithm  Numerous methods may be used to determine if two polygons intersect or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' One method for determining  if two polygons are overlapping uses mathematical equations and is known as the "separating axis  theorem"33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  The separating axis theorem states that if a line divides two convex polygons, they cannot intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' The  separation axis, which is a line, may be thought of as the normal to one of the edges of each polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content="  Using the separating axis theorem, the following procedures can be used to determine if two polygons cross: Determine each polygon A edge's edge normal, then project both polygons onto that value." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Establish the minimum and maximum projections of each polygon onto the normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' If the maximum projection of polygon A is less than the minimum projection of polygon B, or the  other way around, the polygons do not overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' If the projections overlap, repeat the procedure for each edge in polygon B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' If the projections of the polygons onto the separating axes do not overlap, the polygons are  connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  It is likely that non-convex polygons or polygons with holes will not be covered by this theorem, even  though the separating axis theory may be used to determine if two convex polygons overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' In some  situations, it could be necessary to verify for intersection using alternative techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Results and discussion  We construct saturated RSA configurations of polymers (dimer to hexamer) and compare saturation  packing with other findings reported in the literature, particularly in ref2, where authors employed a  different approach and orientation was random, to show the precision and utility of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' We  generate 1000 variations for each particle form using the system size that results in the fastest and densest  packing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Typical monolayer samples (from trimer’s sample) for three different coverages: θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='1, θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='3,  θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='4 and θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5 for trimers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' The collector side length was equal to 50[-].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Fixed boundary conditions  were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Figure shows truncated images of the distribution for a better visibility (20 by 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Using the greatest area (50 by 50) with fixed bounds, most of the results presented later in the study were  achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' We made sure that the adoption of periodic boundary conditions had no discernible impact on  the results that were made (Equivalent of periodic and fixed boundary condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Figure 1 shows the  outcomes of one of the simulations, in which trimers are placed in an area measuring 20 by 20 with a  monomer having a side length of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5 units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Particles are gradually added to the surface as the simulation  progresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  0=10%  0=30%  0=40%  0=50%  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Hexamer units put irrevocably inside a 50 by 50 space are the focus of the RSA algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' (a)  Asymptotic observation of coverage as a function of total simulation duration indicated by t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' (b) The  instantaneous time τ determined by based on coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' The line represents an exponential fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  To look at the development in more details, surface coverage was depicted as a function of simulation time  to the power of 𝑡−1/3 and results are shown in Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Clearly at long simulation times of ~104 coverage  very slowly reaches its asymptotic limit that is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='4968±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='0011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Similarly using same schemes, we arrived  at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5631±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='0002 surface coverage for squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' This surface coverage corresponds very well with the results  reported in the literature for squares 8, 24-30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  The number of attempts needed to add a new particle to the grid (or collector both terms in congruency with  literature has been used here interchangeably) can be known, and this information can be used to model the  blocking of further adsorption through monitoring time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Clearly, as more of the surface is covered, adding  new particles should be more difficult, which can be represented by a lower probability (See Figure 2b)  that is it takes considerably more amount of time for a particle to be added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Adsorption kinetics in a real  experiment typically depends on two variables: the effectiveness of the transport process (primarily  diffusion or convection depending on the experimental setup) that moves the adsorbate from the bulk to the  surface, and the likelihood of catching particles that are nearby 34-39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Authors in other reports40 have  concentrated on the second aspect in this case, which is defined by the blocking function, also known as  the available surface function (ASF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' The simulation makes it simple to obtain it as a ratio of successful  attempts to all RSA attempts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Equivalent to available surface function that is represented in shape of time  simulation is presented in Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Figure 2b shows simulation time as a function of coverage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  statistically, it is evident that more trials are required to attain adsorption because the surface is already  rather packed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' The exponential fit is, τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='0006 exp (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='03 𝜃), thus describing increasing time required to  place an additional point onto the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Discussion on ASF is subject of next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  The main objectives of this work were to determine the maximum random adsorption ratio for squares and  their polymers and compare it to the results for hard circles (spheres).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' That ratio ought to be provided for  an infinite grid area and adsorption duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Although one must deal with constrained simulation durations,  one must also manage the accuracy issue brought on by the finite grid size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Because it is unclear if there  would be any possibility of adsorption after the simulation time, particularly in the case of large grids, the  determination of maximal coverage depends on the RSA kinetics model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' As a result of prior research in the  region 9, 41-42, there have been a number of works in the area, and asymptotically:  𝜃𝑚𝑎𝑥 − 𝜃(𝑡)~𝑡−1  𝐷     eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5  4000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='45  3000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='4  2000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='35  1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='03Regarding the irreversible deposition of discs or squares that are not orientated (formerly known as p = -  1/D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=" Despite controversy, D here specifies the grid's dimension9." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' When adsorbed particles are organised,  the situation is altered9, 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Figure 2a previously in this post showed an example of the results of fitting  Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Asymptotic observations of coverage for squares seem to neatly match Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=" Although  Equation 1 hasn't been definitively proved, its validity has been vigorously defended12 by analytical and  numerical grounds." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=" It should be noted that Equation 1 simplifies to the standard Feder's law8 for isotropic  objects since n equals the number of dimensions." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  For instance, RSA of discs on a two-dimensional plane has D = 2, whereas RSA of rectangles, ellipses, and  other rigid but noticeably anisotropic structures has D = 3 37, 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' It appears that parameter D generally  correlates to the degrees of freedom of a number of packed objects, which has been validated for the random  packing of hyperspheres in higher dimensions 2, 21, not only the integral ones44-45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' The power law (Equation  1) is satisfied for the RSA of polymers examined here, however the exponent -1/D strongly relies on a  polymer length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' The parameter D is about equivalent to 3, which is the value recognised for anisotropic  molecules, for a small number of vertexes such as pentagon and squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' However, as number of vertices  increases parameter D converges to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' This finding is consistent with those made for the RSA of spherical  beads examined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' However, unlike what was shown in the cases of spherical beads46  or generalized  dimers 40, there is no abrupt transition between these two limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  The results are averaged across 10 simulation runs with time t in the order of 5 × 108 for each run in order  to determine parameter D for different polymers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=" These runs' data are not displayed here, and we will go  into more depth about the outcomes in our upcoming paper." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='1 RSA for polymers  In the last part, we laid the foundation for using the RSA approach to create oriented squares and trimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Results showing the behaviour of adsorption at asymptotic limits, the kinetics of the adsorption index (p),  and the relationship between simulation time and coverage were given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Additionally, results and discs were  compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Utilizing the extrapolation method shown in Figure 2a previously, Table 1 generates  saturation densities for various polymer lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Figure 3 displays a sample of saturation densities for  various forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  To arrive at the values reported in Table 1 following equation has been used:  𝜃(𝑡) = 𝜃𝑚𝑎𝑥 + 𝑏/𝑡𝑝     eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='2  When arriving at the values shown in table 1, we gave the data from longer simulation times more weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content="  The approach's possible downside is that each data point is given the same weighting factor, assuming all  values are given the same weight." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Because there are a lot more of these points in the higher part of the  asymptotic area, it is sort of underweighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Therefore, we investigated a novel strategy that introduces a  bias favouring the longer durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' These changes are in line with the accounts in ref12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Saturation density, index, for square-based polymers with a monomer to simulation box length  ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='01 and their respective 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Shape (oriented)  𝜃𝑚𝑎𝑥 [-] (95% confidence  bounds)  p [-]  (95%  confidence  bounds)  b [-]  (95%  confidence  bounds)  Square  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5631(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5629, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5633)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5138 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5125, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5151)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='226 (-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='475, -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='561)  Dimer  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='57 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5697, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5704)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
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+page_content='4595, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='4603)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='805 (-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='819, -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='792)  Trimers  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5621 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5612, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5629)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
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+page_content='4652, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='4668)  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='003 (-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='066, -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='941)  Tetramer  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5558 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
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+page_content='5577)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
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+page_content='5007)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='046 (-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
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+page_content='942)  Pentamer  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5504 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
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+page_content='4624)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
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+page_content='892)  Hexamer  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
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+page_content='501)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='264 (-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='458, -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='069)  Discs  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5470732 As outlined in introduction section, similarly for 2-D aligned squares saturation coverage reported in the  literature is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='56200925, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='562324, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5628, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='556526, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='562527, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='544428, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='562929, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='56230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Our values for  square are very well within range of values reported elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' However, as particles get longer and become  Trimers, saturation has dropped since longer particles require more accessible area for deposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' For dimer  and trimer, the greater aspect ratio of the dimer is projected to result in somewhat higher saturation for  dimers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' These findings are crucial because they suggest that it gets progressively harder for molecules to  adhere to surfaces as they become longer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' an example of superiority of simulation over experiment in giving  researcher a tool to examine parameters hard to measure through experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Results from this study can also be extrapolated to higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' For instance, the efficiency of a  sequential adsorption process with hard materials decreases with increasing size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' It is noteworthy to note  that, as a general rule, the saturation coverage in D dimensions is very well estimated by that in one  dimension raised to power D (for the RSA of spherical particles, 𝜃𝑚𝑎𝑥≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='75 for D = 1, 𝜃𝑚𝑎𝑥≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='55 for D  = 2, 𝜃𝑚𝑎𝑥 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='38 for D = 3, etc)47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Therefore, results obtained here can be extended to 1-D and 3-D cases  with good approximation, for instance for squares for cubes is predicted to lie around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='38 and in 1-D case  around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Square, dimers, and trimers near saturation points for samples with monomer’s side length of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5  and distributed within area of 50 by 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Fixed boundary condition has been applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Figure shows truncated  images of the distribution for a better visibility (20 by 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' For improved visibility, the horizontally oriented  polymers have been coloured blue, while the vertically oriented ones have been painted red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Clearly visually samples experience a bit higher coverage for dimers and less coverage for trimers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Square  Dimers  Trimers  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=" RSA saturation density for polymers, with a line created to guide the viewer's eyes." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Each data  point has an almost imperceptible error bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' For discs, RSA saturation density (red dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=" All the  polymers in Table 1's saturation density changes as a function of simulation duration (a) The dependence  of 𝜃𝑚𝑎𝑥 on length of the polymer (c) Kinetic index as a function of the monomer length." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Figure 4 shows the RSA saturation density as a function of polymer length together with an eye-guiding  line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=" The saturation density coverage first increases somewhat as the polymer's length rises, but as it  continues to grow, it starts to decline." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' This outcome is in line with the outcomes for rectangles with various  aspect ratios and ellipses that have been reported in the literature5, 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Additionally, we discovered that  saturation is somewhat lower for polymers with aspect ratios longer than 2, and we hypothesise that this is  because, as was already noted, longer particles are more difficult to pack efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  The random coverage ratio dropped exponentially with polymer size in earlier investigations refs43, 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' On  a continuous surface, at least two competing variables can affect the maximum random coverage ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  First, there is less chance of finding a large enough uncovered section to place on a collector, making it  harder to separate larger particles than in the lattice case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' The second point is that a polymer globule is more  likely to form a cluster when necessary because it has a greater monomer packing ratio than a group of  individual monomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' For continuous collectors as opposed to lattice ones, this second element is more  important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
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+page_content='55  Sguare  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5  Dimer  Trimer  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='45  Tetramer  A  Hexame  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='4  a  b  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='35  3  4  5  2  X10 4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='65  Monomerlength  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='6  DiscS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='35  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='2 Block function  Knowing how many tries are necessary to add a new particle to the grid allows us to simulate how further  adsorption is blocked over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' It is obvious that when more of the surface is covered, adding more  particles should become more challenging, which may be represented by a decreased probability (as shown  in Figure 2b, which depicts the same trend with time as the dependent variable)  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' The ratio of successful attempts to blocking attempts versus coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' The green line represents  simulation data in b and the dotted line represent polynomial fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' details of the fits are given in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Equation 3 describe the function with a simple second order polynomial perfectly describing the decreasing  trend in probability of a successful adsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  𝐴𝑆𝐹(𝜑) = 𝐶1𝜃2 + 𝐶2𝜃 + 𝐶3 eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='3  In which C1, C2 and C3 are pre-factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Details of the fit of equation 3 is given in the Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Figure 5  shows that the likelihood of adsorption for trimers drops more quickly than for dimers and squares, and the  dimer with respect to the square exhibits the same pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' This is because consecutive adsorption is much  less likely to occur quickly in trimers and dimers than in squares due to their greater aspect ratio and unusual  orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Represents polynomial fit to the data in Figure 5 along with corresponding 95% confidence  bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Shape (oriented)  C1 [-] (95% confidence  bounds)  C2 [-]  (95%  confidence  bounds)  C3 [-]  (95%  confidence  bounds)  Square  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5439 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5296, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5582)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='244 (-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='252, -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='237)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='008 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='007, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='009)  Dimer  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='508 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='484, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='532)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='18 (-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
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+page_content='9986)  In the case of square, dimers, trimers simulations show that C1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5439-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='572 and C2 =-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='244-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='038,  whereas those parameters for hard circles adsorption are analytically derived as C2 =-4 and C1 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='3150 (we  obtained coefficients of C1=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='426  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='071, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='782) and C2=-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='907  (-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='644, -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='169) with aid of our  simulation) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Contrary to discs, the coefficient for squares suggests that they have an easier time adhering  to the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Higher saturation coverage for squares is another manifestation of this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Therefore, dimers values are very close to the values reported for hard discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  According to Figure 5, as coverage levels increase, we get closer to the asymptotic phase, where dynamics  are well understood, and finally the jamming limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Thus, the RSA procedure has now been completely  explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Since it is challenging to conduct adsorption investigations near to the jamming limit51,  measuring the terms of the RSA process is the most effective technique to show that adsorption follows an  RSA process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=" It's crucial to remember that words up to 𝜃2 don't reveal anything about the properties of the  adsorption process (i." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', the degree of irreversibility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' This implies that any experiment that involves the  adsorption of particles that resemble hard discs is susceptible to such an extension (to second order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Our  strategy is also applicable to combinations and non-circular particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='3 Ordering and orientation  We study the presence of any orientational order in a monolayer using the amorphous form of a polymer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Although most of these studies have used a collector surface lattice structure, as in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' 52, such ordering has  been well investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' It could also have an impact on the RSA kinetics mentioned before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Based on the  polymer structure, we offer the following function to measure orientational order in our continuous system:  𝑆(𝜑) =  1  𝑁 ∑  (𝑥𝑖 cos(𝜑) + 𝑦𝑖 sin(𝜑))  𝑁  𝑖=1  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='4  where (xi, yi) are positions along the i-th molecule in a layer for a unit vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' It is clear that 𝑆(𝜑) is an  average scalar product between both the orientation of molecules and the direction determined by an angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  As a result, for a perfectly aligned layer, 𝑆(𝜑) will swing between 0 and 1, with highest values for angles  parallel to molecules and minimum values for angles perpendicular to the alignment direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' 𝑆(𝜑) will  always be a constant and equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5 for pure random alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  For trimers as coverage increases across simulation time, ordering hovered 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='49, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='53,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='50,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='51 as surface  coverage increased from 10 to 30, 40 and 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Clearly ordering in trimer population is very close to random  due to simulation being designed to give equal chance to parallel or vertical orientation of trimers as shown  in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Situation is very similar for dimers as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Figure 4 illustrates how ordering in trimers may be further examined as a function of the radius of the  particle neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Figure 4 was made using a similar idea to the pair correlation function by treating the  trimer centre as a circle with a radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' 𝑆(𝜑)  fluctuates in small regions because dense clusters of  horizontally or vertically oriented trimers are more likely to form, but as the sweeping radius grows, this  fluctuation decreases to a value that is very similar to a randomly oriented arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Dimers also face a  similar set of circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' As a result, at r5, local order in each system vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Small amplitude  fluctuations continue after r=5, although their amplitudes and frequency are higher for trimers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Local ordering as a function of radius for two RSA packings made with dimers and trimers near  their saturation coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' (a) dimers (b) trimers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Similar trends are expected for tetramers due to similarity of behavior we have refrained from exploring  them further here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' As an example, liquid crystals are one type of orientationally organised structure that an  elongated particle (such as polymers here with aspect ratio>2) can produce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' When particle orientations are  chosen at random using a uniform probability distribution for RSA on an infinite collector, the global  orientational order is not expected to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' However, since parallelly aligned particles take up less space,  the formation of local ordered domains is feasible53-54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='4 Radial distribution function  The radial distribution function (G(r)), also known as the pair correlation function, in a system of particles  (such as atoms, molecules, colloids, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=') explains how density changes in response to distance from a  reference particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' G(r) is the radial distribution function 55 obtained from following equation:  𝐺(𝑟) =  1  𝜌 〈∑  𝛿(𝑟 − 𝑟𝑖)  𝑖≠0  〉 eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content="5  The monolayer's first crucial characteristic is the particle autocorrelation." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Squares are assumed to have a  radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='25 and to be treated equally regardless of whether they are made of the same polymer or a  different one in order to compute G(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Figure 5 displays the average structures seen by various RSA  packings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' We consistently saw a peak at a distance of r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5 (right on the edge of the particles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' In other  words, the function reaches its maximum for the closest neighbour, r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5, and then begins to degrade  because of the volume that is lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' The similar trend is seen in the trimer and hexamer, but there are  more peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Hexamer contains additional peaks, for instance, at r=1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5, and 2, while the trimer exhibits  an additional peak at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  As the radius gets bigger, these peaks get weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=" Due to the coverage's randomness, these oscillations  superexponentially vanish21, and after normalisation, the function stabilises at a value of 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' In addition,  when the number of monomers inside the polymer rises, the first peak corresponding to the nearest  neighbour grows progressively sharper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' According to this behaviour, particles that may be seen as a chain  of squares pack more well even if the saturation coverage is smaller for monomers (squares) and short  oligomers (dimers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5  b  a  S( Φ),Dimers  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='- S( Φ),Trimers  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5  +  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5  :  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  L  10  15  20  25  5  10  15  20  25  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Functions of autocorrelation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Behavior autocorrelation function is depicted as a function of radius  for square, dimer, trimer, tetramer, pentamer and hexamer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  The average structure seen by a generic particle of the system described by G(r) displayed in Figure 5,  shows a full agreement with the predicted theoretical regimes found in literature 56-57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' In all cases, we  observe a pronounced peak at a distance r~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5, with the sphere diameter that corresponds to the distance of  the nearest neighbors in contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' For r larger than the diameter, the probability to find neighbors decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  In fluid-like systems, theoretically for 𝜑 ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='55, 56-58 the G(r) is known to oscillate with decreasing  amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Conclusions  For a range of stiff polymers produced using squared monomers, we show the maximum random coverage  or saturation coverage in this paper and contrast our findings with those reported in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' In order  to do this, we enhanced an algorithm that was described in Ref 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' We prove the validity of our method by  calculating the RSA saturation densities of polymers (dimer, trimer, and tetramer) and showing their  consistency with prior findings in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  The RSA model shown here may be extended to squares that may change into rectangles with larger aspect  ratios to incorporate anisotropic particles in future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Moreover, like ref43 it can also include branched  or more flexible polymers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Biological molecules are usually non-spherical, as seen by the previous example,  and when their surface area in contact with the substrate is greatest, they firmly cling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' According to  experimental results, Schaaf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' 59 discovered that the maximum substrate coverage they were able to  achieve during the adsorption of fibrinogen—a non-spherical protein with an aspect ratio of roughly 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='5—  was only about 40%, which was lower than the absorption coverage predicted by the RSA of hard discs—  which is around 55%—and seen in experiments involving reasonably spherical globular proteins8 (A similar  impact was noted for albumin adsorption 60 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Here are some pertinent queries:  How does increasing the aspect ratio affect the saturation coverage of the substrate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  How does the particle shape impact the kinetics over both short and long time periods?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Square  5  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Trimer  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='Hexamer  6  8What are the similarities and differences between equilibrium configurations produced by RSA and  configurations with equivalent surface coverage?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  These questions will get their solutions in upcoming publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=" This study's findings are pertinent since  they considered a variety of particle morphologies, including those of asphaltene, graphene, cellulose  nanocrystals, and kaolinite, among others 61-64." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  The findings are important because they might help to understand how polymers behave when they are  close to surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' For instance, numerous biological processes depend on proteins adhering to different  surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Understanding and having control over how protein molecules attach to surfaces and interact with  them is essential when creating biomaterials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=" For instance, among other things, the production of  biocompatible materials requires decreasing the adsorption of blood proteins to the material's surface." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' It is  generally known that platelet adhesion followed by blood protein adsorption can result in surface-induced  thrombosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' When protein adsorption is prevented or diminished, there is very little platelet adhesion to the  surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Eliminating lysozyme buildup from the surface of contact lenses is another illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' In other  circumstances, we would like to promote the adsorption of some proteins while inhibiting the adsorption  of others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Conflict of interest statement: Author declares no conflict of interest  Data availability statement: The datasets generated during and/or analysed during the current study are  not publicly available but are available from the corresponding author on reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Precise algorithm to generate random sequential adsorption of hard polygons at  saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Physical Review E 2018, 97 (4), 043311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Torquato, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Precise algorithm to generate random sequential addition of hard  hyperspheres at saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Physical Review E 2013, 88 (5), 053312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Stillinger, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Torquato, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Transport, geometrical, and topological properties of  stealthy disordered hyperuniform two-phase systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' the J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' 2016, 145 (24), 244109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Torquato, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Perspective: Basic understanding of condensed phases of matter via packing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  the J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' 2018, 149 (2), 020901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Petrone, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Cieśla, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Random sequential adsorption of oriented rectangles with random aspect  ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Physical Review E 2021, 104 (3), 034903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Abbasi Moud, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Colloidal and Sedimentation Behavior of Kaolinite Suspension in Presence of  Non-Ionic Polyacrylamide (PAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Gels 2022, 8 (12), 807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Shelke, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Random sequential adsorption of n-star objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Surface Science 2016, 644, 34-40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Feder, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Giaever, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Adsorption of ferritin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' journal colloid interface sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' 1980, 78 (1), 144-154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Swendsen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Dynamics of random sequential adsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Physical Review A 1981, 24 (1), 504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Talbot, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Tarjus, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Schaaf, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Unexpected asymptotic behavior in random sequential adsorption  of nonspherical particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Physical Review A 1989, 40 (8), 4808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Vigil, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Ziff, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Random sequential adsorption of unoriented rectangles onto a plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' the  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' 1989, 91 (4), 2599-2602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Viot, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Tarjus, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=", Random sequential addition of unoriented squares: breakdown of Swendsen's  conjecture." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' EPL (Europhysics Letters) 1990, 13 (4), 295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Adamczyk, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Barbasz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Cieśla, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Kinetics of fibrinogen adsorption on hydrophilic substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Langmuir 2010, 26 (14), 11934-11945.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Adamczyk, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Barbasz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Cieśla, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Mechanisms of fibrinogen adsorption at solid substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Langmuir 2011, 27 (11), 6868-6878.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Rabe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Verdes, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Seeger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Understanding protein adsorption phenomena at solid surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  advances Colloid Interface Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' 2011, 162 (1-2), 87-106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Finch, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Hickman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' In A continuum hard-sphere model of protein adsorption, Proceedings  of the ACM Conference on Bioinformatics, Computational Biology and Biomedicine, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' pp 573-575.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Rényi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', On a one-dimensional problem concerning space-filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Hungar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' 1958, 3, 109-127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Cieśla, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Ziff, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Boundary conditions in random sequential adsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Journal of Statistical  Mechanics: Theory and Experiment 2018, 2018 (4), 043302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Cieśla, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Nowak, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Managing numerical errors in random sequential adsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Surface  Science 2016, 651, 182-186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', A fast algorithm for random sequential adsorption of discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' International Journal of  Modern Physics C 1994, 5 (04), 707-715.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Torquato, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Uche, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Stillinger, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Random sequential addition of hard spheres in high Euclidean  dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Physical Review E 2006, 74 (6), 061308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Chen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Holmes-Cerfon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Random sequential adsorption of discs on constant-curvature  surfaces: plane, sphere, hyperboloid, and projective plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' arXiv preprint arXiv:1709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='05029 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Hinrichsen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Feder, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Jøssang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Random packing of disks in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Physical  Review A 1990, 41 (8), 4199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Series expansion and computer simulation studies of random sequential adsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  colloids Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Physicochem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Asp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' 2000, 165 (1-3), 325-343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Brosilow, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Ziff, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Vigil, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Random sequential adsorption of parallel squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Physical Review A 1991, 43 (2), 631.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Blaisdell, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Solomon, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', On random sequential packing in the plane and a conjecture of  Palasti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Journal of Applied Probability 1970, 7 (3), 667-698.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Dickman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Jensen, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Random sequential adsorption: Series and virial expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  the J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' 1991, 94 (12), 8252-8257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Tory, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Jodrey, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Pickard, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Simulation of random sequential adsorption: Efficient  methods and resolution of conflicting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Journal of Theoretical Biology 1983, 102 (3), 439-445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Akeda, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Hori, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', On random sequential packing in two and three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Biometrika 1976,  63 (2), 361-366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Jodrey, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Tory, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Random sequential packing in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Journal of statistical computation and  simulation 1980, 10 (2), 87-93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Moud, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Examination of saturation coverage of polygons using random sequential adsorption  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' arXiv preprint arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='13849 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Adamczyk, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Siwek, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Zembala, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Weroński, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Influence of polydispersity on random  sequential adsorption of spherical particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' journal colloid interface sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' 1997, 185 (1), 236-244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Gottschalk, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Lin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Manocha, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' In OBBTree: A hierarchical structure for rapid  interference detection, Proceedings of the 23rd annual conference on Computer graphics and interactive  techniques, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' pp 171-180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Adamczyk, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Weroński, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Random sequential adsorption of spheroidal particles: kinetics and  jamming limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' the J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' 1996, 105 (13), 5562-5573.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Schaaf, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Talbot, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Surface exclusion effects in adsorption processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' the J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' 1989,  91 (7), 4401-4409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Adamczyk, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Siwek, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Zembala, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Kinetics of localized adsorption of particles on  homogeneous surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' journal colloid interface sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' 1992, 151 (2), 351-369.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Viot, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Tarjus, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Ricci, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Talbot, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Random sequential adsorption of anisotropic particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Jamming limit and asymptotic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' the J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' 1992, 97 (7), 5212-5218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Evans, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Random and cooperative sequential adsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Reviews of modern physics 1993, 65  (4), 1281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Talbot, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Tarjus, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Van Tassel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Viot, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', From car parking to protein adsorption: an overview  of sequential adsorption processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' colloids Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Physicochem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Asp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' 2000, 165 (1-3), 287-324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Cieśla, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Properties of random sequential adsorption of generalized dimers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Physical Review E  2014, 89 (4), 042404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Hinrichsen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Feder, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Jøssang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Geometry of random sequential adsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Journal of  statistical physics 1986, 44 (5), 793-827.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Privman, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Nielaba, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Continuum limit in random sequential adsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Physical  Review B 1991, 43 (4), 3366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Cieśla, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Continuum random sequential adsorption of polymer on a flat and homogeneous  surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Physical Review E 2013, 87 (5), 052401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Ciesla, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Barbasz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Random sequential adsorption on fractals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' the J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' 2012, 137  (4), 044706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Cieśla, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Barbasz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Random packing of spheres in Menger sponge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' the J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' 2013,  138 (21), 214704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Cieśla, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Barbasz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Kinetics of random sequential adsorption of nearly spherically symmetric  particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Physical Review E 2014, 89 (2), 022401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Palásti, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', On some random space filling problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Hung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Sci 1960, 5, 353-  359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Shelke, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Limaye, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Dynamics of Random Sequential Adsorption (RSA) of linear chains  consisting of n circular discs—Role of aspect ratio and departure from convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Surface Science 2015,  637, 1-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Budinski-Petković, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Kozmidis-Luburić, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Jamming configurations for irreversible deposition  on a square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Physica A: Statistical Mechanics and its Applications 1997, 236 (3-4), 211-219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Schaaf, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Talbot, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Kinetics of random sequential adsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' physical rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' 1989, 62 (2),  175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Speedy, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Accurate theory of the hard sphere fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Journal of the Chemical Society, Faraday  Transactions 2: Molecular and Chemical Physics 1977, 73 (5), 714-721.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Lebovka, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Karmazina, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Tarasevich, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Laptev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Random sequential adsorption  of partially oriented linear k-mers on a square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Physical Review E 2011, 84 (6), 061603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Cieśla, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Barbasz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Ordering in fibrinogen layers: A numerical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' colloids Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' B  Biointerfaces 2013, 110, 178-182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Barbasz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Cieśla, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Domain structure created by irreversible adsorption of dimers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' arXiv  preprint arXiv:1301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='4697 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Hansen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' McDonald, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Theory of simple liquids: with applications to soft matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Academic press: 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Richard, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Oger, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Troadec, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Gervois, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Geometrical characterization of hard-sphere  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Physical Review E 1999, 60 (4), 4551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Donev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Torquato, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Stillinger, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Pair correlation function characteristics of nearly jammed  disordered and ordered hard-sphere packings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Physical Review E 2005, 71 (1), 011105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Aste, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Di Matteo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Structural transitions in granular packs: statistical mechanics and statistical  geometry investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' The European Physical Journal B 2008, 64 (3), 511-517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Schaaf, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Déjardin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Johner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Schmitt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Characteristic time scales for the adsorption  process for fibrinogen on silica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Langmuir 1992, 8 (2), 514-517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Mura-Galelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Voegel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Behr, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Bres, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Schaaf, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Adsorption/desorption of human serum  albumin on hydroxyapatite: a critical analysis of the Langmuir model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Proceedings of the National Academy  of Sciences 1991, 88 (13), 5557-5561.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Moud, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Asphaltene induced changes in rheological properties: A review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Fuel 2022, 316,  123372.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Abbasi Moud, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Hatzikiriakos, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Kaolinite colloidal suspensions under the influence of  sodium dodecyl sulfate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' physics Fluid 2022, 34 (1), 013107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Moud, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=', Recent advances in utility of artificial intelligence towards multiscale colloidal based  materials design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' colloid Interface Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' 2022, 47, 100595.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content='  Piette, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Moud, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Poisson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Derakhshandeh, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Hudson, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' Hatzikiriakos, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=',  Rheology of mature fine tailings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
+page_content=' physics Fluid 2022, 34 (6), 063104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQf3Pmp/content/2301.00766v1.pdf'}
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+Broadband three-mode converter and multiplexer based on 
+cascaded symmetric Y-junctions and subwavelength engineered 
+MMI and phase shifters 
+David González-Andradea,*, Irene Olivaresb, Raquel Fernández de Caboc, Jaime Vilasb, Antonio 
+Diasb, Aitor V. Velascoc 
+a Centre de Nanosciences et de Nanotechnologies, CNRS, Université Paris-Saclay, Palaiseau 91120, France 
+b Alcyon Photonics S.L., Madrid 28004, Spain 
+c Instituto de Óptica Daza de Valdés, Consejo Superior de Investigaciones Científicas (CSIC), Madrid 28006, Spain 
+* Corresponding author: david.gonzalez-andrade@c2n.upsaclay.fr 
+ARTICLE INFO 
+ 
+Keywords: 
+Silicon photonics 
+Mode-division 
+multiplexing 
+Subwavelength 
+metamaterial 
+MMI coupler 
+Phase shifter 
+Y-junction 
+ABSTRACT
+ 
+Mode-division multiplexing has emerged as a promising route for increasing 
+transmission capacity while maintaining the same level of on-chip integration. Despite 
+the large number of on-chip mode converters and multiplexers reported for the silicon-
+on-insulator platform, scaling the number of multiplexed modes is still a critical 
+challenge. In this paper, we present a novel three-mode architecture based on 
+multimode interference couplers, passive phase shifters and cascaded symmetric Y-
+junctions. This architecture can readily operate up to the third-order mode by including 
+a single switchable phase shifter. Moreover, we exploit subwavelength grating 
+metamaterials to overcome bandwidth limitations of multimode interference couplers 
+and phase shifters, resulting in a simulated bandwidth of 161 nm with insertion loss 
+and crosstalk below 1.18 dB and -20 dB, respectively.
+1. Introduction 
+The relentless growth of global Internet traffic has been 
+driven in recent years by the emergence of data-hungry 
+services and their mass adoption by an increasingly 
+interconnected society [1-3]. Moreover, the cloud nature 
+of many new applications such as machine learning or 
+artificial intelligence require large data sets to be 
+processed on internal servers or transferred between data 
+centers. This resource-intensive paradigm for accessing, 
+computing, and storing data has led to the creation of 
+hyperscale data centers consisting of thousands of 
+servers located in the same physical facility [4]. To cope 
+with the resulting zetta scale of annual data flow, modern 
+data centers have been relying on optical technologies for 
+both long-haul and few-meter interconnects. Compared 
+to 
+their 
+electronic 
+counterparts, 
+these 
+optical 
+technologies offer higher processing speeds, broader 
+bandwidths and lower latency and energy consumption. 
+Silicon photonics, leveraging the mature fabrication 
+facilities of the microelectronics industry, plays a key 
+role in the optical interconnect industry due to its 
+capacity for high-yield and low-cost mass production of 
+high-performance optoelectronic circuits [5,6]. 
+However, the development of next-generation 
+datacenters for Tbps communications and exascale 
+computing systems 
+is not feasible by scaling 
+infrastructures alone and requires increasingly efficient 
+optical interconnects for short-reach distances [7]. As 
+single-mode transmission approaches its fundamental 
+limits, space-division multiplexing has emerged as a 
+promising way to further improve the transmission 
+capacity of optical interconnects through the use of 
+multicore or multimode waveguides [8]. The latter, 
+which is also called mode-division multiplexing (MDM), 
+has attracted an increasing interest as it leverages the 
+orthogonality of the eigenmodes supported by a single 
+multimode waveguide, thus allowing to maintain the 
+same level of on-chip integration [9,10]. That is, MDM 
+enables encoding different data channels into specific 
+spatial modes, increasing capacity proportionally to the 
+number of modes used. 
+Numerous 
+on-chip 
+mode 
+converters 
+and 
+multiplexers/demultiplexers 
+(MCMD) 
+have 
+been 
+proposed for the silicon-on-insulator (SOI) platform to 
+date. Asymmetric Y-junctions are based on the principle 
+of mode evolution in adiabatic structures, which results 
+in broad operating bandwidths but also in long device 
+lengths [11-13]. The minimum feature size of current 
+lithography processes also has a significant impact in 
+these devices, since the finite resolution at which the tip 
+can be fabricated severely hampers their performance. 
+Asymmetric directional couplers (ADCs) [14], relying 
+on evanescent coupling between adjacent waveguides, 
+are well suited for implementing high-channel count 
+MDM systems, but they typically exhibit narrow 
+bandwidths, and their performance is highly susceptible 
+to fabrication errors. Adiabatic tapers have been 
+employed in the coupling region of ADCs to improve the 
+bandwidth and the resilience against fabrication 
+deviations [15]. MCMDs building upon multimode 
+interference (MMI) couplers and other auxiliary 
+
+components such as phase shifters (PSs) and symmetric 
+Y-junctions have been proposed as well [16,17], yielding 
+low losses and low crosstalk over a relatively broad 
+wavelength range (~100 nm). 
+The patterning of silicon at the subwavelength scale 
+has proven to be a simple yet powerful tool to tailor the 
+medium optical properties while inhibiting diffractive 
+effects [35]. More specifically, subwavelength (SWG) 
+metamaterials 
+can 
+behave 
+as 
+a 
+homogeneous 
+metamaterial that provides flexible dispersion and 
+anisotropy engineering, non-feasible in conventional 
+strip and rib waveguides. These properties have led to the 
+realization 
+of 
+Si 
+devices 
+with 
+unprecedented 
+performance over the past 15 years [36-38]. In the MDM 
+field, MCMDs based on subwavelength pixelated 
+structures have demonstrated ultra-compact footprints 
+[18]. SWGs have also been applied to ADCs and triple-
+waveguide couplers to improve fabrication tolerances 
+and extend the operation bandwidth of conventional 
+counterparts [19-21]. Furthermore, low losses and low 
+crosstalk within ultra-broad bandwidths have also been 
+reported using subwavelength engineered MMI couplers 
+and PSs, and SWG-slot-assisted adiabatic couplers [22-
+26]. 
+Despite the large number of available two-mode 
+MCMDs, scaling the number of multiplexed modes 
+beyond the fundamental and first-order modes is of great 
+importance to multiply the capacity of next-generation 
+datacom systems. Although it is fairly straightforward to 
+extend operation to a larger number of modes in 
+asymmetric Y-junctions and conventional and tapered 
+ADCs [27], three- and four-mode MCMD based on MMI 
+couplers have only recently been reported [28-32]. 
+However, the proposed architectures are still limited by 
+narrow operating bandwidths and high crosstalk values. 
+In this work, we propose a novel MCMD architecture 
+based on a 4×4 MMI, three phase shifters and four 
+symmetric 1×2 Y-junctions arranged in a conventional 
+cascaded configuration. The device operates as a three-
+mode MCDM with passive phase shifters but can readily 
+convert up to the third-order mode by including a single 
+switchable phase shifter. Moreover, we demonstrate loss 
+and crosstalk reduction in a broad bandwidth by SWG-
+engineering of both the MMI coupler and phase shifters. 
+Simulations show operation bandwidth of 161 nm with 
+insertion loss and crosstalk below 1.18 dB and -20 dB, 
+respectively. 
+2. Principle of operation and device design 
+To explain the operation principle and the device design, 
+let us first focus on the nanophotonic structure shown in 
+Fig. 1(a) consisting of a conventional 4×4 MMI, three 
+phase shifters (PS1, PS2 and PS3) and four symmetric 
+1×2 Y-junctions (three identical Y1 and a different one 
+Y2). SWG enhancement of the proposed architecture, 
+shown in Fig. 1(b) and Fig. 1(d) will be discussed in 
+epigraphs 4 and 5. An SOI platform with a thin Si wire 
+surrounded by SiO2 bottom layer and upper cladding are 
+considered. A schematic view of the waveguide cross-
+section is shown in Fig. 1(c) for clarity. 
+In order to illustrate the operation of the MCMD, let 
+us focus on the mode evolution and phase relations in 
+each individual constituent of the MCMD. Here, we aim 
+ 
+Fig. 1. Three-dimensional schematic of the proposed three-mode converter and multiplexer/demultiplexer comprising a 4×4 MMI, three 
+phase shifters and four symmetric Y-junctions implemented with (a) conventional homogeneous and (b) SWG metamaterial waveguides. 
+(c) Cross-sectional view of the SOI strip waveguides with a SiO2 cladding. (c) Top view of the SWG waveguides with their main 
+geometrical parameters. 
+
+(a)
+Wi
+MMI
+PS2
+W
+Y1
+3.
+2W,
+WA
+Y2
+Wps
+tWs
+Y1
+2W,
+P
+2
+PS1<Lpsi
+WMMI
+Y1
+4W1
+LpS2
+4
+2 W,
+PS3
++ Lyl
+LpS3
+1
+Ly2
+Wi
+Wi
+Lyl
+Li
+LMMI
+PS3
+Z
+X
+(b)
+SWG MMI Ws
+WI+
+SPS2
+SPS3
+WR3
+Y1
+D2
+2Wi
+3
+Y2
+Y1
+D
+2W,
+2
+WR2
+Y1
+4WI
+WRI
+4
+2Wi
+Lyl
+Ly2
+Wi
+Ly1
+SPS1
+LsT LsMMI
+LSPS3
+1
+(c)
+(d)
+D
+Si
+H
+Z4
+yt
+SiO2
+y
+W
+xat mode conversion and multiplexing of the first four 
+modes for transverse-electric (TE) polarization, that is, 
+the fundamental mode (TE0), the first-order mode (TE1), 
+the second-order mode (TE2) and the third-order mode 
+(TE3). 
+Our MCMD includes two types of symmetric 
+multimode 1×2 Y-junctions: Y1, with a stem supporting 
+up to two modes; and Y2, with a wider stem supporting 
+up to four modes. In general, multimode symmetric 1×2 
+Y-junctions transform the two in-phase 𝑚𝑡ℎ-order modes 
+in the arms into the (2𝑚)𝑡ℎ-order mode in the stem when 
+𝑚 is even, and into the (2𝑚 + 1)𝑡ℎ-order mode in the 
+steam when 𝑚 is odd [33]. Likewise, two anti-phase 
+𝑚𝑡ℎ-order modes in the arms are transformed into the 
+(2𝑚 + 1)𝑡ℎ-order mode in the stem when 𝑚 is even, and 
+into the (2𝑚)𝑡ℎ-order mode in the stem when 𝑚 is odd. 
+Figure 2(a) illustrates how this principle affects Y1 
+operation. Since only two modes are supported by the Y1 
+stem, a TE0 (red) mode at the stem results in two in-phase 
+TE0 modes at the arms, whereas TE1 (orange) mode at 
+the stem results in two anti-phase TE0 modes at the arms. 
+Figure 2(b) shows the extension of this behavior to four 
+mode operation in Y2. Operation for TE0 (red) and TE1 
+(orange) is the same as in Y1, whereas injection of TE2 
+(green) and TE3 (purple) modes through the stem 
+waveguide generates two anti-phase TE1 or two in-phase 
+TE1 modes at the arms, respectively. Therefore, by 
+cascading Y1 and Y2, and judiciously tailoring the phase 
+relations induced by the rest of the MCMD, mode 
+conversion and multiplexing between up to four modes 
+can be achieved. We will hence study the phase shift 
+induced by the 4×4 MMI coupler, and subsequently 
+design a phase shifter architecture that satisfies the phase 
+distributions imposed by the cascaded Y-junctions. 
+Bachmann et al. already derived a set of equations to 
+calculate the phase relations of 𝑁×𝑁 MMI couplers [34]. 
+At this point, it is important to mention that the definition 
+of the phase in this work is 𝜑 = 𝛽𝑥 − 𝜔𝑡, where 𝛽 is the 
+phase constant (also known as propagation constant), 𝑥 
+is the propagation direction and the term −𝜔𝑡 
+corresponds to the temporal dependence. As in [34] the 
+authors used the opposite phase convention, i.e., = 𝜔𝑡 −
+𝛽𝑥, equations can be rewritten as follows: 
+𝑖 + 𝑗 even: 𝜑𝑖𝑗 = −𝜑0 − 𝜋 −
+𝜋(𝑗−𝑖)(2𝑁−𝑗+𝑖)
+4𝑁
+, 
+(1) 
+𝑖 + 𝑗 odd: 𝜑𝑖𝑗 = −𝜑0 −
+𝜋(𝑗+𝑖−1)(2𝑁−𝑗−𝑖+1)
+4𝑁
+, 
+(2) 
+where 𝜑0 is a constant phase, 𝑖 and 𝑗 are the indices of 
+the 𝑁 inputs and outputs, respectively. Using Eqs. (1) and 
+(2), the phase relations of a 4×4 MMI coupler can be 
+calculated as shown in Table 1. Please note the input and 
+output numbering in Fig. 3. 
+Table 1 
+Calculated phase relations 𝝋𝒊𝒋 of a 4×4 MMI coupler. 
+𝒋 
+𝒊 
+1 
+2 
+3 
+4 
+1 
+−𝜋 
+−3𝜋 4
+⁄  
+−7𝜋 4
+⁄  
+−𝜋 
+2 
+−3𝜋 4
+⁄  
+−𝜋 
+−𝜋 
+−7𝜋 4
+⁄  
+3 
+𝜋 4
+⁄  
+−𝜋 
+−𝜋 
+−3𝜋 4
+⁄  
+4 
+−𝜋 
+𝜋 4
+⁄  
+−3𝜋 4
+⁄  
+−𝜋 
+ 
+We then calculate, for each input port, the resulting 
+phase difference at the two upper output ports (∆𝜑12) and 
+the two lower ports (∆𝜑34) as: 
+∆𝜑12 = 𝜑𝑖1 − 𝜑𝑖2, 
+(3) 
+∆𝜑34 = 𝜑𝑖3 − 𝜑𝑖4, 
+(4) 
+Calculated phase differences are shown in Table 2. 
+Since phase evolution at both the MMIs and Y-splitters 
+are fixed, we then need to design a combination of PSs 
+(placement and phase shift values), that results in the 
+required phase relations. As shown in Figure 1(a), we 
+achieve this goal by including a first phase shifter (PS1) 
+between inputs 3 and 2 of the MMI, with a phase shift of 
+𝜋 2
+⁄ ; a second phase shifter (PS2) between outputs 2 and 
+1, with a phase shift of − 𝜋 4
+⁄ ; and a third phase shifter 
+(PS3) between outputs 3 and 4 with a phase shift of 
+3 𝜋 4
+⁄ . An additional two-mode Y-junction (Y1) is 
+included at MCMD port 2 to satisfy even-order modes 
+phase conditions, as discussed hereunder. 
+Figure 4 shows the operation of the device working 
+in multiplexer configuration, including the value of the 
+phase relations at different locations for clarity. It should 
+be noted that phase values have been calculated with  
+ 
+ 
+Fig. 2. Schematic and principle of operation of a multimode 
+symmetric 1×2 Y-junction for (a) a two-mode stem, and (b) a 
+four-mode stem. 
+ 
+Fig. 3. Schematic of a 4×4 MMI coupler, illustrating port and 
+phase notations. 
+
+(a)
+TEo TEo
+TEo
+Y1
+TE1
+TEo TEo
+y4
+x
+(b)
+TE, TEo
+TE2 TEo
+TE TEo
+Y2
+TE, TEo
+TE,
+TE1
+TE, TEo
+yt
+xInputs (i)
+Outputs (i)
+4
+3
+4×4 MMI
+2
+△34Table 2 
+Calculated phase differences between MMI output pairs for each input. 
+𝒊 
+∆𝝋𝟏𝟐 
+∆𝝋𝟑𝟒 
+1 
+−𝜋 4
+⁄  
+−3𝜋 4
+⁄  
+2 
+𝜋 4
+⁄  
+3𝜋 4
+⁄  
+3 
+5𝜋 4
+⁄  
+−𝜋 4
+⁄  
+4 
+−5𝜋 4
+⁄  
+𝜋 4
+⁄  
+ 
+with respect to the mode phase at the input ports, which 
+is considered to be zero for simplicity. When light is 
+injected through MCMD port 1 [Fig. 4(a)], the 
+combination of all the aforementioned phase relations 
+results in all modes arriving in-phase at the arms of the 
+cascaded Y-junctions. Thus, two in-phase TE0 modes are 
+coupled into Y1 stems, which subsequently generate the 
+desired TE0 mode at the multimode stem waveguide of 
+Y2 (MCMD port 4). 
+When light is injected through MCMD port 2 [Fig. 
+4(b)], the combination of Y1 and PS1 results in 
+simultaneous light coupling to MMI input ports 2 and 3, 
+but with a 𝜋 2
+⁄  phase difference. This in turn generates 
+in-phase modes in the upper arms that are in anti-phase 
+with the two in-phase modes in the lower arms at their 
+arrival at the cascaded Y-junctions. This combination 
+results in TE1 generation at the MCMD output. 
+Finally, when light is injected through MCMD port 3 
+[Fig. 4(c)], that is, MMI input port 4, in-phase modes are 
+generated in the middle arms, which are in anti-phase 
+with the two in-phase modes generated in the top and 
+bottom arms, before the cascaded Y-junctions. This 
+results in anti-phase TE1 modes at the output of Y1 
+stems, which subsequently generate the TE2 mode at the 
+MCDM output. 
+So far, we have only considered passive PSs, that is, 
+PSs with a fixed phase shift. However, if the phase 
+introduced by PS1 is 3𝜋 2
+⁄  instead of 𝜋 2
+⁄ , it is possible 
+to generate the TE3 mode at MCMD output [Fig. 4(d)]. 
+For illustrative purposes, we represent this phase shift by 
+switching the position of the tapers in PS1. This feature 
+opens the possibility of extending MCMD operation to 
+four modes using a single switchable PS. 
+3. Proof-of-concept results 
+To verify the principle of operation explained in the 
+previous section, we firstly optimized each constituent 
+(i.e., MMI, phase shifters and Y-junctions) for a design 
+wavelength of 1550 nm. We chose a standard silicon 
+thickness of 𝐻 = 220 nm and an interconnection 
+waveguide width of 𝑊𝐼 = 400 nm. Thus, symmetric Y-
+junctions are designed with stem widths of 2𝑊𝐼 =
+800 nm for Y1 and 4𝑊𝐼 = 1600 nm for Y2. 
+Geometrical parameters of the 4×4 MMI coupler, the 
+phase shifters and the symmetric Y-junction are 
+summarized in Table 3. 
+In order to evaluate the performance of each 
+constituent element, the figures of merit for the MMI are 
+the excess loss (EL), imbalance (IB) and phase error 
+(PE): 
+EL𝑖 [dB] = −10log10 (∑ |S𝑗𝑖|
+2
+𝑗
+), 
+(5) 
+IB𝑖
+𝑗𝑘 [dB] = 10log10 (|S𝑗𝑖|
+2 |S𝑘𝑖|2
+⁄
+), 
+(6) 
+PE𝑖
+𝑗𝑘[°] = [∠(S𝑗𝑖 S𝑘𝑖
+⁄
+) − 𝜑𝑖𝑑𝑒𝑎𝑙] · 180 π
+⁄ , 
+(7) 
+where S𝑗𝑖 and S𝑘𝑖 are the scattering parameters for input 
+𝑖 and outputs 𝑗 and 𝑘, and 𝜑𝑖𝑑𝑒𝑎𝑙 is the ideal phase 
+relation depending on selected input and output ports as 
+shown in Table 1. The designed 4×4 MMI exhibits EL <  
+ 
+Table 3 
+Geometrical parameters of the three-mode converter and 
+multiplexer/demultiplexer with homogeneous waveguides. 
+Constituent 
+Parameter 
+ 
+Value 
+Waveguides 
+Width 
+𝑊𝐼 
+400 nm 
+MMI 
+Separation 
+𝑊𝑆 
+500 nm 
+Access width 
+𝑊𝐴 
+1.3 µm 
+Taper length 
+𝐿𝑇 
+6 µm 
+MMI width 
+𝑊𝑀𝑀𝐼 7.2 µm 
+MMI length 
+𝐿𝑀𝑀𝐼 91 µm 
+Y1 
+Arm width 
+𝑊𝐼 
+400 nm 
+Arm length 
+𝐿𝑌1 
+5 µm 
+Stem width 
+2𝑊𝐼 
+800 nm 
+Y2 
+Arm width 
+2𝑊𝐼 
+800 nm 
+Arm length 
+𝐿𝑌2 
+20 µm 
+Stem width 
+4𝑊𝐼 
+1.6 µm 
+PS1 
+PS width 
+𝑊𝑃𝑆1 600 nm 
+PS length 
+𝐿𝑃𝑆1 
+2.41 µm 
+PS2 
+PS width 
+𝑊𝑃𝑆2 600 nm 
+PS length 
+𝐿𝑃𝑆2 
+8.38 µm 
+PS3 
+PS width 
+𝑊𝑃𝑆3 600 nm 
+PS length 
+𝐿𝑃𝑆3 
+3.61 µm 
+ 
+Fig. 4. Principle of operation of the proposed three-mode 
+converter and multiplexer/demultiplexer for (a) TE0, (b) TE1, 
+(c) TE2 and (d) TE3 mode multiplexing. 
+
+(a)
+TE, multiplexing
+3
+TEo
+-T元
+一元
+T
+2
+3元/4-元
+f
+F7元/4
+TEo
+T
+.
+(b)
+TE, multiplexing
+3
+T
+TE,
+元
+T
+元/2
+3元/4 一元
+TEo
+0
+0
+3元/4
+0
+1
+0
+0
+(c)
+TE, multiplexing
+TE2
+3
+TEo
+一元
+2
+元/4
+0
+0
+→
+3元／40
+1
+T
+(d)
+TE, multiplexing
+0
+TE3
+3
+0
+0
+0
+F3元/4—元
+TEo
+0
+0
+元/2
+3元/4
+1
+元
+yt
+x0.54 dB, IB < ±0.4 dB 
+and PE < ±0.32° 
+at 
+the 
+wavelength of 𝜆0 = 1550 nm. Regarding the spectral 
+response, 
+EL < 2.15 dB, 
+IB < ±8.1 dB 
+and PE <
+±46.03° are attained in the entire simulated wavelength 
+range (1.45 – 1.65 µm). 
+Designed phase shifters introduce small phase 
+deviations of only 0.12° for PS1, 0.13° for PS2 and 
+0.16° for PS3, with respect to their target phase 
+difference at 1550 nm. However, considering the 
+simulated bandwidth of 200 nm, phase errors increase up 
+to 9.84° for PS1, 22.28° for PS2, and 12.12° for PS3. 
+Symmetric Y-junctions Y1 and Y2 were also 
+designed showing negligible excess losses and power 
+imbalance between output ports at the design 
+wavelength. More specifically, Y1 losses are lower than 
+0.01 dB for both TE0 and TE1 mode operation in the 1.45 
+– 1.65 µm wavelength range. Conversely, calculated 
+excess losses for Y2 are below 0.15 dB for TE0, TE1, TE2 
+and TE3 mode operation within the same bandwidth. 
+Once all elements were optimized, two-dimensional 
+finite-difference time-domain (FDTD) simulations of the 
+whole MCMD were performed by applying the effective 
+index method to the original three-dimensional structure 
+[see Fig. 1(a)]. The simulated field distribution of the 
+three-mode MCMD is shown in Fig. 5(a)-(d), 
+demonstrating the successful implementation of the 
+phase relations described in section 2. 
+Some ripples can be observed for TE0 and TE2 mode 
+multiplexing in the stem waveguide of Y-junction Y2 
+[see Figs. 5(a) and 5(c)], which we attribute to a higher 
+crosstalk between both modes compared to TE1 and TE3 
+mode multiplexing. 
+The transmittance as a function of the wavelength 
+was computed for the complete MCMD [see Fig.5(e)-
+(h)]. At the central wavelength of 𝜆0 = 1550 nm, 
+insertion losses are lower than 0.53 dB, 0.79 dB and 0.59 
+dB for the generation of TE0, TE1 and TE2 modes in the 
+stem waveguide, respectively. Our device also exhibits a 
+low crosstalk at the same wavelength with values below 
+-21.61 dB for TE0, -28.94 dB for TE1 and -21.11 dB for 
+TE2. 
+By tuning the value of PS1 to 3 𝜋 2
+⁄ , TE3 mode 
+(instead of TE1 mode) can be generated. In this case, 
+insertion losses are below 0.75 dB, and the crosstalk is 
+better than -28.79 dB, both at 1550 nm. These results 
+corroborate the higher crosstalk for TE2 mode operation, 
+which leads to a slight ripple in the field distribution at 
+port 4. 
+Regarding performance across the spectrum, insertion 
+losses lower than 1 dB are attained for a 55 nm 
+bandwidth (1542 – 1597 nm), whereas the crosstalk is 
+below -20 dB for a 60 nm bandwidth (1537 –1597 nm) 
+as shown with vertical lines in Fig. 5. These results prove 
+the correct operation of the proposed architecture, but the 
+overall bandwidth is significantly limited by the narrow 
+spectral response of both the MMI and PSs. 
+4. SWG performance enhancement 
+To overcome these bandwidth limitations, we 
+propose the MCMD with SWG metamaterials shown in 
+Fig. 1(b). The design of each of the constituents of the 
+SWG MCMD was performed by individual three-
+dimensional FDTD simulations. The three symmetric Y-
+junction labeled Y1 maintain the same geometrical 
+dimensions as those used for the conventional 
+multiplexer for the arm and stem widths (see Table 3), 
+but arm length was shortened to 𝐿𝑌1 = 2 µm. Y-junction 
+Y2 was slightly redesigned to reduce the crosstalk 
+between TE0 and TE2 modes by increasing the length of 
+the arms to 𝐿𝑌2  =  40 μm. 
+A procedure similar to those already reported in 
+[39,40] was followed for the optimization of the 4×4 
+SWG MMI. We restrict the value of the duty cycle (DC =
+𝑎 Λ
+⁄ ) to 0.5 in order to maximize the minimum feature 
+size for a given period (Λ) [see Fig.1(d)]. We explored 
+then different periods and found that Λ = 222 nm 
+significantly flattens the beat length across the spectrum. 
+Compared to the conventional MMI section design, the 
+width 𝑊𝑆𝑀𝑀𝐼 is increased by 0.8 µm but the length 𝐿𝑆𝑀𝑀𝐼 
+is reduced by more than half to ~41.3 µm. To increase 
+the quality of the interferometric patterns formed in the 
+MMI, the access width is 𝑊𝐵 = 1.7 µm and the 
+ 
+Fig. 5. Electric field amplitude |𝐸| in the XY plane at the middle 
+of the silicon layer for (a) TE0, (b) TE1, (c) TE2 and (d) TE3 
+mode multiplexing. Simulated transmittance to output port 4 as 
+a function of the wavelength when TE0 mode is launched into 
+(e) input port 1, (f) input port 2 with PS1 = 𝜋 2
+⁄ , (g) input port 
+3 and (h) input port 2 with PS1 = 3𝜋 2
+⁄ . Vertical lines indicate 
+the bandwidth where IL < 1 dB (55 nm) and XT < −20 dB (60 
+nm) are achieved for all modes simultaneously. 
+
+(a)
+(b)
+(c)
+(d)
+200
+0.8
+150
+0.6
+X
+0.4
+50
+0.2
+0
+420-2-4 420-2-4420-2-4420-2-4
+y (μm)
+y (μm)
+y (μm)
+y (μm)
+(e)
+Input 1
+(G)
+Input 2 (PS1 = π/2)
+(dB)
+0
+(dB)
+55 nm
+55nm
+Transmittance
+Transmittance
+60 nm
+60 nm
+-20
+-20
+40
+TE
+40
+TE
+1.45
+1.5
+1.55
+1.6
+1.65
+1.45
+1.5
+1.551.61.65
+Wavelength (um)
+Wavelength (um)
+(g)
+(h)
+Input 3
+Input2(PS1=3元/2)
+(dB)
+0
+(dB)
+55 nm
+55 nm
+Transmittance
+Transmittance
+60 nm
+60 nm
+-20
+20
+TE,
+TE
+-40
+40
+TE
+TE
+TE3
+1.45
+1.5
+1.55
+1.6
+1.65
+1.45
+1.5
+1.551.6
+51.65
+Wavelength(um
+Wavelength (um)separation is reduced to 𝑊𝑅 = 0.3 µm. The transition 
+between 
+the 
+interconnection 
+waveguides 
+(𝑊𝐼 =
+400 nm) and the access to the MMI section (𝑊𝐵) is 
+performed by means of adiabatic SWG tapers with a 
+length 𝐿𝑆𝑇 = 13.32 µm. The performance of the 4×4 
+SWG MMI is shown in Fig. 6(a)-(c). Owing to the 
+symmetry of the structure, only the results obtained when 
+injecting light into input ports 1 and 2 are depicted. It is 
+observed that the device exhibits EL < 0.77 dB, IB <
+±1 dB and PE < ±8.02° within a broad bandwidth of 
+200 nm (1.45 – 1.65 µm). 
+To drastically extend the operating bandwidth of the  
+ 
+nanophotonic phase shifters, we build upon the strategy 
+we recently reported in [41] to develop SWG phase 
+shifters SPS1, SPS2 and SPS3. Notwithstanding, here we 
+employ four parallel SWG waveguides of two different 
+widths to implement SPS2 and SPS3. That is, each PS 
+has three identical reference SWG waveguides with 
+width 𝑊𝐷, and one dissimilar SWG waveguide with 
+width 𝑊𝑅. Both the reference and dissimilar waveguides 
+have a length of LSPS. Note that for SPS1 this 
+configuration is not necessary as only two MMI inputs 
+are illuminated for TE1 and TE3 mode generation. 
+Analogous to the 4×4 SWG MMI, a flat phase shift can 
+be achieved by judicious selecting the SWG period and 
+duty cycle. A duty cycle of 0.5 was fixed to maximize 
+MFS, while a period of 200 nm resulted in minimum 
+phase shift deviation. In order to induce 𝜋 4
+⁄ , 𝜋 2
+⁄ , and 
+3𝜋 4
+⁄  phase shifts, we selected respectively 𝑊𝐷2 =
+1.8 µm, 𝑊𝑅2 = 1.6 µm, 𝐿𝑆𝑃𝑆2 = 6.2 µm and 𝐿𝑆𝑇2 =
+3.0 µm for SPS2; 𝑊𝐷1 = 1.8 µm, 𝑊𝑅1 = 1.6 µm, 
+𝐿𝑆𝑃𝑆1 = 16.8 µm and 𝐿𝑆𝑇1 = 3.0 µm for SPS1; and 
+𝑊𝐷3 = 1.8 µm, 𝑊𝑅3 = 1.6 µm, 𝐿𝑆𝑃𝑆3 = 28.2 µm and 
+𝐿𝑆𝑇3 = 3.0 µm for SPS3. The simulated phase shifts are 
+shown in Fig. 6(d). Negligible deviations can be 
+appreciated with phase shift errors as small as 2.29° for 
+SPS1, and 1.15° for SPS2 and SPS3 within the entire 
+1.45 – 1.65 µm wavelength range. 
+5. SWG results 
+The simulation of the entire MCMD is quite 
+resource-intensive and time-consuming due to the device 
+footprint and the need for a fine mesh to simulate SWG-
+based devices. Thus, instead of performing the full 
+device simulation, we leverage the S-parameter matrices 
+calculated during the design process and concatenate all 
+of them using a circuit simulator to obtain the S-
+ 
+Fig. 7. Simulated transmittance as a function of the wavelength of the MCMD with SWG metamaterials when TE0 mode is launched 
+into (a) input port 1, (b) input port 2 with SPS1 = 𝜋 2
+⁄ , (c) input port 3 and (d) input port 2 with SPS1 = 3𝜋 2
+⁄ . Vertical lines indicate 
+the bandwidth where IL < 1 dB (183 nm) and XT < −20 dB (161 nm) are achieved for all modes simultaneously. 
+ 
+Fig. 6. Simulated performance of the 4×4 SWG MMI including 
+(a) excess loss, (b) imbalance and (c) phase error between 
+output ports. (d) Phase error of each SWG PSs as a function of 
+the wavelength. 
+
+(a)
+Input 1
+(b)
+Input 2 (SPS1 = π/2)
+0
+(dB)
+(dB)
+0
+183 nm
+183 nm
+161 nm
+161 nm
+20
+Transmittance
+Transmittance 
+20
+TE
+TEo
+-40
+40
+TE,
+TE,
+-60
+TE,
+-60
+T-80
+TE
+-80
+TE
+1.45
+1.5
+1.55
+1.6
+1.65
+1.45
+1.5
+1.55
+1.6
+1.65
+Wavelength (μm)
+Wavelength (um)
+(c)
+Input 3
+(d)
+Input 2 (SPS1 = 3π/2)
+(dB)
+183 nm
+(dB)
+0
+183 nm
+161 nm
+161 nm
+-20
+20
+Transmittance 
+40
+4
+TE
+-60
+TE
+60
+TH
+TE
+TE
+TE.
+-80
+1.45
+1.5
+1.55
+1.6
+1.65
+1.45
+1.5
+1.55
+1.6
+1.65
+Wavelength (um)
+Wavelength (um)(a)
+(b)
+(dB)
+Input1 (EL)
+(dB)
+—Input 1: 0,/02 (IB12)
+Input2(EL2)
+Input 1: O3/O4 (IB24)
+Imbalance (
+Input 2: O,/0, (IB22)
+2
+Input 2:0,/04 (IB34)
+0
+1.45
+1.5
+1.55
+1.6
+1.65
+1.45
+1.5
+1.551.6
+1.65
+Wavelength (μm)
+Wavelength (um)
+(c)
+(d)
+10
+Input1:012
+10
+SPS2 (元/4rad)
+Input 1: Ap
+SPS1 (元/2rad)
+error
+error
+5
+0
+Phase 
+Input 2: A934
+-5
+SPS1 (3元/2rad)
+Input2:012
+-SPS3 (3元/4rad)
+-10
+-10
+1.45
+1.5
+1.551.6
+1.65
+1.45
+1.5
+1.551.61.65
+Wavelength (um)
+Wavelength (um)parameter matrix and hence the spectral response of the 
+complete device. The circuit simulator enables 
+bidirectional signals to be accurately simulated, 
+including coupling of modes in the single elements. 
+Figure 7 shows the overall transmittance of the SWG 
+MCMD. Insertion losses (ILs) are lower than 0.37 dB, 
+0.47 dB and 0.37 dB for TE0, TE1 and TE2 multiplexing, 
+respectively, at the central wavelength of 𝜆0 =
+1550 nm. Moreover, low crosstalk (XT) is achieved at 
+the same wavelength with values below -21.54 dB for 
+TE0, -32.89 dB for TE1 and -21.24 dB for TE2 
+multiplexing. 
+When SPS1 takes the value of 3 𝜋 4
+⁄ , insertion losses 
+for TE3 multiplexing reach a low value of 0.47 dB at 
+1550 nm, while crosstalk values are lower than -39.48 dB 
+for the same wavelength. 
+This design also shows an excellent performance 
+over a broad bandwidth (BW) of 200 nm with insertion 
+loss lower than 1.18 dB and crosstalk below -16.53 dB. 
+Insertion losses decrease to 1 dB when the bandwidth is 
+restricted to 183 nm (1450 – 1633 nm), whereas a 
+crosstalk below -20 dB is achieved over a 161 nm 
+bandwidth (1489 – 1650 nm). For the sake of 
+comparison, Table 4 summarizes the performance of 
+other three- and four-mode MCMD that are based on 
+MMI couplers and have been reported in the state of the 
+art. To the best of our knowledge, it is the first time such 
+low losses and crosstalk are achieved in an outstanding 
+161 nm wavelength range. 
+ 
+6. Conclusions 
+In this work, we have proposed a novel architecture to 
+scale the number of multiplexed modes of mode 
+converters and multiplexer based on MMI couplers. 
+Unlike 
+other 
+reported 
+architectures 
+that 
+use 
+unconventional 1×4 Y-junctions or 1×3 Ψ-junctions, 
+here we employ symmetric 1×2 Y-junctions arranged in 
+a conventional cascaded configuration. The design 
+methodology was proposed on the basis of a two-
+dimensional model with conventional homogenous 
+components (i.e., without patterning the silicon 
+waveguide). The conventional mode converter and 
+multiplexer features sub-decibel insertion loss and 
+crosstalk better than -20 dB in the 1542 – 1597 nm 
+wavelength range. Once the principle of operation was 
+verified, we redesigned and optimized the mode 
+converter 
+and 
+multiplexer 
+by 
+incorporating 
+subwavelength grating metamaterials to leverage the 
+additional degrees of freedom they introduced into the 
+design. A broad design bandwidth of 161 nm for 
+insertion losses below 1.18 dB and crosstalk lower than 
+-20 dB was confirmed by 3D FDTD simulations, 
+comparing very favorably to state-of-the-art three- and 
+four-mode converters and multiplexers. The crosstalk 
+between TE0 and TE1 modes could be further reduced by 
+including optimized Y-junction geometries that mitigate 
+the effect of the non-perfect tip at the junction [42-44]. 
+We believe that our design strategy will open promising 
+prospects for the development of high-performance 
+mode converters and multiplexer based on MMI couplers 
+with a high channel count. 
+Credit authorship contribution statement 
+David 
+González-Andrade: 
+Conceptualization, 
+Methodology, Software, Validation, Formal analysis, 
+Investigation, Data curation, Writing – original draft, 
+Visualization. Irene Olivares: Methodology, Software, 
+Validation, Formal analysis, Data curation, Writing – 
+review & editing. Raquel Fernández de Cabo: 
+Software, Validation, Data curation, Writing – review & 
+editing. Jaime Vilas: Writing – review & editing. 
+Antonio Dias: Resources, Writing – review & editing, 
+Project administration, Funding acquisition. Aitor V. 
+Velasco: Resources, Writing – review & editing, 
+Supervision, 
+Project 
+administration, 
+Funding 
+acquisition. 
+Declaration of Competing Interest 
+The authors declare that they have no known competing 
+financial interests or personal relationships that could 
+have appeared to influence the work reported in this 
+paper. 
+Acknowledgements 
+This work has been funded in part by the Spanish 
+Ministry of Science and Innovation (MICINN) under 
+grants RTI2018-097957-B-C33, PID2020-115353RA-
+I00; 
+the 
+Spanish 
+State 
+Research 
+Agency 
+(MCIN/AEI/10.13039/501100011033); the Community 
+of Madrid – FEDER funds (S2018/NMT-4326); the 
+European Union – NextGenerationEU through the 
+Recovery, 
+Transformation 
+and 
+Resilience 
+Plan 
+(DIN2020-011488, 
+PTQ2021-011974); 
+and 
+the 
+European Union's Horizon Europe research and 
+innovation program under the Marie Sklodowska-Curie 
+grant agreement Nº 101062518. 
+References 
+[1] 
+R. Sabella, Silicon photonics for 5G and future networks, IEEE 
+J. Sel. Top. Quantum Electron. 26 (2020) 8301611. 
+[2] 
+I. Demirkol, D. Camps-Mur, J. Paradells, M. Combalia, W. 
+Popoola, H. Haas, Powering the Internet of things through light 
+communication, IEEE Commun. Mag. 57 (2019) 107-113. 
+[3] 
+M Filer, J. Gaudette, Y. Yin, Z. Bakhtiari, J. L. Cox, Low-argin 
+optical networking at cloud scale, J. Opt. Commun. Netw. 11 
+(2019) C94-C105. 
+Table 4 
+Performance comparison of state-of-the-art three- and 
+four-MCMD based on MMI couplers. 
+Ref. 
+IL 
+[dB] 
+XT 
+[dB] 
+BW 
+[nm] 
+Length 
+[µm] 
+No. of 
+modes 
+[28] 
+<0.71 
+<-18 
+100 
+400 
+3 
+[29] 
+<0.9 
+<-17 
+40 
+400 
+3 
+[30] 
+<1.1 
+<-10 
+90 
+17.05 
+3 
+[31] 
+<1.3 
+<-15 
+40 
+120 
+3 
+[32] 
+<3.3 
+<-14 
+70 
+173 
+4 
+[32] 
+<2.25 
+<-15 
+140 
+123 
+4 
+This work 
+<1.18 
+<-20 
+161 
+181.14 
+3 
+ 
+
+[4] 
+Q. Cheng, M. Bahadori, M. Glick, S. Rumley, K. Bergman 
+Recent advances in optical technologies for data centers: a 
+review, Optica 5 (2018) 1354-1370. 
+[5] 
+D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. 
+Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. 
+Fédeli, J.-M. Hartmann, J. H. Schmid, D. X. Xu, F. Boeuf, P. 
+O'Brien, G. Z. Mashanovich, M. Nedeljkovic, Roadmap on 
+silicon photonics, J. Opt. 18 (2016) 073003. 
+[6] 
+X. Chen, M. M. Milosevic, S. Stanković, S. Reynolds, T. 
+Domínguez-Bucio, K. Li, D. J. Thomson, F. Gardes, G. T. Reed, 
+The emergence of silicon photonics as a flexible technology 
+platform, Proc. IEEE 106 (2018) 2101-2116. 
+[7] 
+S. Bernabé, Q. Wilmart, K. Hasharoni, K. Hassan, Y Thonnart, 
+P. Tissier, Y. Désières, S. Olivier, T. Tekin, B. Szelag, Silicon 
+photonics for terabit/s communication in data centers and 
+exascale computers, Solid State Electron. 179 (2021) 107928. 
+[8] 
+W. Shi, Y. Tian, A. Gervais, Scaling capacity of fiber-optic 
+transmission systems via silicon photonics, Nanophotonics 9 
+(2020) 4629-4663. 
+[9] 
+C. Li, D. Liu, D. Dai, Multimode silicon photonics, 
+Nanophotonics 8 (2018) 227-247. 
+[10] 
+I. Cristiani, C. Lacava, G. Rademacher, B. J. Puttnam, R. S. 
+Luìs, C. Antonelli, A. Mecozzi, M. Shtaif, D. Cozzolino, L. K. 
+Oxenløwe, J. Wang, Y. Jung, D. J. Richardson, S. 
+Ramachandran, M. Guasoni, K. Krupa, D. Kharenko, A. 
+Tonello, S. Wabnitz, D. B. Philips, D. Faccio, T. G. Eurser, S. 
+Xie, P. St. J. Russell, D. Dai, Y. Yu, P. Petropoulos, F. Gardes, 
+F. Parmiagiani, Roadmap on multimode photonics, J. Opt. 24 
+(2022) 083001. 
+[11] 
+J. B. Driscoll, R. R. Grote, B. Souhan, J. I. Dadap, M. Lu, R. M. 
+Osgood, Asymmetric Y junctions in silicon waveguides for on-
+chip mode-division multiplexing, Opt. Lett. 38 (2013) 1854-
+1856. 
+[12] 
+W. Chen, P. Wang, T. Yang, G. Wang, T. Dai, Y. Zhang, L. 
+Zhou, X. Jiang, J. Yang, Silicon three-mode (de)multiplexer 
+based on cascaded asymmetric Y junctions, Opt. Lett. 41 (2016) 
+2851-2854. 
+[13] 
+T. A. Tran, H. D. T. Nguyen, C. D. Truong, H. T. Nguyen, Y. 
+V. Vu, D. H. Tran, Three-mode multiplexed device based on 
+tilted- branch bus structure using silicon waveguide, Photonics 
+Nanostructures – Fundam. Appl. 35 (2019) 100709. 
+[14] 
+D. Dai, J. Wang, Y. Shi, Silicon mode (de)multiplexer enabling 
+high capacity photonic networks-on-chip with a single-
+wavelength carrier light, Opt. Lett. 38 (2013) 1422-1424. 
+[15] 
+J. Wang, Y. Xuan, M. Qi, H. Huang, Y. Li, M. Li, X. Chen, Z. 
+Sheng, A. Wu, W. Li, X. Wang S. Zou, F. Gan, Broadband and 
+fabrication-tolerant 
+on-chip 
+scalable 
+mode-division 
+multiplexing 
+based 
+on 
+mode-evolution 
+counter-tapered 
+couplers, Opt. Lett. 40 (2015) 1956-1959. 
+[16] 
+T. Uematsu, Y. Ishizaka, Y. Kawaguchi, K. Saitoh, M. Koshiba, 
+Design of a compact two-mode multi/demultiplexer consisting 
+of multimode interference waveguides and a wavelength-
+insensitive phase shifter for mode-division multiplexing 
+transmission, J. Lightwave Technol. 30 (2012) 2421-2426. 
+[17] 
+Y. Li, C. Li, C. Li, B. Cheng, C. Xue, Compact two-mode (de) 
+multiplexer based on symmetric Y-junction and multimode 
+interference waveguides, Opt. Express 22 (2014) 5781-5786. 
+[18] 
+L. F. Frellsen, Y. Ding, O. Sigmund, L. H. Frandsen, Topology 
+optimized mode multiplexing in silicon-on-insulator photonic 
+wire waveguides, Opt. Express 24 (2016) 16866-16873. 
+[19] 
+Z. Jafari, A. Zarifkar, M. Miri, Compact fabrication-tolerant 
+subwavelength-grating-based 
+two-mode 
+division 
+(de)multiplexer, Appl. Opt. 56 (2017) 7311-7319. 
+[20] 
+W. Jiang, J. Hu, S. Mao, H. Zhang, L. Zhou, B. M. A. Rahman, 
+Broadband 
+silicon 
+four-mode 
+(de)multiplexer 
+using 
+subwavelength grating-assisted triple-waveguide couplers, J. 
+Lightwave Technol. 39 (2021) 5042-5047. 
+[21] 
+Y. He, Y. Zhang, Q. Zhu, S. An, R. Cao, X. Guo, C. Qiu, Y. Su, 
+Silicon high-order mode (de)multiplexer on single polarization, 
+J. Lightwave Technol. 36 (2018) 5746-5753. 
+[22] 
+D. González-Andrade, J. G. Wangüemert-Pérez, A. V. Velasco, 
+A. Ortega-Moñux, A. Herrero-Bermello, I. Molina-Fernández, 
+R. Halir, P. Cheben, Ultra-broadband mode converter and 
+multiplexer based on sub-wavelength structures, IEEE 
+Photonics J. 10 (2018) 2201010. 
+[23] 
+D. González-Andrade, A. Dias, J. G. Wangüemert-Pérez, A. 
+Ortega-Moñux, Í. Molina-Fernández, R. Halir, P. Cheben, A. V. 
+Velasco, Experimental demonstration of a broadband mode 
+converter and multiplexer based on subwavelength grating 
+waveguides, Opt. Laser Technol. 129 (2020) 106297. 
+[24] 
+L. Xu, Y. Wang, D. Mao, J. Zhang, Z. Xing, E. El-Fiky, Md. G. 
+Saber, A. Kumar, Y. D’Mello, M. Jacques, D. V. Plant, Ultra-
+broadband and compact two-mode multiplexer based on 
+subwavelength-grating-slot-assisted adiabatic coupler for the 
+silicon-on-insulator platform, J. Lightwave Technol. 37 (2019) 
+5790-5800. 
+[25] 
+L. Zhang, J. Xiao, Compact and broadband mode demultiplexer 
+using a subwavelength grating engineered MMI coupler, J. Opt. 
+Soc. Am. B 38 (2021) 2830-2836. 
+[26] 
+D. González-Andrade, R. Fernández de Cabo, J. Vilas, I. 
+Olivares, A. Dias, J. M. Luque-González, J. G. Wangüemert-
+Pérez, A. Ortega-Moñux, Í. Molina-Fernández, R. Halir, P. 
+Cheben, A. V. Velasco, Mode converter and multiplexer with a 
+subwavelength phase shifter for extended broadband operation, 
+IEEE Photonics Technol. Lett. 33 (2021) 1262-1265. 
+[27] 
+J. Wang, P. Chen, S. Chen, Y. Shi, D. Dai, Improved 8-channel 
+silicon mode demultiplexer with grating polarizers, Opt. 
+Express 22 (2014) 12799. 
+[28] 
+A. T. Tran, D. C. Truong, H. T. Nguyen, Y. V. Vu, A new 
+simulation design of three-mode division (de)multiplexer based 
+on a trident coupler and two cascaded 3×3 MMI silicon 
+waveguides, Opt. Quantum Electron. 50 (2018) 426. 
+[29] 
+C. D. Truong, T. H. Nguyen, Q. T. Pham, M. T. Trinh, K. Vu, 
+Three-mode multiplexer and demultiplexer utilizing trident and 
+multimode couplers, Opt. Commun. 435 (2019) 334-340. 
+[30] 
+Z. Wang, C. Yao, Y. Zhang, Y. Su, Compact silicon three-mode 
+multiplexer by refractive-index manipulation on a multi-mode 
+interferometer, Opt. Express 29 (2021) 13899-13907. 
+[31] 
+Z. L. Hussain, R. S. Fyath, Design and simulation of a compact 
+three-mode (de)multiplexer based on a subwavelength grating 
+multimode interference coupler, Photonics Nanostructures – 
+Fudam. Appl. 47 (2021) 100966. 
+[32] 
+Z. L. Hussain, R. S. Fyath, Design and simulation of 4-mode 
+(de)multiplexers 
+implemented 
+in 
+conventional 
+and 
+subwavelength grating Si/SiO2 platforms, Optik 251 (2022) 
+168449. 
+[33] 
+L. Lu, D. Liu, M. Yan, M. Zhang, On-chip mode converter 
+based 
+on 
+cross-connected 
+subwavelength 
+Y-junctions, 
+Photonics Res. 9 (2021) 43-48. 
+[34] 
+M. Bachmann, P. A. Besse, H Melchior, General self-imaging 
+properties in N×N multimode interference couplers including 
+phase relations, Appl. Opt. 33 (1994) 3905-3911. 
+[35] 
+P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, D. R. Smith, 
+Subwavelength integrated photonics, Nature 560 (2018) 565-
+572. 
+[36] 
+P. Cheben, D.-X. Xu, S. Janz, A. Densmore, Subwavelength 
+waveguide grating for mode conversion and light coupling in 
+integrated optics, Opt. Express 14 (2006) 4695-4702. 
+[37] 
+R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-
+Ramos, J. H. Schmid, J. Lapointe, D.-X. Xu, J. G. Wangüemert-
+Pérez, Í. Molina-Fernández, S. Janz, Waveguide sub-
+
+wavelength structures: a review of principles and applications, 
+Laser Photonics Rev. 9 (2015) 25-49. 
+[38] 
+J. M. Luque-González, A. Sánchez-Postigo, A. Hadij-ElHouati, 
+A. Ortega-Moñux, J. G. Wangüemert-Pérez, J. H. Schmid, P. 
+Cheben, Í. Molina-Fernández, R. Halir, A review of silicon 
+subwavelength gratings: building break-through devices with 
+anisotropic metamaterials, Nanophotonics 10 (2021) 2765-
+2797. 
+[39] 
+R. Halir, P. Cheben, J. M. Luque-González, J. D. Sarmiento-
+Merenguel, J. H. Schmid, G. Wangüemert-Pérez, D.-X. Xu, S. 
+Wang, A. Ortega-Moñux, Í. Molina-Fernández, Ultra-
+broadband nanophotonic beamsplitter using an anisotropic sub-
+wavelength metamaterial, Laser Photonics Rev. 10 (2016) 
+1039-1046. 
+[40] 
+L. Xu, Y. Wang, D. Patel, M. Morsy-Osman, R. Li, M. Hui, M. 
+Parvizi, N. Ben-Hamida, D. V. Plant, Ultra-broadband and ultra-
+compact optical 90° hybrid based on 2×4 MMI coupler with 
+subwavelength gratings on silicon-on-insulator, Optical Fiber 
+Communication Conference (2018), p. M3I.7. 
+[41] 
+D. 
+González-Andrade, 
+J. M. Luque-González, 
+J. 
+G. 
+Wangüemert-Pérez, A. Ortega-Moñux, P. Cheben, Í. Molina-
+Fernández, A. V. Velasco, Ultra-broadband nanophotonic phase 
+shifter based on subwavelength metamaterial waveguides, 
+Photonics Res. 8 (2020), 359-367. 
+[42] 
+R. Fernández de Cabo, D. González-Andrade, P. Cheben, A. V. 
+Velasco, High-performance on-chip silicon beamsplitter based 
+on subwavelength metamaterials for enhanced fabrication 
+tolerance, Nanomaterials 11 (2021) 1304. 
+[43] 
+L. Lu, D. Liu, M. Yan, M. Zhang, Subwavelength adiabatic 
+multimode Y-junctions, Opt. Lett. 44 (2019) 4729-4732. 
+[44] 
+R. Fernández de Cabo, J. Vilas, P. Cheben, A. V. Velasco, D. 
+González-Andrade, Experimental characterization of an ultra-
+broadband dual-mode symmetric Y-junction based on 
+metamaterial waveguides, Opt. Laser Technol. 157 (2023) 
+108742. 
+
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+page_content=' Velascoc  a Centre de Nanosciences et de Nanotechnologies, CNRS, Université Paris-Saclay, Palaiseau 91120, France  b Alcyon Photonics S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=', Madrid 28004, Spain  c Instituto de Óptica Daza de Valdés, Consejo Superior de Investigaciones Científicas (CSIC), Madrid 28006, Spain  Corresponding author: david.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='gonzalez-andrade@c2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='upsaclay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='fr  ARTICLE INFO  Keywords:  Silicon photonics  Mode-division  multiplexing  Subwavelength  metamaterial  MMI coupler  Phase shifter  Y-junction  ABSTRACT  Mode-division multiplexing has emerged as a promising route for increasing  transmission capacity while maintaining the same level of on-chip integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Despite  the large number of on-chip mode converters and multiplexers reported for the silicon-  on-insulator platform, scaling the number of multiplexed modes is still a critical  challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' In this paper, we present a novel three-mode architecture based on  multimode interference couplers, passive phase shifters and cascaded symmetric Y-  junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' This architecture can readily operate up to the third-order mode by including  a single switchable phase shifter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Moreover, we exploit subwavelength grating  metamaterials to overcome bandwidth limitations of multimode interference couplers  and phase shifters, resulting in a simulated bandwidth of 161 nm with insertion loss  and crosstalk below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='18 dB and -20 dB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Introduction  The relentless growth of global Internet traffic has been  driven in recent years by the emergence of data-hungry  services and their mass adoption by an increasingly  interconnected society [1-3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Moreover, the cloud nature  of many new applications such as machine learning or  artificial intelligence require large data sets to be  processed on internal servers or transferred between data  centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' This resource-intensive paradigm for accessing,  computing, and storing data has led to the creation of  hyperscale data centers consisting of thousands of  servers located in the same physical facility [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' To cope  with the resulting zetta scale of annual data flow, modern  data centers have been relying on optical technologies for  both long-haul and few-meter interconnects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Compared  to  their  electronic  counterparts,  these  optical  technologies offer higher processing speeds, broader  bandwidths and lower latency and energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Silicon photonics, leveraging the mature fabrication  facilities of the microelectronics industry, plays a key  role in the optical interconnect industry due to its  capacity for high-yield and low-cost mass production of  high-performance optoelectronic circuits [5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  However, the development of next-generation  datacenters for Tbps communications and exascale  computing systems  is not feasible by scaling  infrastructures alone and requires increasingly efficient  optical interconnects for short-reach distances [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' As  single-mode transmission approaches its fundamental  limits, space-division multiplexing has emerged as a  promising way to further improve the transmission  capacity of optical interconnects through the use of  multicore or multimode waveguides [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' The latter,  which is also called mode-division multiplexing (MDM),  has attracted an increasing interest as it leverages the  orthogonality of the eigenmodes supported by a single  multimode waveguide, thus allowing to maintain the  same level of on-chip integration [9,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' That is, MDM  enables encoding different data channels into specific  spatial modes, increasing capacity proportionally to the  number of modes used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Numerous  on-chip  mode  converters  and  multiplexers/demultiplexers  (MCMD)  have  been  proposed for the silicon-on-insulator (SOI) platform to  date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Asymmetric Y-junctions are based on the principle  of mode evolution in adiabatic structures, which results  in broad operating bandwidths but also in long device  lengths [11-13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' The minimum feature size of current  lithography processes also has a significant impact in  these devices, since the finite resolution at which the tip  can be fabricated severely hampers their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Asymmetric directional couplers (ADCs) [14], relying  on evanescent coupling between adjacent waveguides,  are well suited for implementing high-channel count  MDM systems, but they typically exhibit narrow  bandwidths, and their performance is highly susceptible  to fabrication errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Adiabatic tapers have been  employed in the coupling region of ADCs to improve the  bandwidth and the resilience against fabrication  deviations [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' MCMDs building upon multimode  interference (MMI) couplers and other auxiliary  components such as phase shifters (PSs) and symmetric  Y-junctions have been proposed as well [16,17], yielding  low losses and low crosstalk over a relatively broad  wavelength range (~100 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  The patterning of silicon at the subwavelength scale  has proven to be a simple yet powerful tool to tailor the  medium optical properties while inhibiting diffractive  effects [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' More specifically, subwavelength (SWG)  metamaterials  can  behave  as  a  homogeneous  metamaterial that provides flexible dispersion and  anisotropy engineering, non-feasible in conventional  strip and rib waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' These properties have led to the  realization  of  Si  devices  with  unprecedented  performance over the past 15 years [36-38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' In the MDM  field, MCMDs based on subwavelength pixelated  structures have demonstrated ultra-compact footprints  [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' SWGs have also been applied to ADCs and triple-  waveguide couplers to improve fabrication tolerances  and extend the operation bandwidth of conventional  counterparts [19-21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Furthermore, low losses and low  crosstalk within ultra-broad bandwidths have also been  reported using subwavelength engineered MMI couplers  and PSs, and SWG-slot-assisted adiabatic couplers [22-  26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Despite the large number of available two-mode  MCMDs, scaling the number of multiplexed modes  beyond the fundamental and first-order modes is of great  importance to multiply the capacity of next-generation  datacom systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Although it is fairly straightforward to  extend operation to a larger number of modes in  asymmetric Y-junctions and conventional and tapered  ADCs [27], three- and four-mode MCMD based on MMI  couplers have only recently been reported [28-32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  However, the proposed architectures are still limited by  narrow operating bandwidths and high crosstalk values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  In this work, we propose a novel MCMD architecture  based on a 4×4 MMI, three phase shifters and four  symmetric 1×2 Y-junctions arranged in a conventional  cascaded configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' The device operates as a three-  mode MCDM with passive phase shifters but can readily  convert up to the third-order mode by including a single  switchable phase shifter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Moreover, we demonstrate loss  and crosstalk reduction in a broad bandwidth by SWG-  engineering of both the MMI coupler and phase shifters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Simulations show operation bandwidth of 161 nm with  insertion loss and crosstalk below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='18 dB and -20 dB,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Principle of operation and device design  To explain the operation principle and the device design,  let us first focus on the nanophotonic structure shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 1(a) consisting of a conventional 4×4 MMI, three  phase shifters (PS1, PS2 and PS3) and four symmetric  1×2 Y-junctions (three identical Y1 and a different one  Y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' SWG enhancement of the proposed architecture,  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 1(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 1(d) will be discussed in  epigraphs 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' An SOI platform with a thin Si wire  surrounded by SiO2 bottom layer and upper cladding are  considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' A schematic view of the waveguide cross-  section is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 1(c) for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  In order to illustrate the operation of the MCMD, let  us focus on the mode evolution and phase relations in  each individual constituent of the MCMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Here, we aim  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Three-dimensional schematic of the proposed three-mode converter and multiplexer/demultiplexer comprising a 4×4 MMI, three  phase shifters and four symmetric Y-junctions implemented with (a) conventional homogeneous and (b) SWG metamaterial waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  (c) Cross-sectional view of the SOI strip waveguides with a SiO2 cladding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' (c) Top view of the SWG waveguides with their main  geometrical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  (a)  Wi  MMI  PS2  W  Y1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  2W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  WA  Y2  Wps  tWs  Y1  2W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  P  2  PS1<Lpsi  WMMI  Y1  4W1  LpS2  4  2 W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  PS3  + Lyl  LpS3  1  Ly2  Wi  Wi  Lyl  Li  LMMI  PS3  Z  X  (b)  SWG MMI Ws  WI+  SPS2  SPS3  WR3  Y1  D2  2Wi  3  Y2  Y1  D  2W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  2  WR2  Y1  4WI  WRI  4  2Wi  Lyl  Ly2  Wi  Ly1  SPS1  LsT LsMMI  LSPS3  1  (c)  (d)  D  Si  H  Z4  yt  SiO2  y  W  xat mode conversion and multiplexing of the first four  modes for transverse-electric (TE) polarization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  the fundamental mode (TE0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' the first-order mode (TE1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  the second-order mode (TE2) and the third-order mode  (TE3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Our MCMD includes two types of symmetric  multimode 1×2 Y-junctions: Y1, with a stem supporting  up to two modes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' and Y2, with a wider stem supporting  up to four modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' In general, multimode symmetric 1×2  Y-junctions transform the two in-phase 𝑚𝑡ℎ-order modes  in the arms into the (2𝑚)𝑡ℎ-order mode in the stem when  𝑚 is even, and into the (2𝑚 + 1)𝑡ℎ-order mode in the  steam when 𝑚 is odd [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Likewise, two anti-phase  𝑚𝑡ℎ-order modes in the arms are transformed into the  (2𝑚 + 1)𝑡ℎ-order mode in the stem when 𝑚 is even, and  into the (2𝑚)𝑡ℎ-order mode in the stem when 𝑚 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Figure 2(a) illustrates how this principle affects Y1  operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Since only two modes are supported by the Y1  stem, a TE0 (red) mode at the stem results in two in-phase  TE0 modes at the arms, whereas TE1 (orange) mode at  the stem results in two anti-phase TE0 modes at the arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Figure 2(b) shows the extension of this behavior to four  mode operation in Y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Operation for TE0 (red) and TE1  (orange) is the same as in Y1, whereas injection of TE2  (green) and TE3 (purple) modes through the stem  waveguide generates two anti-phase TE1 or two in-phase  TE1 modes at the arms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Therefore, by  cascading Y1 and Y2, and judiciously tailoring the phase  relations induced by the rest of the MCMD, mode  conversion and multiplexing between up to four modes  can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' We will hence study the phase shift  induced by the 4×4 MMI coupler, and subsequently  design a phase shifter architecture that satisfies the phase  distributions imposed by the cascaded Y-junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Bachmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' already derived a set of equations to  calculate the phase relations of 𝑁×𝑁 MMI couplers [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  At this point, it is important to mention that the definition  of the phase in this work is 𝜑 = 𝛽𝑥 − 𝜔𝑡, where 𝛽 is the  phase constant (also known as propagation constant), 𝑥  is the propagation direction and the term −𝜔𝑡  corresponds to the temporal dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' As in [34] the  authors used the opposite phase convention, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=', = 𝜔𝑡 −  𝛽𝑥, equations can be rewritten as follows:  𝑖 + 𝑗 even: 𝜑𝑖𝑗 = −𝜑0 − 𝜋 −  𝜋(𝑗−𝑖)(2𝑁−𝑗+𝑖)  4𝑁  ,  (1)  𝑖 + 𝑗 odd: 𝜑𝑖𝑗 = −𝜑0 −  𝜋(𝑗+𝑖−1)(2𝑁−𝑗−𝑖+1)  4𝑁  ,  (2)  where 𝜑0 is a constant phase, 𝑖 and 𝑗 are the indices of  the 𝑁 inputs and outputs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' (1) and  (2), the phase relations of a 4×4 MMI coupler can be  calculated as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Please note the input and  output numbering in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Table 1  Calculated phase relations 𝝋𝒊𝒋 of a 4×4 MMI coupler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  𝒋  𝒊  1  2  3  4  1  −𝜋  −3𝜋 4  ⁄  −7𝜋 4  ⁄  −𝜋  2  −3𝜋 4  ⁄  −𝜋  −𝜋  −7𝜋 4  ⁄  3  𝜋 4  ⁄  −𝜋  −𝜋  −3𝜋 4  ⁄  4  −𝜋  𝜋 4  ⁄  −3𝜋 4  ⁄  −𝜋  We then calculate, for each input port, the resulting  phase difference at the two upper output ports (∆𝜑12) and  the two lower ports (∆𝜑34) as:  ∆𝜑12 = 𝜑𝑖1 − 𝜑𝑖2,  (3)  ∆𝜑34 = 𝜑𝑖3 − 𝜑𝑖4,  (4)  Calculated phase differences are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Since phase evolution at both the MMIs and Y-splitters  are fixed, we then need to design a combination of PSs  (placement and phase shift values), that results in the  required phase relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' As shown in Figure 1(a), we  achieve this goal by including a first phase shifter (PS1)  between inputs 3 and 2 of the MMI, with a phase shift of  𝜋 2  ⁄ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' a second phase shifter (PS2) between outputs 2 and  1, with a phase shift of − 𝜋 4  ⁄ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' and a third phase shifter  (PS3) between outputs 3 and 4 with a phase shift of  3 𝜋 4  ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' An additional two-mode Y-junction (Y1) is  included at MCMD port 2 to satisfy even-order modes  phase conditions, as discussed hereunder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Figure 4 shows the operation of the device working  in multiplexer configuration, including the value of the  phase relations at different locations for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' It should  be noted that phase values have been calculated with  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Schematic and principle of operation of a multimode  symmetric 1×2 Y-junction for (a) a two-mode stem, and (b) a  four-mode stem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Schematic of a 4×4 MMI coupler, illustrating port and  phase notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  (a)  TEo TEo  TEo  Y1  TE1  TEo TEo  y4  x  (b)  TE, TEo  TE2 TEo  TE TEo  Y2  TE, TEo  TE,  TE1  TE, TEo  yt  xInputs (i)  Outputs (i)  4  3  4×4 MMI  2  △34Table 2  Calculated phase differences between MMI output pairs for each input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  𝒊  ∆𝝋𝟏𝟐  ∆𝝋𝟑𝟒  1  −𝜋 4  ⁄  −3𝜋 4  ⁄  2  𝜋 4  ⁄  3𝜋 4  ⁄  3  5𝜋 4  ⁄  −𝜋 4  ⁄  4  −5𝜋 4  ⁄  𝜋 4  ⁄  with respect to the mode phase at the input ports, which  is considered to be zero for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' When light is  injected through MCMD port 1 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 4(a)], the  combination of all the aforementioned phase relations  results in all modes arriving in-phase at the arms of the  cascaded Y-junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Thus, two in-phase TE0 modes are  coupled into Y1 stems, which subsequently generate the  desired TE0 mode at the multimode stem waveguide of  Y2 (MCMD port 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  When light is injected through MCMD port 2 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  4(b)], the combination of Y1 and PS1 results in  simultaneous light coupling to MMI input ports 2 and 3,  but with a 𝜋 2  ⁄  phase difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' This in turn generates  in-phase modes in the upper arms that are in anti-phase  with the two in-phase modes in the lower arms at their  arrival at the cascaded Y-junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' This combination  results in TE1 generation at the MCMD output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Finally, when light is injected through MCMD port 3  [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 4(c)], that is, MMI input port 4, in-phase modes are  generated in the middle arms, which are in anti-phase  with the two in-phase modes generated in the top and  bottom arms, before the cascaded Y-junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' This  results in anti-phase TE1 modes at the output of Y1  stems, which subsequently generate the TE2 mode at the  MCDM output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  So far, we have only considered passive PSs, that is,  PSs with a fixed phase shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' However, if the phase  introduced by PS1 is 3𝜋 2  ⁄  instead of 𝜋 2  ⁄ , it is possible  to generate the TE3 mode at MCMD output [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 4(d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  For illustrative purposes, we represent this phase shift by  switching the position of the tapers in PS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' This feature  opens the possibility of extending MCMD operation to  four modes using a single switchable PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Proof-of-concept results  To verify the principle of operation explained in the  previous section, we firstly optimized each constituent  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=', MMI, phase shifters and Y-junctions) for a design  wavelength of 1550 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' We chose a standard silicon  thickness of 𝐻 = 220 nm and an interconnection  waveguide width of 𝑊𝐼 = 400 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Thus, symmetric Y-  junctions are designed with stem widths of 2𝑊𝐼 =  800 nm for Y1 and 4𝑊𝐼 = 1600 nm for Y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Geometrical parameters of the 4×4 MMI coupler, the  phase shifters and the symmetric Y-junction are  summarized in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  In order to evaluate the performance of each  constituent element,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' the figures of merit for the MMI are  the excess loss (EL),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' imbalance (IB) and phase error  (PE):  EL𝑖 [dB] = −10log10 (∑ |S𝑗𝑖|  2  𝑗  ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  (5)  IB𝑖  𝑗𝑘 [dB] = 10log10 (|S𝑗𝑖|  2 |S𝑘𝑖|2  ⁄  ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  (6)  PE𝑖  𝑗𝑘[°] = [∠(S𝑗𝑖 S𝑘𝑖  ⁄  ) − 𝜑𝑖𝑑𝑒𝑎𝑙] · 180 π  ⁄ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  (7)  where S𝑗𝑖 and S𝑘𝑖 are the scattering parameters for input  𝑖 and outputs 𝑗 and 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' and 𝜑𝑖𝑑𝑒𝑎𝑙 is the ideal phase  relation depending on selected input and output ports as  shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' The designed 4×4 MMI exhibits EL <  Table 3  Geometrical parameters of the three-mode converter and  multiplexer/demultiplexer with homogeneous waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Constituent  Parameter  Value  Waveguides  Width  𝑊𝐼  400 nm  MMI  Separation  𝑊𝑆  500 nm  Access width  𝑊𝐴  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='3 µm  Taper length  𝐿𝑇  6 µm  MMI width  𝑊𝑀𝑀𝐼 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='2 µm  MMI length  𝐿𝑀𝑀𝐼 91 µm  Y1  Arm width  𝑊𝐼  400 nm  Arm length  𝐿𝑌1  5 µm  Stem width  2𝑊𝐼  800 nm  Y2  Arm width  2𝑊𝐼  800 nm  Arm length  𝐿𝑌2  20 µm  Stem width  4𝑊𝐼  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='6 µm  PS1  PS width  𝑊𝑃𝑆1 600 nm  PS length  𝐿𝑃𝑆1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='41 µm  PS2  PS width  𝑊𝑃𝑆2 600 nm  PS length  𝐿𝑃𝑆2  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='38 µm  PS3  PS width  𝑊𝑃𝑆3 600 nm  PS length  𝐿𝑃𝑆3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='61 µm  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Principle of operation of the proposed three-mode  converter and multiplexer/demultiplexer for (a) TE0, (b) TE1,  (c) TE2 and (d) TE3 mode multiplexing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  (a)  TE, multiplexing  3  TEo  T元  一元  T  2  3元/4-元  f  F7元/4  TEo  T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  (b)  TE, multiplexing  3  T  TE,  元  T  元/2  3元/4 一元  TEo  0  0  3元/4  0  1  0  0  (c)  TE, multiplexing  TE2  3  TEo  一元  2  元/4  0  0  →  3元／40  1  T  (d)  TE, multiplexing  0  TE3  3  0  0  0  F3元/4—元  TEo  0  0  元/2  3元/4  1  元  yt  x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='54 dB, IB < ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='4 dB  and PE < ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='32°  at  the  wavelength of 𝜆0 = 1550 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Regarding the spectral  response,  EL < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='15 dB,  IB < ±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='1 dB  and PE <  ±46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='03° are attained in the entire simulated wavelength  range (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='45 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='65 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Designed phase shifters introduce small phase  deviations of only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='12° for PS1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='13° for PS2 and  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='16° for PS3, with respect to their target phase  difference at 1550 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' However, considering the  simulated bandwidth of 200 nm, phase errors increase up  to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='84° for PS1, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='28° for PS2, and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='12° for PS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Symmetric Y-junctions Y1 and Y2 were also  designed showing negligible excess losses and power  imbalance between output ports at the design  wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' More specifically, Y1 losses are lower than  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='01 dB for both TE0 and TE1 mode operation in the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='45  – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='65 µm wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Conversely, calculated  excess losses for Y2 are below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='15 dB for TE0, TE1, TE2  and TE3 mode operation within the same bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Once all elements were optimized, two-dimensional  finite-difference time-domain (FDTD) simulations of the  whole MCMD were performed by applying the effective  index method to the original three-dimensional structure  [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 1(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' The simulated field distribution of the  three-mode MCMD is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 5(a)-(d),  demonstrating the successful implementation of the  phase relations described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Some ripples can be observed for TE0 and TE2 mode  multiplexing in the stem waveguide of Y-junction Y2  [see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 5(a) and 5(c)], which we attribute to a higher  crosstalk between both modes compared to TE1 and TE3  mode multiplexing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  The transmittance as a function of the wavelength  was computed for the complete MCMD [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='5(e)-  (h)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' At the central wavelength of 𝜆0 = 1550 nm,  insertion losses are lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='53 dB, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='79 dB and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='59  dB for the generation of TE0, TE1 and TE2 modes in the  stem waveguide, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Our device also exhibits a  low crosstalk at the same wavelength with values below  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='61 dB for TE0, -28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='94 dB for TE1 and -21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='11 dB for  TE2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  By tuning the value of PS1 to 3 𝜋 2  ⁄ , TE3 mode  (instead of TE1 mode) can be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' In this case,  insertion losses are below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='75 dB, and the crosstalk is  better than -28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='79 dB, both at 1550 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' These results  corroborate the higher crosstalk for TE2 mode operation,  which leads to a slight ripple in the field distribution at  port 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Regarding performance across the spectrum, insertion  losses lower than 1 dB are attained for a 55 nm  bandwidth (1542 – 1597 nm), whereas the crosstalk is  below -20 dB for a 60 nm bandwidth (1537 –1597 nm)  as shown with vertical lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' These results prove  the correct operation of the proposed architecture, but the  overall bandwidth is significantly limited by the narrow  spectral response of both the MMI and PSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' SWG performance enhancement  To overcome these bandwidth limitations, we  propose the MCMD with SWG metamaterials shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' The design of each of the constituents of the  SWG MCMD was performed by individual three-  dimensional FDTD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' The three symmetric Y-  junction labeled Y1 maintain the same geometrical  dimensions as those used for the conventional  multiplexer for the arm and stem widths (see Table 3),  but arm length was shortened to 𝐿𝑌1 = 2 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Y-junction  Y2 was slightly redesigned to reduce the crosstalk  between TE0 and TE2 modes by increasing the length of  the arms to 𝐿𝑌2  =  40 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  A procedure similar to those already reported in  [39,40] was followed for the optimization of the 4×4  SWG MMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' We restrict the value of the duty cycle (DC =  𝑎 Λ  ⁄ ) to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='5 in order to maximize the minimum feature  size for a given period (Λ) [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='1(d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' We explored  then different periods and found that Λ = 222 nm  significantly flattens the beat length across the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Compared to the conventional MMI section design, the  width 𝑊𝑆𝑀𝑀𝐼 is increased by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='8 µm but the length 𝐿𝑆𝑀𝑀𝐼  is reduced by more than half to ~41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='3 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' To increase  the quality of the interferometric patterns formed in the  MMI, the access width is 𝑊𝐵 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='7 µm and the  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Electric field amplitude |𝐸| in the XY plane at the middle  of the silicon layer for (a) TE0, (b) TE1, (c) TE2 and (d) TE3  mode multiplexing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Simulated transmittance to output port 4 as  a function of the wavelength when TE0 mode is launched into  (e) input port 1, (f) input port 2 with PS1 = 𝜋 2  ⁄ , (g) input port  3 and (h) input port 2 with PS1 = 3𝜋 2  ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Vertical lines indicate  the bandwidth where IL < 1 dB (55 nm) and XT < −20 dB (60  nm) are achieved for all modes simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  (a)  (b)  (c)  (d)  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='8  150  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='6  X  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='4  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='2  0  420-2-4 420-2-4420-2-4420-2-4  y (μm)  y (μm)  y (μm)  y (μm)  (e)  Input 1  (G)  Input 2 (PS1 = π/2)  (dB)  0  (dB)  55 nm  55nm  Transmittance  Transmittance  60 nm  60 nm  20  20  40  TE  40  TE  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='55  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='65  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='65  Wavelength (um)  Wavelength (um)  (g)  (h)  Input 3  Input2(PS1=3元/2)  (dB)  0  (dB)  55 nm  55 nm  Transmittance  Transmittance  60 nm  60 nm  20  20  TE,  TE  40  40  TE  TE  TE3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='55  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='65  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='6  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='65  Wavelength(um  Wavelength (um)separation is reduced to 𝑊𝑅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='3 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' The transition  between  the  interconnection  waveguides  (𝑊𝐼 =  400 nm) and the access to the MMI section (𝑊𝐵) is  performed by means of adiabatic SWG tapers with a  length 𝐿𝑆𝑇 = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='32 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' The performance of the 4×4  SWG MMI is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 6(a)-(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Owing to the  symmetry of the structure, only the results obtained when  injecting light into input ports 1 and 2 are depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' It is  observed that the device exhibits EL < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='77 dB, IB <  ±1 dB and PE < ±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='02° within a broad bandwidth of  200 nm (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='45 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='65 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  To drastically extend the operating bandwidth of the  nanophotonic phase shifters, we build upon the strategy  we recently reported in [41] to develop SWG phase  shifters SPS1, SPS2 and SPS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Notwithstanding, here we  employ four parallel SWG waveguides of two different  widths to implement SPS2 and SPS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' That is, each PS  has three identical reference SWG waveguides with  width 𝑊𝐷, and one dissimilar SWG waveguide with  width 𝑊𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Both the reference and dissimilar waveguides  have a length of LSPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Note that for SPS1 this  configuration is not necessary as only two MMI inputs  are illuminated for TE1 and TE3 mode generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Analogous to the 4×4 SWG MMI, a flat phase shift can  be achieved by judicious selecting the SWG period and  duty cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' A duty cycle of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='5 was fixed to maximize  MFS, while a period of 200 nm resulted in minimum  phase shift deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' In order to induce 𝜋 4  ⁄ , 𝜋 2  ⁄ , and  3𝜋 4  ⁄  phase shifts, we selected respectively 𝑊𝐷2 =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='8 µm, 𝑊𝑅2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='6 µm, 𝐿𝑆𝑃𝑆2 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='2 µm and 𝐿𝑆𝑇2 =  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='0 µm for SPS2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 𝑊𝐷1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='8 µm, 𝑊𝑅1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='6 µm,  𝐿𝑆𝑃𝑆1 = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='8 µm and 𝐿𝑆𝑇1 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='0 µm for SPS1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' and  𝑊𝐷3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='8 µm, 𝑊𝑅3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='6 µm, 𝐿𝑆𝑃𝑆3 = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='2 µm and  𝐿𝑆𝑇3 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='0 µm for SPS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' The simulated phase shifts are  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 6(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Negligible deviations can be  appreciated with phase shift errors as small as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='29° for  SPS1, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='15° for SPS2 and SPS3 within the entire  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='45 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='65 µm wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' SWG results  The simulation of the entire MCMD is quite  resource-intensive and time-consuming due to the device  footprint and the need for a fine mesh to simulate SWG-  based devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Thus, instead of performing the full  device simulation, we leverage the S-parameter matrices  calculated during the design process and concatenate all  of them using a circuit simulator to obtain the S-  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Simulated transmittance as a function of the wavelength of the MCMD with SWG metamaterials when TE0 mode is launched  into (a) input port 1, (b) input port 2 with SPS1 = 𝜋 2  ⁄ , (c) input port 3 and (d) input port 2 with SPS1 = 3𝜋 2  ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Vertical lines indicate  the bandwidth where IL < 1 dB (183 nm) and XT < −20 dB (161 nm) are achieved for all modes simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Simulated performance of the 4×4 SWG MMI including  (a) excess loss, (b) imbalance and (c) phase error between  output ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' (d) Phase error of each SWG PSs as a function of  the wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  (a)  Input 1  (b)  Input 2 (SPS1 = π/2)  0  (dB)  (dB)  0  183 nm  183 nm  161 nm  161 nm  20  Transmittance  Transmittance  20  TE  TEo  40  40  TE,  TE,  60  TE,  60  T-80  TE  80  TE  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='55  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='65  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='55  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='65  Wavelength (μm)  Wavelength (um)  (c)  Input 3  (d)  Input 2 (SPS1 = 3π/2)  (dB)  183 nm  (dB)  0  183 nm  161 nm  161 nm  20  20  Transmittance  40  4  TE  60  TE  60  TH  TE  TE  TE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  80  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='55  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='65  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='55  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='65  Wavelength (um)  Wavelength (um)(a)  (b)  (dB)  Input1 (EL)  (dB)  —Input 1: 0,/02 (IB12)  Input2(EL2)  Input 1: O3/O4 (IB24)  Imbalance (  Input 2: O,/0, (IB22)  2  Input 2:0,/04 (IB34)  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='55  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='65  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='65  Wavelength (μm)  Wavelength (um)  (c)  (d)  10  Input1:012  10  SPS2 (元/4rad)  Input 1: Ap  SPS1 (元/2rad)  error  error  5  0  Phase  Input 2: A934  5  SPS1 (3元/2rad)  Input2:012  SPS3 (3元/4rad)  10  10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='65  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='65  Wavelength (um)  Wavelength (um)parameter matrix and hence the spectral response of the  complete device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' The circuit simulator enables  bidirectional signals to be accurately simulated,  including coupling of modes in the single elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Figure 7 shows the overall transmittance of the SWG  MCMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Insertion losses (ILs) are lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='37 dB,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='47 dB and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='37 dB for TE0, TE1 and TE2 multiplexing,  respectively, at the central wavelength of 𝜆0 =  1550 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Moreover, low crosstalk (XT) is achieved at  the same wavelength with values below -21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='54 dB for  TE0, -32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='89 dB for TE1 and -21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='24 dB for TE2  multiplexing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  When SPS1 takes the value of 3 𝜋 4  ⁄ , insertion losses  for TE3 multiplexing reach a low value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='47 dB at  1550 nm, while crosstalk values are lower than -39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='48 dB  for the same wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  This design also shows an excellent performance  over a broad bandwidth (BW) of 200 nm with insertion  loss lower than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='18 dB and crosstalk below -16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='53 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Insertion losses decrease to 1 dB when the bandwidth is  restricted to 183 nm (1450 – 1633 nm), whereas a  crosstalk below -20 dB is achieved over a 161 nm  bandwidth (1489 – 1650 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' For the sake of  comparison, Table 4 summarizes the performance of  other three- and four-mode MCMD that are based on  MMI couplers and have been reported in the state of the  art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' To the best of our knowledge, it is the first time such  low losses and crosstalk are achieved in an outstanding  161 nm wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Conclusions  In this work, we have proposed a novel architecture to  scale the number of multiplexed modes of mode  converters and multiplexer based on MMI couplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Unlike  other  reported  architectures  that  use  unconventional 1×4 Y-junctions or 1×3 Ψ-junctions,  here we employ symmetric 1×2 Y-junctions arranged in  a conventional cascaded configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' The design  methodology was proposed on the basis of a two-  dimensional model with conventional homogenous  components (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=', without patterning the silicon  waveguide).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' The conventional mode converter and  multiplexer features sub-decibel insertion loss and  crosstalk better than -20 dB in the 1542 – 1597 nm  wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Once the principle of operation was  verified, we redesigned and optimized the mode  converter  and  multiplexer  by  incorporating  subwavelength grating metamaterials to leverage the  additional degrees of freedom they introduced into the  design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' A broad design bandwidth of 161 nm for  insertion losses below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='18 dB and crosstalk lower than  20 dB was confirmed by 3D FDTD simulations,  comparing very favorably to state-of-the-art three- and  four-mode converters and multiplexers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' The crosstalk  between TE0 and TE1 modes could be further reduced by  including optimized Y-junction geometries that mitigate  the effect of the non-perfect tip at the junction [42-44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  We believe that our design strategy will open promising  prospects for the development of high-performance  mode converters and multiplexer based on MMI couplers  with a high channel count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Credit authorship contribution statement  David  González-Andrade:  Conceptualization,  Methodology, Software, Validation, Formal analysis,  Investigation, Data curation, Writing – original draft,  Visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Irene Olivares: Methodology, Software,  Validation, Formal analysis, Data curation, Writing –  review & editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Raquel Fernández de Cabo:  Software, Validation, Data curation, Writing – review &  editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Jaime Vilas: Writing – review & editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Antonio Dias: Resources, Writing – review & editing,  Project administration, Funding acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Aitor V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Velasco: Resources, Writing – review & editing,  Supervision,  Project  administration,  Funding  acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Declaration of Competing Interest  The authors declare that they have no known competing  financial interests or personal relationships that could  have appeared to influence the work reported in this  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Acknowledgements  This work has been funded in part by the Spanish  Ministry of Science and Innovation (MICINN) under  grants RTI2018-097957-B-C33, PID2020-115353RA-  I00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  the  Spanish  State  Research  Agency  (MCIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
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+page_content=' the Community  of Madrid – FEDER funds (S2018/NMT-4326);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' the  European Union – NextGenerationEU through the  Recovery,  Transformation  and  Resilience  Plan  (DIN2020-011488,  PTQ2021-011974);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content="  and  the  European Union's Horizon Europe research and  innovation program under the Marie Sklodowska-Curie  grant agreement Nº 101062518." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  References  [1]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Sabella, Silicon photonics for 5G and future networks, IEEE  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Sel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Quantum Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 26 (2020) 8301611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [2]  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Demirkol, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Camps-Mur, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Paradells, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Combalia, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Popoola, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Haas, Powering the Internet of things through light  communication, IEEE Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 57 (2019) 107-113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [3]  M Filer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Gaudette, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Yin, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Bakhtiari, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Cox, Low-argin  optical networking at cloud scale, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Netw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 11  (2019) C94-C105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Table 4  Performance comparison of state-of-the-art three- and  four-MCMD based on MMI couplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  IL  [dB]  XT  [dB]  BW  [nm]  Length  [µm]  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' of  modes  [28]  <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='71  <-18  100  400  3  [29]  <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='9  <-17  40  400  3  [30]  <1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='1  <-10  90  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='05  3  [31]  <1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='3  <-15  40  120  3  [32]  <3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='3  <-14  70  173  4  [32]  <2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='25  <-15  140  123  4  This work  <1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='18  <-20  161  181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='14  3  [4]  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Cheng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Bahadori, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Glick, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Rumley, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Bergman  Recent advances in optical technologies for data centers: a  review, Optica 5 (2018) 1354-1370.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [5]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Thomson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Zilkie, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Bowers, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Komljenovic, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Reed, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Vivien, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Marris-Morini, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Cassan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Virot, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Fédeli, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Hartmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Schmid, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Xu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Boeuf, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content="  O'Brien, G." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Mashanovich, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Nedeljkovic, Roadmap on  silicon photonics, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 18 (2016) 073003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [6]  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Chen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Milosevic, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Stanković, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Reynolds, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Domínguez-Bucio, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Li, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Thomson, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Gardes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Reed,  The emergence of silicon photonics as a flexible technology  platform, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' IEEE 106 (2018) 2101-2116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [7]  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Bernabé, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Wilmart, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Hasharoni, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Hassan, Y Thonnart,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Tissier, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Désières, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Olivier, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Tekin, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Szelag, Silicon  photonics for terabit/s communication in data centers and  exascale computers, Solid State Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 179 (2021) 107928.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [8]  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Shi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Tian, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Gervais, Scaling capacity of fiber-optic  transmission systems via silicon photonics, Nanophotonics 9  (2020) 4629-4663.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [9]  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Li, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Liu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Dai, Multimode silicon photonics,  Nanophotonics 8 (2018) 227-247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [10]  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Cristiani, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Lacava, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Rademacher, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Puttnam, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Luìs, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Antonelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Mecozzi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Shtaif, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Cozzolino, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Oxenløwe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Jung, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Richardson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Ramachandran, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Guasoni, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Krupa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Kharenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Tonello, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Wabnitz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Philips, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Faccio, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Eurser, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Xie, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Russell, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Dai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Yu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Petropoulos, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Gardes,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Parmiagiani, Roadmap on multimode photonics, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 24  (2022) 083001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [11]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Driscoll, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Grote, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Souhan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Dadap, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Lu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Osgood, Asymmetric Y junctions in silicon waveguides for on-  chip mode-division multiplexing, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 38 (2013) 1854-  1856.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [12]  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Chen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Wang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Yang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Wang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Dai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Zhou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Yang, Silicon three-mode (de)multiplexer  based on cascaded asymmetric Y junctions, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 41 (2016)  2851-2854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [13]  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Tran, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Nguyen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Truong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Nguyen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Vu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Tran, Three-mode multiplexed device based on  tilted- branch bus structure using silicon waveguide, Photonics  Nanostructures – Fundam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 35 (2019) 100709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [14]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Dai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Shi, Silicon mode (de)multiplexer enabling  high capacity photonic networks-on-chip with a single-  wavelength carrier light, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 38 (2013) 1422-1424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [15]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Xuan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Qi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Huang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Sheng, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Wu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Wang S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Zou, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Gan, Broadband and  fabrication-tolerant  on-chip  scalable  mode-division  multiplexing  based  on  mode-evolution  counter-tapered  couplers, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 40 (2015) 1956-1959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [16]  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Uematsu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Ishizaka, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Kawaguchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Saitoh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Koshiba,  Design of a compact two-mode multi/demultiplexer consisting  of multimode interference waveguides and a wavelength-  insensitive phase shifter for mode-division multiplexing  transmission, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Lightwave Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 30 (2012) 2421-2426.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [17]  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Li, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Cheng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Xue, Compact two-mode (de)  multiplexer based on symmetric Y-junction and multimode  interference waveguides, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Express 22 (2014) 5781-5786.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [18]  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Frellsen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Ding, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Frandsen, Topology  optimized mode multiplexing in silicon-on-insulator photonic  wire waveguides, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Express 24 (2016) 16866-16873.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [19]  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Jafari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
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+page_content=' Miri, Compact fabrication-tolerant  subwavelength-grating-based  two-mode  division  (de)multiplexer, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 56 (2017) 7311-7319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [20]  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Hu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Mao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Zhou, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Rahman,  Broadband  silicon  four-mode  (de)multiplexer  using  subwavelength grating-assisted triple-waveguide couplers, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Lightwave Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 39 (2021) 5042-5047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [21]  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' He, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Zhang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Zhu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' An, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Cao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Guo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Qiu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Su,  Silicon high-order mode (de)multiplexer on single polarization,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Lightwave Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 36 (2018) 5746-5753.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [22]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' González-Andrade, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Wangüemert-Pérez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Velasco,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Ortega-Moñux, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Herrero-Bermello, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Molina-Fernández,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Halir, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Cheben, Ultra-broadband mode converter and  multiplexer based on sub-wavelength structures, IEEE  Photonics J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 10 (2018) 2201010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [23]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' González-Andrade, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Dias, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Wangüemert-Pérez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Ortega-Moñux, Í.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Molina-Fernández, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Halir, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Cheben, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Velasco, Experimental demonstration of a broadband mode  converter and multiplexer based on subwavelength grating  waveguides, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Laser Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 129 (2020) 106297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [24]  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Mao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Xing, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' El-Fiky, Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Saber, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Kumar, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' D’Mello, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Jacques, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Plant, Ultra-  broadband and compact two-mode multiplexer based on  subwavelength-grating-slot-assisted adiabatic coupler for the  silicon-on-insulator platform, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Lightwave Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 37 (2019)  5790-5800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [25]  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Xiao, Compact and broadband mode demultiplexer  using a subwavelength grating engineered MMI coupler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' B 38 (2021) 2830-2836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [26]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' González-Andrade, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Fernández de Cabo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Vilas, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Olivares, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Dias, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Luque-González, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Wangüemert-  Pérez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Ortega-Moñux, Í.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Molina-Fernández, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Halir, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Cheben, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Velasco, Mode converter and multiplexer with a  subwavelength phase shifter for extended broadband operation,  IEEE Photonics Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 33 (2021) 1262-1265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [27]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Wang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Shi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Dai, Improved 8-channel  silicon mode demultiplexer with grating polarizers, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Express 22 (2014) 12799.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [28]  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Tran, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Truong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Nguyen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Vu, A new  simulation design of three-mode division (de)multiplexer based  on a trident coupler and two cascaded 3×3 MMI silicon  waveguides, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Quantum Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 50 (2018) 426.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [29]  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Truong, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Nguyen, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Pham, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Trinh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Vu,  Three-mode multiplexer and demultiplexer utilizing trident and  multimode couplers, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 435 (2019) 334-340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [30]  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Yao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Su, Compact silicon three-mode  multiplexer by refractive-index manipulation on a multi-mode  interferometer, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Express 29 (2021) 13899-13907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [31]  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Hussain, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Fyath, Design and simulation of a compact  three-mode (de)multiplexer based on a subwavelength grating  multimode interference coupler, Photonics Nanostructures –  Fudam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 47 (2021) 100966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [32]  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Hussain, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Fyath, Design and simulation of 4-mode  (de)multiplexers  implemented  in  conventional  and  subwavelength grating Si/SiO2 platforms, Optik 251 (2022)  168449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [33]  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Lu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Yan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Zhang, On-chip mode converter  based  on  cross-connected  subwavelength  Y-junctions,  Photonics Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 9 (2021) 43-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [34]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Bachmann, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Besse, H Melchior, General self-imaging  properties in N×N multimode interference couplers including  phase relations, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 33 (1994) 3905-3911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [35]  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Cheben, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Halir, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Atwater, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Smith,  Subwavelength integrated photonics, Nature 560 (2018) 565-  572.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [36]  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Cheben, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Xu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Janz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Densmore, Subwavelength  waveguide grating for mode conversion and light coupling in  integrated optics, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Express 14 (2006) 4695-4702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [37]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Halir, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Bock, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Cheben, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Ortega-Moñux, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Alonso-  Ramos, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Schmid, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Lapointe, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Wangüemert-  Pérez, Í.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Molina-Fernández, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Janz, Waveguide sub-  wavelength structures: a review of principles and applications,  Laser Photonics Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 9 (2015) 25-49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [38]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Luque-González, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Sánchez-Postigo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Hadij-ElHouati,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Ortega-Moñux, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Wangüemert-Pérez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Schmid, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Cheben, Í.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Molina-Fernández, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Halir, A review of silicon  subwavelength gratings: building break-through devices with  anisotropic metamaterials, Nanophotonics 10 (2021) 2765-  2797.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [39]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Halir, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Cheben, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Luque-González, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Sarmiento-  Merenguel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Schmid, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Wangüemert-Pérez, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Xu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Wang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Ortega-Moñux, Í.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Molina-Fernández, Ultra-  broadband nanophotonic beamsplitter using an anisotropic sub-  wavelength metamaterial, Laser Photonics Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 10 (2016)  1039-1046.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [40]  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Patel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Morsy-Osman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Hui, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Parvizi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Ben-Hamida, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Plant, Ultra-broadband and ultra-  compact optical 90° hybrid based on 2×4 MMI coupler with  subwavelength gratings on silicon-on-insulator, Optical Fiber  Communication Conference (2018), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' M3I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [41]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  González-Andrade,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Luque-González,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Wangüemert-Pérez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Ortega-Moñux, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Cheben, Í.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Molina-  Fernández, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Velasco, Ultra-broadband nanophotonic phase  shifter based on subwavelength metamaterial waveguides,  Photonics Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 8 (2020), 359-367.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [42]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Fernández de Cabo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' González-Andrade, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Cheben, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  Velasco, High-performance on-chip silicon beamsplitter based  on subwavelength metamaterials for enhanced fabrication  tolerance, Nanomaterials 11 (2021) 1304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [43]  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Lu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Yan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Zhang, Subwavelength adiabatic  multimode Y-junctions, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 44 (2019) 4729-4732.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  [44]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Fernández de Cabo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Vilas, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Cheben, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Velasco, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content='  González-Andrade, Experimental characterization of an ultra-  broadband dual-mode symmetric Y-junction based on  metamaterial waveguides, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' Laser Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
+page_content=' 157 (2023)  108742.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFJT4oBgHgl3EQfsi1r/content/2301.11613v1.pdf'}
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+Code-based Cryptography in IoT:
+A HW/SW Co-Design of HQC
+Maximilian Sch¨offel
+Microelectronic Design Research Group
+University of Kaiserslautern
+Kaiserslautern, Germany
+schoeffel@eit.uni-kl.de
+Johannes Feldmann
+Microelectronic Design Research Group
+University of Kaiserslautern
+Kaiserslautern, Germany
+feldmann@eit.uni-kl.de
+Norbert Wehn
+Microelectronic Design Research Group
+University of Kaiserslautern
+Kaiserslautern, Germany
+wehn@eit.uni-kl.de
+Abstract—Recent advances in quantum computing pose a
+serious threat on the security of widely used public-key cryp-
+tosystems. Thus, new post-quantum cryptographic algorithms
+have been proposed as part of the associated US NIST process to
+enable secure, encrypted communication in the age of quantum
+computing. Many hardware accelerators for structured lattice-
+based algorithms have already been published to meet the strict
+power, area and latency requirements of low-power IoT edge de-
+vices. However, the security of these algorithms is still uncertain.
+Currently, many new attacks against the lattice structure are
+investigated to judge on their security. In contrast, code-based
+algorithms, which rely on deeply explored security metrics and
+are appealing candidates in the NIST process, have not yet been
+investigated to the same depth in the context of IoT due to the
+computational complexity and memory footprint of state-of-the-
+art software implementations.
+In this paper, we present to the best of our knowledge
+the first HW/SW co-design based implementation of the code-
+based Hamming Quasi Cyclic Key-Encapsulation Mechanism.
+We profile and evaluate this algorithm in order to explore
+the trade-off between software optimizations, tightly coupled
+hardware acceleration by instruction set extension and modular,
+loosely coupled accelerators. We provide detailed results on
+the energy consumption and performance of our design and
+compare it to existing implementations of lattice- and code-based
+algorithms. The design was implemented in two technologies:
+FPGA and ASIC. Our results show that code-based algorithms
+are valid alternatives in low-power IoT from an implementation
+perspective.
+Index Terms—Post Quantum Cryptography; Key Encapsu-
+lation Mechanism; IoT; Security; RISC-V; ASIC; Hardware
+Implementation; HW/SW co-design; HQC
+I. INTRODUCTION
+Privacy and data integrity are a key requirement in the In-
+ternet of Things (IoT). In many applications such as industrial
+IoT (IIoT), medical and healthcare, online banking, and even
+smart homes, highly sensitive data that should not be altered
+or made available to the public is transmitted over the Internet.
+In the vast majority of cases, the required security is provided
+by a combination of symmetric cryptography and Public Key
+Cryptography (PKC). However, recent advances in quantum
+computing severely compromise the security of the State-of-
+the-Art (SoA) PKC. While they are intractable on conven-
+tional computers, the underlying mathematical problems can
+be solved in polynomial time using Shor’s Algorithms [1]
+once large scale quantum computers become available. This is
+expected to be the case by the end of this decade [2] and thus,
+the US NIST is currently conducting a standardization process
+to find new post-quantum cryptographic (PQC) algorithms.
+The Key Encapsulation Mechanisms (KEMs) in the current,
+third round of the US NIST PQC standardization process rely
+on assumptions about the computational hardness of lattice-,
+code-, or isogeny-based problems. Among these, the structured
+lattice-based algorithms are considered as most promising can-
+didates for future standardization and for IoT applications due
+to their low-complexity computations. However, the structure
+of the lattices used is still the subject of cryptanalysis, and the
+security claims of the developers remain controversial [3].
+Due to the novelty of these algorithms, crypto-agility, i.e.
+the ability to seamlessly replace cryptographic algorithms in
+case that they are vulnerable, is even more important for PQC
+than for SoA cryptography. Code-based algorithms are based
+on different, well studied security assumptions, but have a
+higher computational complexity and larger memory footprints
+than lattice-based algorithms in state-of-the-art implementa-
+tions [4]. To determine if they are a viable alternative in low-
+power IoT environments in case that lattice-based algorithms
+turn out to be vulnerable, hardware implementations are essen-
+tial for a conclusive evaluation and have also been requested
+by the US NIST [5].
+Therefore, in this work we present to the best of our
+knowledge the first HW/SW co-design based implementation
+of the code-based Hamming Quasi Cyclic KEM (HQC) [4].
+Our design deploys a custom RISC-V processor and was im-
+plemented as an application-specific integrated circuit (ASIC)
+and field programmable gate array (FPGA). In summary, the
+new contributions of this work are:
+1) We provide the first ASIC implementation of a code-
+based KEM from the US NIST standardization process,
+which is fully compatible with the NIST C reference
+implementation.
+2) We identify the bottlenecks of the PQC algorithm and
+investigate for each bottleneck the best implementation
+method. We develop and implement software optimiza-
+tions, instruction set extensions, and loosely coupled
+accelerators and provide detailed information about their
+individual benefits and overhead.
+arXiv:2301.04888v1  [cs.CR]  12 Jan 2023
+
+3) We compare the energy consumption, hardware require-
+ments, and latency of our design to SoA implementa-
+tions of lattice- and code-based primitives.
+The results show that our implementation is the most
+efficient design. Furthermore, we show that HQC can be
+implemented with a similar resource utilization as lattice-based
+algorithms while achieving viable performance.
+This paper is structured as follows. In Section II, we briefly
+introduce the working principle of KEMs in general and HQC.
+In Section III, we provide an overview of the related work
+and of the SoA. In Section IV, we identify and evaluate the
+computational bottlenecks of HQC in software. In Section
+V, we present the IoT processing system and the hardware
+implementation of the different accelerators. In Section VI, we
+compare our results with the SoA. In Section VII, we draw a
+conclusion.
+II. BACKGROUND
+KEMs form a public key cryptosystem that is build out
+of three algorithms, Key-Generation (KeyGen), Encapsula-
+tion (Encaps) and Decapsulation (Decaps). Unlike general
+purpose Public Key Encryption Schemes (PKEs), KEMs are
+not thought to perform any application data encryption, but
+are designed to establish a randomly generated shared secret
+between communication partners in cryptographic protocols
+like Transport Layer Security (TLS) similar to the state-of-the-
+art Diffie-Hellmann Key-Exchange. Afterwards, this shared
+secret is used to derive a secret key for de- and encryption of
+application data with fast symmetric cryptographic algorithms
+like Advanced Encryption Standard (AES). KEMs are often
+build out of existing PKEs using transformations like the
+Fujisaki-Okamoto Transform.
+The first code-based PKE was introduced by McEliece in
+1978 and is based on the assumption that the error-correction
+code used is indistinguishable from random codes [6]. Al-
+though the original McEliece cryptosystem, which relied on
+Goppa codes, remains secure to this day, the method of hiding
+the generator matrix of the code in the public key carries
+a potential vulnerability. Attempts to reduce the key size
+by using more structured codes than the original McEliece
+approach have shown that this vulnerability can be exploited
+to crack the cryptosystems in 0.06 seconds [7].
+Therefore, the authors of HQC proposed a novel approach
+which combines two different types of codes:
+1) A decodable [n, k] code C with a fixed, publicly known
+generator matrix G ∈ Fk×n
+2
+and the error correction
+capability δ based on concatenated Reed-Muller (RM)
+and Reed-Solomon (RS) codes.
+2) A random double-circulant [2n, n] code with a publicly
+known parity check matrix h.
+This design rational allows HQC to use significantly smaller
+keys than the other code-based KEM Classic McEliece (2 KB
+vs 255 KB public key size) while still achieving the same
+security metrics.
+Fig. 1 shows how the shared secret ss is established between
+the communication partners using the HQC KEM. HQC uses
+Alice
+Bob
+KeyGen():
+1. h
+$
+←
+− R
+2. sk = (x, y)
+$
+←
+− R2
+3. pk = (h, s = x + h · y)
+Send pk
+Encaps(pk):
+4. m
+$
+←
+− Fk
+2
+5. θ ← G(m)
+6. e
+$
+←
+− R
+7. (r1, r2)
+$
+←
+− R2
+8. u = r1 + h · r2
+9. v = mG + s · r2 + e
+10. c ← (u, v)
+11. d ← H(m)
+12. ss ← K(m, c)
+Send ct = (c, d)
+Decaps(sk,ct)
+13. m′ = C.Decode(v − u · y)
+14. θ′ ← G(m′)
+15. c′ = Encrypt(pk, m′, θ′)
+16. If c ̸= c′or d ̸= H(m′)abort
+17. ss ← K(m′, c)
+Encrypt(pk, m, θ)
+Fig. 1. HQC KEM as defined in [4] with R = F2[X]/(Xn − 1), the hash
+functions G, H, K, the sampling operator
+$
+←− and the KEM’s public key pk,
+private key sk, ciphertext ct and shared secret ss. θ is the seed for the pseudo-
+random number generation during the encryption in Encaps() and Decaps().
+the Keccak-based extendable output function SHAKE as a
+seedexpander of a random generated seed as the scheme
+requires a large amount of random bytes (n = 17669 for the
+smallest parameter set HQC-128). Furthermore, the Keccak-
+based Secure Hash Alorithm 3 (SHA3) [8] is used for the G, H
+and K functions which are required due to the KEM-DEM
+transformation in HQC to construct an IND-CCA2 secure
+KEM.
+The procedure of HQC in short is as follows, a detailed
+description can be found in [4]. First, Alice randomly gener-
+ates the parity check matrix h and the private key sk, from
+which the public key pk is constructed. Here, the polynomials
+x and y which build the secret key are hidden in the public
+key by multiplying h with y and adding x in R. Bob uses
+pk to encrypt his randomly generated message m, which is
+the basis for the shared secret ss. During this encryption, the
+randomly generated vectors r1, r2 and e which have a fixed,
+predefined hamming weight are used to disguise m further.
+The hamming weights are selected in a way such that they
+still allow a correct decryption of m by Alice with respect to
+δ with a very high probability. The ciphertext ct is sent back to
+Alice, who decrypts the message m′ and calculates ss based
+on it.
+The HQC algorithm is available in 3 different parameter
+sets. This paper is focused on the NIST level 1 parameter set
+HQC-128.
+
+III. STATE OF THE ART
+Many works have been published which deal with hard-
+ware accelerations of new PQC primitives. Among these
+publications, the vast majority is focused on accelerators for
+lattice-based algorithms. A cryptographic co-processor was
+implemented in [9] as an ASIC to support various lattice
+based NIST schemes. Fritzmann et al. developed a HW/SW
+based co-design on a RISCV core for the lattice-based scheme
+NewHope [10]. In [11], FrodoKEM, an algorithm which has
+a high security confidentiality due to its less structured lattice,
+was accelerated by using a HW/SW co-design approach.
+In contrast, the code-based KEMs have not yet been in-
+vestigated to the same depth. For BIKE, another code-based
+candidate with a comparable key size to HQC, an FPGA
+implementation has been proposed in [12]. So far, the only
+hardware implementation for HQC was presented by the
+original authors of HQC in [4] and is based on FPGA HLS.
+Therefore, in this work, we present the first HW/SW co-design
+approach of HQC and implement our design both as ASIC and
+FPGA.
+IV. HW/SW CO-DESIGN
+There are three possibilities for implementation:
+1) Software.
+2) Custom processor instructions.
+3) Loosely coupled accelerators.
+In a first step, the execution of the NIST reference software
+was profiled to determine the computational bottlenecks and
+the memory footprint. In a second step, we investigated for
+each bottleneck the most suitable approach to find the optimum
+trade-off between the area, latency, memory footprint, and
+energy consumption. The highest priority was assigned to
+software optimization, as it offers high flexibility without
+additional costs. Then, if this is not efficient, custom processor
+instructions were considered as a second option, since they are
+still flexible and require little additional hardware. Only when
+these two approaches were found to be ineffective a loosely-
+coupled accelerator was considered.
+A. IoT Processing System (IoT-PS)
+Our methodology requires a processing system that allows
+instruction set extensions and the efficient interfacing of
+loosely-coupled accelerators. Therefore, we chose an adaptive
+platform that includes a RISC-V core whose instruction set
+architecture provides the ability to add custom instructions.
+Fig. 2 shows the final architecture of the IoT-PS. Our custom,
+area optimized RISC-V core supports the RV32IC instruction
+set which features additional compressed instructions and,
+therefore, significantly reduces the program size. The Direct
+Memory Access (DMA) controller features a memory copy
+(memcpy) and memory initialization (memset) function, of
+which both are able to operate on byte, half-word, and
+word granularity. The JTAG module provides access to the
+memories and the register file of the RISC-V core. It also can
+be used to start, stop, and reset the IoT-PS. Depending on
+the target platform, the data memory module was either based
+on an SRAM hard macro cell (ASIC) or Block RAM (Xilinx
+FPGA). Block RAM was also used for the instruction memory
+in case of an FPGA implementation. However, for the ASIC
+implementation we used a ROM macro cell, thus the program
+code is available after reset and does not need to be loaded
+via JTAG. The IoT-PS features no peripheral units except the
+I/O controller which is used to communicate via pin toggling.
+RISC-V
+JTAG
+DMA
+HQC Accelerator
+20 KB Instruction Memory
+32 KB Data Memory
+I/O Controller
+AXI4-Lite Interconnect
+Fig. 2. Architecture Overview
+B. Profiling of HQC-128
+The US NIST C reference implementation was used to
+identify the bottlenecks of the HQC-128 execution in our
+setup. The code was compiled with optimization level 2 (O2)
+and simulated cycle-accurately with the RTL model of our
+IoT-PS. Compared to Fig. 2, the size of ROM and RAM had
+to be increased for the analysis due to the large requirements
+of the reference implementation.
+The simulation results are shown in Table I. The specified
+clock cycles in the table refer to the processor cycles that
+the RISC-V core spends within the respective C function
+and excluding the time spent in sub-functions, e.g., gf mul is
+called during the computation of RS-Encode, but not included
+in its reported cycles. For all three KEM-functions, (1) the
+arithmetic in R, (2) the SHAKE-based hashing and (3) mem-
+ory operations are the main contributors to the total execution
+time. On top of that, the sampling operation, the RM-Decoding
+algorithm (4) and the finite field multiplication (5) are further
+contributors to the computation time. The unsigned division,
+which is performed in software, is mostly used during the
+polynomial multiplication in (1).
+In (1), the largest part is accounted by the multiplication of
+the large polynomials (n = 17669 for HQC-128), which are
+represented as bit vectors. This includes the subsequent reduc-
+tion by Xn − 1 of the intermediate result, and is performed,
+for example, in steps 3., 8. and 9. in Fig. 1. The multiplication
+complexity is reduced by the fact that one of the vectors is
+sparse and has a small, known hamming weight w ≤ 75,
+which allows to consider only the non-zero coefficients in
+the sparse polynomial during processing. The execution of the
+multiplication consists mostly of XOR operations for adding
+the binary coefficients of the same degrees and SHIFT / AND
+operations to determine the degree of the intermediate results.
+Due to the high degree of the polynomials, a large number
+of LOAD and STORE instructions is required during the
+computation.
+
+TABLE I
+TOTAL CYCLE COUNT AND SHARE OF IMPORTANT FUNCTIONS OF THE NIST C REFERENCE IMPLEMENTATION ON THE RISC-V, N.A. (NOT
+APPLICABLE) REFERS TO FUNCTIONS WHICH ARE NOT USED IN THIS STEP.
+Function
+Keygen Cycles
+Encaps Cycles
+Decaps Cycles
+Total
+5609k
+13850k
+19903k
+Arithmetic in R
+1540k (27.46%)
+3448k (24.9%)
+4989k (25.1%)
+- Vect Mul
+1528k
+3413k
+4942k
+- Vect Add
+12k
+35k
+47k
+SHAKE
+1854k (33.05%)
+5007k (36.15%)
+5414k (27.2%)
+- Keccak State Permute
+1744k
+4626k
+5005k
+- Keccak Inc Squeeze
+103k
+131k
+154k
+- Keccak Inc Absorb
+7k
+250k
+255k
+RS-RM Code
+n.A.
+26k (0.18%)
+1440k (7.24%)
+- RS-Encode
+n.A.
+26k
+26k
+- RS-Decode
+n.A.
+n.A.
+56k
+- RM-Decode
+n.A.
+n.A.
+1358k
+Sampling
+81k (1.44%)
+155k (1.1%)
+236k (1.18%)
+Memory-Operation
+2071k (36.92%)
+5068k (36.59%)
+7175k (36.05%)
+- memcpy
+2045k
+5021k
+7092k
+- memset
+26k
+47k
+83k
+Rest
+63k (1.12%)
+146k (2.17%)
+649k (3.26%)
+- unsigned division
+49k
+100k
+151k
+- gf mul
+n.A.
+20k
+162k
+Keccak’s permutation function in (2) is the computational
+core of the sponge construction in SHA3 and consists of
+bitwise AND, XOR, and rotate operations on the 25 lanes
+of 64 bits each. The major bottleneck in this permutation is
+the interdependence of the intermediate results which causes
+the contents of the processor registers to be swapped with the
+main memory multiple times during the execution of one of
+the 24 rounds.
+The large overhead of the memory operations in (3) is driven
+by two reasons. First, the RISC-V core supports only one
+outstanding memory read or write access at a time. It waits
+for a slave response before continuing the program execution.
+Second, the reference implementation is not optimized for
+low memory usage, e.g., it often stores multiple copies of
+temporary results, initializes a larger number of arrays, or
+copies parts of arrays to different memory locations.
+TABLE II
+STACK MEMORY AND CODE-SIZE OF THE NIST C REFERENCE
+IMPLEMENTATION OF HQC-128 ON OUR RISC-V CORE.
+Keygen
+Encaps
+Decaps
+Code Size
+10.798 KB
+17.015 KB
+22.378 KB
+Stack Memory
+53.018 KB
+68.714 KB
+77.762 KB
+C. Software Optimization
+In multiple functions, the reference implementation uses
+non-optimal data-types which increases the number of required
+memory accesses and processor instructions. An example of
+this is the comparisons in Step 16 of Fig. 1, which are
+performed on a byte boundary rather than a processor word
+boundary. The memory footprint and computation time was
+further improved by removing redundant arrays which often
+get initialized with zeroes or are the target of memcpy
+operations. The operations are performed with pointers in-
+stead. For the remaining memory operations, the time required
+for memcpy and for array initialization via memset were
+accelerated by using the DMA controller of the platform.
+D. Instruction Set Extension
+The bottleneck of RS-Encoding and -Decoding is caused
+by the multiplication in F28. This operation has only three
+operands, including the generator polynomial, and one return
+value with the size of one byte each. A F28-Unit is added to
+the RISC-V core which is able to perform the operation shown
+in Equation 1, where a = (a15, · · · , a0) and b = (b7, · · · , b0)
+are the input operands, and d = (d14, · · · , d0) is the output.
+This unit is made accessible via both an R-type and an I-type
+custom instruction, where register rs1 is used as operand a,
+register rs2 respectively the immediate value imm is used as
+operand b, and the output d is stored in register rd.
+(a15 · x7 + · · · + a8 · x0) · (b7 · x7 + · · · + b0 · x0)
++(a7 · x7 + · · · + a0 · x0) ⇒ (d14 · x14 + · · · + d0 · x0) (1)
+Using these custom instructions, a multiplication in F28 is
+performed within four clock cycles.
+E. Loosely-Coupled Accelerators
+Fig. 4 shows the block diagram of the loosely-coupled
+HQC accelerators. To enable parallel calculations between
+the processor and the accelerator, the accelerator has both
+an AXI slave and an AXI master interface and fetches its
+calculation inputs (e.g. the polynomials) from the processor’s
+
+Dense Polynomial
+Sparse Polynomial
+coord0
+coord1
+coordw-1
+. . .
+. . .
+word0
+0
+63
+word1
+64
+127
+wordN
+17664
+17228
+. . .
+word0
+coord0
+coord0+63
+wordN
+coord0
++ 17664
+coord0
++17228
+. . .
+word0
+coord1
+coord1+63
+wordN
+coord1
++ 17664
+coord1
++17228
+. . .
+word0
+coordw-1
+Coordw-1+63
+wordN
+coordw-1
++ 17664
+coordw-1
++17228
+. . .
+XOR
+XOR
+XOR
+Intermediate Result
+Fig. 3. Working principle of the polynomial multiplication in R with n = 17669 and 64-bit memory words.
+main memory via the master interface, according to the
+processor command which was previously received via the
+slave interface. Due to the data dependencies between the steps
+in the HQC-KEM, and based on the previous observation that
+the bottleneck in the HQC is driven by memory accesses, we
+decided that enabling parallel read/read or read/write accesses
+is more beneficial than running the dedicated compute units
+in parallel. Therefore, only two SRAMs are used and shared
+between the compute units, and only one of the compute units
+is processing at the same time. The access to the SRAMs and
+the operation mode of the compute units are managed by one
+control unit.
+AXI4-Lite Interconnect
+HQC Control Unit
+SRAM0
+(288 x
+64 Bit)
+SRAM1
+(566 x
+64 Bit)
+R-Unit
+Sampling-Unit
+RM-Decoder
+Keccak-IP
+AXI-Slave
+AXI-Master
+Fig. 4. HQC hardware accelerator.
+The R-Unit implements the addition, multiplication and
+reduction of the polynomials in R. Fig. 3 shows the working
+principle of the polynomial multiplication. The values of
+the sparse polynomial contain the locations of its non-zero
+coordinates. In our implementation, we iterate through the
+multiplication by a word-by-word shift of the dense polyno-
+mial by the coordinates given in the sparse polynomial.
+After the shift, the interim result is XOR-ed with the word
+that is currently stored at the respective location in memory
+and the carry out with respect to the word alignment of the
+memory is calculated. The R-Unit is designed such that only
+two cycles are necessary for calculating a resulting word. In
+the first cycle the address of the dense polynomial and the
+intermediate polynomial are calculated and read based on the
+coordinate, while in the second cycle the new value and the
+carry-out are calculated and written to memory.
+The Sampling-Unit combines the sponge functions squeeze
+and absorb of the incremental version of SHAKE, the permu-
+tation function of Keccak and the rejection-based sampling,
+during which SHAKE functions are used as extendable output
+functions (XOF). For the permutation function, a highspeed
+open-source implementation by the original authors of Kec-
+cak was used, which executes the permutation in 24 clock
+cycles [13].
+As shown in Table I, the vast majority of decoding time
+is spent on the RM-Codes, which employs a Maximum
+Likelihood (ML) algorithm based on the Hadamard Transform
+and a subsequent peak-search for the highest value in the
+transformed codeword. Because HQC uses duplicated RM
+codes, the decoding algorithm must be preceded by another
+transformation function [4]. This transform requires many
+single-bit operations, and thus, the implementation in soft-
+ware is not efficient. Therefore, we adapted the transform
+and the subsequent decoding steps to an efficient hardware-
+implementation.
+V. RESULTS AND COMPARISON
+The IoT-PS presented was implemented in a 22 nm FD-SOI
+technology from GlobalFoundries under worst case Process,
+Voltage and Temperature (PVT) conditions (125 °C, 0.72 V for
+timing; 25 °C, 0.8 V for power). Synthesis is performed with
+the Synopsys DesignCompiler, Place&Route is carried out
+with the Synopsys IC-Compiler. The SRAMs were generated
+by the INVECAS Memory Compiler. Power numbers are
+calculated with back-annotated wiring data. The layout of
+ASIC IP core presented in this work can be seen in Figure 5
+has a size of 0.12 mm2 with an aspect ratio of 1.77 and a
+maximum frequency of 700 MHz. The IoT-PS presented was
+also implemented on a Xilinx Artix xc7a100tcsg324-3 using
+Xilinx Vivado for a better comparison to existing work.
+
+TABLE III
+CYCLE COUNT OF THE HQC-KEM IN OUR SETUP WITH THE DIFFERENT HARDWARE MODULES. THE IMPROVEMENT REFERS TO SPEEDUP WITH
+RESPECT TO THE NIST C REFERENCE IMPLEMENTATION WITHOUT ANY HARDWARE ACCELERATORS.
+Keygen
+Encaps
+Decaps
+Cycles
+Improvement
+Cycles
+Improvement
+Cycles
+Improvement
+Reference
+5609k
+-
+13850k
+19903k
+-
+DMA + SW OPT
+3587k
+36.0%
+7044k
+49.1%
+10851k
+45.5%
++ R-Unit
+1862k
+66.8%
+5183k
+62.6%
+7245k
+63.6%
++ Sampling-Unit
+1623k
+71.1%
+1955k
+85.9%
+5176k
+74.0%
++ RM-Decoder
+n.A.
+n.A.
+n.A.
+n.A.
+9636k
+51.6%
++ F28-Instruction
+n.A.
+n.A.
+7028k
+49.3%
+10722k
+46.1%
++ All Modules
+56k
+98.9%
+131k
+99%
+557k
+97.2%
+Code Size
+1.5 KB
+86.1%
+6.6 KB
+61.2%
+13.362 KB
+40.3%
+Stack Memory
+10 KB
+81.1%
+24 KB
+65.7%
+31 KB
+60.1%
+A. Impact of individual optimizations on the overall run time
+Table III shows the extent to which the presented hardware
+modules accelerate the computation time of the three KEM
+functions. As illustrated, the use of a DMA and the software
+optimization already provide a significant speed up between
+36% and 49% over the reference implementation. This shows
+that the NIST C reference implementation is not meant to
+be used in IoT devices without optimizations. The loosely
+coupled Sampling-Unit is the accelerator that provides the a
+runtime reduction of at least 50% compared to the optimized
+software implementation with DMA in all KEM functions. The
+R-Unit, however, reduces the runtime only between 26.5%
+and 48% depending on the KEM function. The RM-Decoder
+has the least impact on the performance since it is only used
+in Decaps. Also it is able to achieve a runtime reduction of
+11.2%. The F28-Instructions give only a minor speedup on its
+own, but shows its potential in combination with all loosely
+coupled accelerators.
+B. Resource distribution of the individual hardware modules
+Figure 5 shows a qualitative area distribution of the distinct
+modules for ASIC, while Table IV shows a quantitative
+distribution for FPGA. The R-Unit shows the highest area
+efficiency among all loosely coupled accelerators. Although
+this unit has a lower runtime reduction compared to the
+Fig. 5.
+Layout; RISC-V - blue; JTAG - cyan; Interconnect - lime; I/O
+Controller - purple; DMA - pink; Sampling Unit - red; RM-Decoder - orange;
+R-Unit - yellow; HQC Control Unit, SRAM0, SRAM1 - green; ROM - white,
+horizontal strips; RAM - white, cross pattern
+Sampling-Unit, it needs less than 10% of its FPGA resources.
+The RM-Decoder, however, requires fewer resources than the
+R-Unit, but also offers the least performance gain and is only
+used in Decaps. Therefore, it is far less efficient compared to
+both R-Unit and Sampling-Unit. The F28-Unit, which is used
+by the custom instructions, requires only negligible resources.
+However, the decoding of these instructions as well as the
+controlling of the unit requires additional resources which are
+hidden inside the RISC-V core.
+TABLE IV
+RESOURCE UTILIZATION ON FPGA (ARTIX7)
+LUTs
+Registers
+Block RAM
+RISC-V
+2210
+1682
+0
+⌞ F28-Unit
+27
+0
+0
+Interconnect
+2775
+1919
+0
+Memories
+53
+6
+24
+HQC Accelerator
+7920
+2370
+3
+⊢ R-Unit
+565
+117
+0
+⊢RM-Decoder
+435
+63
+0
+⌞Sampling Unit
+5610
+1814
+0
+⌞Keccak Permute
+4685
+1622
+0
+DMA
+489
+412
+0
+JTAG
+452
+546
+0
+I/O Controller
+41
+68
+0
+IoT-PS
+13934
+7003
+27
+C. Comparison to State of the Art
+Table V presents the required clock cycles for the different
+KEM functions of SoA implementations and of our work. We
+use the number of clock cycles as metric rather than absolute
+computation time. This is, for our our work, a pessimistic
+comparison due to the high achievable clock frequency. How-
+ever, even under this assumption, our implementation requires
+a comparable number of clock cycles as the low-latency HQC
+implementation, while it requires less hardware than its low
+area hardware implementation. Compared to BIKE, the other
+code-based KEM, our implementation requires about the same
+amount of clock cycles like the low latency implementation
+while using significantly less hardware resources. The imple-
+mentation of FrodoKEM, which would be an alternative if
+structured lattice-based KEMs like Kyber and NewHope are
+
+.TABLE V
+COMPARISON OF CLOCK CYCLES, FREQUNCY AND FPGA RESOURCES FOR DIFFERENT STATE-OF-THE-ART IMPLEMENTATIONS. HW/SW REFERS TO
+IMPLEMENTATIONS BASED ON HW/SW CO-DESIGN, FULL REFERS TO FULL HARDWARE IMPLEMENTATIONS OF THE RESPECTIVE SCHEME. NEWHOPE
+AND KYBER ARE STRUCTURED-LATTICE BASED ALGORITHMS, FRODOKEM IS BASED ON LESS-STRUCTURED LATTICES, AND THE REMAINING
+IMPLEMENTATIONS ARE BASED ON CODES.
+Implementation
+Keygen
+Encaps
+Decaps
+Frequency
+FPGA Resources
+Target Plattform
+Cycles
+Cycles
+Cycles
+MHz
+LUTs
+FFs
+BRAMs
+HQC (low area, HW) [4]
+630k
+1500k
+2100k
+132
+8.9k
+4k
+14
+FPGA (Xilinx Artix-7)
+HQC (low latency, HW) [4]
+40k
+89k
+190k
+148
+20k
+16k
+12.5
+FPGA (Xilinx Artix-7)
+BIKE (low area, HW) [12]
+2671k
+153k
+1628k
+121
+13k
+5k
+17
+FPGA (Xilinx Artix-7)
+BIKE (low latency, HW) [12]
+259k
+12k
+189k
+96
+53k
+7k
+49
+FPGA (Xilinx Artix-7)
+NewHope (HW/SW) [10]
+357k
+590k
+167k
+n.A.
+11k
+5k
+1
+FPGA (Xilinx Zynq-7000)
+FrodoKEM (HW/SW) [11]
+23.4M
+25.5M
+25.3M
+100
+5.6k
+1.1k
+0
+FPGA (Xilinx Zynq Ultrascale+)
+Kyber (HW/SW) [9]
+75k
+132k
+142k
+72
+n.A.
+n.A.
+n.A.
+ASIC (40nm)
+HQC on Cortex M4 (SW)
+1048k
+2436k
+4001k
+64
+n.A.
+n.A.
+n.A.
+nRF52840
+This Work HQC (DMA+SW OPT)
+3587k
+7044k
+10851k
+700
+n.A.
+n.A.
+n.A.
+ASIC (22nm)
+This Work HQC (HW/SW)
+56k
+131k
+557k
+700
+n.A.
+n.A.
+n.A.
+ASIC (22nm)
+This Work HQC (HW/SW)
+56k
+131k
+557k
+100
+8k
+2.4k
+3
+FPGA (Xilinx Artix-7)
+proven to be vulnerable to attacks, is overall 100 times slower
+than our work.
+Table VI shows the energy consumption of our implementa-
+tion, of HQC on a Cortex M4 processor and of the structured
+lattice-based KYBER, also implemented as an ASIC. As
+can be seen, our implementation requires considerably less
+energy than the pure software on the Cortex M4 and also less
+than KYBER’s ASIC implementation, which, however, was
+implemented on a larger technology node.
+TABLE VI
+COMPARISON OF ENERGY CONSUMPTION. FOR A TRADE-OFF BETWEEN
+POWER AND LATENCY, OUR DESIGN WAS IMPLEMENTED AND SIMULATED
+WITH A 200 MHZ CLOCK.
+Implementation
+Keygen
+Encaps
+Decaps
+µJ
+µJ
+µJ
+Kyber (HW/SW) [9]
+5.97
+9.37
+11.25
+HQC on Cortex M4 (SW)
+500
+1184
+1872
+This Work HQC (HW/SW)
+1.02
+2.41
+7.1
+VI. CONCLUSION
+In this work, we investigated the performance of the code-
+based post-quantum KEM HQC in the context of low power
+IoT system. We presented the first ASIC implementation of a
+code-based US NIST PQC candidate. With a combination of
+software optimizations, instruction set extensions, and loosely
+coupled hardware accelerators, we achieve similar perfor-
+mance to the full SoA hardware implementation of HQC, but
+require significantly less hardware resources and provide more
+flexibility. Compared to SoA implementations of lattice-based
+algorithms, we have shown that code-based algorithms are
+promising alternatives in IoT based applications in terms of
+energy efficiency, computation time, and required hardware.
+ACKNOWLEDGEMENT
+This paper was partly founded by the German Federal
+Ministry of Education and Research as part of the project
+“SIKRIN-KRYPTOV” (16KIS1069).
+REFERENCES
+[1] P. W. Shor, “Algorithms for quantum computation: Discrete logarithms
+and factoring,” in Proceedings of the 35th Annual Symposium on
+Foundations of Computer Science.
+IEEE Computer Society, 1994, p.
+124–134.
+[2] M. Mosca, “Cybersecurity in an Era with Quantum Computers: Will We
+Be Ready?” IEEE Security & Privacy, vol. 16, no. 5, pp. 38–41, 2018.
+[3] Peikert,
+Christopher
+J.
+and
+Bernstein,
+D.J.,
+“CRYSTALS-
+KYBER:
+Round
+3
+Official
+Comments,”
+https://csrc.nist.gov/
+CSRC/media/Projects/post-quantum-cryptography/documents/round-3/
+official-comments/CRYSTALS-KYBER-round3-official-comment.pdf,
+2022, Retrieved 2022-06-07.
+[4] C. A. Melchor, N. Aragon, S. Bettaieb, L. Bidoux, O. Blazy, J.-
+C. Deneuville, P. Gaborit, E. Persichetti, G. Z´emor, and I. Bourges,
+“Hamming Quasi-Cyclic (HQC),” 2021.
+[5] G. Alagic, J. Alperin-Sheriff, D. Apon, D. Cooper, Q. Dang, J. Kelsey,
+Y.-K. Liu, C. Miller, D. Moody, R. Peralta et al., “Status Report on the
+Second Round of the NIST Post-Quantum Cryptography Standardization
+Process,” NIST, Tech. Rep., July, 2020.
+[6] R. J. McEliece, “A public-key cryptosystem based on algebraic coding
+theory,” The Deep Space Network Progress Report, vol. 42-44, pp. 114–
+116, 1978.
+[7] J.-C. Faugere, A. Otmani, L. Perret, and J.-P. Tillich, “Algebraic crypt-
+analysis of McEliece variants with compact keys,” in Annual Interna-
+tional Conference on the Theory and Applications of Cryptographic
+Techniques.
+Springer, 2010, pp. 279–298.
+[8] National Institute of Standards and Technology, “FIPS PUB 202
+-SHA-3 Standard: Permutation-Based Hash and Extendable-Output
+Functions,”
+https://csrc.nist.gov/Projects/post-quantum-cryptography,
+Retrieved 2022-02-17.
+[9] U. Banerjee, T. S. Ukyab, and A. P. Chandrakasan, “Sapphire: A Con-
+figurable Crypto-Processor for Post-Quantum Lattice-Based Protocols,”
+arXiv preprint arXiv:1910.07557, 2019.
+[10] T. Fritzmann, U. Sharif, D. M¨uller-Gritschneder, C. Reinbrecht,
+U. Schlichtmann, and J. Sepulveda, “Towards Reliable and Secure Post-
+Quantum Co-Processors based on RISC-V,” in 2019 Design, Automation
+& Test in Europe Conference & Exhibition (DATE), 2019, pp. 1148–
+1153.
+[11] P. Karl, T. Fritzmann, and G. Sigl, “Hardware Accelerated FrodoKEM
+on RISC-V,” in 2022 25th International Symposium on Design and
+Diagnostics of Electronic Circuits and Systems (DDECS), 2022, pp.
+154–159.
+[12] J. Richter-Brockmann, J. Mono, and T. Guneysu, “Folding BIKE:
+Scalable Hardware Implementation for Reconfigurable Devices,” IEEE
+Transactions on Computers, vol. 71, no. 5, pp. 1204-1215, 2022.
+[13] Bertoni, G. and Daemen, J. and Peeters, M. and Van Assche, G., “Keccak
+in VHDL,” https://keccak.team/hardware.html, 2022, Retrieved 2022-06-
+07.
+
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+page_content='de  Johannes Feldmann  Microelectronic Design Research Group  University of Kaiserslautern  Kaiserslautern, Germany  feldmann@eit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='uni-kl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='de  Norbert Wehn  Microelectronic Design Research Group  University of Kaiserslautern  Kaiserslautern, Germany  wehn@eit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='uni-kl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='de  Abstract—Recent advances in quantum computing pose a  serious threat on the security of widely used public-key cryp-  tosystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Thus, new post-quantum cryptographic algorithms  have been proposed as part of the associated US NIST process to  enable secure, encrypted communication in the age of quantum  computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Many hardware accelerators for structured lattice-  based algorithms have already been published to meet the strict  power, area and latency requirements of low-power IoT edge de-  vices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' However, the security of these algorithms is still uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  Currently, many new attacks against the lattice structure are  investigated to judge on their security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' In contrast, code-based  algorithms, which rely on deeply explored security metrics and  are appealing candidates in the NIST process, have not yet been  investigated to the same depth in the context of IoT due to the  computational complexity and memory footprint of state-of-the-  art software implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  In this paper, we present to the best of our knowledge  the first HW/SW co-design based implementation of the code-  based Hamming Quasi Cyclic Key-Encapsulation Mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  We profile and evaluate this algorithm in order to explore  the trade-off between software optimizations, tightly coupled  hardware acceleration by instruction set extension and modular,  loosely coupled accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' We provide detailed results on  the energy consumption and performance of our design and  compare it to existing implementations of lattice- and code-based  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The design was implemented in two technologies:  FPGA and ASIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Our results show that code-based algorithms  are valid alternatives in low-power IoT from an implementation  perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  Index Terms—Post Quantum Cryptography;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Key Encapsu-  lation Mechanism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' IoT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Security;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' RISC-V;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' ASIC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Hardware  Implementation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' HW/SW co-design;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' HQC  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' INTRODUCTION  Privacy and data integrity are a key requirement in the In-  ternet of Things (IoT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' In many applications such as industrial  IoT (IIoT), medical and healthcare, online banking, and even  smart homes, highly sensitive data that should not be altered  or made available to the public is transmitted over the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  In the vast majority of cases, the required security is provided  by a combination of symmetric cryptography and Public Key  Cryptography (PKC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' However, recent advances in quantum  computing severely compromise the security of the State-of-  the-Art (SoA) PKC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' While they are intractable on conven-  tional computers, the underlying mathematical problems can  be solved in polynomial time using Shor’s Algorithms [1]  once large scale quantum computers become available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' This is  expected to be the case by the end of this decade [2] and thus,  the US NIST is currently conducting a standardization process  to find new post-quantum cryptographic (PQC) algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  The Key Encapsulation Mechanisms (KEMs) in the current,  third round of the US NIST PQC standardization process rely  on assumptions about the computational hardness of lattice-,  code-, or isogeny-based problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Among these, the structured  lattice-based algorithms are considered as most promising can-  didates for future standardization and for IoT applications due  to their low-complexity computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' However, the structure  of the lattices used is still the subject of cryptanalysis, and the  security claims of the developers remain controversial [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  Due to the novelty of these algorithms, crypto-agility, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  the ability to seamlessly replace cryptographic algorithms in  case that they are vulnerable, is even more important for PQC  than for SoA cryptography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Code-based algorithms are based  on different, well studied security assumptions, but have a  higher computational complexity and larger memory footprints  than lattice-based algorithms in state-of-the-art implementa-  tions [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' To determine if they are a viable alternative in low-  power IoT environments in case that lattice-based algorithms  turn out to be vulnerable, hardware implementations are essen-  tial for a conclusive evaluation and have also been requested  by the US NIST [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  Therefore, in this work we present to the best of our  knowledge the first HW/SW co-design based implementation  of the code-based Hamming Quasi Cyclic KEM (HQC) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  Our design deploys a custom RISC-V processor and was im-  plemented as an application-specific integrated circuit (ASIC)  and field programmable gate array (FPGA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' In summary, the  new contributions of this work are:  1) We provide the first ASIC implementation of a code-  based KEM from the US NIST standardization process,  which is fully compatible with the NIST C reference  implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  2) We identify the bottlenecks of the PQC algorithm and  investigate for each bottleneck the best implementation  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' We develop and implement software optimiza-  tions, instruction set extensions, and loosely coupled  accelerators and provide detailed information about their  individual benefits and overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='04888v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='CR]  12 Jan 2023  3) We compare the energy consumption, hardware require-  ments, and latency of our design to SoA implementa-  tions of lattice- and code-based primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  The results show that our implementation is the most  efficient design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Furthermore, we show that HQC can be  implemented with a similar resource utilization as lattice-based  algorithms while achieving viable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  This paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' In Section II, we briefly  introduce the working principle of KEMs in general and HQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  In Section III, we provide an overview of the related work  and of the SoA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' In Section IV, we identify and evaluate the  computational bottlenecks of HQC in software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' In Section  V, we present the IoT processing system and the hardware  implementation of the different accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' In Section VI, we  compare our results with the SoA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' In Section VII, we draw a  conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' BACKGROUND  KEMs form a public key cryptosystem that is build out  of three algorithms, Key-Generation (KeyGen), Encapsula-  tion (Encaps) and Decapsulation (Decaps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Unlike general  purpose Public Key Encryption Schemes (PKEs), KEMs are  not thought to perform any application data encryption, but  are designed to establish a randomly generated shared secret  between communication partners in cryptographic protocols  like Transport Layer Security (TLS) similar to the state-of-the-  art Diffie-Hellmann Key-Exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Afterwards, this shared  secret is used to derive a secret key for de- and encryption of  application data with fast symmetric cryptographic algorithms  like Advanced Encryption Standard (AES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' KEMs are often  build out of existing PKEs using transformations like the  Fujisaki-Okamoto Transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  The first code-based PKE was introduced by McEliece in  1978 and is based on the assumption that the error-correction  code used is indistinguishable from random codes [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Al-  though the original McEliece cryptosystem, which relied on  Goppa codes, remains secure to this day, the method of hiding  the generator matrix of the code in the public key carries  a potential vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Attempts to reduce the key size  by using more structured codes than the original McEliece  approach have shown that this vulnerability can be exploited  to crack the cryptosystems in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='06 seconds [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  Therefore, the authors of HQC proposed a novel approach  which combines two different types of codes:  1) A decodable [n, k] code C with a fixed, publicly known  generator matrix G ∈ Fk×n  2  and the error correction  capability δ based on concatenated Reed-Muller (RM)  and Reed-Solomon (RS) codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  2) A random double-circulant [2n, n] code with a publicly  known parity check matrix h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  This design rational allows HQC to use significantly smaller  keys than the other code-based KEM Classic McEliece (2 KB  vs 255 KB public key size) while still achieving the same  security metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' 1 shows how the shared secret ss is established between  the communication partners using the HQC KEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' HQC uses  Alice  Bob  KeyGen():  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' h  $  ←  − R  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' sk = (x, y)  $  ←  − R2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' pk = (h, s = x + h · y)  Send pk  Encaps(pk):  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' m  $  ←  − Fk  2  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' θ ← G(m)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' e  $  ←  − R  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' (r1, r2)  $  ←  − R2  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' u = r1 + h · r2  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' v = mG + s · r2 + e  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' c ← (u, v)  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' d ← H(m)  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' ss ← K(m, c)  Send ct = (c, d)  Decaps(sk,ct)  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' m′ = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='Decode(v − u · y)  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' θ′ ← G(m′)  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' c′ = Encrypt(pk, m′, θ′)  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' If c ̸= c′or d ̸= H(m′)abort  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' ss ← K(m′, c)  Encrypt(pk, m, θ)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' HQC KEM as defined in [4] with R = F2[X]/(Xn − 1), the hash  functions G, H, K, the sampling operator  $  ←− and the KEM’s public key pk,  private key sk, ciphertext ct and shared secret ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' θ is the seed for the pseudo-  random number generation during the encryption in Encaps() and Decaps().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  the Keccak-based extendable output function SHAKE as a  seedexpander of a random generated seed as the scheme  requires a large amount of random bytes (n = 17669 for the  smallest parameter set HQC-128).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Furthermore, the Keccak-  based Secure Hash Alorithm 3 (SHA3) [8] is used for the G, H  and K functions which are required due to the KEM-DEM  transformation in HQC to construct an IND-CCA2 secure  KEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  The procedure of HQC in short is as follows, a detailed  description can be found in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' First, Alice randomly gener-  ates the parity check matrix h and the private key sk, from  which the public key pk is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Here, the polynomials  x and y which build the secret key are hidden in the public  key by multiplying h with y and adding x in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Bob uses  pk to encrypt his randomly generated message m, which is  the basis for the shared secret ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' During this encryption, the  randomly generated vectors r1, r2 and e which have a fixed,  predefined hamming weight are used to disguise m further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  The hamming weights are selected in a way such that they  still allow a correct decryption of m by Alice with respect to  δ with a very high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The ciphertext ct is sent back to  Alice, who decrypts the message m′ and calculates ss based  on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  The HQC algorithm is available in 3 different parameter  sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' This paper is focused on the NIST level 1 parameter set  HQC-128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' STATE OF THE ART  Many works have been published which deal with hard-  ware accelerations of new PQC primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Among these  publications, the vast majority is focused on accelerators for  lattice-based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' A cryptographic co-processor was  implemented in [9] as an ASIC to support various lattice  based NIST schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Fritzmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' developed a HW/SW  based co-design on a RISCV core for the lattice-based scheme  NewHope [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' In [11], FrodoKEM, an algorithm which has  a high security confidentiality due to its less structured lattice,  was accelerated by using a HW/SW co-design approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  In contrast, the code-based KEMs have not yet been in-  vestigated to the same depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' For BIKE, another code-based  candidate with a comparable key size to HQC, an FPGA  implementation has been proposed in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' So far, the only  hardware implementation for HQC was presented by the  original authors of HQC in [4] and is based on FPGA HLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  Therefore, in this work, we present the first HW/SW co-design  approach of HQC and implement our design both as ASIC and  FPGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' HW/SW CO-DESIGN  There are three possibilities for implementation:  1) Software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  2) Custom processor instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  3) Loosely coupled accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  In a first step, the execution of the NIST reference software  was profiled to determine the computational bottlenecks and  the memory footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' In a second step, we investigated for  each bottleneck the most suitable approach to find the optimum  trade-off between the area, latency, memory footprint, and  energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The highest priority was assigned to  software optimization, as it offers high flexibility without  additional costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Then, if this is not efficient, custom processor  instructions were considered as a second option, since they are  still flexible and require little additional hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Only when  these two approaches were found to be ineffective a loosely-  coupled accelerator was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' IoT Processing System (IoT-PS)  Our methodology requires a processing system that allows  instruction set extensions and the efficient interfacing of  loosely-coupled accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Therefore, we chose an adaptive  platform that includes a RISC-V core whose instruction set  architecture provides the ability to add custom instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' 2 shows the final architecture of the IoT-PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Our custom,  area optimized RISC-V core supports the RV32IC instruction  set which features additional compressed instructions and,  therefore, significantly reduces the program size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The Direct  Memory Access (DMA) controller features a memory copy  (memcpy) and memory initialization (memset) function, of  which both are able to operate on byte, half-word, and  word granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The JTAG module provides access to the  memories and the register file of the RISC-V core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' It also can  be used to start, stop, and reset the IoT-PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Depending on  the target platform, the data memory module was either based  on an SRAM hard macro cell (ASIC) or Block RAM (Xilinx  FPGA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Block RAM was also used for the instruction memory  in case of an FPGA implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' However, for the ASIC  implementation we used a ROM macro cell, thus the program  code is available after reset and does not need to be loaded  via JTAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The IoT-PS features no peripheral units except the  I/O controller which is used to communicate via pin toggling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  RISC-V  JTAG  DMA  HQC Accelerator  20 KB Instruction Memory  32 KB Data Memory  I/O Controller  AXI4-Lite Interconnect  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Architecture Overview  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Profiling of HQC-128  The US NIST C reference implementation was used to  identify the bottlenecks of the HQC-128 execution in our  setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The code was compiled with optimization level 2 (O2)  and simulated cycle-accurately with the RTL model of our  IoT-PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Compared to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' 2, the size of ROM and RAM had  to be increased for the analysis due to the large requirements  of the reference implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  The simulation results are shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The specified  clock cycles in the table refer to the processor cycles that  the RISC-V core spends within the respective C function  and excluding the time spent in sub-functions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=', gf mul is  called during the computation of RS-Encode, but not included  in its reported cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' For all three KEM-functions, (1) the  arithmetic in R, (2) the SHAKE-based hashing and (3) mem-  ory operations are the main contributors to the total execution  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' On top of that, the sampling operation, the RM-Decoding  algorithm (4) and the finite field multiplication (5) are further  contributors to the computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The unsigned division,  which is performed in software, is mostly used during the  polynomial multiplication in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  In (1), the largest part is accounted by the multiplication of  the large polynomials (n = 17669 for HQC-128), which are  represented as bit vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' This includes the subsequent reduc-  tion by Xn − 1 of the intermediate result, and is performed,  for example, in steps 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=', 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The multiplication  complexity is reduced by the fact that one of the vectors is  sparse and has a small, known hamming weight w ≤ 75,  which allows to consider only the non-zero coefficients in  the sparse polynomial during processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The execution of the  multiplication consists mostly of XOR operations for adding  the binary coefficients of the same degrees and SHIFT / AND  operations to determine the degree of the intermediate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  Due to the high degree of the polynomials, a large number  of LOAD and STORE instructions is required during the  computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  TABLE I  TOTAL CYCLE COUNT AND SHARE OF IMPORTANT FUNCTIONS OF THE NIST C REFERENCE IMPLEMENTATION ON THE RISC-V, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' (NOT  APPLICABLE) REFERS TO FUNCTIONS WHICH ARE NOT USED IN THIS STEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  Function  Keygen Cycles  Encaps Cycles  Decaps Cycles  Total  5609k  13850k  19903k  Arithmetic in R  1540k (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='46%)  3448k (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='9%)  4989k (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='1%)  Vect Mul  1528k  3413k  4942k  Vect Add  12k  35k  47k  SHAKE  1854k (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='05%)  5007k (36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='15%)  5414k (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='2%)  Keccak State Permute  1744k  4626k  5005k  Keccak Inc Squeeze  103k  131k  154k  Keccak Inc Absorb  7k  250k  255k  RS-RM Code  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  26k (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='18%)  1440k (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='24%)  RS-Encode  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  26k  26k  RS-Decode  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  56k  RM-Decode  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  1358k  Sampling  81k (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='44%)  155k (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='1%)  236k (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='18%)  Memory-Operation  2071k (36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='92%)  5068k (36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='59%)  7175k (36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='05%)  memcpy  2045k  5021k  7092k  memset  26k  47k  83k  Rest  63k (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='12%)  146k (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='17%)  649k (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='26%)  unsigned division  49k  100k  151k  gf mul  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  20k  162k  Keccak’s permutation function in (2) is the computational  core of the sponge construction in SHA3 and consists of  bitwise AND, XOR, and rotate operations on the 25 lanes  of 64 bits each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The major bottleneck in this permutation is  the interdependence of the intermediate results which causes  the contents of the processor registers to be swapped with the  main memory multiple times during the execution of one of  the 24 rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  The large overhead of the memory operations in (3) is driven  by two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' First, the RISC-V core supports only one  outstanding memory read or write access at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' It waits  for a slave response before continuing the program execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  Second, the reference implementation is not optimized for  low memory usage, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=', it often stores multiple copies of  temporary results, initializes a larger number of arrays, or  copies parts of arrays to different memory locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  TABLE II  STACK MEMORY AND CODE-SIZE OF THE NIST C REFERENCE  IMPLEMENTATION OF HQC-128 ON OUR RISC-V CORE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  Keygen  Encaps  Decaps  Code Size  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='798 KB  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='015 KB  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='378 KB  Stack Memory  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='018 KB  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='714 KB  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='762 KB  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Software Optimization  In multiple functions, the reference implementation uses  non-optimal data-types which increases the number of required  memory accesses and processor instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' An example of  this is the comparisons in Step 16 of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' 1, which are  performed on a byte boundary rather than a processor word  boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The memory footprint and computation time was  further improved by removing redundant arrays which often  get initialized with zeroes or are the target of memcpy  operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The operations are performed with pointers in-  stead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' For the remaining memory operations, the time required  for memcpy and for array initialization via memset were  accelerated by using the DMA controller of the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Instruction Set Extension  The bottleneck of RS-Encoding and -Decoding is caused  by the multiplication in F28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' This operation has only three  operands, including the generator polynomial, and one return  value with the size of one byte each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' A F28-Unit is added to  the RISC-V core which is able to perform the operation shown  in Equation 1, where a = (a15, · · · , a0) and b = (b7, · · · , b0)  are the input operands, and d = (d14, · · · , d0) is the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  This unit is made accessible via both an R-type and an I-type  custom instruction, where register rs1 is used as operand a,  register rs2 respectively the immediate value imm is used as  operand b, and the output d is stored in register rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  (a15 · x7 + · · · + a8 · x0) · (b7 · x7 + · · · + b0 · x0)  +(a7 · x7 + · · · + a0 · x0) ⇒ (d14 · x14 + · · · + d0 · x0) (1)  Using these custom instructions, a multiplication in F28 is  performed within four clock cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Loosely-Coupled Accelerators  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' 4 shows the block diagram of the loosely-coupled  HQC accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' To enable parallel calculations between  the processor and the accelerator, the accelerator has both  an AXI slave and an AXI master interface and fetches its  calculation inputs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' the polynomials) from the processor’s  Dense Polynomial  Sparse Polynomial  coord0  coord1  coordw-1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  word0  0  63  word1  64  127  wordN  17664  17228  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  word0  coord0  coord0+63  wordN  coord0  + 17664  coord0  +17228  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  word0  coord1  coord1+63  wordN  coord1  + 17664  coord1  +17228  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  word0  coordw-1  Coordw-1+63  wordN  coordw-1  + 17664  coordw-1  +17228  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  XOR  XOR  XOR  Intermediate Result  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Working principle of the polynomial multiplication in R with n = 17669 and 64-bit memory words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  main memory via the master interface, according to the  processor command which was previously received via the  slave interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Due to the data dependencies between the steps  in the HQC-KEM, and based on the previous observation that  the bottleneck in the HQC is driven by memory accesses, we  decided that enabling parallel read/read or read/write accesses  is more beneficial than running the dedicated compute units  in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Therefore, only two SRAMs are used and shared  between the compute units, and only one of the compute units  is processing at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The access to the SRAMs and  the operation mode of the compute units are managed by one  control unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  AXI4-Lite Interconnect  HQC Control Unit  SRAM0  (288 x  64 Bit)  SRAM1  (566 x  64 Bit)  R-Unit  Sampling-Unit  RM-Decoder  Keccak-IP  AXI-Slave  AXI-Master  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' HQC hardware accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  The R-Unit implements the addition, multiplication and  reduction of the polynomials in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' 3 shows the working  principle of the polynomial multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The values of  the sparse polynomial contain the locations of its non-zero  coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' In our implementation, we iterate through the  multiplication by a word-by-word shift of the dense polyno-  mial by the coordinates given in the sparse polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  After the shift, the interim result is XOR-ed with the word  that is currently stored at the respective location in memory  and the carry out with respect to the word alignment of the  memory is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The R-Unit is designed such that only  two cycles are necessary for calculating a resulting word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' In  the first cycle the address of the dense polynomial and the  intermediate polynomial are calculated and read based on the  coordinate, while in the second cycle the new value and the  carry-out are calculated and written to memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  The Sampling-Unit combines the sponge functions squeeze  and absorb of the incremental version of SHAKE, the permu-  tation function of Keccak and the rejection-based sampling,  during which SHAKE functions are used as extendable output  functions (XOF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' For the permutation function, a highspeed  open-source implementation by the original authors of Kec-  cak was used, which executes the permutation in 24 clock  cycles [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  As shown in Table I, the vast majority of decoding time  is spent on the RM-Codes, which employs a Maximum  Likelihood (ML) algorithm based on the Hadamard Transform  and a subsequent peak-search for the highest value in the  transformed codeword.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Because HQC uses duplicated RM  codes, the decoding algorithm must be preceded by another  transformation function [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' This transform requires many  single-bit operations, and thus, the implementation in soft-  ware is not efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Therefore, we adapted the transform  and the subsequent decoding steps to an efficient hardware-  implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' RESULTS AND COMPARISON  The IoT-PS presented was implemented in a 22 nm FD-SOI  technology from GlobalFoundries under worst case Process,  Voltage and Temperature (PVT) conditions (125 °C, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='72 V for  timing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' 25 °C, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='8 V for power).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Synthesis is performed with  the Synopsys DesignCompiler, Place&Route is carried out  with the Synopsys IC-Compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The SRAMs were generated  by the INVECAS Memory Compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Power numbers are  calculated with back-annotated wiring data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The layout of  ASIC IP core presented in this work can be seen in Figure 5  has a size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='12 mm2 with an aspect ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='77 and a  maximum frequency of 700 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The IoT-PS presented was  also implemented on a Xilinx Artix xc7a100tcsg324-3 using  Xilinx Vivado for a better comparison to existing work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  TABLE III  CYCLE COUNT OF THE HQC-KEM IN OUR SETUP WITH THE DIFFERENT HARDWARE MODULES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' THE IMPROVEMENT REFERS TO SPEEDUP WITH  RESPECT TO THE NIST C REFERENCE IMPLEMENTATION WITHOUT ANY HARDWARE ACCELERATORS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  Keygen  Encaps  Decaps  Cycles  Improvement  Cycles  Improvement  Cycles  Improvement  Reference  5609k 13850k  19903k DMA + SW OPT  3587k  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='0%  7044k  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='1%  10851k  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='5%  + R-Unit  1862k  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='8%  5183k  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='6%  7245k  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='6%  + Sampling-Unit  1623k  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='1%  1955k  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='9%  5176k  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='0%  + RM-Decoder  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
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+page_content='1%  24 KB  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
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+page_content='1%  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Impact of individual optimizations on the overall run time  Table III shows the extent to which the presented hardware  modules accelerate the computation time of the three KEM  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' As illustrated, the use of a DMA and the software  optimization already provide a significant speed up between  36% and 49% over the reference implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' This shows  that the NIST C reference implementation is not meant to  be used in IoT devices without optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The loosely  coupled Sampling-Unit is the accelerator that provides the a  runtime reduction of at least 50% compared to the optimized  software implementation with DMA in all KEM functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The  R-Unit, however, reduces the runtime only between 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='5%  and 48% depending on the KEM function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The RM-Decoder  has the least impact on the performance since it is only used  in Decaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Also it is able to achieve a runtime reduction of  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The F28-Instructions give only a minor speedup on its  own, but shows its potential in combination with all loosely  coupled accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Resource distribution of the individual hardware modules  Figure 5 shows a qualitative area distribution of the distinct  modules for ASIC, while Table IV shows a quantitative  distribution for FPGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The R-Unit shows the highest area  efficiency among all loosely coupled accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Although  this unit has a lower runtime reduction compared to the  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  Layout;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' RISC-V - blue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' JTAG - cyan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Interconnect - lime;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' I/O  Controller - purple;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' DMA - pink;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Sampling Unit - red;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' RM-Decoder - orange;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  R-Unit - yellow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' HQC Control Unit, SRAM0, SRAM1 - green;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' ROM - white,  horizontal strips;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' RAM - white, cross pattern  Sampling-Unit, it needs less than 10% of its FPGA resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  The RM-Decoder, however, requires fewer resources than the  R-Unit, but also offers the least performance gain and is only  used in Decaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Therefore, it is far less efficient compared to  both R-Unit and Sampling-Unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The F28-Unit, which is used  by the custom instructions, requires only negligible resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  However, the decoding of these instructions as well as the  controlling of the unit requires additional resources which are  hidden inside the RISC-V core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  TABLE IV  RESOURCE UTILIZATION ON FPGA (ARTIX7)  LUTs  Registers  Block RAM  RISC-V  2210  1682  0  ⌞ F28-Unit  27  0  0  Interconnect  2775  1919  0  Memories  53  6  24  HQC Accelerator  7920  2370  3  ⊢ R-Unit  565  117  0  ⊢RM-Decoder  435  63  0  ⌞Sampling Unit  5610  1814  0  ⌞Keccak Permute  4685  1622  0  DMA  489  412  0  JTAG  452  546  0  I/O Controller  41  68  0  IoT-PS  13934  7003  27  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Comparison to State of the Art  Table V presents the required clock cycles for the different  KEM functions of SoA implementations and of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' We  use the number of clock cycles as metric rather than absolute  computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' This is, for our our work, a pessimistic  comparison due to the high achievable clock frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' How-  ever, even under this assumption, our implementation requires  a comparable number of clock cycles as the low-latency HQC  implementation, while it requires less hardware than its low  area hardware implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Compared to BIKE, the other  code-based KEM, our implementation requires about the same  amount of clock cycles like the low latency implementation  while using significantly less hardware resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' The imple-  mentation of FrodoKEM, which would be an alternative if  structured lattice-based KEMs like Kyber and NewHope are  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='TABLE V  COMPARISON OF CLOCK CYCLES, FREQUNCY AND FPGA RESOURCES FOR DIFFERENT STATE-OF-THE-ART IMPLEMENTATIONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' HW/SW REFERS TO  IMPLEMENTATIONS BASED ON HW/SW CO-DESIGN, FULL REFERS TO FULL HARDWARE IMPLEMENTATIONS OF THE RESPECTIVE SCHEME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' NEWHOPE  AND KYBER ARE STRUCTURED-LATTICE BASED ALGORITHMS, FRODOKEM IS BASED ON LESS-STRUCTURED LATTICES, AND THE REMAINING  IMPLEMENTATIONS ARE BASED ON CODES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  Implementation  Keygen  Encaps  Decaps  Frequency  FPGA Resources  Target Plattform  Cycles  Cycles  Cycles  MHz  LUTs  FFs  BRAMs  HQC (low area, HW) [4]  630k  1500k  2100k  132  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='9k  4k  14  FPGA (Xilinx Artix-7)  HQC (low latency, HW) [4]  40k  89k  190k  148  20k  16k  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='5  FPGA (Xilinx Artix-7)  BIKE (low area, HW) [12]  2671k  153k  1628k  121  13k  5k  17  FPGA (Xilinx Artix-7)  BIKE (low latency, HW) [12]  259k  12k  189k  96  53k  7k  49  FPGA (Xilinx Artix-7)  NewHope (HW/SW) [10]  357k  590k  167k  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  11k  5k  1  FPGA (Xilinx Zynq-7000)  FrodoKEM (HW/SW) [11]  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='4M  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='5M  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='3M  100  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='6k  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='1k  0  FPGA (Xilinx Zynq Ultrascale+)  Kyber (HW/SW) [9]  75k  132k  142k  72  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  ASIC (40nm)  HQC on Cortex M4 (SW)  1048k  2436k  4001k  64  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  nRF52840  This Work HQC (DMA+SW OPT)  3587k  7044k  10851k  700  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  ASIC (22nm)  This Work HQC (HW/SW)  56k  131k  557k  700  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  ASIC (22nm)  This Work HQC (HW/SW)  56k  131k  557k  100  8k  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='4k  3  FPGA (Xilinx Artix-7)  proven to be vulnerable to attacks, is overall 100 times slower  than our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  Table VI shows the energy consumption of our implementa-  tion, of HQC on a Cortex M4 processor and of the structured  lattice-based KYBER, also implemented as an ASIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' As  can be seen, our implementation requires considerably less  energy than the pure software on the Cortex M4 and also less  than KYBER’s ASIC implementation, which, however, was  implemented on a larger technology node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  TABLE VI  COMPARISON OF ENERGY CONSUMPTION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' FOR A TRADE-OFF BETWEEN  POWER AND LATENCY, OUR DESIGN WAS IMPLEMENTED AND SIMULATED  WITH A 200 MHZ CLOCK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  Implementation  Keygen  Encaps  Decaps  µJ  µJ  µJ  Kyber (HW/SW) [9]  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='97  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='37  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='25  HQC on Cortex M4 (SW)  500  1184  1872  This Work HQC (HW/SW)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='02  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='41  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='1  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' CONCLUSION  In this work, we investigated the performance of the code-  based post-quantum KEM HQC in the context of low power  IoT system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' We presented the first ASIC implementation of a  code-based US NIST PQC candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' With a combination of  software optimizations, instruction set extensions, and loosely  coupled hardware accelerators, we achieve similar perfor-  mance to the full SoA hardware implementation of HQC, but  require significantly less hardware resources and provide more  flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Compared to SoA implementations of lattice-based  algorithms, we have shown that code-based algorithms are  promising alternatives in IoT based applications in terms of  energy efficiency, computation time, and required hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  ACKNOWLEDGEMENT  This paper was partly founded by the German Federal  Ministry of Education and Research as part of the project  “SIKRIN-KRYPTOV” (16KIS1069).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  REFERENCES  [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Shor, “Algorithms for quantum computation: Discrete logarithms  and factoring,” in Proceedings of the 35th Annual Symposium on  Foundations of Computer Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  IEEE Computer Society, 1994, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  124–134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Mosca, “Cybersecurity in an Era with Quantum Computers: Will We  Be Ready?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' IEEE Security & Privacy, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' 16, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' 38–41, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  [3] Peikert,  Christopher  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  and  Bernstein,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=',  “CRYSTALS-  KYBER:  Round  3  Official  Comments,”  https://csrc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='gov/  CSRC/media/Projects/post-quantum-cryptography/documents/round-3/  official-comments/CRYSTALS-KYBER-round3-official-comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='pdf,  2022, Retrieved 2022-06-07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  [4] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Melchor, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Aragon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Bettaieb, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Bidoux, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Blazy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='-  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Deneuville, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Gaborit, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Persichetti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Z´emor, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Bourges,  “Hamming Quasi-Cyclic (HQC),” 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  [5] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Alagic, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Alperin-Sheriff, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Apon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Cooper, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Dang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Kelsey,  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Miller, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Moody, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Peralta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=', “Status Report on the  Second Round of the NIST Post-Quantum Cryptography Standardization  Process,” NIST, Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=', July, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  [6] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' McEliece, “A public-key cryptosystem based on algebraic coding  theory,” The Deep Space Network Progress Report, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' 42-44, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' 114–  116, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Faugere, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Otmani, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Perret, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Tillich, “Algebraic crypt-  analysis of McEliece variants with compact keys,” in Annual Interna-  tional Conference on the Theory and Applications of Cryptographic  Techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  Springer, 2010, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' 279–298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  [8] National Institute of Standards and Technology, “FIPS PUB 202  SHA-3 Standard: Permutation-Based Hash and Extendable-Output  Functions,”  https://csrc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='gov/Projects/post-quantum-cryptography,  Retrieved 2022-02-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  [9] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Banerjee, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Ukyab, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Chandrakasan, “Sapphire: A Con-  figurable Crypto-Processor for Post-Quantum Lattice-Based Protocols,”  arXiv preprint arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='07557, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  [10] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Fritzmann, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Sharif, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' M¨uller-Gritschneder, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Reinbrecht,  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Schlichtmann, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Sepulveda, “Towards Reliable and Secure Post-  Quantum Co-Processors based on RISC-V,” in 2019 Design, Automation  & Test in Europe Conference & Exhibition (DATE), 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' 1148–  1153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  [11] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Karl, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Fritzmann, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Sigl, “Hardware Accelerated FrodoKEM  on RISC-V,” in 2022 25th International Symposium on Design and  Diagnostics of Electronic Circuits and Systems (DDECS), 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  154–159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  [12] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Richter-Brockmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Mono, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' Guneysu, “Folding BIKE:  Scalable Hardware Implementation for Reconfigurable Devices,” IEEE  Transactions on Computers, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' 71, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' 1204-1215, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='  [13] Bertoni, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' and Daemen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' and Peeters, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=' and Van Assche, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content=', “Keccak  in VHDL,” https://keccak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
+page_content='team/hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfFwxN/content/2301.04888v1.pdf'}
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+arXiv:2301.02661v1  [q-bio.QM]  5 Jan 2023
+Evaluating Evasion Strategies in Zebrafish Larvae
+Yusheng Jiaoa, Brendan Colverta,b, Yi Mana,c, Matthew J. McHenryd, and Eva Kanso∗a,*
+aAerospace and Mechanical Engineering, University of Southern California, 854 Downey way, Los Angeles, California 90089, USA
+bDepartment of Bioengineering, University of California, San Diego, 9500 Gilman Dr, La Jolla, CA 92093, USA
+cDepartment of Mechanics and Engineering Science at College of Engineering and LTCS, Peking University, Beijing 100871, P. R. China.
+dDepartment of Ecology and Evolutionary Biology, University of California, Irvine, 321 Steinhaus Hall, Irvine, CA 92697, USA
+January 10, 2023
+Abstract
+An effective evasion strategy allows prey to survive encoun-
+ters with predators.
+Prey are generally thought to escape
+in a direction that is either random or serves to maximize
+the minimum distance from the predator.
+Here we intro-
+duce a comprehensive approach to determine the most likely
+evasion strategy among multiple hypotheses and the role of
+biomechanical constraints on the escape response of prey fish.
+Through a consideration of six strategies with sensorimo-
+tor noise and previous kinematic measurements, our analy-
+sis shows that zebrafish larvae generally escape in a direc-
+tion orthogonal to the predator’s heading. By sensing only
+the predator’s heading, this orthogonal strategy maximizes
+the distance from fast-moving predators, and, when operating
+within the biomechanical constraints of the escape response,
+it provides the best predictions of prey behavior among all
+alternatives. This work demonstrates a framework for resolv-
+ing the strategic basis of evastion in predator-prey interac-
+tions, which could be applied to a broad diversity of animals.
+Keywords— Predator-prey interactions | Probabilistic modeling |
+Inference | Fish C-start | Fluid-structure interactions | Hydrody-
+namics
+Author contributions: EK secured funds and designed and supervised
+research.
+YJ, BC, YM, and EK performed research.
+YJ, BC, YM,
+MJM, and EK analyzed results. YJ, YM, and EK wrote the paper and
+YJ, BC, YM, MJM, and EK revised and edited it.
+Author declaration: The authors declare no conflict of interest.
+Introduction
+The abilities to sense and evade predators are central to the survival
+of a diversity of prey species. The timing, speed, and direction of
+a prey’s escape reflect the animal’s evasion strategy, which is form-
+ulated by its neurophysiology and biomechanics [1]. Despite the
+fundamental importance of predator encounters, resolving a prey’s
+strategy is experimentally challenging due to the variability inher-
+ent to animal behavior. Predators vary in their approach toward
+prey and the ability of the prey to respond is filtered through the
+environment and the animal’s physiology, which may additionally
+introduce noise in sensing, integration, and motor response. The
+aims of the present study are to develop an analytical approach
+that is capable of resolving prey strategy from kinematic measure-
+ments and to use that approach to test classic theory on strategy
+in fish predator-prey interactions.
+The interactions between an individual predator fish and prey
+fish offer a classic system for the study of evasion strategy. Fish
+evade predators with a stereotypical ‘C-start’ response, character-
+ized by the fish body bending into a preparatory ‘C’ shape, followed
+by a rapid acceleration as the body unfolds with largely planar mo-
+tion [2]. Fish escape behavior inspired an application of differential
+game theory to determine the optimal strategy of prey [3]. The
+distance-optimal strategy is the solution to the ‘homicidal chauf-
+feur’ game where prey move in the direction that maximizes the
+∗To
+whom
+correspondence
+should
+be
+addressed.
+E-mail:
+kanso@usc.edu
+closest distance achieved by a predator that maintains a constant
+velocity [4]. The distance-optimal strategy has been invoked to ex-
+plain the escape responses in animals as divergent as cockroaches
+[5], crickets [6], shrimp [7], frogs [8], salamanders [9], crabs [10],
+and a variety of fish species [11, 12]. This strategy is generally con-
+sidered the primary alternative to escaping in a random direction,
+known as the protean strategy, which offers the tactical benefit of
+confusing the predator [13, 14, 15, 16].
+The present study con-
+siders whether previously-measured escape kinematics in zebrafish
+larvae [17, 18] are consistent with distance-optimal, pure-protean,
+or alternative strategies.
+The zebrafish (Danio rerio) larva is a compelling system for
+investigating evasion because it has served as a model for the neu-
+rophysiology and biomechanics of the C-start. Its small size, lack
+of pigmentation, and amenability to genetic manipulation have fa-
+cilitated applications of functional imaging and optogenetics in ze-
+brafish to observe and manipulate the sensory and motor circuits
+responsible for visually-mediated escapes [19, 20, 21].
+A combi-
+nation of high-speed kinematics, flow visualization, and computa-
+tional fluid dynamics have revealed a comprehensive accounting of
+the fluid forces that propel the escape response of zebrafish lar-
+vae [22, 23, 24, 25, 26]. We incorporate these findings into a con-
+sideration of the biomechanical constraints on the escape strategy.
+We adopt a multi-pronged approach for testing the evasion
+strategy in larval zebrafish. Using a strong-inference technique [27],
+we mathematically define models for six strategies, both with and
+without sensorimotor noise (Fig. 1). We then proceed to evaluate
+these model predictions against previous measurements of escape
+kinematics [17] to determine the strategy that most-likely describes
+those observations. Finally, a consideration of the escape hydro-
+dynamics and fluid-structure interactions allows us to evaluate the
+constraints on these strategies. These measures combine to offer
+a general framework for evaluating evasion strategies in predator-
+prey encounters.
+Results
+Our description of the major results is organized around four main
+themes: (1) experimental data of the evasion kinematics of larval
+zebrafish and their descriptive statistics, (2) mathematical defini-
+tions of the evasion strategies, (3) formulation of the analytical
+approach and testing of evasion strategies, with and without sen-
+sorimotor noise, and (4) evaluation of the effects of biomechanical
+constraints on evasion.
+Experimental measurements of escape kinematics
+We analyzed anew a large experimental dataset for the kinematics
+of escape responses in zebrafish larvae that were previously pub-
+lished [17, 18]. Larvae were exposed to a robotic predator, consist-
+ing of a sacrificed adult zebrafish (of fixed size) controlled with a
+motor to move through an aquarium of otherwise still water. As
+detailed previously [17, 18], larvae were largely motionless prior to
+the escape response that was stimulated by the presentation of the
+predator.
+The recorded responses of larvae were compiled from
+numerous experiments, each of which elicited a modest number
+1
+
+V
+−λ
+d
+φ
+prey frame of reference
+predator
+prey
+Contralateral
+θ
+Orthogonal
+θ
+Distance-optimal
+θ
+Parallel
+θ
+Antipodal
+θ
+ψ
+B
+A
+Fig. 1. Evasion strategies. (A) Schematic shows the predator position (d, φ) and heading ψ in the prey’s frame of reference. (B) The change θ in prey heading direction
+at evasion as predicted by five evasion strategies: distance-optimal, prey makes a turn that maximizes the shortest distance from predator; orthogonal, prey turns to the
+direction orthogonal to the predator heading in order to flee the path of the predator; parallel, prey turns to align with the predator heading direction; antipodal, prey turns in the
+opposite direction of the predator angular position; and contralateral, prey turns left or right by 90◦ depending on the predator angular position. Strategies are distinguished
+by color.
+of responses. Larvae were excluded from the analysis if they re-
+sponded within a few body lengths, or a few seconds after, another
+responding larva. The 3D kinematics of larvae were compiled in the
+predator’s frame-of-reference to yield a cloud of responses anterior
+to the robotic predator.
+The speed of the robotic predator was set to a constant equal to
+2, 11, or 20 cm·s−1 to reflect the speed range of a typical foraging
+predator [22]. This ensured a repeatable stimulus that elicited a
+fast C-start response from the larvae [18, 17]. High-speed kinemat-
+ics recorded a total of 699 evasion instances: Nslow = 251 for the
+slow-moving predator, Nmid = 233 for the mid-speed predator, and
+Nfast = 215 for the fast-moving predator (Fig. 2).
+Experimental analysis
+From the previous kinematic measurements, we presently calcu-
+lated the predator distance d, angular position φ ∈ [0, 2π), and
+heading ψ ∈ [−π, π) in the prey’s frame of reference at the onset of
+the C-start escape response, and we calculated the change in the
+prey’s orientation θ ∈ [−π, π) as it completed the C-start escape
+response (Fig. 1A and SI, Fig. S1B). In our analysis, θ captures the
+rotation of the entire fish body, that is, the change in prey heading,
+which is not the same as the change in the body angular position
+employed previously [17, 18]. We clearly distinguish between the
+prey’s sensing of the predator angular position φ and heading ψ,
+which are often confused in empirical studies of evasion [28, 29, 12].
+In addition to the predator’s actual heading direction ψ, we con-
+sidered that the prey perceives λ, the deviation of the predator’s
+heading from the angular position φ, given by λ = ψ − (φ + π),
+λ ∈ [−π, π) (see Fig. 1A and SI, Fig. S1B, C). The predatory stim-
+ulus is said to be sinistral if λ > 0, that is, predator is headed to
+the left of where it appears in the prey’s visual field, and dextral
+otherwise.
+Table 1. Sensory Requirements of Evasion Strategies
+Predator state
+angular
+heading
+Complexity
+position
+heading
+deviation
+speed
+of sensing
+φ
+ψ
+λ
+V
+Distance-optimal
+·
+⃝
+◦
+⃝
+most
+Orthogonal
+·
+⃝
+◦
+·
+Parallel
+·
+⃝
+·
+·
+�
+Antipodal
+⃝
+·
+·
+·
+Contralateral
+◦
+·
+·
+·
+least
+⃝ exact value
+◦ interval value
+· not needed
+Descriptive statistics of kinematic measurements
+We found no correlation between the prey’s escape direction θ and
+its distance d from the predator at the onset of evasion (SI, Fig.
+S3). However, we did find a clear correlation between the escape
+direction θ and the angular position φ in instances where the preda-
+tor appears in the prey’s visual field (SI, Fig. S5). The data also
+showed a correlation between θ and the predator heading ψ, when
+partitioned based on whether the predator’s heading is sinistral
+(λ > 0) or dextral (λ < 0), relative to its angular position φ (SI,
+Fig. S7). Importantly, although the distributions of φ, ψ, and θ
+varied with predator speed V , the correlations between θ and φ
+and between θ and ψ were qualitatively similar for all V (SI, Figs.
+S5 and S7), suggesting that for the range of speeds considered, V
+can be treated as a model parameter, rather than a variable that
+fundamentally changed the evasion behavior.
+In sum, our statistical analysis (SI, Figs. S2-S7, Table S1) in-
+dicates that the escape direction θ depends on the prey’s sensing
+of the predator’s angular position φ, heading ψ, and deviation be-
+tween them λ, but does not disambiguate which stimuli determine
+the escape direction and the behavioral rules that best explain the
+data.
+Definition of evasion strategies
+We next define the six fish evasion strategies: distance-optimal, or-
+thogonal, parallel, antipodal, contralateral, and pure-protean. We
+index these strategies with an integer n = 1, . . . , 6 in the order listed
+above. In all strategies, we ignore the prey biomechanics and treat
+both the predator and prey as point masses equipped with head-
+ing directions. Therefore, the prey’s strategy is demonstrated by
+the direction of its escape θ. However, these escape direction vary
+among the strategies depending on the relative position and head-
+ing of predator (Fig. 1B). We rate the strategies by their complexity
+of sensing (Table 1), which is a relative measure that increases with
+the number of geometric parameters that must be accurately de-
+termined to execute the escape in the direction predicted by the
+strategy. By this metric, the sensing of exact quantities is more
+complex than interval quantities.
+Distance-optimal evasion strategy
+A distance-optimal evasion strategy considers that the prey’s ob-
+jective, once it detects the predator, is to maximize its minimum
+future distance from the predator [3, 30].
+Accordingly, the prey
+should head in the direction θ relative to its pre-evasion heading
+(see SI, section 2),
+θ = f(1)(ψ, λ; χ) =
+� ψ − χ,
+sinistral: λ ∈ [0, π),
+ψ + χ,
+dextral: λ ∈ (−π, 0),
+(1)
+where χ = cos−1(U/V ) is an angle that depends on the ratio U/V
+of prey speed U to predator speed V . For U > V , χ = 0. We treat χ
+2
+
+2cm/s
+11cm/s
+20cm/s
+head
+tail
+before
+after
+1cm
+slow predator 
+mid-speed predator 
+fast predator
+Fig. 2. Experimental measurements of zebrafish larvae evasion in response to robotic predator. Zebrafish larvae were randomly placed in a tank with an approaching robotic
+predator driven at three speeds: V = 2, 11 and 20 cm·s−1. They were mostly straight and motionless until exhibiting a fast C-start evasion response to the predator [17, 18].
+Each experiment involved a single predator-prey encounter. The experiment was repeated to collect three large datasets of size Nslow = 251, Nmid = 233, Nfast = 215
+for the slow, mid-speed, and fast moving predator, respectively. Evasion instances are superimposed for visualization purposes. For each evasion instance, we calculated,
+in the predator frame of reference, the position and orientation of the prey at the onset of evasion (gray macebells where the head represents the prey’s position and spike
+represents its orientation). The change in prey’s orientation θ induced by the C-start evasion response (see inset) is shown in colored macebells. Color is used only for
+illustration purposes and not to be confused with the color code used in Figs. 1, 3, 4, 6 to distinguish between evasion strategies.
+as a model parameter rather than a variable. This distinction may
+not be important for the prey, but it is relevant for our subsequent
+analysis of this strategy.
+Orthogonal evasion strategy
+We propose a simpler evasion strategy where the prey turns 90o
+away from the heading direction ψ of the predator,
+θ = f(2)(ψ, λ) =
+� ψ − π/2,
+sinistral: λ ∈ [0, π),
+ψ + π/2,
+dextral: λ ∈ (−π, 0).
+(2)
+This strategy is equivalent to the distance-optimal strategy in the
+fast predator limit U/V → 0, but may determine θ without the
+need to sense the predator speed V .
+Parallel evasion strategy
+For a slow predator U/V
+≥ 1, the optimal strategy is for the
+prey to reorient itself in the direction of the predator heading,
+θ = f(3)(ψ) = ψ, which can be readily deduced by setting χ = 0
+in (1).
+The major disadvantage of this strategy is that it could
+place the predator in the blind spot of the prey’s visual field.
+Antipodal evasion strategy
+Empirical observations [31, 28] suggest that the prey might follow
+an antipodal strategy by reorienting its heading θ in the direction
+opposite to the angular position φ where the predator appears in
+its visual field, without any account for the predator heading ψ,
+θ = f(4)(φ) =
+� φ + π,
+left stimulus: φ ∈ [0, π),
+φ − π,
+right stimulus: φ ∈ (π, 2π).
+(3)
+Contralateral evasion strategy
+A similar but simpler strategy, called contralateral, was suggested
+in [17] when the prey is approached by the predator from either
+side. Accordingly, the prey escapes by turning 90o either to the
+‘left’ or ‘right’ of its own pre-evasion heading,
+θ = f(5)(φ) =
+� −π/2,
+left stimulus: φ ∈ [0, π),
+π/2,
+right stimulus: φ ∈ (π, 2π).
+(4)
+Pure-protean evasion strategy
+The pure-protean strategy suggests that the evasion response θ is
+random, independent of the predator state, with a uniform prob-
+ability of moving in any particular direction.
+This strategy is
+best expressed in a probabilistic manner, where the probability
+density function (PDF) is uniform with equal probability density
+p(6)(θ) = 1/(2π) of obtaining any change in orientation θ.
+Testing evasion strategies
+We developed a method for evaluating evasion strategies in terms
+of their ability to explain the experimental observations (Fig. 2).
+Although the pure-protean strategy is not supported by our exper-
+imental data (see SI, Figs. S5-S7), the data exhibits some level of
+randomness, as indicated by the variability in the location of the
+predator at the onset of evasion, but potentially also due to inher-
+ent sensorimotor noise in the prey’s perception of the predator and
+its execution of the evasion response. Our approach accounts for
+this variation in evaluating, comparing, and ranking the hypotheti-
+cal evasion strategies. To emphasize the generality of our approach,
+we express it in terms of a generic stimulus s and response r, with-
+out reference to the specific degrees of freedom that these vectors
+encompass.
+For the zebrafish larvae, r is simply θ, but s varies
+depending on the strategy; theoretically, it could encompass all or
+any combination of the variables that define the predator state d,
+φ, ψ, λ and V .
+To examine how well the probabilistic strategy models fit the
+experimental data, we interpreted the latter from a probabilistic
+perspective. An experimental dataset generates N samples (si, ri),
+i = 1, . . . N, from a joint PDF, denoted by po(s, r), whose exact
+form is unknown. An evasion behavior follows a conditional PDF
+po(r|s) = po(s, r)/po(s), which is related to the joint PDF po(s, r)
+and the PDF po(s) of stimuli that elicit an escape response via the
+Law of Total Probability [32]. Unfortunately, po(s, r) and po(s) are
+unknown, and only discrete samples of these PDFs are available
+from experiments, thus the need for further modeling and analysis.
+Probabilistic models under precise vs. noisy sensing
+and response
+We distinguish between the actual predator state s and the prey’s
+sensing ˆs of the predator state. Similarly, we distinguish between
+3
+
+strategy
+r=f(n)(s)
+perception
+noise σS
+response
+noise σR
+predator
+state
+s
+prey
+response
+r
+ˆs
+ˆr
+ˆ
+ˆ
+prey response θ
+predator angle φ
+predator angle φ
+predator angle φ
+predator angle φ
+predator angle φ
+prey response θ
+180◦
+−180◦
+0◦
+360◦
+0◦
+180◦
+φ
+Experiment
+frequency
+0
+max
+Probablistic models
+180◦
+−180◦
+0◦
+0◦
+360◦
+180◦
+0◦
+360◦
+180◦
+0◦
+360◦
+180◦
+0◦
+360◦
+180◦
+0◦
+360◦
+180◦
+Contralateral
+Antipodal
+Parallel
+Distance-optimal
+Orthogonal
+B
+C
+A
+D
+Fig. 3. Model predictions in response to experimentally observed predator states. (A) Bivariate histogram of (φ, θ) from experimental data. Darker color means larger
+fraction of data points in that area of the (φ, θ) space. (B) Bivariate histogram based on the evasion models (Eqs 1–4) with no noise; model predictions θi in response to
+experimentally observed predator states φi, ψi, λi, i = 1 . . . , N, where N = 699 is the size of the combined data. (C) Schematic illustration of how noise in sensing and
+response is built into the evasion models. (D) Bivariate histogram using realizations from the noisy evasion models (Eq. 5) under optimal noise levels.
+the actual escape heading r and the prey’s desired escape heading
+ˆr. If the prey’s sensing and response are precise, we get ˆs = s and
+ˆr = r. However, the sensorimotor modalities underlying evasion
+are often noisy: the prey may perceive a noisy version ˆs of the
+predator’s state s and its desired response ˆr may be altered by
+noisy execution or environmental conditions to yield r.
+Each evasion strategy n, save the pure-protean, defines a desired
+escape response ˆr given a perceived predatory stimulus ˆs and can
+be expressed as a conditional PDF using the Dirac-delta function
+p(n)(ˆr|ˆs) = δ �
+ˆr − f (n)(ˆs)�
+. The joint PDF p(n)(s, r) formed based
+on evasion strategy n follows from the Law of Total Probability
+p(n)(s, r) =
+� �
+p(r|ˆr)p(n)(ˆr|ˆs)p(ˆs|s)po(s)dˆs dˆr.
+(5)
+Here, p(ˆs|s) and p(r|ˆr) model the noise in the prey’s sensing and
+response. In the case of precise sensing and response, (5) reduces
+to
+p(n)(s, r) = δ �
+r − f (n)(s)�
+po(s).
+(6)
+In the following, we treat each case separately.
+Evaluating evasion strategies under precise sensing
+and response
+To obtain samples of the evasion response predicted by (6), we use
+as input the distribution of the empirically-observed stimuli si, and
+we construct a dataset (si, r(n)
+i
+= f (n)(si)) for each strategy. For
+each predator speed, we arrive at five datasets representing theo-
+retical predictions of the prey’s evasion response according to the
+distance-optimal, orthogonal, parallel, antipodal, and contralat-
+eral strategies.
+Bivariate histograms in the (φ, θ)-plane for each
+strategy based on the dataset combining all predator speeds are
+shown in Fig. 3B. The histograms represent discrete cross-sections
+of p(n)(s, r) and can be used to estimate the joint probability of
+obtaining a predatory stimulus φi and prey response θi.
+Com-
+pared to the histogram obtained from experiments (Fig. 3A), the
+contralateral and antipodal strategies form straight lines because
+the predicted θ(n)
+i
+are uniquely determined by the predator angu-
+lar position φi, while the other distributions are spread out due to
+their dependency on the predator heading ψi and λi.
+To measure the difference between model predictions and exper-
+imental data, we estimated numerically the Kullback-Leibler (K-L)
+divergence DKL, which quantifies the entropy of p(n)(s, r) relative
+to po(s, r), using the method in [33]; see SI, S5.
+Results of the
+K-L divergence are shown in Fig. 4A for all five strategies applied
+to the slow, mid-speed, and fast predator, as well as the combined
+data. The actual K-L divergence is always non-negative; the nega-
+tive values are due to discrete estimation of the PDF. In each of the
+four datasets, the distance-optimal and orthogonal strategies yield
+the lowest estimates of the K-L divergence, implying that, of all
+five evasion strategies, they give the closest predictions of the prey
+escape response. The distance-optimal strategy performs slightly
+better for the slow and mid-speed predator while the orthogonal is
+more advantageous for the fast predator and when considering all
+data combined. The antipodal strategy also gives relatively low K-
+L divergence estimates. The parallel and contralateral strategies,
+whose K-L divergence estimates are significantly higher than the
+other strategies, have the worst fit to experimental data across all
+predator speeds.
+Modeling noise in sensing and response
+We next introduced sensing and response noise according to (5).
+To model sensing noise, we considered ˆs to be normally-distributed
+around the actual state of the predator s, with dispersion σS, and
+to model response noise, we considered r to be normally-distributed
+around the desired response ˆr, with dispersion σR. Substituting
+the noise models p(ˆs|s; σS) and p(r|ˆr; σR) into (5), and recalling
+that p(n)(ˆr|ˆs) = δ �
+ˆr − f (n)(ˆs)�
+, we arrived, for each evasion strat-
+egy n, at a probabilistic model that depends on the noise parame-
+ters σ = {σS, σR} (see SI, section 4). Specifically, we used a von
+Mises distribution (normal distribution on the circle) for θ, φ and λ
+with noise parameters σΘ, σΦ and σΛ; we let the noise on ψ follow
+from ψ = φ + λ + π (see SI, section 4).
+At zero noise, the von
+Mises distribution converges to a Dirac-delta function at the mean
+value; when the noise level is high, it approaches a circular uniform
+distribution with constant PDF 1/(2π) in any escape direction.
+Limit of high noise levels
+If the response noise σΘ is large, any evasion direction is predicted
+with equal probability density 1/(2π), irrespective of the strat-
+egy or the sensing noise, that is, all strategies become essentially
+equivalent to the pure-protean strategy. On the other hand, if the
+response is precise σΘ = 0, but the noise in sensing the preda-
+tor’s angular position σΦ is large, all strategies, except the con-
+tralateral, converge to the pure-protean strategy; the contralateral
+strategy predicts θ = ±π/2 with equal probability. If the prey’s
+4
+
+all
+slow
+mid-speed
+fast
+K-L divergence estimate
+0
+1
+2
+1.5
+0.5
+∆AIC/N
+all
+slow
+mid-speed
+fast
+−0.2
+0.8
+0.6
+0.4
+0.2
+0
+Contralateral
+Orthogonal
+Antipodal
+Parallel
+Distance-optimal
+A
+B
+Fig. 4. Evaluation of precise and noisy evasion strategies. (A) K-L divergence estimate from precise model predictions to experiment data is computed separately for each
+dataset (slow, mid-speed and fast predator) and for all data combined. The K-L values for the distance-optimal and orthogonal strategies are the lowest, indicating better fit
+to data. (B) AIC difference (∆AIC = AIC-AICmin), normalized by the respective sample size of each dataset. For each dataset, we used bootstrap method to construct 200
+distinct datasets (by sampling with repetition) of equal size to the original dataset. We optimized each of 200 sets, evaluated the corresponding AIC, and computed the mean
+and standard deviation of ∆AIC. The orthogonal strategy has the lowest ∆AIC, indicating that it is the most parsimonious strategy and best explains the data.
+α1
+−α2
+β
+b2 b1
+A
+C
+Nair et al. 2015
+model (massless)
+model (neutral)
+time
+0
+0.25T
+0.5T
+0.75T
+T
+0◦
+middle link orientation β
+20◦
+40◦
+60◦
+80◦
+−20◦
+head/tail angles
+0◦
+α2
+α1
+90◦
+135◦
+45◦
+−45◦
+−90◦
+stage 1
+stage 2
+stage 3
+B
+maximum
+bending αmax
+D
+Voesenek et al. 2019
+model (massless)
+model (neutral)
+prey response θ
+maximum bending angle αmax
+100◦
+50◦
+0◦
+25◦
+75◦
+125◦
+Nair et al. 2015
+E
+180◦
+−180◦
+180◦
+−180◦
+deformation angle α1
+deformation angle α2
+0◦
+0◦
+stage 2
+stage 1
+αmax
+A
+B
+F
+0
+1
+−1
+Fig. 5. Biomechanics of fish C-start response. (A) Larval zebrafish bends its body into a C-shape to initiate a fast start. (B) Three fish model, with α1 and α2 representing
+the fish body shape and β the overall body orientation relative to the straight pre-evasion direction. (C) Experimental data of shape changes of larval zebrafish during evasion
+taken from Ref.[34] and processed to represent body deformations in terms of head and tail rotations α1, α2 (gray dots) then fitted by third-order fourier series (black lines).
+(D) Experimental data of overall body rotation for the same evasion instance shown in C (gray dots and black line). Predictions based on fish model, taking as input the
+shape changes in C are shown in blue lines (solid line for massless and dashed line for neutrally-buoyant fish). (E) The sequence of shape changes in C forms a curve C
+in the shape space (α1, α2) (black line). The experimental curve C is approximated by an ellipse (red) of axes A, B along α1 = ±α2 directions. Colormap represents
+curl2[A1, A2] of fish model, which predicts larger turns for curves that encompass solely positive (orange) or negative (blue) values. (F) By varying A and calculating B that
+maximizes the turn, we get a mapping from maximum bending angle αmax =
+√
+2A to turning angle θ for massless and neutrally buoyant fish (blue lines) that form an upper
+and lower bounds on the experimental data set of Ref. [23]. Both numerical and experimental data show that the C-start mechanics limits larval zebrafish to turning angles
+θ ≲ 100◦.
+response and sensing of the predator angular position are both pre-
+cise σΘ = σΦ = 0, but the prey’s sensing of the predator’s heading
+is very noisy (σΛ large), the antipodal and contralateral strategies
+do not get affected while the parallel strategy becomes protean.
+Interestingly, in this case, the distance-optimal strategy predicts
+higher probability of evasion in directions opposite to the predator
+location spanning a range of 2χ (see SI, section 4, Fig. S9). That
+is, the distance-optimal strategy becomes a noisy variant of the an-
+tipodal strategy. For χ = π/2, the orthogonal strategy with large
+noise on λ converges to the antipodal strategy with uniform noise
+spanning a range of π on either φ or θ.
+Optimizing noise levels in sensing and response
+For each noisy evasion strategy, we calculated the noise parame-
+ters σ = {σS, σR} that maximize the total likelihood L of the
+model given an experimental dataset, or equivalently minimize the
+negative log-likelihood function NLL (see SI, section 6)
+NLL = − ln L (σ|(r|s); n) = −
+�
+i
+ln p(n)(ri|si; σ),
+(7)
+where p(n)(ri|si; σ) is the conditional PDF of obtaining a response
+ri given stimulus si for strategy n at noise level σ. The optimal
+noise parameters σ∗ are given by
+σ∗ = arg min
+σ NLL.
+(8)
+5
+
+We solved this optimization problem numerically in the range
+σΦ, σΛ, σΘ ∈ (0, π) (SI, Fig. S6). In Fig. 3D, we plot realizations
+generated from the five probabilistic evasion models p(n)(s, r; σ∗)
+at the optimal noise values corresponding to the dataset of all
+predator speeds combined.
+Compared to the deterministic pre-
+dictions in Fig. 3B, all five distributions appear closer to the ex-
+perimental data in Fig. 3A.
+Evaluating strategies under optimal noise parameters
+To evaluate how well each optimized strategy describes the exper-
+imental data, we applied the Akaike information criterion (AIC)
+defined as [35]
+AIC = 2K − 2 ln L(σ∗|(s, r); n)
+(9)
+where K is the number of model parameters in each strategy. AIC
+considers both the goodness of fit represented by the likelihood
+function, and the complexity of the model: if two models have
+the same likelihood to explain the data, the criterion favors the
+simpler model. For example, for the antipodal strategy, we have
+two noise parameters σS ≡ {σΦ} and σR ≡ {σΘ}, thus K = 2;
+whereas for the orthogonal strategy, we have three noise parameters
+σS ≡ {σΦ, σΛ} and σR ≡ {σΘ}, and the orthogonal strategy is
+deemed more complex than the antipodal strategy.
+We used bootstrapping to probe the accuracy of our evaluation
+of the noisy strategies.
+Starting from each dataset (e.g., that of
+the fast predator), we constructed 200 distinct datasets of equal
+size to the original dataset (e.g., Nfast) by random sampling with
+repetition. We solved the optimization problem 200 times and ob-
+tained 200 values of σ∗ per strategy for each dataset. We calculated
+the likelihood value L and evaluated the AIC for all 200 optimal
+noise values, thus obtaining a distribution of AIC values for each
+strategy and predator speed. The mean and standard deviation of
+the distributions of AIC values, minus the lowest mean value and
+normalized by the size of the respective dataset (Fig. 4B) show
+that strategies with lower mean values of the AIC better fit the
+experimental data.
+The results based on the AIC evaluation of the probabilis-
+tic strategies in the presence of sensory and response noise are
+mostly consistent with the results based on the K-L divergence
+(Fig. 4A) for precise sensing and response, but with marked differ-
+ences. The orthogonal strategy ranks the highest in every dataset;
+the distance-optimal strategy is slightly behind, in second place, in
+all but the slow predator dataset where the antipodal strategy ranks
+second. The difference between the orthogonal, distance-optimal,
+and antipodal strategies is most distinguishable in the case of the
+fast predator. The contralateral and parallel strategies come last
+in all datasets and are least descriptive of experimental data.
+Further analysis of distance-optimal strategy
+While the predator speed was controlled at V = 2, 11, 20 cm s−1,
+the zebrafish larvae were almost identical in all experiments, im-
+plying that the speed ratio U/V varied drastically between evasion
+instances: for the fast predator, this ratio is up to 10 times that of
+the slow predator. If the prey were to sense and use the speed ratio
+to implement the distance-optimal strategy, we would expect the
+best performance to appear at different values of χ = cos−1(U/V )
+depending on predator speed. To test this, we evaluated this strat-
+egy for the slow, mid, and fast predator as a function of χ ∈ [0, 90◦]
+under both precise and noisy sensing and response (SI, section 7,
+Fig.
+S14).
+We found that the K-L divergence decreased as χ
+increased and reached a minimum near χ = 75◦ independent of
+predator speed. Similarly, the NLL dropped as χ increased until it
+reached a minimum at, or close to, χ = 90◦. These results suggest
+that, even if following the distance-optimal strategy, the prey does
+not rely on real-time and accurate measurements of the speed ratio
+U/V , but favors the limit of large predator speed (χ → 90◦), where
+the distance-optimal strategy converges to the orthogonal strategy.
+The same conclusion can be reached by examining the values
+of the optimized noise parameters. In the range 20◦ ≲ χ ≲ 75◦,
+the optimizer mostly selects the largest possible value of σΛ = π to
+best fit the data. This high level of optimized noise indicates that
+λ is not an effective sensory cue in the distance-optimal strategy,
+and that the prey is unlikely to use this strategy at moderate χ
+values (see SI, section7, Fig. S14).
+Evaluating the biomechanical constraints on es-
+cape strategy
+To complete our evaluation of fish evasion strategies, we consid-
+ered the biomechanics of the C-start response. In [34], the motion
+of a zebrafish larvae undergoing a C-start maneuver starting from
+a straight motionless configuration was recorded using high-speed
+photography, and the time evolution of each segment of the fish
+body from the onset of evasion at time t = 0 to after the com-
+pletion of the C-start response at t = T = 25ms was measured.
+We developed a mathematical model of the biomechanics of these
+events and incorporated that model into our analysis.
+We reinterpreted the experimental measurements in the context
+of a three-link fish, head, middle, and tail (Fig. 5B), and we ex-
+tracted from experimental measurements the fish orientation β(t)
+and rotations α1(t) and α2(t) of the head and tail relative to the
+middle segment (see SI, section 8-9).
+The time evolution of the
+zebrafish body during evasion follows the three archetypal stages
+of the C-start response: in stage 1, the fish curls its body to one
+side, rapidly unfurls its body in stage 2, and begins its undulatory
+swimming in stage 3 (Fig. 5C-D).
+A larger number of C-start maneuvers were recorded in [23],
+albeit only measuring the maximum degree of body bending αmax
+and the net change in heading θ = β(T) − β(0) induced by the
+C-start maneuver (Fig. 5F). These results show that the change
+in heading direction θ correlates strongly with the degree of body
+bending [23]. In all recorded maneuvers, the change in body orien-
+tation barely reaches 100◦.
+Physics-based modeling of the C-start response
+To shed light on the relationship between body deformations and
+change in heading θ during evasion, we employed a physics-based
+model of a three-link fish in potential flow [36, 37]. Experimen-
+tal and computational flow analysis had shown that the C-start
+maneuver is dominated by unsteady, pressure-based exchange of
+momentum between the fish and surrounding fluid, with negligible
+contributions from fluid viscosity and shed vorticity [26, 38]. The
+potential flow model captures these unsteady pressure forces via
+the added mass effect (see SI, S8). The fish model is composed of
+three identical prolate spheroids (of major and minor axes a and b)
+such that the head and tail are free to rotate relative to the middle
+link (Fig. 5B); as before, body deformations are described by the
+angles α1(t), α2(t) representing the relative head and tail rotations
+as a function of time t, and body orientation β(t) is the angle be-
+tween the middle section and an inertial direction taken along the
+direction of the initially-straight fish.
+From consideration of momentum balance on the fish-fluid sys-
+tem, we arrived at an equation governing the rate of change of body
+orientation [39, 40, 37] (see SI, section 8)
+˙β = A1(α1, α2) ˙α1 + A2(α1, α2) ˙α2,
+(10)
+where A1 and A2 are nonlinear functions of body deformations
+α1(t), α2(t); A1 and A2 also depend on fish geometry and fluid
+and body densities (ρf and ρb). For ρb = ρf, the fish is neutrally-
+buoyant. When the fluid forces are dominate, the fish can be con-
+sidered massless and ρb is set to zero. Body rotations are propor-
+tional to the line integral of (10) over a curve C describing body
+deformations in the shape space (α1, α2). For a closed curve C,
+this line integral can be rewritten, using Stokes theorem, as an
+area integral over the region of the (α1, α2) space enclosed by C,
+θ = β(T) − β(0) =
+� � �∂A2
+∂α1
+− ∂A1
+∂α2
+�
+dα1dα2.
+(11)
+The scalar field curl2([A1, A2]) ≡ ∂A2/∂α1 − ∂A1/∂α2 is shown
+in Fig. 5E as a colormap over the entire shape space (α1, α2). To
+maximize the turning angle θ, a straight fish should deform its body
+following a closed curve C that encompasses either non-positive or
+non-negative values of curl2([A1, A2]), i.e., either blue or orange
+regions of the shape space. Closed curves in the orange region lead
+6
+
+−0.2
+0.6
+0.2
+0
+−0.4
+0.4
+0.8
+K-L divergence estimate
+combined
+slow
+mid-speed
+fast
+∆AIC/N
+−0.2
+0.5
+0.4
+0.2
+0.1
+0
+0.3
+−0.1
+combined
+slow
+mid-speed
+fast
+Orthogonal
+Distance-optimal
+original
+constrained
+Antipodal
+B
+A
+Fig. 6. Evaluation of the constrained strategies that consider the physical constraint on turning. Results are shown for the three best-performing models. (A) The
+K-L divergence estimates of the constrained models with precise sensing and response (hollow bars), shown with the results of the original models (solid bars, from Fig. 4A).
+In all four datasets, the constrained models provide discernibly lower K-L divergence estimates, thus better fit to experimental data than the original models, except the
+distance-optimal strategy for the mid-speed predator. The orthogonal strategy improved the most after imposing the constraint, making it fit the data best in all datasets.
+(B) The normalized relative AIC for the constrained models with optimized noise in sensing and response, compared to results using the original models in Fig. 4B. The
+orthogonal strategy still provides best fit to all datasets, marked by the lowest AIC scores, and its advantage over the second best model is more noticeable after imposing
+the constraint. The constrained antipodal strategy performs comparable to or even better than the constrained distance-optimal strategy.
+to turning counter-clockwise.
+By symmetry, diagonally-opposite
+curves in the blue region lead to turning clockwise. Theoretically,
+the simplest curve for turning is a circle or an ellipse in the shape
+space of major axis A aligned with α1 = α2, for which the maxi-
+mum bending angle is αmax =
+√
+2A (Fig. 5E). Corresponding fish
+shape deformations and body rotations β(t) are discussed in SI
+(S8-9, Figs. S15-S16).
+Comparing
+model predictions
+to
+C-start
+induced
+turning of the fish body
+We represented the empirical time evolution of shape deformations
+(α1(t), α2(t)) (Fig. 5C) onto the shape space (Fig. 5E). Interest-
+ingly, the curve C (black line) traced by the actual fish follows
+closely the elliptic curve (red line) predicted by the model as best
+for turning. Moreover, when taking the empirical values of α1(t)
+and α2(t) as input to the physics-based model in (10), the resulting
+predictions of β(t) (blue lines in Fig. 5D) follow closely the empiri-
+cal values of β(t) (black line), especially during the first stage of the
+C-start response, where vorticity is negligible; note that while the
+neutrally buoyant model (dashed blue line) deviates slightly from
+empirical observations in stage 2, the massless fish model (solid blue
+line) performs remarkably well way into stage 3, indicating that in-
+deed unsteady pressure forces dominate the C-start maneuver, as
+previously predicted [26].
+We next considered a family of shape changes following the el-
+liptic curve in Fig. 5E by varying A such that αmax =
+√
+2A varied
+from 0 to 120◦. This upper limit on αmax corresponds to a maxi-
+mum bending angle without causing the head and tail of the model
+fish to cross each other, and is consistent with the experimental
+observations of [23]. Using (11), we computed, for each αmax, the
+value of B that optimizes the change in orientation θ, thus creating
+a map from αmax to θ. We compare these model-predictions (blue
+lines) to experimental data [23] (black dots) in Fig. 5F. As before,
+we considered massless and neutrally-buoyant fish. The theoretical
+predictions behave nearly as upper and lower limits to experimen-
+tal data.
+As observed previously [23], turning in the model fish
+barely reaches 100◦ even when the three-link fish bends its body
+to the extreme of the head and tail touching. This indicates that
+the biomechanics of the C-start maneuver imposes an upper limit
+on achievable heading directions θ.
+Constrained evasion strategies
+We next incorporated the physical constraints on θ imposed by the
+C-start biomechanics into our evasion strategies. To this end, we
+mapped the response angle θ(n)
+i
+predicted by evasion strategy n
+onto the interval [0, 100◦] using the quadratic mapping
+θ →
+�
+1 −
+�
+1 − θmax
+π
+� |θ|
+π
+�
+θ.
+(12)
+Small turns get less constrained whereas large turns are limited
+to the maximum angle θmax = 100◦ allowable by the fish biome-
+chanics.
+We applied this constraint to the three most plausible
+strategies: distance-optimal, orthogonal, and antipodal. For each
+constrained strategy, we repeated the analysis presented above un-
+der precise and noisy sensing and response. Results of the K-L di-
+vergence and AIC analysis for the constrained strategies are shown
+in Fig. 6.
+Compared to the unconstrained strategies, penalizing
+large turns makes all three strategies fit better the experimental
+data across all datasets, with or without added noise, with the ex-
+ception of the distance-optimal strategy for the mid-speed preda-
+tor. Under precise sensing and response, the relative ranking of the
+constrained strategies (Fig. 6A) is similar to the original ranking
+(Fig. 4A), with the distinction that the orthogonal strategy at slow
+and mid-speed predator speed surpasses the distance-optimal strat-
+egy and becomes the best ranking model. Under noisy sensing and
+response, the antipodal strategy ranks higher than the distance-
+optimal strategy in all but the slow predator dataset (Fig. 6B).
+Importantly, whether precise or noisy, the orthogonal strategy fits
+the experimental data better than the other two in all datasets.
+Discussion
+We developed a comprehensive framework for resolving evasion
+strategy from kinematic measurements.
+Our approach considers
+multiple hypotheses, each defined mathematically (Fig. 1), that
+address the role of sensorimotor noise (Fig.
+3) and incorporate
+the effects of biomechanical constraints (Figs. 5–6). Importantly,
+our approach provides a rigorous methodology, rooted in strong-
+inference principles [27], for revealing the strategy that best fits pre-
+vious kinematic measurements of zebrafish larvae. This approach
+eliminates bias towards a particular hypothetical strategy, as done
+in a previous study that favored the contralateral strategy from the
+dataset presently analyzed [17].
+We found that the responses of zebrafish larvae to evade a preda-
+tor are best-characterized by the orthogonal strategy (Fig. 4). This
+finding challenges the notion that a prey aims either to solely con-
+fuse, or maximize its distance from, the predator with its escape
+[16]. The kinematics of zebrafish do not exhibit the uniform distri-
+bution of escape direction θ characteristic of a pure-protean strat-
+egy (SI, Fig. S2E) [13, 14, 15]. Instead, larvae exhibited correla-
+tions between θ and predator state, including angular position φ
+(SI, Fig. S5) and heading ψ (SI, Fig. S7). The distance-optimal
+strategy is more predictive of zebrafish kinematics, but is inferior
+to the orthogonal strategy, based on K-L divergence and the AIC
+scores (Figs. 4 and 6). Therefore, zebrafish larvae do not conform
+to the classic dichotomy of models for prey strategy. Although the
+prevailing patterns favor an orthogonal strategy, variation about
+the predictions for this hypothesis allows for the possibility of a
+mixed strategy that could hinder a predator’s ability to anticipate
+7
+
+the prey’s direction.
+These results are relevant to predator-prey
+encounters, and hence the ecology, of fish species and reflect the ad-
+vantages and constraints of the prey’s neurophysiology and biome-
+chanics.
+The distance-optimal strategy requires sensing that may exceed
+the abilities of larval fish. This strategy requires detection of the
+speed of the approaching predator (Table 1), but larval fish possess
+poor visual acuity, compared to adult fish, due to a relatively small
+number of retinal cells [41].
+It has been demonstrated that the
+escape is triggered by a threshold diameter of a looming visual
+stimulus, which may be simulated as a circle with an expanding
+diameter [19]. A looming stimulus alone does not offer the means
+to differentiate between threats that are small and fast or large and
+slow. Therefore, the visual system of larval fish may offer a sensory
+constraint on its ability to perform the distance-optimal strategy.
+A more sophisticated visual system could allow for additional cues
+to gauge the speed or size of a predator, but the processing time
+necessary to formulate a distance-optimal response may still pose
+a liability in evasion speed compared with the orthogonal strategy.
+The orthogonal strategy merely requires an estimate of the
+predator’s heading and offers tactical benefits relative to many of
+the alternatives. This strategy is equivalent to the distance-optimal
+strategy for a high-speed approach (U/V ≪ 1) and therefore suc-
+ceeds in maximizing the prey’s distance from a fast predator at
+reduced sensing requirements (Table 1). It is the fastest predators
+that likely present the greatest threat to the prey. The orthogo-
+nal strategy offers an additional tactical advantage by evading in
+a direction that is challenging for a fast-approaching predator to
+follow because, in order for the predator to execute such large turn
+at high speed, it needs a large turning radius, which could increase
+its distance from the prey even further.
+The predictions of the orthogonal strategy improved in their fit
+to measured kinematics when we considered constraints imposed by
+the biomechanics of the C-start (Fig. 6). In particular, our model
+of a three-link fish in potential flow accurately describes the rela-
+tionship between the change in fish shape and its turning motion
+during evasion (Fig. 5). By mapping maximum bending angle to
+turning angle, the model predicted an upper limit (around 100◦) on
+achievable turning motion, consistent with the maximum angle ob-
+served in zebrafish exposed to a lateral looming stimulus [19, 23].
+The improvement in model predictions that included mechanics
+demonstrates the influence of the constraints imposed by the prey
+biomechanics and its interaction with the fluid environment on the
+evasion strategy of zebrafish larvae.
+Comparing the K-L divergence and AIC values across slow, in-
+termediate, and fast predators, the prevalence of the orthogonal
+strategy is clearest in the case of the fast predator (Fig. 4 and
+Fig. 6). This feature can be related to the fact that a weak stimu-
+lus (slow predator) is more likely to trigger an escape response via
+the less predictable, long-latency neural pathway [42, 43], as op-
+posed to the fast pathway with minimal latency between perceived
+danger and motor response [44]. An untangling of these features
+requires a deeper investigation of how our analytical framework
+relates to the neurophysiology underlying zebrafish evasion.
+Our study combined tools from information theory and proba-
+bilistic methods with behavioral evasion models and physics-based
+models of the C-start biomechanics to develop a comprehensive
+analytical approach and thereby determine the evasion strategy of
+zebrafish larvae. Aside from the details of the biomechanics model,
+nothing about our approach is specific to the study of fish. Our
+analysis could be applied to the myriad of studies that have mea-
+sured escape responses relative to a predator’s approach in a diver-
+sity of animals [5, 6, 7, 8, 9, 10]. This approach may therefore be
+applied broadly to the study of predator-prey encounters to reveal
+the strategic basis of this fundamental aspect of animal behavior.
+Acknowledgement
+E.K. acknowledges support from the Of-
+fice of Naval Research (ONR) Grants N00014-22-1-2655, N00014-
+19-1-2035, N00014-17-1-2062, and N00014-14-1-0421; the National
+Science Foundation (NSF) Grants RAISE IOS-2034043, CBET-
+2100209, and INSPIRE MCB-1608744; the National Institutes of
+Health (NIH) Grant R01 HL 153622-01A1; the Army Research Of-
+fice (ARO) Grant W911NF-16-1-0074.
+This research started in
+summer 2018 at the Summer Graduate School on Mathematical
+Analysis of Behavior organized by Ann Hermundstad, Vivek Ja-
+yaraman, Eva Kanso, and L. Mahadevan. The school was jointly
+supported by the Mathematical Science Research Institute (MSRI)
+and the Howard-Hugh Medical Institute (HHMI) Janelia Research
+Campus, and was held at Janelia. BC, YM, and EK acknowledge
+support from Janelia and would like to thank Sashank Pisupati and
+Ann Hermundstad for helpful discussions.
+References
+[1] Jos. J. Schall and Eric R. Pianka. Evolution of escape behavior
+diversity. The American Naturalist, 115(4):551–566, 1980.
+[2] D Weihs.
+The mechanism of rapid starting of slender fish.
+Biorheology, 10(3):343–350, 1973.
+[3] D. Weihs and P.W. Webb.
+Optimal avoidance and evasion
+tactics in predator-prey interactions. Journal of Theoretical
+Biology, 106(2):189 – 206, 1984.
+[4] Rufus Isaacs. Differential games: a mathematical theory with
+applications to warfare and pursuit, control and optimization.
+Courier Corporation, 1999.
+[5] Paolo Domenici, David Booth, Jonathan M Blagburn, and
+Jonathan P Bacon. Cockroaches keep predators guessing by
+using preferred escape trajectories. Curr. Biol., 18(22):1792–
+1796, 2008.
+[6] Jérôme Casas and Thomas Steinmann.
+Predator-induced
+flow disturbances alert prey, from the onset of an attack.
+Proceedings of the Royal Society B: Biological Sciences,
+281(1790):20141083, 2014.
+[7] Stephen A Arnott, DOUGLAS M Neil, and Alan D Ansell.
+Escape trajectories of the brown shrimp crangon crangon, and
+a theoretical consideration of initial escape angles from preda-
+tors. J. Exp. Biol., 202(2):193–209, 1999.
+[8] Rafe M Brown and Douglas H Taylor. Compensatory escape
+mode trade-offs between swimming performance and maneu-
+vering behavior through larval ontogeny of the wood frog, rana
+sylvatica. Copeia, pages 1–7, 1995.
+[9] Emanuel Azizi and Tobias Landberg. Effects of metamorphosis
+on the aquatic escape response of the two-lined salamander
+(eurycea bislineata). J. Exp. Biol., 205(6):841–849, 2002.
+[10] Patrick B Woodbury.
+The geometry of predator avoidance
+by the blue crab, callinectes sapidus rathbun.
+Ani. Behav.,
+34:28–37, 1986.
+[11] José E Trujillo, Ian Bouyoucos, William J Rayment, Paolo
+Domenici, Serge Planes, Jodie L Rummer, and Bridie JM Al-
+lan.
+Escape response kinematics in two species of tropical
+shark: short escape latencies and high turning performance.
+Journal of Experimental Biology, 225(22):jeb243973, 2022.
+[12] Paolo Domenici. The visually mediated escape response in fish:
+predicting prey responsiveness and the locomotor behaviour
+of predators and prey. Marine and Freshwater Behaviour and
+Physiology, 35(1-2):87–110, 2002.
+[13] D. A. Humphries and P. M. Driver. Protean defence by prey
+animals. Oecologia, 5(4):285–302, 1970.
+[14] Peter M Driver and David Andrew Humphries. Protean be-
+haviour. Clarendon Press, 1988.
+[15] Talia Y Moore, Kimberly L Cooper, Andrew A Biewener, and
+Ramanarayan Vasudevan. Unpredictability of escape trajec-
+tory explains predator evasion ability and microhabitat pref-
+erence of desert rodents. Nature comm., 8(1):1–9, 2017.
+[16] Paolo Domenici, Jonathan M Blagburn, and Jonathan P Ba-
+con.
+Animal escapology ii:
+escape trajectory case studies.
+Journal of Experimental biology, 214(15):2474–2494, 2011.
+[17] Arjun Nair, Kelsey Changsing,
+William J. Stewart,
+and
+Matthew J. McHenry.
+Fish prey change strategy with the
+direction of a threat. Proceedings of the Royal Society B: Bi-
+ological Sciences, 284(1857):20170393, 2017.
+8
+
+[18] William
+J.
+Stewart,
+Arjun
+Nair,
+Houshuo
+Jiang,
+and
+Matthew J. McHenry.
+Prey fish escape by sensing the
+bow wave of a predator.
+Journal of Experimental Biology,
+217(24):4328–4336, 2014.
+[19] Timothy W Dunn, Christoph Gebhardt, Eva A Naumann,
+Clemens Riegler, Misha B Ahrens, Florian Engert, and Filippo
+Del Bene. Neural circuits underlying visually evoked escapes
+in larval zebrafish. Neuron, 89(3):613–628, 2016.
+[20] Alix MB Lacoste, David Schoppik, Drew N Robson, Martin
+Haesemeyer, Ruben Portugues, Jennifer M Li, Owen Randlett,
+Caroline L Wee, Florian Engert, and Alexander F Schier. A
+convergent and essential interneuron pathway for mauthner-
+cell-mediated escapes. Curr. Biol., 25(11):1526–1534, 2015.
+[21] Shin-ichi Higashijima, Mark A Masino, Gail Mandel, and
+Joseph R Fetcho. Imaging neuronal activity during zebrafish
+behavior with a genetically encoded calcium indicator. J. neu-
+rophysiol., 90(6):3986–3997, 2003.
+[22] William J. Stewart, Gilberto S. Cardenas, and Matthew J.
+McHenry. Zebrafish larvae evade predators by sensing water
+flow. Journal of Experimental Biology, 216(3):388–398, 2013.
+[23] Cees J. Voesenek, Remco P. M. Pieters, Florian T. Muijres,
+and Johan L. van Leeuwen. Reorientation and propulsion in
+fast-starting zebrafish larvae: an inverse dynamics analysis.
+Journal of Experimental Biology, 222(14), 2019.
+[24] Cees J. Voesenek, Florian T. Muijres, and Johan L. van
+Leeuwen. Biomechanics of swimming in developing larval fish.
+Journal of Experimental Biology, 221(1), 2018.
+[25] Ulrike K. Müller and Johan L. van Leeuwen.
+Swimming of
+larval zebrafish: ontogeny of body waves and implications for
+locomotory development.
+Journal of Experimental Biology,
+207(5):853–868, 2004.
+[26] M. Gazzola, W. M. Van Rees, and P. Koumoutsakos.
+C-
+start: optimal start of larval fish. Journal of Fluid Mechanics,
+698:5–18, 2012.
+[27] John R Platt. Strong inference: Certain systematic methods
+of scientific thinking may produce much more rapid progress
+than others. Science, 146(3642):347–353, 1964.
+[28] PAOLO Domenici and ROBERT W Blake. Escape trajectories
+in angelfish (pterophyllum eimekei). Journal of Experimental
+Biology, 177(1):253–272, 1993.
+[29] Robert C Eaton and Deanna S Emberley. How stimulus di-
+rection determines the trajectory of the mauthner-initiated es-
+cape response in a teleost fish. Journal of Experimental Biol-
+ogy, 161(1):469–487, 1991.
+[30] Alberto Soto, William J. Stewart, and Matthew J. McHenry.
+When Optimal Strategy Matters to Prey Fish.
+Int. Comp.
+Biol., 55(1):110–120, 05 2015.
+[31] Robert C Eaton, William A Lavender, and Chris M Wieland.
+Identification of mauthner-initiated response patterns in gold-
+fish: evidence from simultaneous cinematography and electro-
+physiology. Journal of Comparative Physiology, 144(4):521–
+531, 1981.
+[32] Mark J Schervish. Theory of statistics. Springer Science &
+Business Media, 2012.
+[33] F. Perez-Cruz. Kullback-leibler divergence estimation of con-
+tinuous distributions. In 2008 IEEE International Symposium
+on Information Theory, pages 1666–1670, 2008.
+[34] Arjun Nair, Grigor Azatian, and Matthew J. McHenry. The
+kinematics of directional control in the fast start of zebrafish
+larvae. Journal of Experimental Biology, 218(24):3996–4004,
+2015.
+[35] H. Akaike.
+A new look at the statistical model identifica-
+tion.
+IEEE Transactions on Automatic Control, 19(6):716–
+723, 1974.
+[36] Eva Kanso and Jerrold E Marsden.
+Optimal motion of an
+articulated body in a perfect fluid. In Proceedings of the 44th
+IEEE Conference on Decision and Control, pages 2511–2516.
+IEEE, 2005.
+[37] Yusheng Jiao, Feng Ling, Sina Heydari, Nicolas Heess, Josh
+Merel, and Eva Kanso. Learning to swim in potential flow.
+Physical Review Fluids, 6(5):050505, 2021.
+[38] Ulrike K Muller, Jos GM van den Boogaart, and Johan L van
+Leeuwen. Flow patterns of larval fish: undulatory swimming
+in the intermediate flow regime. Journal of Experimental Bi-
+ology, 211(2):196–205, 2008.
+[39] Eva Kanso, Jerrold E Marsden, Clarence W Rowley, and
+Juan B Melli-Huber. Locomotion of articulated bodies in a
+perfect fluid.
+Journal of Nonlinear Science, 15(4):255–289,
+2005.
+[40] Ross L Hatton and Howie Choset. Connection vector fields
+and optimized coordinates for swimming systems at low and
+high reynolds numbers. In ASME 2010 Dynamic Systems and
+Control Conference, pages 817–824. American Society of Me-
+chanical Engineers Digital Collection, 2010.
+[41] Marion F Haug, Oliver Biehlmaier, Kaspar P Mueller, and
+Stephan CF Neuhauss. Visual acuity in larval zebrafish: be-
+havior and histology. Front. Zool., 7(1):1–7, 2010.
+[42] Harold A Burgess and Michael Granato.
+Sensorimotor gat-
+ing in larval zebrafish. Journal of Neuroscience, 27(18):4984–
+4994, 2007.
+[43] Gwyneth M Card. Escape behaviors in insects. Current Opin-
+ion in Neurobiology, 22(2):180–186, 2012. Neuroethology.
+[44] J. A. WALKER, C. K. GHALAMBOR, O. L. GRISET,
+D. McKENNEY, and D. N. REZNICK. Do faster starts in-
+crease the probability of evading predators? Functional Ecol-
+ogy, 19(5):808–815, 2005.
+9
+
diff --git a/DtE0T4oBgHgl3EQfygLM/content/tmp_files/load_file.txt b/DtE0T4oBgHgl3EQfygLM/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf,len=685
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='02661v1  [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='QM]  5 Jan 2023  Evaluating Evasion Strategies in Zebrafish Larvae  Yusheng Jiaoa, Brendan Colverta,b, Yi Mana,c, Matthew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' McHenryd, and Eva Kanso∗a,*  aAerospace and Mechanical Engineering, University of Southern California, 854 Downey way, Los Angeles, California 90089, USA  bDepartment of Bioengineering, University of California, San Diego, 9500 Gilman Dr, La Jolla, CA 92093, USA  cDepartment of Mechanics and Engineering Science at College of Engineering and LTCS, Peking University, Beijing 100871, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  dDepartment of Ecology and Evolutionary Biology, University of California, Irvine, 321 Steinhaus Hall, Irvine, CA 92697, USA  January 10, 2023  Abstract  An effective evasion strategy allows prey to survive encoun-  ters with predators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Prey are generally thought to escape  in a direction that is either random or serves to maximize  the minimum distance from the predator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Here we intro-  duce a comprehensive approach to determine the most likely  evasion strategy among multiple hypotheses and the role of  biomechanical constraints on the escape response of prey fish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Through a consideration of six strategies with sensorimo-  tor noise and previous kinematic measurements, our analy-  sis shows that zebrafish larvae generally escape in a direc-  tion orthogonal to the predator’s heading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' By sensing only  the predator’s heading, this orthogonal strategy maximizes  the distance from fast-moving predators, and, when operating  within the biomechanical constraints of the escape response,  it provides the best predictions of prey behavior among all  alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' This work demonstrates a framework for resolv-  ing the strategic basis of evastion in predator-prey interac-  tions, which could be applied to a broad diversity of animals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Keywords— Predator-prey interactions | Probabilistic modeling |  Inference | Fish C-start | Fluid-structure interactions | Hydrody-  namics  Author contributions: EK secured funds and designed and supervised  research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  YJ, BC, YM, and EK performed research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  YJ, BC, YM,  MJM, and EK analyzed results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' YJ, YM, and EK wrote the paper and  YJ, BC, YM, MJM, and EK revised and edited it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Author declaration: The authors declare no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Introduction  The abilities to sense and evade predators are central to the survival  of a diversity of prey species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The timing, speed, and direction of  a prey’s escape reflect the animal’s evasion strategy, which is form-  ulated by its neurophysiology and biomechanics [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Despite the  fundamental importance of predator encounters, resolving a prey’s  strategy is experimentally challenging due to the variability inher-  ent to animal behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Predators vary in their approach toward  prey and the ability of the prey to respond is filtered through the  environment and the animal’s physiology, which may additionally  introduce noise in sensing, integration, and motor response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The  aims of the present study are to develop an analytical approach  that is capable of resolving prey strategy from kinematic measure-  ments and to use that approach to test classic theory on strategy  in fish predator-prey interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  The interactions between an individual predator fish and prey  fish offer a classic system for the study of evasion strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Fish  evade predators with a stereotypical ‘C-start’ response, character-  ized by the fish body bending into a preparatory ‘C’ shape, followed  by a rapid acceleration as the body unfolds with largely planar mo-  tion [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Fish escape behavior inspired an application of differential  game theory to determine the optimal strategy of prey [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The  distance-optimal strategy is the solution to the ‘homicidal chauf-  feur’ game where prey move in the direction that maximizes the  ∗To  whom  correspondence  should  be  addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  E-mail:  kanso@usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='edu  closest distance achieved by a predator that maintains a constant  velocity [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The distance-optimal strategy has been invoked to ex-  plain the escape responses in animals as divergent as cockroaches  [5], crickets [6], shrimp [7], frogs [8], salamanders [9], crabs [10],  and a variety of fish species [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' This strategy is generally con-  sidered the primary alternative to escaping in a random direction,  known as the protean strategy, which offers the tactical benefit of  confusing the predator [13, 14, 15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  The present study con-  siders whether previously-measured escape kinematics in zebrafish  larvae [17, 18] are consistent with distance-optimal, pure-protean,  or alternative strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  The zebrafish (Danio rerio) larva is a compelling system for  investigating evasion because it has served as a model for the neu-  rophysiology and biomechanics of the C-start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Its small size, lack  of pigmentation, and amenability to genetic manipulation have fa-  cilitated applications of functional imaging and optogenetics in ze-  brafish to observe and manipulate the sensory and motor circuits  responsible for visually-mediated escapes [19, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  A combi-  nation of high-speed kinematics, flow visualization, and computa-  tional fluid dynamics have revealed a comprehensive accounting of  the fluid forces that propel the escape response of zebrafish lar-  vae [22, 23, 24, 25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' We incorporate these findings into a con-  sideration of the biomechanical constraints on the escape strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  We adopt a multi-pronged approach for testing the evasion  strategy in larval zebrafish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Using a strong-inference technique [27],  we mathematically define models for six strategies, both with and  without sensorimotor noise (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' We then proceed to evaluate  these model predictions against previous measurements of escape  kinematics [17] to determine the strategy that most-likely describes  those observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Finally, a consideration of the escape hydro-  dynamics and fluid-structure interactions allows us to evaluate the  constraints on these strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' These measures combine to offer  a general framework for evaluating evasion strategies in predator-  prey encounters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Results  Our description of the major results is organized around four main  themes: (1) experimental data of the evasion kinematics of larval  zebrafish and their descriptive statistics, (2) mathematical defini-  tions of the evasion strategies, (3) formulation of the analytical  approach and testing of evasion strategies, with and without sen-  sorimotor noise, and (4) evaluation of the effects of biomechanical  constraints on evasion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Experimental measurements of escape kinematics  We analyzed anew a large experimental dataset for the kinematics  of escape responses in zebrafish larvae that were previously pub-  lished [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Larvae were exposed to a robotic predator, consist-  ing of a sacrificed adult zebrafish (of fixed size) controlled with a  motor to move through an aquarium of otherwise still water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' As  detailed previously [17, 18], larvae were largely motionless prior to  the escape response that was stimulated by the presentation of the  predator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  The recorded responses of larvae were compiled from  numerous experiments, each of which elicited a modest number  1  V  −λ  d  φ  prey frame of reference  predator  prey  Contralateral  θ  Orthogonal  θ  Distance-optimal  θ  Parallel  θ  Antipodal  θ  ψ  B  A  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Evasion strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' (A) Schematic shows the predator position (d, φ) and heading ψ in the prey’s frame of reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' (B) The change θ in prey heading direction  at evasion as predicted by five evasion strategies: distance-optimal, prey makes a turn that maximizes the shortest distance from predator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' orthogonal, prey turns to the  direction orthogonal to the predator heading in order to flee the path of the predator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' parallel, prey turns to align with the predator heading direction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' antipodal, prey turns in the  opposite direction of the predator angular position;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' and contralateral, prey turns left or right by 90◦ depending on the predator angular position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Strategies are distinguished  by color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  of responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Larvae were excluded from the analysis if they re-  sponded within a few body lengths, or a few seconds after, another  responding larva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The 3D kinematics of larvae were compiled in the  predator’s frame-of-reference to yield a cloud of responses anterior  to the robotic predator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  The speed of the robotic predator was set to a constant equal to  2, 11, or 20 cm·s−1 to reflect the speed range of a typical foraging  predator [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' This ensured a repeatable stimulus that elicited a  fast C-start response from the larvae [18, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' High-speed kinemat-  ics recorded a total of 699 evasion instances: Nslow = 251 for the  slow-moving predator, Nmid = 233 for the mid-speed predator, and  Nfast = 215 for the fast-moving predator (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Experimental analysis  From the previous kinematic measurements, we presently calcu-  lated the predator distance d, angular position φ ∈ [0, 2π), and  heading ψ ∈ [−π, π) in the prey’s frame of reference at the onset of  the C-start escape response, and we calculated the change in the  prey’s orientation θ ∈ [−π, π) as it completed the C-start escape  response (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 1A and SI, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' S1B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' In our analysis, θ captures the  rotation of the entire fish body, that is, the change in prey heading,  which is not the same as the change in the body angular position  employed previously [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' We clearly distinguish between the  prey’s sensing of the predator angular position φ and heading ψ,  which are often confused in empirical studies of evasion [28, 29, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  In addition to the predator’s actual heading direction ψ, we con-  sidered that the prey perceives λ, the deviation of the predator’s  heading from the angular position φ, given by λ = ψ − (φ + π),  λ ∈ [−π, π) (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 1A and SI, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' S1B, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The predatory stim-  ulus is said to be sinistral if λ > 0, that is, predator is headed to  the left of where it appears in the prey’s visual field, and dextral  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Sensory Requirements of Evasion Strategies  Predator state  angular  heading  Complexity  position  heading  deviation  speed  of sensing  φ  ψ  λ  V  Distance-optimal ⃝ ⃝  most  Orthogonal ⃝ Parallel ⃝ \uf8e6�  Antipodal  ⃝ Contralateral least  ⃝ exact value  interval value  not needed  Descriptive statistics of kinematic measurements  We found no correlation between the prey’s escape direction θ and  its distance d from the predator at the onset of evasion (SI, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' However, we did find a clear correlation between the escape  direction θ and the angular position φ in instances where the preda-  tor appears in the prey’s visual field (SI, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The data also  showed a correlation between θ and the predator heading ψ, when  partitioned based on whether the predator’s heading is sinistral  (λ > 0) or dextral (λ < 0), relative to its angular position φ (SI,  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Importantly, although the distributions of φ, ψ, and θ  varied with predator speed V , the correlations between θ and φ  and between θ and ψ were qualitatively similar for all V (SI, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  S5 and S7), suggesting that for the range of speeds considered, V  can be treated as a model parameter, rather than a variable that  fundamentally changed the evasion behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  In sum, our statistical analysis (SI, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' S2-S7, Table S1) in-  dicates that the escape direction θ depends on the prey’s sensing  of the predator’s angular position φ, heading ψ, and deviation be-  tween them λ, but does not disambiguate which stimuli determine  the escape direction and the behavioral rules that best explain the  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Definition of evasion strategies  We next define the six fish evasion strategies: distance-optimal, or-  thogonal, parallel, antipodal, contralateral, and pure-protean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' We  index these strategies with an integer n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' , 6 in the order listed  above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' In all strategies, we ignore the prey biomechanics and treat  both the predator and prey as point masses equipped with head-  ing directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Therefore, the prey’s strategy is demonstrated by  the direction of its escape θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' However, these escape direction vary  among the strategies depending on the relative position and head-  ing of predator (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 1B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' We rate the strategies by their complexity  of sensing (Table 1), which is a relative measure that increases with  the number of geometric parameters that must be accurately de-  termined to execute the escape in the direction predicted by the  strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' By this metric, the sensing of exact quantities is more  complex than interval quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Distance-optimal evasion strategy  A distance-optimal evasion strategy considers that the prey’s ob-  jective, once it detects the predator, is to maximize its minimum  future distance from the predator [3, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Accordingly, the prey  should head in the direction θ relative to its pre-evasion heading  (see SI, section 2),  θ = f(1)(ψ, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' χ) =  � ψ − χ,  sinistral: λ ∈ [0, π),  ψ + χ,  dextral: λ ∈ (−π, 0),  (1)  where χ = cos−1(U/V ) is an angle that depends on the ratio U/V  of prey speed U to predator speed V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' For U > V , χ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' We treat χ  2  2cm/s  11cm/s  20cm/s  head  tail  before  after  1cm  slow predator  mid-speed predator  fast predator  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Experimental measurements of zebrafish larvae evasion in response to robotic predator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Zebrafish larvae were randomly placed in a tank with an approaching robotic  predator driven at three speeds: V = 2, 11 and 20 cm·s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' They were mostly straight and motionless until exhibiting a fast C-start evasion response to the predator [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Each experiment involved a single predator-prey encounter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The experiment was repeated to collect three large datasets of size Nslow = 251, Nmid = 233, Nfast = 215  for the slow, mid-speed, and fast moving predator, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Evasion instances are superimposed for visualization purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' For each evasion instance, we calculated,  in the predator frame of reference, the position and orientation of the prey at the onset of evasion (gray macebells where the head represents the prey’s position and spike  represents its orientation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The change in prey’s orientation θ induced by the C-start evasion response (see inset) is shown in colored macebells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Color is used only for  illustration purposes and not to be confused with the color code used in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 1, 3, 4, 6 to distinguish between evasion strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  as a model parameter rather than a variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' This distinction may  not be important for the prey, but it is relevant for our subsequent  analysis of this strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Orthogonal evasion strategy  We propose a simpler evasion strategy where the prey turns 90o  away from the heading direction ψ of the predator,  θ = f(2)(ψ, λ) =  � ψ − π/2,  sinistral: λ ∈ [0, π),  ψ + π/2,  dextral: λ ∈ (−π, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  (2)  This strategy is equivalent to the distance-optimal strategy in the  fast predator limit U/V → 0, but may determine θ without the  need to sense the predator speed V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Parallel evasion strategy  For a slow predator U/V  ≥ 1, the optimal strategy is for the  prey to reorient itself in the direction of the predator heading,  θ = f(3)(ψ) = ψ, which can be readily deduced by setting χ = 0  in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  The major disadvantage of this strategy is that it could  place the predator in the blind spot of the prey’s visual field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Antipodal evasion strategy  Empirical observations [31, 28] suggest that the prey might follow  an antipodal strategy by reorienting its heading θ in the direction  opposite to the angular position φ where the predator appears in  its visual field, without any account for the predator heading ψ,  θ = f(4)(φ) =  � φ + π,  left stimulus: φ ∈ [0, π),  φ − π,  right stimulus: φ ∈ (π, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  (3)  Contralateral evasion strategy  A similar but simpler strategy, called contralateral, was suggested  in [17] when the prey is approached by the predator from either  side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Accordingly, the prey escapes by turning 90o either to the  ‘left’ or ‘right’ of its own pre-evasion heading,  θ = f(5)(φ) =  � −π/2,  left stimulus: φ ∈ [0, π),  π/2,  right stimulus: φ ∈ (π, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  (4)  Pure-protean evasion strategy  The pure-protean strategy suggests that the evasion response θ is  random, independent of the predator state, with a uniform prob-  ability of moving in any particular direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  This strategy is  best expressed in a probabilistic manner, where the probability  density function (PDF) is uniform with equal probability density  p(6)(θ) = 1/(2π) of obtaining any change in orientation θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Testing evasion strategies  We developed a method for evaluating evasion strategies in terms  of their ability to explain the experimental observations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Although the pure-protean strategy is not supported by our exper-  imental data (see SI, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' S5-S7), the data exhibits some level of  randomness, as indicated by the variability in the location of the  predator at the onset of evasion, but potentially also due to inher-  ent sensorimotor noise in the prey’s perception of the predator and  its execution of the evasion response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Our approach accounts for  this variation in evaluating, comparing, and ranking the hypotheti-  cal evasion strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' To emphasize the generality of our approach,  we express it in terms of a generic stimulus s and response r, with-  out reference to the specific degrees of freedom that these vectors  encompass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  For the zebrafish larvae, r is simply θ, but s varies  depending on the strategy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' theoretically, it could encompass all or  any combination of the variables that define the predator state d,  φ, ψ, λ and V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  To examine how well the probabilistic strategy models fit the  experimental data, we interpreted the latter from a probabilistic  perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' An experimental dataset generates N samples (si, ri),  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' N, from a joint PDF, denoted by po(s, r), whose exact  form is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' An evasion behavior follows a conditional PDF  po(r|s) = po(s, r)/po(s), which is related to the joint PDF po(s, r)  and the PDF po(s) of stimuli that elicit an escape response via the  Law of Total Probability [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Unfortunately, po(s, r) and po(s) are  unknown, and only discrete samples of these PDFs are available  from experiments, thus the need for further modeling and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Probabilistic models under precise vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' noisy sensing  and response  We distinguish between the actual predator state s and the prey’s  sensing ˆs of the predator state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' we distinguish between  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='strategy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='r=f(n)(s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='perception  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='noise σS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='response  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='noise σR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='predator  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='state  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='prey  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='response  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='ˆs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='ˆr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='ˆ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='ˆ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='prey response θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='predator angle φ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='predator angle φ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='predator angle φ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='predator angle φ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='predator angle φ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='prey response θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='180◦  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
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+page_content='φ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='Experiment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='frequency  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
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+page_content='Probablistic models  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='180◦  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
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+page_content='180◦  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
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+page_content='180◦  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
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+page_content='Contralateral  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='Antipodal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='Parallel  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='Distance-optimal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='Orthogonal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='B  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='C  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Model predictions in response to experimentally observed predator states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' (A) Bivariate histogram of (φ, θ) from experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Darker color means larger  fraction of data points in that area of the (φ, θ) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' (B) Bivariate histogram based on the evasion models (Eqs 1–4) with no noise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' model predictions θi in response to  experimentally observed predator states φi, ψi, λi, i = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' , N, where N = 699 is the size of the combined data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' (C) Schematic illustration of how noise in sensing and  response is built into the evasion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' (D) Bivariate histogram using realizations from the noisy evasion models (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 5) under optimal noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  the actual escape heading r and the prey’s desired escape heading  ˆr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' If the prey’s sensing and response are precise, we get ˆs = s and  ˆr = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' However, the sensorimotor modalities underlying evasion  are often noisy: the prey may perceive a noisy version ˆs of the  predator’s state s and its desired response ˆr may be altered by  noisy execution or environmental conditions to yield r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Each evasion strategy n, save the pure-protean, defines a desired  escape response ˆr given a perceived predatory stimulus ˆs and can  be expressed as a conditional PDF using the Dirac-delta function  p(n)(ˆr|ˆs) = δ �  ˆr − f (n)(ˆs)�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The joint PDF p(n)(s, r) formed based  on evasion strategy n follows from the Law of Total Probability  p(n)(s, r) =  � �  p(r|ˆr)p(n)(ˆr|ˆs)p(ˆs|s)po(s)dˆs dˆr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  (5)  Here, p(ˆs|s) and p(r|ˆr) model the noise in the prey’s sensing and  response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' In the case of precise sensing and response, (5) reduces  to  p(n)(s, r) = δ �  r − f (n)(s)�  po(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  (6)  In the following, we treat each case separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Evaluating evasion strategies under precise sensing  and response  To obtain samples of the evasion response predicted by (6), we use  as input the distribution of the empirically-observed stimuli si, and  we construct a dataset (si, r(n)  i  = f (n)(si)) for each strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' For  each predator speed, we arrive at five datasets representing theo-  retical predictions of the prey’s evasion response according to the  distance-optimal, orthogonal, parallel, antipodal, and contralat-  eral strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Bivariate histograms in the (φ, θ)-plane for each  strategy based on the dataset combining all predator speeds are  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 3B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The histograms represent discrete cross-sections  of p(n)(s, r) and can be used to estimate the joint probability of  obtaining a predatory stimulus φi and prey response θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Com-  pared to the histogram obtained from experiments (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 3A), the  contralateral and antipodal strategies form straight lines because  the predicted θ(n)  i  are uniquely determined by the predator angu-  lar position φi, while the other distributions are spread out due to  their dependency on the predator heading ψi and λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  To measure the difference between model predictions and exper-  imental data, we estimated numerically the Kullback-Leibler (K-L)  divergence DKL, which quantifies the entropy of p(n)(s, r) relative  to po(s, r), using the method in [33];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' see SI, S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Results of the  K-L divergence are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 4A for all five strategies applied  to the slow, mid-speed, and fast predator, as well as the combined  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The actual K-L divergence is always non-negative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' the nega-  tive values are due to discrete estimation of the PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' In each of the  four datasets, the distance-optimal and orthogonal strategies yield  the lowest estimates of the K-L divergence, implying that, of all  five evasion strategies, they give the closest predictions of the prey  escape response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The distance-optimal strategy performs slightly  better for the slow and mid-speed predator while the orthogonal is  more advantageous for the fast predator and when considering all  data combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The antipodal strategy also gives relatively low K-  L divergence estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The parallel and contralateral strategies,  whose K-L divergence estimates are significantly higher than the  other strategies, have the worst fit to experimental data across all  predator speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Modeling noise in sensing and response  We next introduced sensing and response noise according to (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  To model sensing noise, we considered ˆs to be normally-distributed  around the actual state of the predator s, with dispersion σS, and  to model response noise, we considered r to be normally-distributed  around the desired response ˆr, with dispersion σR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Substituting  the noise models p(ˆs|s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' σS) and p(r|ˆr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' σR) into (5), and recalling  that p(n)(ˆr|ˆs) = δ �  ˆr − f (n)(ˆs)�  , we arrived, for each evasion strat-  egy n, at a probabilistic model that depends on the noise parame-  ters σ = {σS, σR} (see SI, section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Specifically, we used a von  Mises distribution (normal distribution on the circle) for θ, φ and λ  with noise parameters σΘ, σΦ and σΛ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' we let the noise on ψ follow  from ψ = φ + λ + π (see SI, section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  At zero noise, the von  Mises distribution converges to a Dirac-delta function at the mean  value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' when the noise level is high, it approaches a circular uniform  distribution with constant PDF 1/(2π) in any escape direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Limit of high noise levels  If the response noise σΘ is large, any evasion direction is predicted  with equal probability density 1/(2π), irrespective of the strat-  egy or the sensing noise, that is, all strategies become essentially  equivalent to the pure-protean strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' On the other hand, if the  response is precise σΘ = 0, but the noise in sensing the preda-  tor’s angular position σΦ is large, all strategies, except the con-  tralateral, converge to the pure-protean strategy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' the contralateral  strategy predicts θ = ±π/2 with equal probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' If the prey’s  4  all  slow  mid-speed  fast  K-L divergence estimate  0  1  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='5  ∆AIC/N  all  slow  mid-speed  fast  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='2  0  Contralateral  Orthogonal  Antipodal  Parallel  Distance-optimal  A  B  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Evaluation of precise and noisy evasion strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' (A) K-L divergence estimate from precise model predictions to experiment data is computed separately for each  dataset (slow, mid-speed and fast predator) and for all data combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The K-L values for the distance-optimal and orthogonal strategies are the lowest, indicating better fit  to data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' (B) AIC difference (∆AIC = AIC-AICmin), normalized by the respective sample size of each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' For each dataset, we used bootstrap method to construct 200  distinct datasets (by sampling with repetition) of equal size to the original dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' We optimized each of 200 sets, evaluated the corresponding AIC, and computed the mean  and standard deviation of ∆AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The orthogonal strategy has the lowest ∆AIC, indicating that it is the most parsimonious strategy and best explains the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  α1  −α2  β  b2 b1  A  C  Nair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 2015  model (massless)  model (neutral)  time  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='25T  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='5T  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='75T  T  0◦  middle link orientation β  20◦  40◦  60◦  80◦  −20◦  head/tail angles  0◦  α2  α1  90◦  135◦  45◦  −45◦  −90◦  stage 1  stage 2  stage 3  B  maximum  bending αmax  D  Voesenek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 2019  model (massless)  model (neutral)  prey response θ  maximum bending angle αmax  100◦  50◦  0◦  25◦  75◦  125◦  Nair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 2015  E  180◦  −180◦  180◦  −180◦  deformation angle α1  deformation angle α2  0◦  0◦  stage 2  stage 1  αmax  A  B  F  0  1  −1  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Biomechanics of fish C-start response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' (A) Larval zebrafish bends its body into a C-shape to initiate a fast start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' (B) Three fish model, with α1 and α2 representing  the fish body shape and β the overall body orientation relative to the straight pre-evasion direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' (C) Experimental data of shape changes of larval zebrafish during evasion  taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' [34] and processed to represent body deformations in terms of head and tail rotations α1, α2 (gray dots) then fitted by third-order fourier series (black lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  (D) Experimental data of overall body rotation for the same evasion instance shown in C (gray dots and black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Predictions based on fish model, taking as input the  shape changes in C are shown in blue lines (solid line for massless and dashed line for neutrally-buoyant fish).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' (E) The sequence of shape changes in C forms a curve C  in the shape space (α1, α2) (black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The experimental curve C is approximated by an ellipse (red) of axes A, B along α1 = ±α2 directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Colormap represents  curl2[A1, A2] of fish model, which predicts larger turns for curves that encompass solely positive (orange) or negative (blue) values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' (F) By varying A and calculating B that  maximizes the turn, we get a mapping from maximum bending angle αmax =  √  2A to turning angle θ for massless and neutrally buoyant fish (blue lines) that form an upper  and lower bounds on the experimental data set of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Both numerical and experimental data show that the C-start mechanics limits larval zebrafish to turning angles  θ ≲ 100◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  response and sensing of the predator angular position are both pre-  cise σΘ = σΦ = 0, but the prey’s sensing of the predator’s heading  is very noisy (σΛ large), the antipodal and contralateral strategies  do not get affected while the parallel strategy becomes protean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Interestingly, in this case, the distance-optimal strategy predicts  higher probability of evasion in directions opposite to the predator  location spanning a range of 2χ (see SI, section 4, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' That  is, the distance-optimal strategy becomes a noisy variant of the an-  tipodal strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' For χ = π/2, the orthogonal strategy with large  noise on λ converges to the antipodal strategy with uniform noise  spanning a range of π on either φ or θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Optimizing noise levels in sensing and response  For each noisy evasion strategy, we calculated the noise parame-  ters σ = {σS, σR} that maximize the total likelihood L of the  model given an experimental dataset, or equivalently minimize the  negative log-likelihood function NLL (see SI, section 6)  NLL = − ln L (σ|(r|s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' n) = −  �  i  ln p(n)(ri|si;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' σ),  (7)  where p(n)(ri|si;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' σ) is the conditional PDF of obtaining a response  ri given stimulus si for strategy n at noise level σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The optimal  noise parameters σ∗ are given by  σ∗ = arg min  σ NLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  (8)  5  We solved this optimization problem numerically in the range  σΦ, σΛ, σΘ ∈ (0, π) (SI, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 3D, we plot realizations  generated from the five probabilistic evasion models p(n)(s, r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' σ∗)  at the optimal noise values corresponding to the dataset of all  predator speeds combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Compared to the deterministic pre-  dictions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 3B, all five distributions appear closer to the ex-  perimental data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 3A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Evaluating strategies under optimal noise parameters  To evaluate how well each optimized strategy describes the exper-  imental data, we applied the Akaike information criterion (AIC)  defined as [35]  AIC = 2K − 2 ln L(σ∗|(s, r);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' n)  (9)  where K is the number of model parameters in each strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' AIC  considers both the goodness of fit represented by the likelihood  function, and the complexity of the model: if two models have  the same likelihood to explain the data, the criterion favors the  simpler model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' For example, for the antipodal strategy, we have  two noise parameters σS ≡ {σΦ} and σR ≡ {σΘ}, thus K = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  whereas for the orthogonal strategy, we have three noise parameters  σS ≡ {σΦ, σΛ} and σR ≡ {σΘ}, and the orthogonal strategy is  deemed more complex than the antipodal strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  We used bootstrapping to probe the accuracy of our evaluation  of the noisy strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Starting from each dataset (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=', that of  the fast predator), we constructed 200 distinct datasets of equal  size to the original dataset (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=', Nfast) by random sampling with  repetition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' We solved the optimization problem 200 times and ob-  tained 200 values of σ∗ per strategy for each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' We calculated  the likelihood value L and evaluated the AIC for all 200 optimal  noise values, thus obtaining a distribution of AIC values for each  strategy and predator speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The mean and standard deviation of  the distributions of AIC values, minus the lowest mean value and  normalized by the size of the respective dataset (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 4B) show  that strategies with lower mean values of the AIC better fit the  experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  The results based on the AIC evaluation of the probabilis-  tic strategies in the presence of sensory and response noise are  mostly consistent with the results based on the K-L divergence  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 4A) for precise sensing and response, but with marked differ-  ences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The orthogonal strategy ranks the highest in every dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  the distance-optimal strategy is slightly behind, in second place, in  all but the slow predator dataset where the antipodal strategy ranks  second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The difference between the orthogonal, distance-optimal,  and antipodal strategies is most distinguishable in the case of the  fast predator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The contralateral and parallel strategies come last  in all datasets and are least descriptive of experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Further analysis of distance-optimal strategy  While the predator speed was controlled at V = 2, 11, 20 cm s−1,  the zebrafish larvae were almost identical in all experiments, im-  plying that the speed ratio U/V varied drastically between evasion  instances: for the fast predator, this ratio is up to 10 times that of  the slow predator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' If the prey were to sense and use the speed ratio  to implement the distance-optimal strategy, we would expect the  best performance to appear at different values of χ = cos−1(U/V )  depending on predator speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' To test this, we evaluated this strat-  egy for the slow, mid, and fast predator as a function of χ ∈ [0, 90◦]  under both precise and noisy sensing and response (SI, section 7,  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  S14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  We found that the K-L divergence decreased as χ  increased and reached a minimum near χ = 75◦ independent of  predator speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Similarly, the NLL dropped as χ increased until it  reached a minimum at, or close to, χ = 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' These results suggest  that, even if following the distance-optimal strategy, the prey does  not rely on real-time and accurate measurements of the speed ratio  U/V , but favors the limit of large predator speed (χ → 90◦), where  the distance-optimal strategy converges to the orthogonal strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  The same conclusion can be reached by examining the values  of the optimized noise parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' In the range 20◦ ≲ χ ≲ 75◦,  the optimizer mostly selects the largest possible value of σΛ = π to  best fit the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' This high level of optimized noise indicates that  λ is not an effective sensory cue in the distance-optimal strategy,  and that the prey is unlikely to use this strategy at moderate χ  values (see SI, section7, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' S14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Evaluating the biomechanical constraints on es-  cape strategy  To complete our evaluation of fish evasion strategies, we consid-  ered the biomechanics of the C-start response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' In [34], the motion  of a zebrafish larvae undergoing a C-start maneuver starting from  a straight motionless configuration was recorded using high-speed  photography, and the time evolution of each segment of the fish  body from the onset of evasion at time t = 0 to after the com-  pletion of the C-start response at t = T = 25ms was measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  We developed a mathematical model of the biomechanics of these  events and incorporated that model into our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  We reinterpreted the experimental measurements in the context  of a three-link fish, head, middle, and tail (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 5B), and we ex-  tracted from experimental measurements the fish orientation β(t)  and rotations α1(t) and α2(t) of the head and tail relative to the  middle segment (see SI, section 8-9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  The time evolution of the  zebrafish body during evasion follows the three archetypal stages  of the C-start response: in stage 1, the fish curls its body to one  side, rapidly unfurls its body in stage 2, and begins its undulatory  swimming in stage 3 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 5C-D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  A larger number of C-start maneuvers were recorded in [23],  albeit only measuring the maximum degree of body bending αmax  and the net change in heading θ = β(T) − β(0) induced by the  C-start maneuver (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 5F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' These results show that the change  in heading direction θ correlates strongly with the degree of body  bending [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' In all recorded maneuvers, the change in body orien-  tation barely reaches 100◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Physics-based modeling of the C-start response  To shed light on the relationship between body deformations and  change in heading θ during evasion, we employed a physics-based  model of a three-link fish in potential flow [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Experimen-  tal and computational flow analysis had shown that the C-start  maneuver is dominated by unsteady, pressure-based exchange of  momentum between the fish and surrounding fluid, with negligible  contributions from fluid viscosity and shed vorticity [26, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The  potential flow model captures these unsteady pressure forces via  the added mass effect (see SI, S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The fish model is composed of  three identical prolate spheroids (of major and minor axes a and b)  such that the head and tail are free to rotate relative to the middle  link (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 5B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' as before, body deformations are described by the  angles α1(t), α2(t) representing the relative head and tail rotations  as a function of time t, and body orientation β(t) is the angle be-  tween the middle section and an inertial direction taken along the  direction of the initially-straight fish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  From consideration of momentum balance on the fish-fluid sys-  tem, we arrived at an equation governing the rate of change of body  orientation [39, 40, 37] (see SI, section 8)  ˙β = A1(α1, α2) ˙α1 + A2(α1, α2) ˙α2,  (10)  where A1 and A2 are nonlinear functions of body deformations  α1(t), α2(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' A1 and A2 also depend on fish geometry and fluid  and body densities (ρf and ρb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' For ρb = ρf, the fish is neutrally-  buoyant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' When the fluid forces are dominate, the fish can be con-  sidered massless and ρb is set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Body rotations are propor-  tional to the line integral of (10) over a curve C describing body  deformations in the shape space (α1, α2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' For a closed curve C,  this line integral can be rewritten, using Stokes theorem, as an  area integral over the region of the (α1, α2) space enclosed by C,  θ = β(T) − β(0) =  � � �∂A2  ∂α1  − ∂A1  ∂α2  �  dα1dα2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  (11)  The scalar field curl2([A1, A2]) ≡ ∂A2/∂α1 − ∂A1/∂α2 is shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 5E as a colormap over the entire shape space (α1, α2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' To  maximize the turning angle θ, a straight fish should deform its body  following a closed curve C that encompasses either non-positive or  non-negative values of curl2([A1, A2]), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=', either blue or orange  regions of the shape space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Closed curves in the orange region lead  6  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='2  0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='8  K-L divergence estimate  combined  slow  mid-speed  fast  ∆AIC/N  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='3  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='1  combined  slow  mid-speed  fast  Orthogonal  Distance-optimal  original  constrained  Antipodal  B  A  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Evaluation of the constrained strategies that consider the physical constraint on turning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Results are shown for the three best-performing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' (A) The  K-L divergence estimates of the constrained models with precise sensing and response (hollow bars), shown with the results of the original models (solid bars, from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 4A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  In all four datasets, the constrained models provide discernibly lower K-L divergence estimates, thus better fit to experimental data than the original models, except the  distance-optimal strategy for the mid-speed predator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The orthogonal strategy improved the most after imposing the constraint, making it fit the data best in all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  (B) The normalized relative AIC for the constrained models with optimized noise in sensing and response, compared to results using the original models in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 4B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The  orthogonal strategy still provides best fit to all datasets, marked by the lowest AIC scores, and its advantage over the second best model is more noticeable after imposing  the constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The constrained antipodal strategy performs comparable to or even better than the constrained distance-optimal strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  to turning counter-clockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  By symmetry, diagonally-opposite  curves in the blue region lead to turning clockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Theoretically,  the simplest curve for turning is a circle or an ellipse in the shape  space of major axis A aligned with α1 = α2, for which the maxi-  mum bending angle is αmax =  √  2A (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 5E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Corresponding fish  shape deformations and body rotations β(t) are discussed in SI  (S8-9, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' S15-S16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Comparing  model predictions  to  C-start  induced  turning of the fish body  We represented the empirical time evolution of shape deformations  (α1(t), α2(t)) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 5C) onto the shape space (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 5E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Interest-  ingly, the curve C (black line) traced by the actual fish follows  closely the elliptic curve (red line) predicted by the model as best  for turning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Moreover, when taking the empirical values of α1(t)  and α2(t) as input to the physics-based model in (10), the resulting  predictions of β(t) (blue lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 5D) follow closely the empiri-  cal values of β(t) (black line), especially during the first stage of the  C-start response, where vorticity is negligible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' note that while the  neutrally buoyant model (dashed blue line) deviates slightly from  empirical observations in stage 2, the massless fish model (solid blue  line) performs remarkably well way into stage 3, indicating that in-  deed unsteady pressure forces dominate the C-start maneuver, as  previously predicted [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  We next considered a family of shape changes following the el-  liptic curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 5E by varying A such that αmax =  √  2A varied  from 0 to 120◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' This upper limit on αmax corresponds to a maxi-  mum bending angle without causing the head and tail of the model  fish to cross each other, and is consistent with the experimental  observations of [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Using (11), we computed, for each αmax, the  value of B that optimizes the change in orientation θ, thus creating  a map from αmax to θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' We compare these model-predictions (blue  lines) to experimental data [23] (black dots) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 5F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' As before,  we considered massless and neutrally-buoyant fish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The theoretical  predictions behave nearly as upper and lower limits to experimen-  tal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  As observed previously [23], turning in the model fish  barely reaches 100◦ even when the three-link fish bends its body  to the extreme of the head and tail touching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' This indicates that  the biomechanics of the C-start maneuver imposes an upper limit  on achievable heading directions θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Constrained evasion strategies  We next incorporated the physical constraints on θ imposed by the  C-start biomechanics into our evasion strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' To this end, we  mapped the response angle θ(n)  i  predicted by evasion strategy n  onto the interval [0, 100◦] using the quadratic mapping  θ →  �  1 −  �  1 − θmax  π  � |θ|  π  �  θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  (12)  Small turns get less constrained whereas large turns are limited  to the maximum angle θmax = 100◦ allowable by the fish biome-  chanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  We applied this constraint to the three most plausible  strategies: distance-optimal, orthogonal, and antipodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' For each  constrained strategy, we repeated the analysis presented above un-  der precise and noisy sensing and response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Results of the K-L di-  vergence and AIC analysis for the constrained strategies are shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Compared to the unconstrained strategies, penalizing  large turns makes all three strategies fit better the experimental  data across all datasets, with or without added noise, with the ex-  ception of the distance-optimal strategy for the mid-speed preda-  tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Under precise sensing and response, the relative ranking of the  constrained strategies (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 6A) is similar to the original ranking  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 4A), with the distinction that the orthogonal strategy at slow  and mid-speed predator speed surpasses the distance-optimal strat-  egy and becomes the best ranking model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Under noisy sensing and  response, the antipodal strategy ranks higher than the distance-  optimal strategy in all but the slow predator dataset (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 6B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Importantly, whether precise or noisy, the orthogonal strategy fits  the experimental data better than the other two in all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Discussion  We developed a comprehensive framework for resolving evasion  strategy from kinematic measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Our approach considers  multiple hypotheses, each defined mathematically (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 1), that  address the role of sensorimotor noise (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  3) and incorporate  the effects of biomechanical constraints (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 5–6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Importantly,  our approach provides a rigorous methodology, rooted in strong-  inference principles [27], for revealing the strategy that best fits pre-  vious kinematic measurements of zebrafish larvae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' This approach  eliminates bias towards a particular hypothetical strategy, as done  in a previous study that favored the contralateral strategy from the  dataset presently analyzed [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  We found that the responses of zebrafish larvae to evade a preda-  tor are best-characterized by the orthogonal strategy (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' This  finding challenges the notion that a prey aims either to solely con-  fuse, or maximize its distance from, the predator with its escape  [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The kinematics of zebrafish do not exhibit the uniform distri-  bution of escape direction θ characteristic of a pure-protean strat-  egy (SI, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' S2E) [13, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Instead, larvae exhibited correla-  tions between θ and predator state, including angular position φ  (SI, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' S5) and heading ψ (SI, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The distance-optimal  strategy is more predictive of zebrafish kinematics, but is inferior  to the orthogonal strategy, based on K-L divergence and the AIC  scores (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 4 and 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Therefore, zebrafish larvae do not conform  to the classic dichotomy of models for prey strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Although the  prevailing patterns favor an orthogonal strategy, variation about  the predictions for this hypothesis allows for the possibility of a  mixed strategy that could hinder a predator’s ability to anticipate  7  the prey’s direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  These results are relevant to predator-prey  encounters, and hence the ecology, of fish species and reflect the ad-  vantages and constraints of the prey’s neurophysiology and biome-  chanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  The distance-optimal strategy requires sensing that may exceed  the abilities of larval fish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' This strategy requires detection of the  speed of the approaching predator (Table 1), but larval fish possess  poor visual acuity, compared to adult fish, due to a relatively small  number of retinal cells [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  It has been demonstrated that the  escape is triggered by a threshold diameter of a looming visual  stimulus, which may be simulated as a circle with an expanding  diameter [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' A looming stimulus alone does not offer the means  to differentiate between threats that are small and fast or large and  slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Therefore, the visual system of larval fish may offer a sensory  constraint on its ability to perform the distance-optimal strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  A more sophisticated visual system could allow for additional cues  to gauge the speed or size of a predator, but the processing time  necessary to formulate a distance-optimal response may still pose  a liability in evasion speed compared with the orthogonal strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  The orthogonal strategy merely requires an estimate of the  predator’s heading and offers tactical benefits relative to many of  the alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' This strategy is equivalent to the distance-optimal  strategy for a high-speed approach (U/V ≪ 1) and therefore suc-  ceeds in maximizing the prey’s distance from a fast predator at  reduced sensing requirements (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' It is the fastest predators  that likely present the greatest threat to the prey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The orthogo-  nal strategy offers an additional tactical advantage by evading in  a direction that is challenging for a fast-approaching predator to  follow because, in order for the predator to execute such large turn  at high speed, it needs a large turning radius, which could increase  its distance from the prey even further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  The predictions of the orthogonal strategy improved in their fit  to measured kinematics when we considered constraints imposed by  the biomechanics of the C-start (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' In particular, our model  of a three-link fish in potential flow accurately describes the rela-  tionship between the change in fish shape and its turning motion  during evasion (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' By mapping maximum bending angle to  turning angle, the model predicted an upper limit (around 100◦) on  achievable turning motion, consistent with the maximum angle ob-  served in zebrafish exposed to a lateral looming stimulus [19, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  The improvement in model predictions that included mechanics  demonstrates the influence of the constraints imposed by the prey  biomechanics and its interaction with the fluid environment on the  evasion strategy of zebrafish larvae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Comparing the K-L divergence and AIC values across slow, in-  termediate, and fast predators, the prevalence of the orthogonal  strategy is clearest in the case of the fast predator (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 4 and  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' This feature can be related to the fact that a weak stimu-  lus (slow predator) is more likely to trigger an escape response via  the less predictable, long-latency neural pathway [42, 43], as op-  posed to the fast pathway with minimal latency between perceived  danger and motor response [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' An untangling of these features  requires a deeper investigation of how our analytical framework  relates to the neurophysiology underlying zebrafish evasion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Our study combined tools from information theory and proba-  bilistic methods with behavioral evasion models and physics-based  models of the C-start biomechanics to develop a comprehensive  analytical approach and thereby determine the evasion strategy of  zebrafish larvae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Aside from the details of the biomechanics model,  nothing about our approach is specific to the study of fish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Our  analysis could be applied to the myriad of studies that have mea-  sured escape responses relative to a predator’s approach in a diver-  sity of animals [5, 6, 7, 8, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' This approach may therefore be  applied broadly to the study of predator-prey encounters to reveal  the strategic basis of this fundamental aspect of animal behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Acknowledgement  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' acknowledges support from the Of-  fice of Naval Research (ONR) Grants N00014-22-1-2655, N00014-  19-1-2035, N00014-17-1-2062, and N00014-14-1-0421;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' the National  Science Foundation (NSF) Grants RAISE IOS-2034043, CBET-  2100209, and INSPIRE MCB-1608744;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' the National Institutes of  Health (NIH) Grant R01 HL 153622-01A1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' the Army Research Of-  fice (ARO) Grant W911NF-16-1-0074.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  This research started in  summer 2018 at the Summer Graduate School on Mathematical  Analysis of Behavior organized by Ann Hermundstad, Vivek Ja-  yaraman, Eva Kanso, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Mahadevan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The school was jointly  supported by the Mathematical Science Research Institute (MSRI)  and the Howard-Hugh Medical Institute (HHMI) Janelia Research  Campus, and was held at Janelia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' BC, YM, and EK acknowledge  support from Janelia and would like to thank Sashank Pisupati and  Ann Hermundstad for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  References  [1] Jos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Schall and Eric R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Pianka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Evolution of escape behavior  diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The American Naturalist, 115(4):551–566, 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [2] D Weihs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  The mechanism of rapid starting of slender fish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Biorheology, 10(3):343–350, 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Weihs and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Webb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Optimal avoidance and evasion  tactics in predator-prey interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Journal of Theoretical  Biology, 106(2):189 – 206, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [4] Rufus Isaacs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Differential games: a mathematical theory with  applications to warfare and pursuit, control and optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Courier Corporation, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [5] Paolo Domenici, David Booth, Jonathan M Blagburn, and  Jonathan P Bacon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Cockroaches keep predators guessing by  using preferred escape trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Curr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=', 18(22):1792–  1796, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [6] Jérôme Casas and Thomas Steinmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Predator-induced  flow disturbances alert prey, from the onset of an attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Proceedings of the Royal Society B: Biological Sciences,  281(1790):20141083, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [7] Stephen A Arnott, DOUGLAS M Neil, and Alan D Ansell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Escape trajectories of the brown shrimp crangon crangon, and  a theoretical consideration of initial escape angles from preda-  tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=', 202(2):193–209, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [8] Rafe M Brown and Douglas H Taylor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Compensatory escape  mode trade-offs between swimming performance and maneu-  vering behavior through larval ontogeny of the wood frog, rana  sylvatica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Copeia, pages 1–7, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [9] Emanuel Azizi and Tobias Landberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Effects of metamorphosis  on the aquatic escape response of the two-lined salamander  (eurycea bislineata).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=', 205(6):841–849, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [10] Patrick B Woodbury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  The geometry of predator avoidance  by the blue crab, callinectes sapidus rathbun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Ani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Behav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=',  34:28–37, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [11] José E Trujillo, Ian Bouyoucos, William J Rayment, Paolo  Domenici, Serge Planes, Jodie L Rummer, and Bridie JM Al-  lan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Escape response kinematics in two species of tropical  shark: short escape latencies and high turning performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Journal of Experimental Biology, 225(22):jeb243973, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [12] Paolo Domenici.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The visually mediated escape response in fish:  predicting prey responsiveness and the locomotor behaviour  of predators and prey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Marine and Freshwater Behaviour and  Physiology, 35(1-2):87–110, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [13] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Humphries and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Protean defence by prey  animals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Oecologia, 5(4):285–302, 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [14] Peter M Driver and David Andrew Humphries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Protean be-  haviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Clarendon Press, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [15] Talia Y Moore, Kimberly L Cooper, Andrew A Biewener, and  Ramanarayan Vasudevan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Unpredictability of escape trajec-  tory explains predator evasion ability and microhabitat pref-  erence of desert rodents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Nature comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=', 8(1):1–9, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [16] Paolo Domenici, Jonathan M Blagburn, and Jonathan P Ba-  con.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Animal escapology ii:  escape trajectory case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Journal of Experimental biology, 214(15):2474–2494, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [17] Arjun Nair, Kelsey Changsing,  William J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Stewart,  and  Matthew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' McHenry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Fish prey change strategy with the  direction of a threat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Proceedings of the Royal Society B: Bi-  ological Sciences, 284(1857):20170393, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  8  [18] William  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Stewart,  Arjun  Nair,  Houshuo  Jiang,  and  Matthew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' McHenry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Prey fish escape by sensing the  bow wave of a predator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Journal of Experimental Biology,  217(24):4328–4336, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [19] Timothy W Dunn, Christoph Gebhardt, Eva A Naumann,  Clemens Riegler, Misha B Ahrens, Florian Engert, and Filippo  Del Bene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Neural circuits underlying visually evoked escapes  in larval zebrafish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Neuron, 89(3):613–628, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [20] Alix MB Lacoste, David Schoppik, Drew N Robson, Martin  Haesemeyer, Ruben Portugues, Jennifer M Li, Owen Randlett,  Caroline L Wee, Florian Engert, and Alexander F Schier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' A  convergent and essential interneuron pathway for mauthner-  cell-mediated escapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Curr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=', 25(11):1526–1534, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [21] Shin-ichi Higashijima, Mark A Masino, Gail Mandel, and  Joseph R Fetcho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Imaging neuronal activity during zebrafish  behavior with a genetically encoded calcium indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' neu-  rophysiol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=', 90(6):3986–3997, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [22] William J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Stewart, Gilberto S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Cardenas, and Matthew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  McHenry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Zebrafish larvae evade predators by sensing water  flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Journal of Experimental Biology, 216(3):388–398, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [23] Cees J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Voesenek, Remco P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Pieters, Florian T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Muijres,  and Johan L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' van Leeuwen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Reorientation and propulsion in  fast-starting zebrafish larvae: an inverse dynamics analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Journal of Experimental Biology, 222(14), 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [24] Cees J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Voesenek, Florian T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Muijres, and Johan L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' van  Leeuwen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Biomechanics of swimming in developing larval fish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Journal of Experimental Biology, 221(1), 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [25] Ulrike K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Müller and Johan L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' van Leeuwen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Swimming of  larval zebrafish: ontogeny of body waves and implications for  locomotory development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Journal of Experimental Biology,  207(5):853–868, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [26] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Gazzola, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Van Rees, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Koumoutsakos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  C-  start: optimal start of larval fish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Journal of Fluid Mechanics,  698:5–18, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [27] John R Platt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Strong inference: Certain systematic methods  of scientific thinking may produce much more rapid progress  than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Science, 146(3642):347–353, 1964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [28] PAOLO Domenici and ROBERT W Blake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Escape trajectories  in angelfish (pterophyllum eimekei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Journal of Experimental  Biology, 177(1):253–272, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [29] Robert C Eaton and Deanna S Emberley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' How stimulus di-  rection determines the trajectory of the mauthner-initiated es-  cape response in a teleost fish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Journal of Experimental Biol-  ogy, 161(1):469–487, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [30] Alberto Soto, William J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Stewart, and Matthew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' McHenry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  When Optimal Strategy Matters to Prey Fish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=', 55(1):110–120, 05 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [31] Robert C Eaton, William A Lavender, and Chris M Wieland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Identification of mauthner-initiated response patterns in gold-  fish: evidence from simultaneous cinematography and electro-  physiology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Journal of Comparative Physiology, 144(4):521–  531, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [32] Mark J Schervish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Theory of statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Springer Science &  Business Media, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [33] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Perez-Cruz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Kullback-leibler divergence estimation of con-  tinuous distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' In 2008 IEEE International Symposium  on Information Theory, pages 1666–1670, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [34] Arjun Nair, Grigor Azatian, and Matthew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' McHenry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' The  kinematics of directional control in the fast start of zebrafish  larvae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Journal of Experimental Biology, 218(24):3996–4004,  2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [35] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Akaike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  A new look at the statistical model identifica-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  IEEE Transactions on Automatic Control, 19(6):716–  723, 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [36] Eva Kanso and Jerrold E Marsden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Optimal motion of an  articulated body in a perfect fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' In Proceedings of the 44th  IEEE Conference on Decision and Control, pages 2511–2516.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  IEEE, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [37] Yusheng Jiao, Feng Ling, Sina Heydari, Nicolas Heess, Josh  Merel, and Eva Kanso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Learning to swim in potential flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Physical Review Fluids, 6(5):050505, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [38] Ulrike K Muller, Jos GM van den Boogaart, and Johan L van  Leeuwen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Flow patterns of larval fish: undulatory swimming  in the intermediate flow regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Journal of Experimental Bi-  ology, 211(2):196–205, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [39] Eva Kanso, Jerrold E Marsden, Clarence W Rowley, and  Juan B Melli-Huber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Locomotion of articulated bodies in a  perfect fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Journal of Nonlinear Science, 15(4):255–289,  2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [40] Ross L Hatton and Howie Choset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Connection vector fields  and optimized coordinates for swimming systems at low and  high reynolds numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' In ASME 2010 Dynamic Systems and  Control Conference, pages 817–824.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' American Society of Me-  chanical Engineers Digital Collection, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [41] Marion F Haug, Oliver Biehlmaier, Kaspar P Mueller, and  Stephan CF Neuhauss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Visual acuity in larval zebrafish: be-  havior and histology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Zool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=', 7(1):1–7, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [42] Harold A Burgess and Michael Granato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  Sensorimotor gat-  ing in larval zebrafish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Journal of Neuroscience, 27(18):4984–  4994, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [43] Gwyneth M Card.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Escape behaviors in insects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Current Opin-  ion in Neurobiology, 22(2):180–186, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Neuroethology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  [44] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' WALKER, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' GHALAMBOR, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' GRISET,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' McKENNEY, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' REZNICK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Do faster starts in-  crease the probability of evading predators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content=' Functional Ecol-  ogy, 19(5):808–815, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
+page_content='  9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfygLM/content/2301.02661v1.pdf'}
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+LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL
+NOTES
+CHENGYANG SHAO
+In this note, we propose several unsolved problems concerning the irrotational oscillation of a water
+droplet under zero gravity. We will derive the governing equation of this physical model, and convert it
+to a quasilinear dispersive partial differential equation defined on the sphere, which formally resembles the
+capillary water waves equation but describes oscillation defined on curved manifold instead. Three types of
+unsolved mathematical problems related to this model will be discussed in observation of hydrodynamical
+experiments under zero gravity1: (1) Strichartz type inequalities for the linearized problem (2) existence of
+periodic solutons (3) normal form reduction and generic lifespan estimate. It is pointed out that all of these
+problems are closely related to certain Diophantine equations, especially the third one.
+1. Capillary Spherical Water Waves Equation: Derivation
+1.1. Water Waves Equation for a Bounded Water Drop. Comparing to gravity water waves problems,
+the governing equation for a spherical droplet of water under zero gravity takes a very different form. At
+a first glance it looks similar to the water waves systems as mentioned above, but some crucial differences
+do arise after careful analysis. To the author’s knowledge, besides those dealing with generic free-boundary
+Euler equation ( [14], [36]), the only reference on this problem is Beyer-G¨unther [9], in which the local
+well-posedness of the equation is proved using a Nash-Moser type implicit function theorem. We will briefly
+discribe known results for gravity water waves problems in the next subsection.
+To start with, let us pose the following assumptions on the fluid motion that we try to describe:
+• (A1) The perfect, irrotational fluid of constant density ρ0 occupies a smooth, compact region in R3.
+• (A2) There is no gravity or any other external force in presence.
+• (A3) The air-fluid interface is governed by the Young-Laplace law, and the effect of air flow is
+neglected.
+We assume that the boundary of the fluid region has the topological type of a smooth compact orientable
+surface M, and is described by a time-dependent embedding ι(t, ·) : M → R3. We will denote a point on M
+by x, the image of M under ι(t, ·) by Mt, and the region enclosed by Mt by Ωt. The outer normal will be
+denoted by N(ι). We also write ¯∇ for the flat connection on R3.
+Adopting assumption (A3), we have the Young-Laplace equation:
+σ0H(ι) = pi − pe,
+where H(ι) is the (scalar) mean curvature of the embedding, σ0 is the surface tension coefficient (which is
+assumed to be a constant), and pi, pe are respectively the inner and exterior air pressure at the boundary;
+they are scalar functions on the boundary and we assume that pe is a constant. Under assumptions (A1) and
+(A2), we obtain Bernoulli’s equation, sometimes referred as the pressure balance condition, on the evolving
+1There are numbers of visual materials on such experiments conducted by astronauts. See for example https://www.youtube.
+com/watch?v=H_qPWZbxFl8&t or https://www.youtube.com/watch?v=e6Faq1AmISI&t.
+1
+arXiv:2301.00115v1  [math.AP]  31 Dec 2022
+
+2
+CHENGYANG SHAO
+surface:
+(1.1)
+∂Φ
+∂t
+����
+Mt
++ 1
+2| ¯∇Φ|Mt|2 − pe = −σ0
+ρ0
+H(ι),
+where Φ is the velocity potential of the velocity field of the air. Note that Φ is determined up to a function in
+t, so we shall leave the constant pe around for convenience reason that will be explained shortly. According
+to assumption (A1), the function Φ is a harmonic function within the region Ωt, so it is uniquely determined
+by its boundary value, and the velocity field within Ωt is ¯∇Φ. The kinematic equation on the free boundary
+Mt is naturally obtained as
+(1.2)
+∂ι
+∂t · N(ι) = ¯∇Φ|Mt · N(ι).
+Finally, we would like to discuss the conservation laws for (1.1)-(1.2). The preservation of volume Vol(Ωt) =
+Vol(Ω0) is a consequence of incompressibility. The system describes an Eulerian flow without any external
+force, so the center of mass moves at a uniform speed along a fixed direction, i.e.
+(1.3)
+1
+Vol(Ω0)
+�
+Ωt
+PdVol(P) = V0t + C0,
+with Vol being the Lebesgue measure, P marking points in R3, V0 and C0 being the velocity and starting
+position of center of mass respectively. Furthermore, the total momentum is conserved, and since the flow is
+a potential incompressible one, the conservation of total momentum is expressed as
+(1.4)
+�
+Mt
+ρ0ΦN(ι)dArea(Mt) ≡ ρ0Vol(Ω0)V0.
+Most importantly, it is not surprising that (1.1)-(1.2) is a Hamilton system (the Zakharov formulation for
+water waves; see Zakharov [43]), with Hamiltonian
+(1.5)
+σ0Area(ι) + 1
+2
+�
+Ωt
+ρ0| ¯∇Φ|2dVol = σ0Area(Mt) + 1
+2
+�
+Mt
+ρ0Φ|Mt
+� ¯∇Φ|Mt · N(ι)
+�
+dArea,
+i.e. potential proportional to surface area plus kinetic energy of the fluid.
+1.2. Converting to a Differential System. It is not hard to verify that the system (1.1)-(1.2) is invariant
+if ι is composed with a diffeomorphism of M; we may thus regard it as a geometric flow. If we are only
+interested in perturbation near a given configuration, we may reduce system (1.1)-(1.2) to a non-degenerate
+dispersive differential system concerning two scalar functions defined on M, just as Beyer and G¨unther did
+in [9]. In fact, during a short time of evolution, the interface can be represented as the graph of a function
+defined on the initial surface: if ι0 : M → R3 is a fixed embedding close to the initial embedding ι(0, x), we
+may assume that ι(t, x) = ι0(x) + ζ(t, x)N0(x), where ζ is a scalar “height” function defined on M0 and N0
+is the outer normal vector field of M0.
+With this observation, we shall transform the system (1.1)&(1.2) into a non-local system of two real scalar
+functions (ζ, φ) defined on M, where ζ is the “height” function described as above, and φ(t, x) = Φ(t, ι(t, x))
+is the boundary value of the velocity potential, pulled back to the underlying manifold M.
+The operator
+Bζ : φ → ¯∇Φ|Mt
+maps the pulled-back Dirichlet boundary value φ to the boundary value of the gradient of Φ. We shall write
+(D[ζ]φ)N(ι)
+
+LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL NOTES
+3
+Figure 1. The shape of the surface
+for its normal part, where D[ζ] is the Dirichlet-Neumann operator corresponding to the region enclosed by
+the image of ι0 + ζN0. Thus
+∂ζ
+∂t N0 · N(ι) = D[ζ]φ.
+We also need to calculate the restriction of ∂tΦ on Mt in terms of φ and ι. By the chain rule,
+∂Φ
+∂t
+����
+Mt
+= ∂φ
+∂t − ¯∇Φ|Mt · ∂ι
+∂t
+= ∂φ
+∂t −
+� ¯∇Φ|Mt · N0
+� ∂ζ
+∂t
+= ∂φ
+∂t −
+1
+N0 · N(ι)
+� ¯∇Φ|Mt · N0
+�
+· D[ζ]φ.
+We thus arrive at the following nonlinear system:
+(EQ(M))
+�
+�
+�
+�
+�
+�
+�
+∂ζ
+∂t =
+1
+N0 · N(ι)D[ζ]φ,
+∂φ
+∂t =
+1
+N0 · N(ι) (Bζφ · N0) · D[ζ]φ − 1
+2|Bζφ|2 − σ0
+ρ0
+H(ι) + pe,
+where ι = ι0 + ζN0.
+Remark 1.
+We may obtain an explicit expression of Bζφ = ¯∇Φ|Mt in terms of φ (together with the
+connection ∇0 on the fixed embedding ι0(M)), just as standard references did for Euclidean or periodic
+water waves, but that is not necessary for our discussion at the moment. It is important to keep in mind
+that the preservation of volume and conservation of total momentum (1.3)-(1.4) convert to integral equalities
+of (ζ, φ). These additional restrictions are not obvious from the differential equations (EQ(M)), though they
+can be deduced from (EQ(M)) since they are just rephrase of the original physical laws (1.1)-(1.2).
+For M = S2, the case that we shall discuss in detail, we refer to the system as the capillary spherical water
+waves equation. To simplify our discussion, we shall be working under the center of mass frame, and require
+the eigenmode Π(0)φ to vanish for all t. This could be easily accomplished by absorbing the eigenmode
+into φ since the equation is invariant by a shift of φ. In a word, from now on, we will be focusing on the
+
+NS+ 0
+1
+1
+1
+-
+-
+-
+-
+1
+-
+-
+y4
+CHENGYANG SHAO
+non-dimensional capillary spherical water waves equation
+(EQ)
+�
+�
+�
+�
+�
+�
+�
+∂ζ
+∂t =
+1
+N0 · N(ι)D[ζ]φ,
+∂φ
+∂t =
+1
+N0 · N(ι) (Bζφ · N0) · D[ζ]φ − 1
+2|Bζφ|2 − H(ι),
+where ι = (1 + ζ)ι0, and Π(0)φ ≡ 0. We assume that the total volume of the fluid is 4π/3, so that the
+preservation of volume is expressed as
+(1.6)
+1
+3
+�
+S2(1 + ζ)3dµ0 ≡ 4π
+3 ,
+where µ0 is the standard measure on S2. The inertial movement of center of mass (1.3) and conservation of
+total momentum (1.4) under our center of mass frame are expressed respectively as
+(1.7)
+�
+S2(1 + ζ)4N0dµ0 = 0,
+�
+S2 φN(ι)dµ(ι) = 0,
+where µ(ι) is the induced surface measure. Further, the Hamiltonian of the system is
+(1.8)
+H[ζ, φ] = Area(ι) + 1
+2
+�
+S2 φ · D[ζ]φ · dµ(ι),
+and for a solution (ζ, φ) there holds H[ζ, φ] ≡ 4π.
+Up to this point, we are still working within the realm of well-established frameworks. We already know
+that the general free-boundary Euler equation is locally well-posed due to the work of [14] or [36], and due
+to the curl equation the curl free condition persists during the evolution. On the other hand, the Cauchy
+problem of system (1.1) and (1.2) is known to be locally well-posed, due to Beyer and G¨unther in [9].
+They used an iteration argument very similar to a Nash-Moser type argument in the sense that it involves
+multiple scales of Banach spaces and “tame” maps in a certain sense. Finally, it is not hard to transplant
+the potential-theoretic argument of Wu [40] to prove the local well-posedness.
+To sum up, we already know that the system (EQ(M)) for a compact orientable surface M (hence (EQ)
+specifically) is locally well-posed. But this is all we can assert for the motion of a water droplet under zero
+gravity. In the following part of this note, we will propose several questions and conjectures concerning the
+long-time behaviour of water droplets under zero gravity.
+1.3. Previous Works on Water Waves. There has already been several different approaches to describe
+the motion of perfect fluid with free boundary. One is to consider the motion of perfect fluid occupying an
+arbitrary domain in R2 or R3, either with or without surface tension. The motion is described by a free
+boundary value problem of Euler equation. This generic approach was employed by Countand-Shkoller [14]
+and Shatah-Zeng [36]. Both groups proved the local-wellposedness of the problem. This approach has the
+advantage of being very general, applicable to all geometric shapes of the fluid.
+On the other hand, when coming to potential flows of a perfect fluid, the curl free property results in dis-
+persive nature of the problem. The motion of a curl free perfect fluid under gravity and a free boundary value
+condition is usually referred to as the gravity water waves problem. The first breakthroughs in understanding
+local well-posedness were works of Wu [39] [40], who proved local well-posedness of the gravity water waves
+equation without any smallness assumption. Lannes [27] extended this to more generic bottom shapes. Taking
+surface tension into account, the problem becomes gravity-capillary water waves. Schweizer [32] proved local
+well-posedness with small Cauchy data of the gravity-capillary water waves problem, and Ming-Zhang [30]
+proved local well-posedness without smallness assumption. Alazard-Metevier [2] and Alazard-Burq-Zuily [3]
+
+LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL NOTES
+5
+used para-differential calculus to obtain the optimal regularity for local well-posedness of the water waves
+equation, either with or without surface tension.
+For discussion of long time behavior, it is important to take into account the dispersive nature of the
+problem. For linear dispersive properties, there has been work of Christianson-Hur-Staffilani [13]. For gravity
+water waves living in R2, works on lifespan estimate include Wu [41] and Hunter-Ifrim-Tataru [22] (almost
+global result), Ionescu-Pusateri [24] and Alazard-Delort [4] and Ifrim-Tataru [23] (global result). For gravity
+water waves living in R3, there are works of Germain-Masmoudi-Shatah [18] and Wu [42] (no surface tension),
+Germain-Masmoudi-Shatah [19] (no gravity), Deng-Ionescu-Pausader-Pusateri [15] (gravity-capillary water
+waves) and Wang [38] (gravity-capillary water waves with finite depth). These results all employed different
+forms of decay estimates derived from dispersive properties.
+As for long time behavior of periodic water waves, Berti-Delort [6] considered gravity-capillary water waves
+defined on T1, Berti-Feola-Pusateri [7] considered gravity water waves defined on T1, Ionescu-Pusateri [26]
+considered gravity-capillary water waves defined on T2, and obtained an estimate on the lifespan beyond
+standard energy method. All of the three groups used para-differential calculus and suitable normal form
+reduction; the results of Berti-Delort and Ionescu-Pusateri were proved for physical data of full Lebesgue
+measure.
+To sum up, all results on the gravity water waves problem listed above are concerned with an equation for
+two scalar functions defined on a fixed flat manifold, being one of the following: R1, R2, T1, T2, sometimes
+called the “bottom” of the fluid. These two functions represents the geometry of the liquid-gas interface and
+the boundary value of the velocity potential, respectively. The manifold itself is considered as the bottom
+of the container in which all dynamics are performed. We observe that the differential equation (EQ(M)) is,
+mathematically, fundamentally different from the water waves equations that have been well-studied.
+2. Initial Notes on Unsolved Problems
+In this section, we propose unsolved problems related to the spherical capillary water waves system
+(EQ(M)) with M = S2. Not surprisingly, these problems all have deep backgrounds in number theory.
+2.1. Linearization Around the Static Solution. We are mostly interested in the stability of the static
+solution of (EQ(M)). A static solution should be a fluid region whose shape stays still, with motion being
+a mere shift within the space.
+In this case, we have pe = 0 since the reference is relatively static with
+respect to the air. Moreover, the velocity field ¯∇Φ and “potential of acceleration” ∂Φ/∂t must both be
+spatially uniform, so that the left-hand-side of the pressure balance condition (1.1) is a function of t alone.
+It follows that ι(M) is always a compact embedded surface of constant mean curvature, hence in fact always
+an Euclidean sphere by the Alexandrov sphere theorem (see [29]), and we may just take M = S2. Moreover,
+since ι(S2) should enclose a constant volume by incompressibility, the radius of that sphere does not change.
+After suitable scaling, we may assume that the radius is always 1, and ρ0 = 1, σ0 = 1, to make (1.1)-(1.2)
+non-dimensional. Finally, by choosing the center of mass frame, we may simply assume that the spatial shift
+is always zero, so that the velocity potential Φ ≡ Φ0, a real constant. It is harmless to fix it to be zero.
+Thus, under our convention, a static solution of (1.1)-(1.2) takes the form
+(2.1)
+�
+ι(t, x)
+Φ(t, x)
+�
+=
+�
+ι0(x)
+0
+�
+,
+where a ∈ R3 is a constant vector, and ι0 is the standard embedding of S2 as ∂B(0, 1) ⊂ R3. Equivalently,
+this means that a static solution of (EQ(M)) under our convention must be (ζ, φ) = (0, 0). Note here that
+the Gauss map of ι0 coincides with itself.
+
+6
+CHENGYANG SHAO
+We can now start our perturbation analysis around a static solution at the linear level. Let E(n) be the
+space of spherical harmonics of order n, normalized according to the standard surface measure on S2. In
+particular, E(1) is spanned by three components of N0. Let Π(n) be the orthogonal projection on L2(S2)
+onto E(n), Π≤n be the orthogonal projection on L2(S2) onto �
+k≤n E(k), Π≥n be the orthogonal projection
+on L2(S2) onto �
+k≥n E(k). For ι = (1 + ζ)ι0, the linearization of −H(ι) around the sphere ζ ≡ 0 is ∆ζ + 2ζ,
+where ∆ is the Laplacian on the sphere S2; cf. the standard formula for the second variation of area in [5].
+Then H′(ι0) acts on E(n) as the multiplier −(n − 1)(n + 2). Note that even if we consider the dimensional
+form of (EQ), there will only be an additional scaling factor σ0/(ρ0R2), where R is the radius of the sphere.
+On the other hand, the following solution formula for the Dirichlet problem on B(0, 1) is well-known: if
+f ∈ L2(S2), then the harmonic function in B(0, 1) with Dirichlet boundary value f is determined by
+u(r, ω) =
+�
+n≥0
+rn(Π(n)f)(ω),
+where (r, ω) is the spherical coordinate in R3. Thus the Dirichlet-Neumann operator D[ι0] acts on E(n) as the
+multiplier n. Note again that even if we consider the dimensional form (EQ), there will only be an additional
+scaling factor R−1.
+Thus, setting
+u = Π(0)ζ + Π(1)ζ +
+�
+n≥2
+�
+(n − 1)(n + 2) · Π(n)ζ + i
+�
+n≥1
+√n · Π(n)φ,
+we find that the linearization of (EQ) around the static solution (2.1) is a linear dispersive equation
+(2.2)
+∂u
+∂t + iΛu = 0,
+where the 3/2-order elliptic operator Λ is given by a multiplier
+Λ =
+�
+n≥2
+�
+n(n − 1)(n + 2) · Π(n) =:
+�
+n≥0
+Λ(n)Π(n).
+Note that (ζ, φ) is completely determined by u. At the linear level, there must hold Π(0)u ≡ 0 because the
+first variation of volume must be zero; and Π(1)u ≡ 0 because of the conservation laws (1.7).
+Let us also re-write the original nonlinear system (EQ) into a form that better illustrates its perturbative
+nature. For simplicity, we use O(u⊗k) to abbreviate a quantity that can be controlled by k-linear expressions
+in u, and disregard its continuity properties for the moment. For example, ∥u∥2
+H1 + ∥u∥4
+H2 is an expression
+of order O(u⊗2) when u → 0.
+Since the operator Λ acts degenerately on E(0) ⊕ E(1), we should be more careful about the eigenmodes
+Π(0)u and Π(1)u. The volume preservation equation (1.6) implies ∂tΠ(0)ζ = O(u⊗2). Projecting (EQ) to
+E(1), which is spanned by the components of N0, we obtain ∂tΠ(1)ζ = Π(1)φ + O(u⊗2) = O(u⊗2) since the
+conservation law (1.7) implies Π(1)φ = O(u⊗2); and ∂tΠ(1)φ = O(u⊗2) since H′(ι0) = −∆ − 2 annihilates
+E(1). We can thus formally re-write the nonlinear system (EQ) as the following:
+(2.3)
+∂u
+∂t + iΛu = N(u),
+with N(u) = O(u⊗2) vanishing quadratically as u → 0. Note that we are disregarding all regularity problems
+at the moment.
+2.2. Question at Linear Level. At the linear level, our first unanswered question is
+
+LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL NOTES
+7
+Question 1. Does the solution of the linear capillary spherical water waves equation (2.2) satisfy a Strichartz
+type estimate of the form
+∥eitΛf∥Lp
+T Lq
+x ≲T ∥f∥Hs,
+where Lp
+T Lq
+x = Lp([0, T]; Lq(S2)), and the admissible indices (p, q) and s should be determined?
+Answer to Question 1 should be important in understanding the dispersive nature of linear capillary
+spherical water waves. For Schr¨odinger equation on a compact manifold, a widly cited result was obtained
+by Burq-G´erard-Tzvetkov [12]:
+Theorem 2.1. On a general compact Riemannian manifold (M d, g), there holds
+∥eit∆gf∥Lp
+t Lq
+x([0,T ]×M) ≲T ∥f∥H1/p(M),
+where
+2
+p + d
+q = d
+2,
+p > 2.
+The authors used a time-localization argument for the parametrix of ∂t − i∆g to prove this result. For the
+sphere Sd, this inequality is not optimal. The authors further used a Bourgain space argument to obtain the
+optimal Strichartz inequality:
+Theorem 2.2. Let (Sd, g) be the standard n-dimensional sphere. For a function f ∈ C∞(Sd), there holds
+the Strichartz inequality
+∥eit∆gf∥Lp
+t Lq
+x([0,T ]×M) ≲T ∥f∥Hs(M),
+s > s0(d)
+where
+s0(2) = 1
+8,
+s0(d) = d
+4 − 1
+2, d ≥ 3.
+Furthermore these inequalities are optimal in the sense that the Sobolev index s cannot be less than or equal
+to s0(d).
+The proof is the consequence of two propositions. The first one is the “decoupling inequality on compact
+manifolds”, in particular the following result proved by Sogge [33]:
+Proposition 2.1. Let Πk be the spectral projection to eigenspaces with eigenvalues in [k2, (k + 1)2] on Sd.
+Then there holds
+∥Πk∥L2→Lq ≤ Cqns(q),
+where
+s(q) =
+�
+d−1
+2
+�
+1
+2 − 1
+2q
+�
+,
+2 ≤ q ≤ 2(d+1)
+d−1
+d−1
+2
+− d
+q ,
+2(d+1)
+d−1
+≤ q ≤ ∞.
+These estimates are sharp in the following sense: if hk is a zonal spherical harmonic function of degree k on
+Sd, then as k → ∞,
+∥hk∥Lq ≃ Cqks(q)∥hk∥L2.
+The second one is a Bourgain space embedding result:
+Proposition 2.2. For a function f ∈ C∞
+0 (R × Sd), define the Bourgain space norm
+∥f(t, x)∥Xs,b :=
+��⟨∂t + i∆g⟩bf(t, x)
+��
+L2
+t Hsx .
+Then for b > 1/2 and s > s0(d), there holds
+∥f∥L4(R×Sd) ≤ Cs,b∥f∥Xs,b.
+
+8
+CHENGYANG SHAO
+The key ingredient for proving this proposition is the following number-theoretic result:
+#{(p, q) ∈ N2 : p2 + q2 = A} = O(Aε).
+As for optimality of the Strichartz inequality, the authors of [12] implemented standard results of Gauss
+sums.
+The parametrix and Bourgain space argument can be repeated without essential change for the linear
+capillary spherical water waves equation (2.2), but this time the Bourgain space argument would be more
+complicated: the Bourgain space norm is now
+∥f(t, x)∥Xs,b :=
+��⟨∂t + iΛ⟩bf(t, x)
+��
+L2
+t Hsx ,
+and the embedding result becomes
+∥f∥L4(R×S2) ≤ Cs,b∥f∥Xs,b
+for f ∈ C∞
+0 (R × S2) and all s > 3ρ/8 + 1/8, b > 1/2, where ρ is the infimum of all exponents ρ′ such that
+when A → ∞, the number
+#
+�
+(n1, n2) ∈ N2 : 1
+2 ≤ n2
+n1
+≤ 2, |Λ(n1) + Λ(n2) − A| ≤ 1
+2
+�
+≤ Cρ′Aρ′.
+Some basic analytic number theory implies ρ = 1/3, and thus the range of s is s > 1/4. Surprisingly this
+index is not better than that predicted by the parametrix method. It remains unknown whether this index
+could be further optimized.
+To close this subsection, we note that the capillary spherical water wave lives on a compact region, so the
+dispersion does not take away energy from a locality to infinity. This is a crucial difference between waves
+on compact regions and waves in Euclidean spaces. In particular, we do not expect decay estimate for eitΛf.
+For the nonlinear problem (EQ), techniques like vector field method (Klainerman-Sobolev type inequalities)
+do not apply.
+2.3. Rotationally Symmetric Solutions: Bifurcation Analysis. Illuminated by observations in hydro-
+dynamical experiments under zero gravity, and suggested by the existence of standing gravity capillary water
+waves due to Alazard-Baldi [1], we propose the following conjecture:
+Conjecture 2.1. There is a Cantor family of small amplitude periodic solutions to the spherical capillary
+water waves system (EQ).
+Let us conduct the bifurcation analysis that suggests why this conjecture should be true. By time rescaling,
+we aim to find solution (ζ, φ, ω0) of the following system that is 2π-peiodic in t:
+(2.4)
+�
+�
+�
+�
+�
+�
+�
+ω0
+∂ζ
+∂t =
+1
+N0 · N(ι)D[ζ]φ,
+ω0
+∂φ
+∂t =
+1
+N0 · N(ι) (Bζφ · N0) · D[ζ]φ − 1
+2|Bζφ|2 − H(ι),
+together with the conservation laws (1.6)-(1.8). Here we refer ω0 > 0 as the fundamental frequency. The
+linearization of this system at the equilibrium (ζ, φ) = (0, 0) is
+(2.5)
+Lω0
+�
+ζ
+φ
+�
+:=
+�
+ω0∂t
+−D[0]
+−∆ − 2
+ω0∂t
+� �
+ζ
+φ
+�
+= 0,
+Π(0)ζ = Π(1)ζ = Π(1)φ = 0.
+We restrict to rotationally symmetric solutions of the system: that is, water droplets which are always
+rotationally symmetric with a fixed axis. In addition, we require ζ to be even in t and φ to be odd in t. The
+
+LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL NOTES
+9
+solution thus should take the form
+ζ(t, x) =
+�
+j,n≥0
+ζjn cos(jt)Yn(x),
+φ(t, x) =
+�
+j≥1,n≥0
+φjn sin(jt)Yn(x),
+where Yn is the n’th zonal spherical harmonic, i.e. the (unique) normalized spherical harmonic of degree n
+that is axially symmetric. In spherical coordinates this means that Yn(θ, ϕ) = Pn(cos θ), where Pn is the
+n’th Legendre polynomial. Since φ0n are irrelevant we fix them to be 0. Then
+Lω0
+�
+ζ
+φ
+�
+=
+�
+j,n≥0
+�
+(−ω0jζjn − nφjn) sin(jt)Yn(x)
+((n − 1)(n + 2)ζjn + ω0jφjn) cos(jt)Yn(x)
+�
+.
+In order that (ζ, φ)T ∈ KerLω0, at the level n = 0, we must have ζj0 = φj0 = 0 for all j ≥ 0. At the level
+n = 1, we have ζ01 = φ01 = 0, and for j ≥ 1 there holds ω0jζj1 − φj1 = 0 and ω0jφj1 = 0, so ζj1 = φj1 = 0
+for all j ≥ 0. Hence ζjn, φjn can be nonzero only for j ≥ 1 and n ≥ 2.
+Consequently, Lω0 has a one-dimensional kernel if and only if the Diophantine equation
+(2.6)
+ω2
+0j2 = n(n − 1)(n + 2),
+j ≥ 1, n ≥ 2
+has exactly one solution (j0, n0). If ω0 has this property, then at the linear level, the lowest frequency of
+oscillation is
+ω0j0 =
+�
+n0(n0 − 1)(n0 + 2).
+We look into this Diophantine equation. Obviously ω2
+0 has to be a rational number. The equation is
+closely related to a family of elliptic curves over Q:
+Ec : y2 = x(x − c)(x + 2c) = x3 + c2x2 − 2c2x,
+c ∈ N.
+If we set a/b = ω2
+0 (irreducible fraction), then integral solutions of (2.6) are in 1-1 correspondence with
+integral points with natural number coordinates on the elliptic curve Eab, under the following map:
+(j0, n0) → (abn0, a2bj0) ∈ Eab.
+Thus we just need to find natural numbers a, b such that there is a unique (up to negation) integral point
+(x, y) ∈ Eab, where x > 0 is divided by ab, and y is divided by a2b. We seek for n0 as small as possible with
+such property, which gives lowest frequency of oscillation as small as possible. For ab = 1, · · · , 50, we find
+that if ab = 15, then the integral points on elliptic curve E15 (up to negation of Mordel-Weil group) are
+(−30, 0), (−5, 50), (0, 0), (15, 0), (24, 108), (90, 900).
+The only point (x, y) with ab|x and ab2|y is (90,900), which gives n0 = 6, and the lowest frequency Λ(n0) =
+�
+n0(n0 − 1)(n0 + 2) of oscillation is 4
+√
+15 ≃ 15.49 · · · .
+There are other choices of a, b. We list down the value of ab below 50, the corresponding n0 and the lowest
+frequency Λ(n0):
+ab
+n0
+Λ(n0)
+15
+6
+4
+√
+15
+17
+49
+4
+√
+323
+22
+9
+6
+√
+22
+26
+50
+15
+√
+78
+42
+7
+3
+√
+42
+46
+576
+2040
+√
+46
+50
+25
+90
+√
+2
+
+10
+CHENGYANG SHAO
+See Appendix A for the MAGMA code used to find these values. This list suggests that n0 = 6 might
+be the smallest order that meets the requirement, but this remains unproved. We summarize these into the
+following number-theoretic question:
+Question 2. For the family of elliptic curves
+Eab : y2 = x(x − ab)(x + 2ab),
+a, b ∈ N,
+how many choices of a, b ∈ N are there such that, there is exactly one integral point (x, y) ∈ Eab with x, y > 0
+and ab|x, a2b|y? For such a, b and integral point (x, y), is the minimal value of x/(ab) exactly 6, or is it
+smaller?
+Of course, a complete answer of Question 2 should imply very clear understanding of periodic solutions
+of the spherical capillary water waves equation constructed using bifurcation analysis. But at this moment
+we are satisfied with existence, so we may pick any ω0 = a/b and (j0, n0) that meets the requirement, for
+example the simplest case n0 = 6, and any of the following choices of ω0 and j0:
+ω0 =
+√
+15, j0 = 4;
+ω0 =
+�
+1
+15, j0 = 60;
+ω0 =
+�
+3
+5, j0 = 20;
+ω0 =
+�
+5
+3, j0 = 12.
+We thus refine our conjecture as follows:
+Conjecture 2.2. Let (n0, j0) be a pair of natural numbers with n0 ≥ 2, j0 ≥ 1, and set ω0 =
+�
+Λ(n0)/j0.
+Suppose that the only natural number solution of the Diophantine equation
+ω2
+0j0 = Λ(n0)2 = n0(n0 − 1)(n0 + 1)
+is (j0, n0). Then there is a Cantor set with positive measure of parameters ω, clustered near ω0, such that
+the spherical capillary water waves equation (2.4) admits small amplitude periodic solution with frequency ω.
+The counterpart of Conjecture 2.2 for gravity capillary standing water waves was proved by Alazard-
+Baldi [1] using a Nash-Moser type theorem. The key technique in their proof was to find a conjugation of
+the linearized operator of the gravity capillary water waves system on T1 to an operator of the form
+ω∂t + iT + iλ1|Dx|1/2 + iλ−1|Dx|−1/2 + Operator of order ≤ −3
+2,
+where T is an elliptic Fourier multiplier of order 3/2, and λ1, λ−1 are real constants. The frequency ω lives
+in a Cantor type set that clusters around a given frequency so that the kernel of the linearized operator
+is 1-dimensional. With this conjugation, they were able to find periodic solutions of linearized problems
+required by Nash-Moser iteration.
+It is expected that this technique could be transplanted to the equation (2.4), since our analysis for (2.6)
+suggests that the 1-dimensional kernel requirement for bifurcation analysis is met. It seems that the greatest
+technical issue is to find a suitable conjugation that takes the linearized operator of (2.4) to an operator of
+the form
+ω∂t + i(T3/2 + T1/2 + T−1/2) + Operator of order ≤ −3
+2,
+where each Tk is a real Fourier multiplier acting on spherical harmonics. The difficulty is that, since we are
+working with pseudo-differential operators on S2, the formulas of symbolic calculus are not as neat as those
+on flat spaces. It seems necessary to implement some global harmonic analysis for compact homogeneous
+spaces, e.g. extension of results collected in Ruzhansky-Turunen [31]. Unfortunately, those results do not
+include pseudo-differential operators with “rough coefficients” and para-differential operators, so it seems
+necessary to re-write the whole theory.
+
+LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL NOTES
+11
+2.4. Number-Theoretic Obstruction with Normal Form Reduction. As pointed out in Section 1,
+the system (2.3) is locally well-posed due to a result in [9]. General well-posedness results [14], [36] for free
+boundary value problem of Euler equation also apply. As for lifespan estimate for initial data ε-close to
+the static solution (2.1), it should not be hard to conclude that the lifespan should be bounded below by
+1/ε. The result relates to the fact that the sphere is a stable critical point of the area functional, cf. [5].
+This is nothing new: a suitable energy inequality should imply it. However, although the clue is clear, the
+implementation is far from standard since we are working on a compact manifold. A rigorous proof still calls
+for hard technicalities.
+Now we will be looking into the nonlinear equation (2.3) for its longer time behavior. Although appearing
+similar to the well-studied water waves equation in e.g. [26], [27], [39], [40], there is a crucial difference between
+the dispersive relation in (2.3) and the well-studied water waves equations: the dispersive relation exhibits
+a strong rigidity property, i.e. the arbitrary physical constants enter into the dispersive relation Λ only as
+scaling factors. For the gravity-capillary water waves, the linear dispersive relation reads
+�
+g|∇| + σ|∇|3,
+where g is the gravitational constant and σ is the surface tension coefficient. For the Klein-Gordon equation
+on a Riemannian manifold, the linear dispersive relation reads
+�
+−∆ + m2,
+where m is the mass. In [26], Ionescu and Pusateri referred such dispersive relations as having non-degenerate
+dependence on physical parameter, while in our context it is appropriate to refer to the dependence as
+degenerate. We will see that this crucial difference brings about severe obstructions for the long-time well-
+posedness of the system.
+Following the idea of Delort and Szeftel [16], we look for a normal form reduction of (2.3) and explain
+why the rigidity property could cause obstructions. Not surprisingly, the obstruction is due to resonances,
+and strongly relates to the solvablity of a Diophantine equation. Delort and Szeftel cast a normal form
+reduction to the small-initial-data problem of quasilinear Klein-Gordon equation on the sphere and obtained
+an estimate on the lifespan longer than the one provided by standard energy method. After their work, normal
+form reduction has been used by mathematicians to understand water waves on flat tori, for example [6], [7]
+and [26]. The idea was inspired by the normal form reduction method introduced by Shatah [35]: for a
+quadratic perturbation of a linear dispersive equation
+∂tu + iLu = N(u) = O(u⊗2),
+using a new variable u + B(u, ¯u) with a suitably chosen quadratic addendum B(u, ¯u) can possibly eliminate
+the quadratic part of N, thus extending the lifespan estimate beyond the standard 1/ε.
+So we shall write the quadraticr part of N(u) as
+�
+n3≥0
+Π(n3)
+�
+� �
+n1≥0
+�
+n2≥0
+M1
+�
+Π(n1)u, Π(n2)u
+�
++ M2
+�
+Π(n1)u, Π(n2)¯u
+�
++ M3
+�
+Π(n1)¯u, Π(n2)¯u
+�
+�
+� ,
+where M1, M2, M3 are complex bi-linear operators, following the argument of Section 4 in [16]. They are
+independent of t since the right-hand-side of the equation does not depend on t explicitly. Let’s look for a
+diffeomorphism
+u → v := u + B[u, u]
+
+12
+CHENGYANG SHAO
+in the function space C∞(S2), where B is a bilinear operator, so that the equation (2.3) with quadratic
+nonlinearity reduces to an equation with cubic nonlinearity.
+The B[u, u] is supposed to take the form
+B[u, u] = B1[u, u] + B2[u, u] + B3[u, u], with
+B1[u, u] =
+�
+n3≥0
+�
+n1,n2≥0
+b1(n1, n2, n3)Π(n3)M1
+�
+Π(n1)u, Π(n2)u
+�
+,
+B2[u, u] =
+�
+n3≥0
+�
+n1,n2≥0
+b2(n1, n2, n3)Π(n3)M2
+�
+Π(n1)u, Π(n2)¯u
+�
+,
+B3[u, u] =
+�
+n3≥0
+�
+n1,n2≥0
+b3(n1, n2, n3)Π(n3)M3
+�
+Π(n1)¯u, Π(n2)¯u
+�
+,
+where the bj(n1, n2, n3)’s are complex numbers to be determined. Implementing (2.3), we find
+(∂t + iΛ)(u + B[u, u])
+= N(u) +
+�
+n3≥0
+�
+n1,n2≥0
+b1(n1, n2, n3)Π(n3)M1
+�
+Π(n1)∂tu, Π(n2)u
+�
++
+�
+n3≥0
+�
+n1,n2≥0
+b1(n1, n2, n3)Π(n3)M1
+�
+Π(n1)u, Π(n2)∂tu
+�
++ (similar terms)
++
+�
+n3≥0
+�
+n1,n2≥0
+iΛ(n3)b1(n1, n2, n3)Π(n3)M1
+�
+Π(n1)u, Π(n2)u
+�
+= N(u) +
+�
+n3≥0
+�
+min(n1,n2)≤1
+i [Λ(n3) − Λ(n1) − Λ(n2)] b1(n1, n2, n3)Π(n3)M1
+�
+Π(n1)u, Π(n2)u
+�
++
+�
+n3≥0
+�
+n1,n2≥2
+i [Λ(n3) − Λ(n1) − Λ(n2)] b1(n1, n2, n3)Π(n3)M1
+�
+Π(n1)u, Π(n2)u
+�
++ (similar terms) + O(u⊗3).
+We aim to eliminate most of the second order portions of N(u). The coefficients bj(n1, n2, n3) are fixed as
+follows:
+(2.7)
+b1(n1, n2, n3) = i [Λ(n3) − Λ(n1) − Λ(n2)]−1 ,
+n1, n2, n3 ≥ 2
+b2(n1, n2, n3) = i [Λ(n3) − Λ(n1) + Λ(n2)]−1 ,
+n1, n2, n3 ≥ 2
+b3(n1, n2, n3) = i [Λ(n3) + Λ(n1) + Λ(n2)]−1 ,
+n1, n2, n3 ≥ 2
+b1,2,3(n1, n2, n3) = 0,
+if Λ(n3) ± Λ(n1) ± Λ(n2) = 0 or min(n1, n2, n3) ≤ 1,
+then a large portion of the second order part of Π≥2N (u) will be eliminated. In fact, for n1, n2, n3 ≥ 2, if
+Λ(n3) ± Λ(n1) ± Λ(n2) ̸= 0, then the term
+Π(n3)
+�
+� �
+n1,n2≥2
+M1
+�
+Π(n1)u, Π(n2)u
+�
++ M2
+�
+Π(n1)u, Π(n2)¯u
+�
++ M3
+�
+Π(n1)¯u, Π(n2)¯u
+�
+�
+�
+is cancelled out. On the other hand, by the volume preservation equality (1.6) and conservation law (1.7),
+there holds Π(0)u = O(u⊗2), Π(1)u = O(u⊗2), so the low-low interaction
+Π≥2
+�
+�
+�
+min(n1,n2)≤1
+M1
+�
+Π(n1)u, Π(n2)u
+�
++ M2
+�
+Π(n1)u, Π(n2)¯u
+�
++ M3
+�
+Π(n1)¯u, Π(n2)¯u
+�
+�
+�
+is automatically O(u⊗3).
+
+LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL NOTES
+13
+Thus the existence and continuity of the normal form B[u, u] depends on the property of the 3-way
+resonance equation
+(2.8)
+Λ(n3) − Λ(n1) − Λ(n2) = 0,
+n1, n2, n3 ≥ 2.
+which is equivalent to the Diophantine equation
+(2.9)
+[F(n1) + F(n2) − F(n3)]2 − 4F(n1)F(n2) = 0,
+n1, n2, n3 ≥ 2,
+where F(X) = X(X − 1)(X + 2).
+If the tuple (n1, n2, n3) is non-resonant, i.e. it is such that b1,2,3(n1, n2, n3) ̸= 0, then some elementary
+number theoretic argument will give a lower bound on |b1,2,3(n1, n2, n3)| in terms of a negative power (can
+be fixed as −9/2) of n1, n2, n3. This is usually referred as small divisor estimate.
+To study the distribution of resonant frequencies, we propose the following unsolved question:
+Question 3. Does the Diophantine equation (2.9) have finitely many solutions?
+However, the Diophantine equation (2.9) does admit non-trivial solutions (5, 5, 8) and (10, 10, 16). In other
+words, the second order terms e.g.
+Π(8)M1(Π(5)u · Π(5)u),
+Π(5)M1(Π(8)u · Π(5)¯u),
+in the quadratic part of N(u) cannot be eliminated by normal form reduction. On the other hand, it seems
+to be very hard to determine whether (2.9) still admits any other solution. We have the following proposition
+(the author would like to thank Professor Bjorn Poonen for the proof):
+Proposition 2.3. The Diophantine equation (2.9) has no solution with n1 ≤ 104 other than (5, 5, 8) and
+(10, 10, 16).
+The proof of this proposition is computer-aided. The key point is to use the so-called Runge’s method to
+show that if (n1, n2, n3) is a solution, then there must hold n2 = O(n2
+1). For a given n1, this reduces the proof
+to numerical verification for finitely many possibilities. The algorithm can of course be further optimized,
+but due to some algebraic geometric considerations, it is reasonable to conjecture that the solution of (2.9)
+should be very rare. In fact, there are two ways of viewing the problem. We observe that if (n1, n2, n3) is
+a solution, then F(n1)F(n2) must be a square, and the square free part of F(n1), F(n2), F(n3) must be the
+same. Further reduction turns the problem into finding integral points on a family of elliptic curves
+Y 2 = cF(X),
+c is square-free,
+which is of course difficult, but since Siegel’s theorem asserts that there are only finitely many integer points
+on an elliptic curve over Q, it is reasonable to conjecture that there are not “too many” solutions to (2.9).
+We may also view the problem as finding integral (rational) points on a given algebraic surface. The complex
+projective surface corresponding to (2.9) is given by
+V : [X(X − W)(X + 2W) + Y (Y − W)(Y + 2W) − Z(Z − W)(Z + 2W)]2
+= 4X(X − W)(X + 2W)Y (Y − W)(Y + 2W),
+where [X, Y, Z, W] is the homogeneous coordinate on CP3. With the aid of computer, we obtain
+Proposition 2.4. The complex projective surface V ⊂ CP3 has Kodaira dimension 2 (i.e. it is of general
+type under Kodaira-Enriquez classification), and its first Betti number is 0.
+See Appendix A for the code.
+
+14
+CHENGYANG SHAO
+The first part of the proposition suggests that the rational points of V should be localized on finitely
+many algebraic curves laying on V; nevertheless, this seemingly simple suggestion is indeed a special case of
+the Bomberi-Lang conjecture, a hard problem in number theory (its planar case is known as the celebrated
+Faltings’s theorem). The second part suggests that the rational points of V should be rare since its Albanese
+variety is a single point. But these are just heuristics that solutions to (2.9) should be rare. In general,
+determining the solvability of a given Diophantine equation is very difficult 2, as number theorists and
+arithmetic geometers generally believe.
+The reason that such issues do not occur for water waves in the flat setting or nonlinear Klein-Gordon
+equations is twofold. First of all, the resonance equation is easily understood even in the degenerate case in
+the flat setting. For example, the capillary water waves without gravity on T2 has dispersive relation |∇|1/2,
+and the 3-way resonance equation is
+4�
+k2
+1 + k2
+2 +
+4�
+l2
+1 + l2
+2 =
+4�
+m2
+1 + m2
+2,
+with the additional requirement m = k + l. We already know that the resonance equation has no non-trivial
+solution at all, cf. [7]. But even without using m = k +l, we would be able to conclude that there are at most
+finitely many non-trivial solutions from the celebrated Faltings’s theorem on rational points on high-genus
+algebraic projective curves (although this is like using a sledge hammer to crack a nut). Secondly, for the non-
+degenrate case, for example the gravity-capillary waves, the dispersive relation reads
+�
+g|∇| + σ|∇|3, so if the
+ratio σ/g is a transcedental number then the 3-way resonance equation has no solution. Furthermore, using
+some elementary calculus and a measure-theoretic argument, it can be shown, not without technicalities, that
+the resonances admit certain small-divisor estimates for almost all parameters. This is exactly the argument
+employed by Delort-Szeftle [16], Berti-Delort [6] and Ionescu-Pusateri [26], so that their results were stated for
+almost all parameters. These parameters are, roughly speaking, badly approximated by algebraic numbers.
+However, the resonance equation (2.8) is inhomogeneous and allows no arbitrary physical parameter at
+all. Furthermore, since product of spherical harmonics are no longer spherical harmonics in general, Fourier
+series techniques employed by [7] [6] [26] that works for the torus are never valid for S2; for example, we
+cannot simply assume n3 = n1 + n2 in (2.8), as already illustrated by the solutions (5,5,8) (10,10,16). These
+are the crucial differences between the capillary spherical water waves and all known results for water waves
+in the flat setting.
+2.5. Heuristics for Lifespan Estimate. To summarize, almost global lifespan estimate of (2.3) depends
+on the difficult number theoretic question 3. Before it is fully resolved, we can only expect partial results
+regarding the normal form transformation.
+If there are only finitely many solutions to the Diophantine equation (2.9), then under the normal form
+reduction u → v = u + B[u, u] with coefficients given by (2.7), the equation (2.3) is transformed into the
+following system:
+∂
+∂tΠcv = O(v⊗2),
+∂
+∂t(1 − Πc)v = O(v⊗3),
+where Πc is the orthogonal projection to �
+n3 E(n3) ⊂ L2(S2), with n3 being either 0 or 1, or exhausting the
+third component of all nontrivial solutions of (2.9).
+2For example, the seemingly simple Diophantine equation x3 + y3 + z3
+=
+42 is in fact a puzzle of more than 60
+years, and its first solution was found recently by Booker-Sutherland [11].
+It is of extremely large magnitude:
+42 =
+(−80 538 738 812 075 974)3 + 80 435 758 145 817 5153 + 12 602 123 297 335 6313.
+Another example is the equation of
+same type x3 + y3 + z3 = 3. Beyond the easily found solutions (1, 1, 1), (4, 4, -5), (4, -5, 4), (-5, 4, 4), the next solution reads
+(569 936 821 221 962 380 720, −569 936 821 113 563 493 509, −472 715 493 453 327 032).
+
+LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL NOTES
+15
+There is no reasonable assertion to be made if the conjecture fails. However, if the conjecture does hold
+true, then we can expect that the lifespan estimate for ε-Cauchy data of (EQ) goes beyond ε−1, as what we
+expect for gravity water waves in the periodic setting, e.g. in [26]:
+Conjecture 2.3. If the Diophantine equation (2.9) has only finitely many solutions, then there is some
+α > 0 such that for ε-Cauchy data of (EQ), the lifespan goes beyond ε−(1+α) as ε → 0.
+Let’s explain the heuristic as follows. The argument we aim to implement is the standard “continuous
+induction method”, i.e. for some suitably large s and K and suitable α > 0, assuming T = ε−(1+α) and
+supt∈[0,T ] ∥v∥Hs(g0) ≤ Kε, we try to prove a better bound supt∈[0,T ] ∥v∥Hs(g0) ≤ Kε/2.
+Here g0 is the
+standard metric on S2. It is intuitive to expect such a result for the cubic equation ∂t(1−Πc)v = O(v⊗3). As
+for the quadratic equation ∂tΠcv = O(v⊗2), it is crucial to implement the conservation of energy H[ζ, φ] ≡ 4π
+for a solution. We summarize it as
+Proposition 2.5. Fix T > 0. Let u be a smooth solution of (2.3) and v = u + B[u, u] be as above. Suppose
+for some suitably large s and K, there holds
+sup
+t∈[0,T ]
+∥v∥Hs(g0) ≤ Kε
+with ε sufficiently small. Then there is in fact a better bound for the low frequency part Πcv:
+sup
+t∈[0,T ]
+∥Πcv∥Hs(g0) ≤ Kε/4.
+Proof. We consider the “approximate” energy functional
+H0[ζ, φ] = 4π +
+�
+S2 2ζ · dµ0 + 1
+2
+�
+S2(|∇0ζ|2 + 2|ζ|2)dµ0 + 1
+2
+�
+S2
+���|∇1/2
+0
+|φ
+���
+2
+dµ0,
+where µ0 is the standard area measure on S2 and ∇0 is the standard connection on S2. This is nothing but
+the quadratic approximation to H[ζ, φ] in (1.8) at (0, 0), so there holds H0[ζ, φ] = H[ζ, φ] + O(u⊗3). Using
+volume preservation (1.6), we obtain
+�
+S2 ζdµ0 = −∥ζ∥2
+L2(g0) + O(ζ⊗3), so summarizing we have
+(2.10)
+1
+2
+�
+S2(|∇0ζ|2 − 2|ζ|2)dµ0 + 1
+2
+�
+S2
+���|∇1/2
+0
+|φ
+���
+2
+dµ0 = O(u⊗3).
+Note that we used the conservation law H[ζ, φ] ≡ 4π. By spectral calculus on S2, we have
+∥ζ∥2
+H1(g0) ≃ ∥Π(0)ζ∥2
+L2(g0) + ∥Π(1)ζ∥2
+L2(g0) +
+�
+S2(|∇0ζ|2 − 2|ζ|2)dµ0,
+and by volume preservation and conservation of momentum (1.7) we find
+(2.11)
+∥ζ∥2
+H1(g0) ≃
+�
+S2(|∇0ζ|2 − 2|ζ|2)dµ0 + O(ζ⊗4).
+Now, for some N0 > 0 relating to the loss of regularity caused by B, we may choose s >> 2N0. Then if
+∥v∥Hs(g0) ≤ Kε, it follows that ∥u∥Hs−N0(g0) ≤ K′ε, so by (2.10) and (2.11) we have
+∥u∥2
+L2(g0) ≃ ∥ζ∥2
+H1(g0) + ∥φ∥2
+H1/2(g0) ≤ K′ε3.
+Thus
+∥v∥L2(g0) ≤ C∥u∥L2(g0) + C∥u∥2
+HN0(g0)
+≤ Cε3/2 + K′ε2.
+Since the spectrum of Πcv is bounded, by Bernstein type inequality we have
+∥Πcv∥Hs(g0) ≤ C∥v∥L2(g0) ≤ K′ε3/2(1 + ε1/2).
+
+16
+CHENGYANG SHAO
+If ε is sufficiently small then this implies ∥Πcv∥Hs(g0) ≤ Kε/4.
+□
+We point out that the above proof is independent from the magnitude of the lifespan T, so it is always
+applicable as long as the cubic equation ∂t(1 − Πc)v = O(v⊗3) is well-understood. There are two crucial
+points in the proof of Proposition 2.5: the conservation of energy, and that the projection Πc is of finite
+rank, so that a Bernstein type inequality holds. The last fact holds only if there are finitely many 3-way
+resonances, i.e. there are only finitely many solutions to the Diophantine equation (2.8).
+Finally, we propose an even more ambitious conjecture concerning global dynamical properties of spherical
+water droplets, which is again illuminated by observation in hydrodynamical experiments under zero gravity,
+and also suggested by the results of Berti-Montalto [8]:
+Conjecture 2.4. If the Diophantine equation (2.9) has only finitely many solutions, then a KAM type result
+holds for (EQ): there is a family of infinitely many quasi-periodic solutions of (EQ), depending on a parameter
+which takes value in a Cantor-type set.
+Appendix A. MAGMA Code
+MAGMA is a large, well-supported software package designed for computations in algebra, number theory,
+algebraic geometry, and algebraic combinatorics. In this appendix, we give the MAGMA code used to conduct
+computations on Diophantine equations related to the spherical capillary water waves system.
+A.1. Integral Points on Elliptic Curve. We can find all integral points on a given elliptic curve over
+Q using MAGMA. For a monic cubic polynomial f(x), the function EllipticCurve(f) creates the elliptic
+curve
+E : y2 = f(x),
+and the function IntegralPoints(E) returns a sequence containing all the integral points on E under the
+homogeneous coordinate of QP2, modulo negation. We use this to find out all integral points on the elliptic
+curve
+Ec : y2 = x3 + cx2 − 2c2x = x(x − c)(x + 2c).
+for natural number c ≤ 50. The MAGMA code is listed below, which excludes all the c’s such that there are
+only trivial integral points {(−c, 0), (0, 0), (c, 0)} on Ec.
+> Qx<x> := PolynomialRing(Rationals());
+> for c in [1..50] do
+> E := EllipticCurve(x^3+c*x^2-2*c^2*x);
+> S, reps := IntegralPoints(E);
+> if # S gt 3 then
+> print c, E;
+> print S;
+> end if;
+> end for;
+2 Elliptic Curve defined by y^2 = x^3 + 2*x^2 - 8*x over Rational Field
+[ (-4 : 0 : 1), (-2 : 4 : 1), (-1 : -3 : 1), (0 : 0 : 1), (2 : 0 : 1), (4 : 8 : 1), (8 : -24
+: 1), (50 : 360 : 1) ]
+8 Elliptic Curve defined by y^2 = x^3 + 8*x^2 - 128*x over Rational Field
+[ (-16 : 0 : 1), (-8 : 32 : 1), (-4 : -24 : 1), (0 : 0 : 1), (8 : 0 : 1), (9 : -15 : 1), (16
+: 64 : 1), (32 : -192 : 1), (200 : 2880 : 1) ]
+
+LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL NOTES
+17
+13 Elliptic Curve defined by y^2 = x^3 + 13*x^2 - 338*x over Rational Field
+[ (-26 : 0 : 1), (0 : 0 : 1), (13 : 0 : 1), (121 : 1386 : 1) ]
+15 Elliptic Curve defined by y^2 = x^3 + 15*x^2 - 450*x over Rational Field
+[ (-30 : 0 : 1), (-5 : -50 : 1), (0 : 0 : 1), (15 : 0 : 1), (24 : 108 : 1), (90 : -900 : 1)
+]
+17 Elliptic Curve defined by y^2 = x^3 + 17*x^2 - 578*x over Rational Field
+[ (-34 : 0 : 1), (-32 : -56 : 1), (0 : 0 : 1), (17 : 0 : 1), (833 : 24276 : 1) ]
+18 Elliptic Curve defined by y^2 = x^3 + 18*x^2 - 648*x over Rational Field
+[ (-36 : 0 : 1), (-32 : -80 : 1), (-18 : 108 : 1), (-9 : -81 : 1), (0 : 0 : 1), (18 : 0 : 1),
+(36 : 216 : 1), (72 : -648 : 1), (450 : 9720 : 1) ]
+22 Elliptic Curve defined by y^2 = x^3 + 22*x^2 - 968*x over Rational Field
+[ (-44 : 0 : 1), (-32 : -144 : 1), (0 : 0 : 1), (22 : 0 : 1), (198 : 2904 : 1) ]
+23 Elliptic Curve defined by y^2 = x^3 + 23*x^2 - 1058*x over Rational Field
+[ (-46 : 0 : 1), (0 : 0 : 1), (23 : 0 : 1), (50 : -360 : 1) ]
+26 Elliptic Curve defined by y^2 = x^3 + 26*x^2 - 1352*x over Rational Field
+[ (-52 : 0 : 1), (-49 : -105 : 1), (0 : 0 : 1), (26 : 0 : 1), (1300 : 47320 : 1) ]
+30 Elliptic Curve defined by y^2 = x^3 + 30*x^2 - 1800*x over Rational Field
+[ (-60 : 0 : 1), (-50 : 200 : 1), (-45 : -225 : 1), (-24 : -216 : 1), (-20 : 200 : 1), (-6
+: 108 : 1), (0 : 0 : 1), (30 : 0 : 1), (36 : 144 : 1), (40 : -200 :
+1), (75 : -675 : 1), (90 : 900 : 1), (300 : 5400 : 1), (324 : -6048 : 1), (480 : -10800 : 1),
+(7290 : 623700 : 1), (10830 : -1128600 : 1), (226875 : 108070875 : 1) ]
+32 Elliptic Curve defined by y^2 = x^3 + 32*x^2 - 2048*x over Rational Field
+[ (-64 : 0 : 1), (-32 : 256 : 1), (-16 : -192 : 1), (0 : 0 : 1), (32 : 0 : 1), (36 : -120 :
+1), (64 : 512 : 1), (128 : -1536 : 1), (800 : 23040 : 1) ]
+33 Elliptic Curve defined by y^2 = x^3 + 33*x^2 - 2178*x over Rational Field
+[ (-66 : 0 : 1), (0 : 0 : 1), (33 : 0 : 1), (81 : -756 : 1) ]
+35 Elliptic Curve defined by y^2 = x^3 + 35*x^2 - 2450*x over Rational Field
+[ (-70 : 0 : 1), (-49 : 294 : 1), (-45 : -300 : 1), (-40 : 300 : 1), (-14 : -196 : 1), (0 :
+0 : 1), (35 : 0 : 1), (50 : 300 : 1), (175 : -2450 : 1), (224 : 3528 : 1), (280 : -4900 : 1),
+(4410 : -294000 : 1), (14450 : -1739100 : 1) ]
+39 Elliptic Curve defined by y^2 = x^3 + 39*x^2 - 3042*x over Rational Field
+[ (-78 : 0 : 1), (0 : 0 : 1), (39 : 0 : 1), (147 : -1890 : 1) ]
+42 Elliptic Curve defined by y^2 = x^3 + 42*x^2 - 3528*x over Rational Field
+[ (-84 : 0 : 1), (-56 : -392 : 1), (-12 : 216 : 1), (0 : 0 : 1), (42 : 0 : 1), (63 : -441 :
+1), (294 : 5292 : 1) ]
+43 Elliptic Curve defined by y^2 = x^3 + 43*x^2 - 3698*x over Rational Field
+[ (-86 : 0 : 1), (-32 : -360 : 1), (0 : 0 : 1), (43 : 0 : 1) ]
+46 Elliptic Curve defined by y^2 = x^3 + 46*x^2 - 4232*x over Rational Field
+[ (-92 : 0 : 1), (0 : 0 : 1), (46 : 0 : 1), (26496 : 4316640 : 1) ]
+50 Elliptic Curve defined by y^2 = x^3 + 50*x^2 - 5000*x over Rational Field
+[ (-100 : 0 : 1), (-50 : 500 : 1), (-25 : -375 : 1), (-4 : 144 : 1), (0 : 0 : 1), (50 : 0 :
+1), (100 : 1000 : 1), (200 : -3000 : 1), (1250 : 45000 : 1) ]
+Running Magma V2.27-7.
+Seed: 821911319; Total time: 2.430 seconds; Total memory usage: 85.16MB.
+
+18
+CHENGYANG SHAO
+A.2. Classification of Projective Surface. Some basic geometric parameters of the complex projective
+surface
+V : [X(X − W)(X + 2W) + Y (Y − W)(Y + 2W) − Z(Z − W)(Z + 2W)]2
+= 4X(X − W)(X + 2W)Y (Y − W)(Y + 2W)
+in CP3 can be computed using MAGMA. The author would like to thank Professor Bjorn Poonen for intro-
+ducing MAGMA and providing the code listed below.
+> Q:=Rationals();
+> P<x,y,z,t>:=ProjectiveSpace(Q,3);
+> fx:=x*(x-t)*(x+2*t);
+> fy:=y*(y-t)*(y+2*t);
+> fz:=z*(z-t)*(z+2*t);
+> V:=Surface(P,(fz-fx-fy)^2-4*fx*fy);
+> KodairaEnriquesType(V);
+2 0 General type
+Running Magma V2.27-7.
+Seed: 989287753; Total time: 0.650 seconds; Total memory usage: 32.09MB.
+Note that the Kodaira dimension is invariant regardless of the choice of base field, so it is legitimate to
+choose the base field to be Q in the above code. The variable t is used to homogenize the equation. The
+function KodairaEnriquesType(V) returns three values for the given projective surface V : the first is the
+Kodaira dimension, the second is irrelevant when the Kodaira dimension is not −∞, 1 or 0, and the third is
+the Kodaira-Enriquez classification of the surface X.
+References
+[1] Alazard, T., & Baldi, P. (2015). Gravity capillary standing water waves. Archive for Rational Mechanics and Analysis,
+217(3), 741-830.
+[2] Alazard, T., & M´etivier, G. (2009). Paralinearization of the Dirichlet to Neumann operator, and regularity of three-
+dimensional water waves. Communications in Partial Differential Equations, 34(12), 1632-1704.
+[3] Alazard, T., Burq, N., & Zuily, C.. (2011). On the water-wave equations with surface tension. Duke Mathematical Journal,
+158(3), 413-499.
+[4] Alazard, D. & Delort, J. (2015). Global solutions and asymptotic behavior for two dimensional gravity water waves. Ann.
+Sci. ´Ec. Norm. Sup´er., 48, (2015), no. 5, 1149-1238.
+[5] Barbosa, J. L., & do Carmo, M. (2012). Stability of hypersurfaces with constant mean curvature. In Manfredo P. do
+Carmo–Selected Papers (pp. 221-235). Springer, Berlin, Heidelberg.
+[6] Berti, M., & Delort, J. M. (2018). Almost global solutions of capillary-gravity water waves equations on the circle. Springer
+International Publishing.
+[7] Berti M, Feola R, Pusateri F., Birkhoff normal form and long time existence for periodic gravity water waves. arXiv preprint,
+arXiv:1810.11549. 2018 Oct 26.
+[8] Berti, M., & Montalto, R. (2020). Quasi-periodic standing wave solutions of gravity-capillary water waves (Vol. 263, No.
+1273). American mathematical society.
+[9] Beyer, K., & G¨unther, M. (1998). On the Cauchy problem for a capillary drop. Part I: irrotational motion. Mathematical
+methods in the applied sciences, 21(12), 1149-1183.
+[10] Bombieri, E.,& Pila, J. (1989). The number of integral points on arcs and ovals. Duke Mathematical Journal, 59(2), 337–357.
+[11] Booker, A. R., & Sutherland, A. V. (2021). On a question of Mordell. Proceedings of the National Academy of Sciences,
+118(11).
+[12] Burq N, G´erard P, & Tzvetkov N. Strichartz inequalities and the nonlinear Schr¨odinger equation on compact manifolds.
+American Journal of Mathematics. 2004;126(3):569-605.
+[13] Christianson, H., Hur, V.M. & Staffilani, G. (2010). Strichartz estimates for the water-wave problem with surface tension.
+Communications in Partial Differential Equations, 35(12), pp.2195-2252.
+
+LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL NOTES
+19
+[14] Coutand, D., & Shkoller, S. (2007). Well-posedness of the free-surface incompressible Euler equations with or without
+surface tension. Journal of the American Mathematical Society, 20(3), 829-930.
+[15] Deng, Y., Ionescu, A. D., Pausader, B., & Pusateri, F. (2017). Global solutions of the gravity-capillary water-wave system
+in three dimensions. Acta Mathematica, 219(2), 213-402.
+[16] Delort, J.-M., & Szeftel, J. (2004). Long-time existence for small data nonlinear klein-gordon equations on tori and spheres.
+International Mathematics Research Notices (37), 1897-1966.
+[17] Delort, J. M., & Imekraz, R. (2017). Long-time existence for the semilinear Klein–Gordon equation on a compact boundary-
+less Riemannian manifold. Communications in Partial Differential Equations, 42(3), 388-416.
+[18] Germain, P., Masmoudi, N., & Shatah, J. (2012). Global solutions for the gravity water waves equation in dimension 3.
+Annals of Mathematics, 175: 691–754, 2012.
+[19] Germain, P., Masmoudi, N., & Shatah, J. (2015). Global solutions for capillary waves equation in dimension 3. Comm.
+Pure Appl. Math., 68(4): 625–687, 2015
+[20] H¨ormander, L. (1988). Pseudo-differential operaors of type 1, 1. Communications in partial differential equations, 13(9),
+1085-1111.
+[21] H¨ormander, L. (1997). Lectures on nonlinear hyperbolic differential equations (Vol. 26). Springer Science & Business Media.
+[22] Hunter, J. K., Ifrim, M., & Tataru, D. (2016). Two dimensional water waves in holomorphic coordinates. Communications
+in Mathematical Physics, 346(2), 483-552.
+[23] Ifrim, M., & Tataru, D. (2014). Two dimensional water waves in holomorphic coordinates II: global solutions. arXiv preprint
+arXiv:1404.7583.
+[24] Ionescu, A. D., & Pusateri, F. (2015). Global solutions for the gravity water waves system in 2d. Inventiones mathematicae,
+199(3), 653-804.
+[25] Ionescu, A. D., & Pusateri, F. (2018). Recent advances on the global regularity for irrotational water waves. Philosophical
+Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 376(2111), 20170089.
+[26] Ionescu, A. D., & Pusateri, F. (2019). Long-time existence for multi-dimensional periodic water waves. Geometric and
+Functional Analysis, 29(3), 811-870.
+[27] Lannes, D. (2005). Well-posedness of the water-waves equations. Journal of the American Mathematical Society, 18(3),
+605-654.
+[28] Lannes, D. (2013). The water waves problem. Mathematical analysis and asymptotics. Mathematical Surveys and Mono-
+graphs, Vol. 188. American Mathematical Society, Providence, RI, 2013.
+[29] Magnanini, R., & Poggesi, G. (2019). On the stability for Alexandrov’s Soap Bubble theorem. Journal d’Analyse
+Math´ematique, 139(1), 179-205.
+[30] Ming M, & Zhang Z. Well-posedness of the water-wave problem with surface tension. Journal de math´ematiques pures et
+appliqu´ees. 2009 Nov 1;92(5):429-55.
+[31] Ruzhansky, M., & Turunen, V. (2009). Pseudo-differential operators and symmetries: background analysis and advanced
+topics (Vol. 2). Springer Science & Business Media.
+[32] Schweizer, B. (2005). On the three-dimensional Euler equations with a free boundary subject to surface tension. Annales
+de l’institut Henri Poincar´e C, Analyse non lin´eaire, 22(6), 753-781.
+[33] Sogge, C. D. (1988). Concerning the Lp norm of spectral clusters for second-order elliptic operators on compact manifolds.
+Journal of functional analysis, 77(1), 123-138.
+[34] Shao, C. (2020). Long Time Behavior of a Quasilinear Hyperbolic System Modelling Elastic Membranes. https://arxiv.
+org/abs/2010.10663.
+[35] Shatah, J. (1985) Normal forms and quadratic Klein-Gordon equations, Comm. Pure Appl. Math. 38, 685-696.
+[36] Shatah, J., & Zeng, C. (2008). Geometry and a priori estimates for free boundary problems of the Euler’s equation.
+Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences,
+61(5), 698-744.
+[37] Taylor, M. (2013). Partial differential equations II: Qualitative studies of linear equations (Vol. 116). Springer Science &
+Business Media.
+[38] Wang, X. (2020) Global regularity for the 3d finite depth capillary water waves. Annales Scientifiques de l’´Ecole Normale
+Sup´erieure, 53(4): 847–943, 2020.
+[39] Wu, S. (1997). Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Inventiones mathematicae, 130(1),
+39-72.
+[40] Wu, S. (1999). Well-posedness in Sobolev spaces of the full water wave problem in 3-D. Journal of the American Mathe-
+matical Society, 12(2), 445-495.
+
+20
+CHENGYANG SHAO
+[41] Wu, S. (2009). Almost global wellposedness of the 2-D full water wave problem. Inventiones mathematicae, 177(1), pp.45-135.
+[42] Wu, S. (2011). Global wellposedness of the 3-D full water wave problem. Inventiones mathematicae, 184(1), 125-220.
+[43] Zakharov, V. E. (1968). Stability of periodic waves of finite amplitude on the surface of a deep fluid. Journal of Applied
+Mechanics and Technical Physics, 9(2), 190-194.
+
diff --git a/LtAyT4oBgHgl3EQfT_ca/content/tmp_files/load_file.txt b/LtAyT4oBgHgl3EQfT_ca/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..a63891a2a55d51c8f1bed5c74b060f5064bf8904
--- /dev/null
+++ b/LtAyT4oBgHgl3EQfT_ca/content/tmp_files/load_file.txt
@@ -0,0 +1,895 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf,len=894
+page_content='LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL  NOTES  CHENGYANG SHAO  In this note, we propose several unsolved problems concerning the irrotational oscillation of a water  droplet under zero gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' We will derive the governing equation of this physical model, and convert it  to a quasilinear dispersive partial differential equation defined on the sphere, which formally resembles the  capillary water waves equation but describes oscillation defined on curved manifold instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Three types of  unsolved mathematical problems related to this model will be discussed in observation of hydrodynamical  experiments under zero gravity1: (1) Strichartz type inequalities for the linearized problem (2) existence of  periodic solutons (3) normal form reduction and generic lifespan estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' It is pointed out that all of these  problems are closely related to certain Diophantine equations, especially the third one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Capillary Spherical Water Waves Equation: Derivation  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Water Waves Equation for a Bounded Water Drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Comparing to gravity water waves problems,  the governing equation for a spherical droplet of water under zero gravity takes a very different form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' At  a first glance it looks similar to the water waves systems as mentioned above, but some crucial differences  do arise after careful analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' To the author’s knowledge, besides those dealing with generic free-boundary  Euler equation ( [14], [36]), the only reference on this problem is Beyer-G¨unther [9], in which the local  well-posedness of the equation is proved using a Nash-Moser type implicit function theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' We will briefly  discribe known results for gravity water waves problems in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  To start with, let us pose the following assumptions on the fluid motion that we try to describe:  (A1) The perfect, irrotational fluid of constant density ρ0 occupies a smooth, compact region in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  (A2) There is no gravity or any other external force in presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  (A3) The air-fluid interface is governed by the Young-Laplace law, and the effect of air flow is  neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  We assume that the boundary of the fluid region has the topological type of a smooth compact orientable  surface M, and is described by a time-dependent embedding ι(t, ·) : M → R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' We will denote a point on M  by x, the image of M under ι(t, ·) by Mt, and the region enclosed by Mt by Ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The outer normal will be  denoted by N(ι).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' We also write ¯∇ for the flat connection on R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Adopting assumption (A3), we have the Young-Laplace equation:  σ0H(ι) = pi − pe,  where H(ι) is the (scalar) mean curvature of the embedding, σ0 is the surface tension coefficient (which is  assumed to be a constant), and pi, pe are respectively the inner and exterior air pressure at the boundary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  they are scalar functions on the boundary and we assume that pe is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Under assumptions (A1) and  (A2), we obtain Bernoulli’s equation, sometimes referred as the pressure balance condition, on the evolving  1There are numbers of visual materials on such experiments conducted by astronauts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' See for example https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  com/watch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='v=H_qPWZbxFl8&t or https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='com/watch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='v=e6Faq1AmISI&t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='00115v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='AP]  31 Dec 2022  2  CHENGYANG SHAO  surface:  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='1)  ∂Φ  ∂t  ����  Mt  + 1  2| ¯∇Φ|Mt|2 − pe = −σ0  ρ0  H(ι),  where Φ is the velocity potential of the velocity field of the air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Note that Φ is determined up to a function in  t, so we shall leave the constant pe around for convenience reason that will be explained shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' According  to assumption (A1), the function Φ is a harmonic function within the region Ωt, so it is uniquely determined  by its boundary value, and the velocity field within Ωt is ¯∇Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The kinematic equation on the free boundary  Mt is naturally obtained as  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='2)  ∂ι  ∂t · N(ι) = ¯∇Φ|Mt · N(ι).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Finally, we would like to discuss the conservation laws for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The preservation of volume Vol(Ωt) =  Vol(Ω0) is a consequence of incompressibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The system describes an Eulerian flow without any external  force, so the center of mass moves at a uniform speed along a fixed direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='3)  1  Vol(Ω0)  �  Ωt  PdVol(P) = V0t + C0,  with Vol being the Lebesgue measure, P marking points in R3, V0 and C0 being the velocity and starting  position of center of mass respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Furthermore, the total momentum is conserved, and since the flow is  a potential incompressible one, the conservation of total momentum is expressed as  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='4)  �  Mt  ρ0ΦN(ι)dArea(Mt) ≡ ρ0Vol(Ω0)V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Most importantly, it is not surprising that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='2) is a Hamilton system (the Zakharov formulation for  water waves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' see Zakharov [43]), with Hamiltonian  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='5)  σ0Area(ι) + 1  2  �  Ωt  ρ0| ¯∇Φ|2dVol = σ0Area(Mt) + 1  2  �  Mt  ρ0Φ|Mt  � ¯∇Φ|Mt · N(ι)  �  dArea,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' potential proportional to surface area plus kinetic energy of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Converting to a Differential System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' It is not hard to verify that the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='2) is invariant  if ι is composed with a diffeomorphism of M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' we may thus regard it as a geometric flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' If we are only  interested in perturbation near a given configuration, we may reduce system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='2) to a non-degenerate  dispersive differential system concerning two scalar functions defined on M, just as Beyer and G¨unther did  in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' In fact, during a short time of evolution, the interface can be represented as the graph of a function  defined on the initial surface: if ι0 : M → R3 is a fixed embedding close to the initial embedding ι(0, x), we  may assume that ι(t, x) = ι0(x) + ζ(t, x)N0(x), where ζ is a scalar “height” function defined on M0 and N0  is the outer normal vector field of M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  With this observation, we shall transform the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='1)&(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='2) into a non-local system of two real scalar  functions (ζ, φ) defined on M, where ζ is the “height” function described as above, and φ(t, x) = Φ(t, ι(t, x))  is the boundary value of the velocity potential, pulled back to the underlying manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  The operator  Bζ : φ → ¯∇Φ|Mt  maps the pulled-back Dirichlet boundary value φ to the boundary value of the gradient of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' We shall write  (D[ζ]φ)N(ι)  LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL NOTES  3  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The shape of the surface  for its normal part, where D[ζ] is the Dirichlet-Neumann operator corresponding to the region enclosed by  the image of ι0 + ζN0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Thus  ∂ζ  ∂t N0 · N(ι) = D[ζ]φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  We also need to calculate the restriction of ∂tΦ on Mt in terms of φ and ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' By the chain rule,  ∂Φ  ∂t  ����  Mt  = ∂φ  ∂t − ¯∇Φ|Mt · ∂ι  ∂t  = ∂φ  ∂t −  � ¯∇Φ|Mt · N0  � ∂ζ  ∂t  = ∂φ  ∂t −  1  N0 · N(ι)  � ¯∇Φ|Mt · N0  �  D[ζ]φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  We thus arrive at the following nonlinear system:  (EQ(M))  �  �  �  �  �  �  �  ∂ζ  ∂t =  1  N0 · N(ι)D[ζ]φ,  ∂φ  ∂t =  1  N0 · N(ι) (Bζφ · N0) · D[ζ]φ − 1  2|Bζφ|2 − σ0  ρ0  H(ι) + pe,  where ι = ι0 + ζN0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  We may obtain an explicit expression of Bζφ = ¯∇Φ|Mt in terms of φ (together with the  connection ∇0 on the fixed embedding ι0(M)), just as standard references did for Euclidean or periodic  water waves, but that is not necessary for our discussion at the moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' It is important to keep in mind  that the preservation of volume and conservation of total momentum (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='3)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='4) convert to integral equalities  of (ζ, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' These additional restrictions are not obvious from the differential equations (EQ(M)), though they  can be deduced from (EQ(M)) since they are just rephrase of the original physical laws (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  For M = S2, the case that we shall discuss in detail, we refer to the system as the capillary spherical water  waves equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' To simplify our discussion, we shall be working under the center of mass frame, and require  the eigenmode Π(0)φ to vanish for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' This could be easily accomplished by absorbing the eigenmode  into φ since the equation is invariant by a shift of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' In a word, from now on, we will be focusing on the  NS+ 0  1  1  1 1 y4  CHENGYANG SHAO  non-dimensional capillary spherical water waves equation  (EQ)  �  �  �  �  �  �  �  ∂ζ  ∂t =  1  N0 · N(ι)D[ζ]φ,  ∂φ  ∂t =  1  N0 · N(ι) (Bζφ · N0) · D[ζ]φ − 1  2|Bζφ|2 − H(ι),  where ι = (1 + ζ)ι0, and Π(0)φ ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' We assume that the total volume of the fluid is 4π/3, so that the  preservation of volume is expressed as  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='6)  1  3  �  S2(1 + ζ)3dµ0 ≡ 4π  3 ,  where µ0 is the standard measure on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The inertial movement of center of mass (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='3) and conservation of  total momentum (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='4) under our center of mass frame are expressed respectively as  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='7)  �  S2(1 + ζ)4N0dµ0 = 0,  �  S2 φN(ι)dµ(ι) = 0,  where µ(ι) is the induced surface measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Further, the Hamiltonian of the system is  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='8)  H[ζ, φ] = Area(ι) + 1  2  �  S2 φ · D[ζ]φ · dµ(ι),  and for a solution (ζ, φ) there holds H[ζ, φ] ≡ 4π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Up to this point, we are still working within the realm of well-established frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' We already know  that the general free-boundary Euler equation is locally well-posed due to the work of [14] or [36], and due  to the curl equation the curl free condition persists during the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' On the other hand, the Cauchy  problem of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='2) is known to be locally well-posed, due to Beyer and G¨unther in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  They used an iteration argument very similar to a Nash-Moser type argument in the sense that it involves  multiple scales of Banach spaces and “tame” maps in a certain sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Finally, it is not hard to transplant  the potential-theoretic argument of Wu [40] to prove the local well-posedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  To sum up, we already know that the system (EQ(M)) for a compact orientable surface M (hence (EQ)  specifically) is locally well-posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' But this is all we can assert for the motion of a water droplet under zero  gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' In the following part of this note, we will propose several questions and conjectures concerning the  long-time behaviour of water droplets under zero gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Previous Works on Water Waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' There has already been several different approaches to describe  the motion of perfect fluid with free boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' One is to consider the motion of perfect fluid occupying an  arbitrary domain in R2 or R3, either with or without surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The motion is described by a free  boundary value problem of Euler equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' This generic approach was employed by Countand-Shkoller [14]  and Shatah-Zeng [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Both groups proved the local-wellposedness of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' This approach has the  advantage of being very general, applicable to all geometric shapes of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  On the other hand, when coming to potential flows of a perfect fluid, the curl free property results in dis-  persive nature of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The motion of a curl free perfect fluid under gravity and a free boundary value  condition is usually referred to as the gravity water waves problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The first breakthroughs in understanding  local well-posedness were works of Wu [39] [40], who proved local well-posedness of the gravity water waves  equation without any smallness assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Lannes [27] extended this to more generic bottom shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Taking  surface tension into account, the problem becomes gravity-capillary water waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Schweizer [32] proved local  well-posedness with small Cauchy data of the gravity-capillary water waves problem, and Ming-Zhang [30]  proved local well-posedness without smallness assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Alazard-Metevier [2] and Alazard-Burq-Zuily [3]  LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL NOTES  5  used para-differential calculus to obtain the optimal regularity for local well-posedness of the water waves  equation, either with or without surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  For discussion of long time behavior, it is important to take into account the dispersive nature of the  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' For linear dispersive properties, there has been work of Christianson-Hur-Staffilani [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' For gravity  water waves living in R2, works on lifespan estimate include Wu [41] and Hunter-Ifrim-Tataru [22] (almost  global result), Ionescu-Pusateri [24] and Alazard-Delort [4] and Ifrim-Tataru [23] (global result).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' For gravity  water waves living in R3, there are works of Germain-Masmoudi-Shatah [18] and Wu [42] (no surface tension),  Germain-Masmoudi-Shatah [19] (no gravity), Deng-Ionescu-Pausader-Pusateri [15] (gravity-capillary water  waves) and Wang [38] (gravity-capillary water waves with finite depth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' These results all employed different  forms of decay estimates derived from dispersive properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  As for long time behavior of periodic water waves, Berti-Delort [6] considered gravity-capillary water waves  defined on T1, Berti-Feola-Pusateri [7] considered gravity water waves defined on T1, Ionescu-Pusateri [26]  considered gravity-capillary water waves defined on T2, and obtained an estimate on the lifespan beyond  standard energy method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' All of the three groups used para-differential calculus and suitable normal form  reduction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' the results of Berti-Delort and Ionescu-Pusateri were proved for physical data of full Lebesgue  measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  To sum up, all results on the gravity water waves problem listed above are concerned with an equation for  two scalar functions defined on a fixed flat manifold, being one of the following: R1, R2, T1, T2, sometimes  called the “bottom” of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' These two functions represents the geometry of the liquid-gas interface and  the boundary value of the velocity potential, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The manifold itself is considered as the bottom  of the container in which all dynamics are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' We observe that the differential equation (EQ(M)) is,  mathematically, fundamentally different from the water waves equations that have been well-studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Initial Notes on Unsolved Problems  In this section, we propose unsolved problems related to the spherical capillary water waves system  (EQ(M)) with M = S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Not surprisingly, these problems all have deep backgrounds in number theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Linearization Around the Static Solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' We are mostly interested in the stability of the static  solution of (EQ(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' A static solution should be a fluid region whose shape stays still, with motion being  a mere shift within the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  In this case, we have pe = 0 since the reference is relatively static with  respect to the air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Moreover, the velocity field ¯∇Φ and “potential of acceleration” ∂Φ/∂t must both be  spatially uniform, so that the left-hand-side of the pressure balance condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='1) is a function of t alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  It follows that ι(M) is always a compact embedded surface of constant mean curvature, hence in fact always  an Euclidean sphere by the Alexandrov sphere theorem (see [29]), and we may just take M = S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Moreover,  since ι(S2) should enclose a constant volume by incompressibility, the radius of that sphere does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  After suitable scaling, we may assume that the radius is always 1, and ρ0 = 1, σ0 = 1, to make (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='2)  non-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Finally, by choosing the center of mass frame, we may simply assume that the spatial shift  is always zero, so that the velocity potential Φ ≡ Φ0, a real constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' It is harmless to fix it to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Thus, under our convention, a static solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='2) takes the form  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='1)  �  ι(t, x)  Φ(t, x)  �  =  �  ι0(x)  0  �  ,  where a ∈ R3 is a constant vector, and ι0 is the standard embedding of S2 as ∂B(0, 1) ⊂ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Equivalently,  this means that a static solution of (EQ(M)) under our convention must be (ζ, φ) = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Note here that  the Gauss map of ι0 coincides with itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  6  CHENGYANG SHAO  We can now start our perturbation analysis around a static solution at the linear level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Let E(n) be the  space of spherical harmonics of order n, normalized according to the standard surface measure on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' In  particular, E(1) is spanned by three components of N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Let Π(n) be the orthogonal projection on L2(S2)  onto E(n), Π≤n be the orthogonal projection on L2(S2) onto �  k≤n E(k), Π≥n be the orthogonal projection  on L2(S2) onto �  k≥n E(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' For ι = (1 + ζ)ι0, the linearization of −H(ι) around the sphere ζ ≡ 0 is ∆ζ + 2ζ,  where ∆ is the Laplacian on the sphere S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' the standard formula for the second variation of area in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Then H′(ι0) acts on E(n) as the multiplier −(n − 1)(n + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Note that even if we consider the dimensional  form of (EQ), there will only be an additional scaling factor σ0/(ρ0R2), where R is the radius of the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  On the other hand, the following solution formula for the Dirichlet problem on B(0, 1) is well-known: if  f ∈ L2(S2), then the harmonic function in B(0, 1) with Dirichlet boundary value f is determined by  u(r, ω) =  �  n≥0  rn(Π(n)f)(ω),  where (r, ω) is the spherical coordinate in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Thus the Dirichlet-Neumann operator D[ι0] acts on E(n) as the  multiplier n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Note again that even if we consider the dimensional form (EQ), there will only be an additional  scaling factor R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Thus, setting  u = Π(0)ζ + Π(1)ζ +  �  n≥2  �  (n − 1)(n + 2) · Π(n)ζ + i  �  n≥1  √n · Π(n)φ,  we find that the linearization of (EQ) around the static solution (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='1) is a linear dispersive equation  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='2)  ∂u  ∂t + iΛu = 0,  where the 3/2-order elliptic operator Λ is given by a multiplier  Λ =  �  n≥2  �  n(n − 1)(n + 2) · Π(n) =:  �  n≥0  Λ(n)Π(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Note that (ζ, φ) is completely determined by u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' At the linear level, there must hold Π(0)u ≡ 0 because the  first variation of volume must be zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' and Π(1)u ≡ 0 because of the conservation laws (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Let us also re-write the original nonlinear system (EQ) into a form that better illustrates its perturbative  nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' For simplicity, we use O(u⊗k) to abbreviate a quantity that can be controlled by k-linear expressions  in u, and disregard its continuity properties for the moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' For example, ∥u∥2  H1 + ∥u∥4  H2 is an expression  of order O(u⊗2) when u → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Since the operator Λ acts degenerately on E(0) ⊕ E(1), we should be more careful about the eigenmodes  Π(0)u and Π(1)u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The volume preservation equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='6) implies ∂tΠ(0)ζ = O(u⊗2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Projecting (EQ) to  E(1), which is spanned by the components of N0, we obtain ∂tΠ(1)ζ = Π(1)φ + O(u⊗2) = O(u⊗2) since the  conservation law (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='7) implies Π(1)φ = O(u⊗2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' and ∂tΠ(1)φ = O(u⊗2) since H′(ι0) = −∆ − 2 annihilates  E(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' We can thus formally re-write the nonlinear system (EQ) as the following:  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='3)  ∂u  ∂t + iΛu = N(u),  with N(u) = O(u⊗2) vanishing quadratically as u → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Note that we are disregarding all regularity problems  at the moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Question at Linear Level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' At the linear level, our first unanswered question is  LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL NOTES  7  Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Does the solution of the linear capillary spherical water waves equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='2) satisfy a Strichartz  type estimate of the form  ∥eitΛf∥Lp  T Lq  x ≲T ∥f∥Hs,  where Lp  T Lq  x = Lp([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Lq(S2)), and the admissible indices (p, q) and s should be determined?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Answer to Question 1 should be important in understanding the dispersive nature of linear capillary  spherical water waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' For Schr¨odinger equation on a compact manifold, a widly cited result was obtained  by Burq-G´erard-Tzvetkov [12]:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' On a general compact Riemannian manifold (M d, g), there holds  ∥eit∆gf∥Lp  t Lq  x([0,T ]×M) ≲T ∥f∥H1/p(M),  where  2  p + d  q = d  2,  p > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  The authors used a time-localization argument for the parametrix of ∂t − i∆g to prove this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' For the  sphere Sd, this inequality is not optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The authors further used a Bourgain space argument to obtain the  optimal Strichartz inequality:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Let (Sd, g) be the standard n-dimensional sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' For a function f ∈ C∞(Sd), there holds  the Strichartz inequality  ∥eit∆gf∥Lp  t Lq  x([0,T ]×M) ≲T ∥f∥Hs(M),  s > s0(d)  where  s0(2) = 1  8,  s0(d) = d  4 − 1  2, d ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Furthermore these inequalities are optimal in the sense that the Sobolev index s cannot be less than or equal  to s0(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  The proof is the consequence of two propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The first one is the “decoupling inequality on compact  manifolds”, in particular the following result proved by Sogge [33]:  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Let Πk be the spectral projection to eigenspaces with eigenvalues in [k2, (k + 1)2] on Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Then there holds  ∥Πk∥L2→Lq ≤ Cqns(q),  where  s(q) =  �  d−1  2  �  1  2 − 1  2q  �  ,  2 ≤ q ≤ 2(d+1)  d−1  d−1  2  − d  q ,  2(d+1)  d−1  ≤ q ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  These estimates are sharp in the following sense: if hk is a zonal spherical harmonic function of degree k on  Sd, then as k → ∞,  ∥hk∥Lq ≃ Cqks(q)∥hk∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  The second one is a Bourgain space embedding result:  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' For a function f ∈ C∞  0 (R × Sd), define the Bourgain space norm  ∥f(t, x)∥Xs,b :=  ��⟨∂t + i∆g⟩bf(t, x)  ��  L2  t Hsx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Then for b > 1/2 and s > s0(d), there holds  ∥f∥L4(R×Sd) ≤ Cs,b∥f∥Xs,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  8  CHENGYANG SHAO  The key ingredient for proving this proposition is the following number-theoretic result:  #{(p, q) ∈ N2 : p2 + q2 = A} = O(Aε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  As for optimality of the Strichartz inequality, the authors of [12] implemented standard results of Gauss  sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  The parametrix and Bourgain space argument can be repeated without essential change for the linear  capillary spherical water waves equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='2), but this time the Bourgain space argument would be more  complicated: the Bourgain space norm is now  ∥f(t, x)∥Xs,b :=  ��⟨∂t + iΛ⟩bf(t, x)  ��  L2  t Hsx ,  and the embedding result becomes  ∥f∥L4(R×S2) ≤ Cs,b∥f∥Xs,b  for f ∈ C∞  0 (R × S2) and all s > 3ρ/8 + 1/8, b > 1/2, where ρ is the infimum of all exponents ρ′ such that  when A → ∞, the number  #  �  (n1, n2) ∈ N2 : 1  2 ≤ n2  n1  ≤ 2, |Λ(n1) + Λ(n2) − A| ≤ 1  2  �  ≤ Cρ′Aρ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Some basic analytic number theory implies ρ = 1/3, and thus the range of s is s > 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Surprisingly this  index is not better than that predicted by the parametrix method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' It remains unknown whether this index  could be further optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  To close this subsection, we note that the capillary spherical water wave lives on a compact region, so the  dispersion does not take away energy from a locality to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' This is a crucial difference between waves  on compact regions and waves in Euclidean spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' In particular, we do not expect decay estimate for eitΛf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  For the nonlinear problem (EQ), techniques like vector field method (Klainerman-Sobolev type inequalities)  do not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Rotationally Symmetric Solutions: Bifurcation Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Illuminated by observations in hydro-  dynamical experiments under zero gravity, and suggested by the existence of standing gravity capillary water  waves due to Alazard-Baldi [1], we propose the following conjecture:  Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' There is a Cantor family of small amplitude periodic solutions to the spherical capillary  water waves system (EQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Let us conduct the bifurcation analysis that suggests why this conjecture should be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' By time rescaling,  we aim to find solution (ζ, φ, ω0) of the following system that is 2π-peiodic in t:  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='4)  �  �  �  �  �  �  �  ω0  ∂ζ  ∂t =  1  N0 · N(ι)D[ζ]φ,  ω0  ∂φ  ∂t =  1  N0 · N(ι) (Bζφ · N0) · D[ζ]φ − 1  2|Bζφ|2 − H(ι),  together with the conservation laws (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='6)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Here we refer ω0 > 0 as the fundamental frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The  linearization of this system at the equilibrium (ζ, φ) = (0, 0) is  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='5)  Lω0  �  ζ  φ  �  :=  �  ω0∂t  −D[0]  −∆ − 2  ω0∂t  � �  ζ  φ  �  = 0,  Π(0)ζ = Π(1)ζ = Π(1)φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  We restrict to rotationally symmetric solutions of the system: that is, water droplets which are always  rotationally symmetric with a fixed axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' In addition, we require ζ to be even in t and φ to be odd in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The  LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL NOTES  9  solution thus should take the form  ζ(t, x) =  �  j,n≥0  ζjn cos(jt)Yn(x),  φ(t, x) =  �  j≥1,n≥0  φjn sin(jt)Yn(x),  where Yn is the n’th zonal spherical harmonic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' the (unique) normalized spherical harmonic of degree n  that is axially symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' In spherical coordinates this means that Yn(θ, ϕ) = Pn(cos θ), where Pn is the  n’th Legendre polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Since φ0n are irrelevant we fix them to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Then  Lω0  �  ζ  φ  �  =  �  j,n≥0  �  (−ω0jζjn − nφjn) sin(jt)Yn(x)  ((n − 1)(n + 2)ζjn + ω0jφjn) cos(jt)Yn(x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  In order that (ζ, φ)T ∈ KerLω0, at the level n = 0, we must have ζj0 = φj0 = 0 for all j ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' At the level  n = 1, we have ζ01 = φ01 = 0, and for j ≥ 1 there holds ω0jζj1 − φj1 = 0 and ω0jφj1 = 0, so ζj1 = φj1 = 0  for all j ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Hence ζjn, φjn can be nonzero only for j ≥ 1 and n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Consequently, Lω0 has a one-dimensional kernel if and only if the Diophantine equation  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='6)  ω2  0j2 = n(n − 1)(n + 2),  j ≥ 1, n ≥ 2  has exactly one solution (j0, n0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' If ω0 has this property, then at the linear level, the lowest frequency of  oscillation is  ω0j0 =  �  n0(n0 − 1)(n0 + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  We look into this Diophantine equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Obviously ω2  0 has to be a rational number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The equation is  closely related to a family of elliptic curves over Q:  Ec : y2 = x(x − c)(x + 2c) = x3 + c2x2 − 2c2x,  c ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  If we set a/b = ω2  0 (irreducible fraction), then integral solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='6) are in 1-1 correspondence with  integral points with natural number coordinates on the elliptic curve Eab, under the following map:  (j0, n0) → (abn0, a2bj0) ∈ Eab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Thus we just need to find natural numbers a, b such that there is a unique (up to negation) integral point  (x, y) ∈ Eab, where x > 0 is divided by ab, and y is divided by a2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' We seek for n0 as small as possible with  such property, which gives lowest frequency of oscillation as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' For ab = 1, · · · , 50, we find  that if ab = 15, then the integral points on elliptic curve E15 (up to negation of Mordel-Weil group) are  (−30, 0), (−5, 50), (0, 0), (15, 0), (24, 108), (90, 900).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  The only point (x, y) with ab|x and ab2|y is (90,900), which gives n0 = 6, and the lowest frequency Λ(n0) =  �  n0(n0 − 1)(n0 + 2) of oscillation is 4  √  15 ≃ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='49 · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  There are other choices of a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' We list down the value of ab below 50, the corresponding n0 and the lowest  frequency Λ(n0):  ab  n0  Λ(n0)  15  6  4  √  15  17  49  4  √  323  22  9  6  √  22  26  50  15  √  78  42  7  3  √  42  46  576  2040  √  46  50  25  90  √  2  10  CHENGYANG SHAO  See Appendix A for the MAGMA code used to find these values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' This list suggests that n0 = 6 might  be the smallest order that meets the requirement, but this remains unproved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' We summarize these into the  following number-theoretic question:  Question 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' For the family of elliptic curves  Eab : y2 = x(x − ab)(x + 2ab),  a, b ∈ N,  how many choices of a, b ∈ N are there such that, there is exactly one integral point (x, y) ∈ Eab with x, y > 0  and ab|x, a2b|y?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' For such a, b and integral point (x, y), is the minimal value of x/(ab) exactly 6, or is it  smaller?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Of course, a complete answer of Question 2 should imply very clear understanding of periodic solutions  of the spherical capillary water waves equation constructed using bifurcation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' But at this moment  we are satisfied with existence, so we may pick any ω0 = a/b and (j0, n0) that meets the requirement, for  example the simplest case n0 = 6, and any of the following choices of ω0 and j0:  ω0 =  √  15, j0 = 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  ω0 =  �  1  15, j0 = 60;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  ω0 =  �  3  5, j0 = 20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  ω0 =  �  5  3, j0 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  We thus refine our conjecture as follows:  Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Let (n0, j0) be a pair of natural numbers with n0 ≥ 2, j0 ≥ 1, and set ω0 =  �  Λ(n0)/j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Suppose that the only natural number solution of the Diophantine equation  ω2  0j0 = Λ(n0)2 = n0(n0 − 1)(n0 + 1)  is (j0, n0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Then there is a Cantor set with positive measure of parameters ω, clustered near ω0, such that  the spherical capillary water waves equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='4) admits small amplitude periodic solution with frequency ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  The counterpart of Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='2 for gravity capillary standing water waves was proved by Alazard-  Baldi [1] using a Nash-Moser type theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The key technique in their proof was to find a conjugation of  the linearized operator of the gravity capillary water waves system on T1 to an operator of the form  ω∂t + iT + iλ1|Dx|1/2 + iλ−1|Dx|−1/2 + Operator of order ≤ −3  2,  where T is an elliptic Fourier multiplier of order 3/2, and λ1, λ−1 are real constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The frequency ω lives  in a Cantor type set that clusters around a given frequency so that the kernel of the linearized operator  is 1-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' With this conjugation, they were able to find periodic solutions of linearized problems  required by Nash-Moser iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  It is expected that this technique could be transplanted to the equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='4), since our analysis for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='6)  suggests that the 1-dimensional kernel requirement for bifurcation analysis is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' It seems that the greatest  technical issue is to find a suitable conjugation that takes the linearized operator of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='4) to an operator of  the form  ω∂t + i(T3/2 + T1/2 + T−1/2) + Operator of order ≤ −3  2,  where each Tk is a real Fourier multiplier acting on spherical harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The difficulty is that, since we are  working with pseudo-differential operators on S2, the formulas of symbolic calculus are not as neat as those  on flat spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' It seems necessary to implement some global harmonic analysis for compact homogeneous  spaces, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' extension of results collected in Ruzhansky-Turunen [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Unfortunately, those results do not  include pseudo-differential operators with “rough coefficients” and para-differential operators, so it seems  necessary to re-write the whole theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL NOTES  11  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Number-Theoretic Obstruction with Normal Form Reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' As pointed out in Section 1,  the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='3) is locally well-posed due to a result in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' General well-posedness results [14], [36] for free  boundary value problem of Euler equation also apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' As for lifespan estimate for initial data ε-close to  the static solution (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='1), it should not be hard to conclude that the lifespan should be bounded below by  1/ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The result relates to the fact that the sphere is a stable critical point of the area functional, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  This is nothing new: a suitable energy inequality should imply it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' However, although the clue is clear, the  implementation is far from standard since we are working on a compact manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' A rigorous proof still calls  for hard technicalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Now we will be looking into the nonlinear equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='3) for its longer time behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Although appearing  similar to the well-studied water waves equation in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' [26], [27], [39], [40], there is a crucial difference between  the dispersive relation in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='3) and the well-studied water waves equations: the dispersive relation exhibits  a strong rigidity property, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' the arbitrary physical constants enter into the dispersive relation Λ only as  scaling factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' For the gravity-capillary water waves, the linear dispersive relation reads  �  g|∇| + σ|∇|3,  where g is the gravitational constant and σ is the surface tension coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' For the Klein-Gordon equation  on a Riemannian manifold, the linear dispersive relation reads  �  −∆ + m2,  where m is the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' In [26], Ionescu and Pusateri referred such dispersive relations as having non-degenerate  dependence on physical parameter, while in our context it is appropriate to refer to the dependence as  degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' We will see that this crucial difference brings about severe obstructions for the long-time well-  posedness of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Following the idea of Delort and Szeftel [16], we look for a normal form reduction of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='3) and explain  why the rigidity property could cause obstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Not surprisingly, the obstruction is due to resonances,  and strongly relates to the solvablity of a Diophantine equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Delort and Szeftel cast a normal form  reduction to the small-initial-data problem of quasilinear Klein-Gordon equation on the sphere and obtained  an estimate on the lifespan longer than the one provided by standard energy method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' After their work, normal  form reduction has been used by mathematicians to understand water waves on flat tori, for example [6], [7]  and [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The idea was inspired by the normal form reduction method introduced by Shatah [35]: for a  quadratic perturbation of a linear dispersive equation  ∂tu + iLu = N(u) = O(u⊗2),  using a new variable u + B(u, ¯u) with a suitably chosen quadratic addendum B(u, ¯u) can possibly eliminate  the quadratic part of N, thus extending the lifespan estimate beyond the standard 1/ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  So we shall write the quadraticr part of N(u) as  �  n3≥0  Π(n3)  �  � �  n1≥0  �  n2≥0  M1  �  Π(n1)u, Π(n2)u  �  + M2  �  Π(n1)u, Π(n2)¯u  �  + M3  �  Π(n1)¯u, Π(n2)¯u  �  �  � ,  where M1, M2, M3 are complex bi-linear operators, following the argument of Section 4 in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' They are  independent of t since the right-hand-side of the equation does not depend on t explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Let’s look for a  diffeomorphism  u → v := u + B[u, u]  12  CHENGYANG SHAO  in the function space C∞(S2), where B is a bilinear operator, so that the equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='3) with quadratic  nonlinearity reduces to an equation with cubic nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  The B[u, u] is supposed to take the form  B[u, u] = B1[u, u] + B2[u, u] + B3[u, u], with  B1[u, u] =  �  n3≥0  �  n1,n2≥0  b1(n1, n2, n3)Π(n3)M1  �  Π(n1)u, Π(n2)u  �  ,  B2[u, u] =  �  n3≥0  �  n1,n2≥0  b2(n1, n2, n3)Π(n3)M2  �  Π(n1)u, Π(n2)¯u  �  ,  B3[u, u] =  �  n3≥0  �  n1,n2≥0  b3(n1, n2, n3)Π(n3)M3  �  Π(n1)¯u, Π(n2)¯u  �  ,  where the bj(n1, n2, n3)’s are complex numbers to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Implementing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' we find  (∂t + iΛ)(u + B[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' u])  = N(u) +  �  n3≥0  �  n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='n2≥0  b1(n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' n3)Π(n3)M1  �  Π(n1)∂tu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Π(n2)u  �  +  �  n3≥0  �  n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='n2≥0  b1(n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' n3)Π(n3)M1  �  Π(n1)u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Π(n2)∂tu  �  + (similar terms)  +  �  n3≥0  �  n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='n2≥0  iΛ(n3)b1(n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' n3)Π(n3)M1  �  Π(n1)u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Π(n2)u  �  = N(u) +  �  n3≥0  �  min(n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='n2)≤1  i [Λ(n3) − Λ(n1) − Λ(n2)] b1(n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' n3)Π(n3)M1  �  Π(n1)u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Π(n2)u  �  +  �  n3≥0  �  n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='n2≥2  i [Λ(n3) − Λ(n1) − Λ(n2)] b1(n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' n3)Π(n3)M1  �  Π(n1)u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Π(n2)u  �  + (similar terms) + O(u⊗3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  We aim to eliminate most of the second order portions of N(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The coefficients bj(n1, n2, n3) are fixed as  follows:  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='7)  b1(n1, n2, n3) = i [Λ(n3) − Λ(n1) − Λ(n2)]−1 ,  n1, n2, n3 ≥ 2  b2(n1, n2, n3) = i [Λ(n3) − Λ(n1) + Λ(n2)]−1 ,  n1, n2, n3 ≥ 2  b3(n1, n2, n3) = i [Λ(n3) + Λ(n1) + Λ(n2)]−1 ,  n1, n2, n3 ≥ 2  b1,2,3(n1, n2, n3) = 0,  if Λ(n3) ± Λ(n1) ± Λ(n2) = 0 or min(n1, n2, n3) ≤ 1,  then a large portion of the second order part of Π≥2N (u) will be eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' In fact, for n1, n2, n3 ≥ 2, if  Λ(n3) ± Λ(n1) ± Λ(n2) ̸= 0, then the term  Π(n3)  �  � �  n1,n2≥2  M1  �  Π(n1)u, Π(n2)u  �  + M2  �  Π(n1)u, Π(n2)¯u  �  + M3  �  Π(n1)¯u, Π(n2)¯u  �  �  �  is cancelled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' On the other hand, by the volume preservation equality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='6) and conservation law (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='7),  there holds Π(0)u = O(u⊗2), Π(1)u = O(u⊗2), so the low-low interaction  Π≥2  �  �  �  min(n1,n2)≤1  M1  �  Π(n1)u, Π(n2)u  �  + M2  �  Π(n1)u, Π(n2)¯u  �  + M3  �  Π(n1)¯u, Π(n2)¯u  �  �  �  is automatically O(u⊗3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL NOTES  13  Thus the existence and continuity of the normal form B[u, u] depends on the property of the 3-way  resonance equation  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='8)  Λ(n3) − Λ(n1) − Λ(n2) = 0,  n1, n2, n3 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  which is equivalent to the Diophantine equation  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='9)  [F(n1) + F(n2) − F(n3)]2 − 4F(n1)F(n2) = 0,  n1, n2, n3 ≥ 2,  where F(X) = X(X − 1)(X + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  If the tuple (n1, n2, n3) is non-resonant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' it is such that b1,2,3(n1, n2, n3) ̸= 0, then some elementary  number theoretic argument will give a lower bound on |b1,2,3(n1, n2, n3)| in terms of a negative power (can  be fixed as −9/2) of n1, n2, n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' This is usually referred as small divisor estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  To study the distribution of resonant frequencies, we propose the following unsolved question:  Question 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Does the Diophantine equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='9) have finitely many solutions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  However, the Diophantine equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='9) does admit non-trivial solutions (5, 5, 8) and (10, 10, 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' In other  words, the second order terms e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Π(8)M1(Π(5)u · Π(5)u),  Π(5)M1(Π(8)u · Π(5)¯u),  in the quadratic part of N(u) cannot be eliminated by normal form reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' On the other hand, it seems  to be very hard to determine whether (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='9) still admits any other solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' We have the following proposition  (the author would like to thank Professor Bjorn Poonen for the proof):  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The Diophantine equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='9) has no solution with n1 ≤ 104 other than (5, 5, 8) and  (10, 10, 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  The proof of this proposition is computer-aided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The key point is to use the so-called Runge’s method to  show that if (n1, n2, n3) is a solution, then there must hold n2 = O(n2  1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' For a given n1, this reduces the proof  to numerical verification for finitely many possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The algorithm can of course be further optimized,  but due to some algebraic geometric considerations, it is reasonable to conjecture that the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='9)  should be very rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' In fact, there are two ways of viewing the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' We observe that if (n1, n2, n3) is  a solution, then F(n1)F(n2) must be a square, and the square free part of F(n1), F(n2), F(n3) must be the  same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Further reduction turns the problem into finding integral points on a family of elliptic curves  Y 2 = cF(X),  c is square-free,  which is of course difficult, but since Siegel’s theorem asserts that there are only finitely many integer points  on an elliptic curve over Q, it is reasonable to conjecture that there are not “too many” solutions to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  We may also view the problem as finding integral (rational) points on a given algebraic surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The complex  projective surface corresponding to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='9) is given by  V : [X(X − W)(X + 2W) + Y (Y − W)(Y + 2W) − Z(Z − W)(Z + 2W)]2  = 4X(X − W)(X + 2W)Y (Y − W)(Y + 2W),  where [X, Y, Z, W] is the homogeneous coordinate on CP3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' With the aid of computer, we obtain  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The complex projective surface V ⊂ CP3 has Kodaira dimension 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' it is of general  type under Kodaira-Enriquez classification), and its first Betti number is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  See Appendix A for the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  14  CHENGYANG SHAO  The first part of the proposition suggests that the rational points of V should be localized on finitely  many algebraic curves laying on V;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' nevertheless, this seemingly simple suggestion is indeed a special case of  the Bomberi-Lang conjecture, a hard problem in number theory (its planar case is known as the celebrated  Faltings’s theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The second part suggests that the rational points of V should be rare since its Albanese  variety is a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' But these are just heuristics that solutions to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='9) should be rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' In general,  determining the solvability of a given Diophantine equation is very difficult 2, as number theorists and  arithmetic geometers generally believe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  The reason that such issues do not occur for water waves in the flat setting or nonlinear Klein-Gordon  equations is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' First of all, the resonance equation is easily understood even in the degenerate case in  the flat setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' For example, the capillary water waves without gravity on T2 has dispersive relation |∇|1/2,  and the 3-way resonance equation is  4�  k2  1 + k2  2 +  4�  l2  1 + l2  2 =  4�  m2  1 + m2  2,  with the additional requirement m = k + l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' We already know that the resonance equation has no non-trivial  solution at all, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' But even without using m = k +l, we would be able to conclude that there are at most  finitely many non-trivial solutions from the celebrated Faltings’s theorem on rational points on high-genus  algebraic projective curves (although this is like using a sledge hammer to crack a nut).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Secondly, for the non-  degenrate case, for example the gravity-capillary waves, the dispersive relation reads  �  g|∇| + σ|∇|3, so if the  ratio σ/g is a transcedental number then the 3-way resonance equation has no solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Furthermore, using  some elementary calculus and a measure-theoretic argument, it can be shown, not without technicalities, that  the resonances admit certain small-divisor estimates for almost all parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' This is exactly the argument  employed by Delort-Szeftle [16], Berti-Delort [6] and Ionescu-Pusateri [26], so that their results were stated for  almost all parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' These parameters are, roughly speaking, badly approximated by algebraic numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  However, the resonance equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='8) is inhomogeneous and allows no arbitrary physical parameter at  all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Furthermore, since product of spherical harmonics are no longer spherical harmonics in general, Fourier  series techniques employed by [7] [6] [26] that works for the torus are never valid for S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' for example, we  cannot simply assume n3 = n1 + n2 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='8), as already illustrated by the solutions (5,5,8) (10,10,16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' These  are the crucial differences between the capillary spherical water waves and all known results for water waves  in the flat setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Heuristics for Lifespan Estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' To summarize, almost global lifespan estimate of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='3) depends  on the difficult number theoretic question 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Before it is fully resolved, we can only expect partial results  regarding the normal form transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  If there are only finitely many solutions to the Diophantine equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='9), then under the normal form  reduction u → v = u + B[u, u] with coefficients given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='7), the equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='3) is transformed into the  following system:  ∂  ∂tΠcv = O(v⊗2),  ∂  ∂t(1 − Πc)v = O(v⊗3),  where Πc is the orthogonal projection to �  n3 E(n3) ⊂ L2(S2), with n3 being either 0 or 1, or exhausting the  third component of all nontrivial solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  2For example, the seemingly simple Diophantine equation x3 + y3 + z3  =  42 is in fact a puzzle of more than 60  years, and its first solution was found recently by Booker-Sutherland [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  It is of extremely large magnitude:  42 =  (−80 538 738 812 075 974)3 + 80 435 758 145 817 5153 + 12 602 123 297 335 6313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Another example is the equation of  same type x3 + y3 + z3 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Beyond the easily found solutions (1, 1, 1), (4, 4, -5), (4, -5, 4), (-5, 4, 4), the next solution reads  (569 936 821 221 962 380 720, −569 936 821 113 563 493 509, −472 715 493 453 327 032).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL NOTES  15  There is no reasonable assertion to be made if the conjecture fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' However, if the conjecture does hold  true, then we can expect that the lifespan estimate for ε-Cauchy data of (EQ) goes beyond ε−1, as what we  expect for gravity water waves in the periodic setting, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' in [26]:  Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' If the Diophantine equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='9) has only finitely many solutions, then there is some  α > 0 such that for ε-Cauchy data of (EQ), the lifespan goes beyond ε−(1+α) as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Let’s explain the heuristic as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The argument we aim to implement is the standard “continuous  induction method”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' for some suitably large s and K and suitable α > 0, assuming T = ε−(1+α) and  supt∈[0,T ] ∥v∥Hs(g0) ≤ Kε, we try to prove a better bound supt∈[0,T ] ∥v∥Hs(g0) ≤ Kε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Here g0 is the  standard metric on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' It is intuitive to expect such a result for the cubic equation ∂t(1−Πc)v = O(v⊗3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' As  for the quadratic equation ∂tΠcv = O(v⊗2), it is crucial to implement the conservation of energy H[ζ, φ] ≡ 4π  for a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' We summarize it as  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Fix T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Let u be a smooth solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='3) and v = u + B[u, u] be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Suppose  for some suitably large s and K, there holds  sup  t∈[0,T ]  ∥v∥Hs(g0) ≤ Kε  with ε sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Then there is in fact a better bound for the low frequency part Πcv:  sup  t∈[0,T ]  ∥Πcv∥Hs(g0) ≤ Kε/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' We consider the “approximate” energy functional  H0[ζ, φ] = 4π +  �  S2 2ζ · dµ0 + 1  2  �  S2(|∇0ζ|2 + 2|ζ|2)dµ0 + 1  2  �  S2  ���|∇1/2  0  |φ  ���  2  dµ0,  where µ0 is the standard area measure on S2 and ∇0 is the standard connection on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' This is nothing but  the quadratic approximation to H[ζ, φ] in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='8) at (0, 0), so there holds H0[ζ, φ] = H[ζ, φ] + O(u⊗3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Using  volume preservation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='6), we obtain  �  S2 ζdµ0 = −∥ζ∥2  L2(g0) + O(ζ⊗3), so summarizing we have  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='10)  1  2  �  S2(|∇0ζ|2 − 2|ζ|2)dµ0 + 1  2  �  S2  ���|∇1/2  0  |φ  ���  2  dµ0 = O(u⊗3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Note that we used the conservation law H[ζ, φ] ≡ 4π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' By spectral calculus on S2, we have  ∥ζ∥2  H1(g0) ≃ ∥Π(0)ζ∥2  L2(g0) + ∥Π(1)ζ∥2  L2(g0) +  �  S2(|∇0ζ|2 − 2|ζ|2)dµ0,  and by volume preservation and conservation of momentum (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='7) we find  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='11)  ∥ζ∥2  H1(g0) ≃  �  S2(|∇0ζ|2 − 2|ζ|2)dµ0 + O(ζ⊗4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Now, for some N0 > 0 relating to the loss of regularity caused by B, we may choose s >> 2N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Then if  ∥v∥Hs(g0) ≤ Kε, it follows that ∥u∥Hs−N0(g0) ≤ K′ε, so by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='11) we have  ∥u∥2  L2(g0) ≃ ∥ζ∥2  H1(g0) + ∥φ∥2  H1/2(g0) ≤ K′ε3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Thus  ∥v∥L2(g0) ≤ C∥u∥L2(g0) + C∥u∥2  HN0(g0)  ≤ Cε3/2 + K′ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Since the spectrum of Πcv is bounded, by Bernstein type inequality we have  ∥Πcv∥Hs(g0) ≤ C∥v∥L2(g0) ≤ K′ε3/2(1 + ε1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  16  CHENGYANG SHAO  If ε is sufficiently small then this implies ∥Πcv∥Hs(g0) ≤ Kε/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  □  We point out that the above proof is independent from the magnitude of the lifespan T, so it is always  applicable as long as the cubic equation ∂t(1 − Πc)v = O(v⊗3) is well-understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' There are two crucial  points in the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='5: the conservation of energy, and that the projection Πc is of finite  rank, so that a Bernstein type inequality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The last fact holds only if there are finitely many 3-way  resonances, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' there are only finitely many solutions to the Diophantine equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Finally, we propose an even more ambitious conjecture concerning global dynamical properties of spherical  water droplets, which is again illuminated by observation in hydrodynamical experiments under zero gravity,  and also suggested by the results of Berti-Montalto [8]:  Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' If the Diophantine equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='9) has only finitely many solutions, then a KAM type result  holds for (EQ): there is a family of infinitely many quasi-periodic solutions of (EQ), depending on a parameter  which takes value in a Cantor-type set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' MAGMA Code  MAGMA is a large, well-supported software package designed for computations in algebra, number theory,  algebraic geometry, and algebraic combinatorics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' In this appendix, we give the MAGMA code used to conduct  computations on Diophantine equations related to the spherical capillary water waves system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Integral Points on Elliptic Curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' We can find all integral points on a given elliptic curve over  Q using MAGMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' For a monic cubic polynomial f(x), the function EllipticCurve(f) creates the elliptic  curve  E : y2 = f(x),  and the function IntegralPoints(E) returns a sequence containing all the integral points on E under the  homogeneous coordinate of QP2, modulo negation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' We use this to find out all integral points on the elliptic  curve  Ec : y2 = x3 + cx2 − 2c2x = x(x − c)(x + 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  for natural number c ≤ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The MAGMA code is listed below, which excludes all the c’s such that there are  only trivial integral points {(−c, 0), (0, 0), (c, 0)} on Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  > Qx<x> := PolynomialRing(Rationals());' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  > for c in [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='.50] do  > E := EllipticCurve(x^3+c*x^2-2*c^2*x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  > S, reps := IntegralPoints(E);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  > if # S gt 3 then  > print c, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  > print S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  > end if;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  > end for;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  2 Elliptic Curve defined by y^2 = x^3 + 2*x^2 - 8*x over Rational Field  [ (-4 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-2 : 4 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-1 : -3 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (0 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (4 : 8 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (8 : -24  : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (50 : 360 : 1) ]  8 Elliptic Curve defined by y^2 = x^3 + 8*x^2 - 128*x over Rational Field  [ (-16 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-8 : 32 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-4 : -24 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (0 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (8 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (9 : -15 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (16  : 64 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (32 : -192 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (200 : 2880 : 1) ]  LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL NOTES  17  13 Elliptic Curve defined by y^2 = x^3 + 13*x^2 - 338*x over Rational Field  [ (-26 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (0 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (13 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (121 : 1386 : 1) ]  15 Elliptic Curve defined by y^2 = x^3 + 15*x^2 - 450*x over Rational Field  [ (-30 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-5 : -50 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (0 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (15 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (24 : 108 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (90 : -900 : 1)  ]  17 Elliptic Curve defined by y^2 = x^3 + 17*x^2 - 578*x over Rational Field  [ (-34 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-32 : -56 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (0 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (17 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (833 : 24276 : 1) ]  18 Elliptic Curve defined by y^2 = x^3 + 18*x^2 - 648*x over Rational Field  [ (-36 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-32 : -80 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-18 : 108 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-9 : -81 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (0 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (18 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  (36 : 216 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (72 : -648 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (450 : 9720 : 1) ]  22 Elliptic Curve defined by y^2 = x^3 + 22*x^2 - 968*x over Rational Field  [ (-44 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-32 : -144 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (0 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (22 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (198 : 2904 : 1) ]  23 Elliptic Curve defined by y^2 = x^3 + 23*x^2 - 1058*x over Rational Field  [ (-46 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (0 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (23 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (50 : -360 : 1) ]  26 Elliptic Curve defined by y^2 = x^3 + 26*x^2 - 1352*x over Rational Field  [ (-52 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-49 : -105 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (0 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (26 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (1300 : 47320 : 1) ]  30 Elliptic Curve defined by y^2 = x^3 + 30*x^2 - 1800*x over Rational Field  [ (-60 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-50 : 200 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-45 : -225 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-24 : -216 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-20 : 200 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-6  : 108 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (0 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (30 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (36 : 144 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (40 : -200 :  1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (75 : -675 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (90 : 900 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (300 : 5400 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (324 : -6048 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (480 : -10800 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  (7290 : 623700 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (10830 : -1128600 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (226875 : 108070875 : 1) ]  32 Elliptic Curve defined by y^2 = x^3 + 32*x^2 - 2048*x over Rational Field  [ (-64 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-32 : 256 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-16 : -192 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (0 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (32 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (36 : -120 :  1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (64 : 512 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (128 : -1536 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (800 : 23040 : 1) ]  33 Elliptic Curve defined by y^2 = x^3 + 33*x^2 - 2178*x over Rational Field  [ (-66 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (0 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (33 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (81 : -756 : 1) ]  35 Elliptic Curve defined by y^2 = x^3 + 35*x^2 - 2450*x over Rational Field  [ (-70 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-49 : 294 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-45 : -300 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-40 : 300 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-14 : -196 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (0 :  0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (35 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (50 : 300 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (175 : -2450 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (224 : 3528 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (280 : -4900 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  (4410 : -294000 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (14450 : -1739100 : 1) ]  39 Elliptic Curve defined by y^2 = x^3 + 39*x^2 - 3042*x over Rational Field  [ (-78 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (0 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (39 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (147 : -1890 : 1) ]  42 Elliptic Curve defined by y^2 = x^3 + 42*x^2 - 3528*x over Rational Field  [ (-84 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-56 : -392 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-12 : 216 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (0 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (42 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (63 : -441 :  1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (294 : 5292 : 1) ]  43 Elliptic Curve defined by y^2 = x^3 + 43*x^2 - 3698*x over Rational Field  [ (-86 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-32 : -360 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (0 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (43 : 0 : 1) ]  46 Elliptic Curve defined by y^2 = x^3 + 46*x^2 - 4232*x over Rational Field  [ (-92 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (0 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (46 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (26496 : 4316640 : 1) ]  50 Elliptic Curve defined by y^2 = x^3 + 50*x^2 - 5000*x over Rational Field  [ (-100 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-50 : 500 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-25 : -375 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (-4 : 144 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (0 : 0 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (50 : 0 :  1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (100 : 1000 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (200 : -3000 : 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (1250 : 45000 : 1) ]  Running Magma V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='27-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Seed: 821911319;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Total time: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='430 seconds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Total memory usage: 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='16MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  18  CHENGYANG SHAO  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Classification of Projective Surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Some basic geometric parameters of the complex projective  surface  V : [X(X − W)(X + 2W) + Y (Y − W)(Y + 2W) − Z(Z − W)(Z + 2W)]2  = 4X(X − W)(X + 2W)Y (Y − W)(Y + 2W)  in CP3 can be computed using MAGMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The author would like to thank Professor Bjorn Poonen for intro-  ducing MAGMA and providing the code listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  > Q:=Rationals();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  > P<x,y,z,t>:=ProjectiveSpace(Q,3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  > fx:=x*(x-t)*(x+2*t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  > fy:=y*(y-t)*(y+2*t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  > fz:=z*(z-t)*(z+2*t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  > V:=Surface(P,(fz-fx-fy)^2-4*fx*fy);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  > KodairaEnriquesType(V);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  2 0 General type  Running Magma V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='27-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Seed: 989287753;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Total time: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='650 seconds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Total memory usage: 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='09MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Note that the Kodaira dimension is invariant regardless of the choice of base field, so it is legitimate to  choose the base field to be Q in the above code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The variable t is used to homogenize the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The  function KodairaEnriquesType(V) returns three values for the given projective surface V : the first is the  Kodaira dimension, the second is irrelevant when the Kodaira dimension is not −∞, 1 or 0, and the third is  the Kodaira-Enriquez classification of the surface X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  References  [1] Alazard, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', & Baldi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Gravity capillary standing water waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Archive for Rational Mechanics and Analysis,  217(3), 741-830.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [2] Alazard, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', & M´etivier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Paralinearization of the Dirichlet to Neumann operator, and regularity of three-  dimensional water waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Communications in Partial Differential Equations, 34(12), 1632-1704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [3] Alazard, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', Burq, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', & Zuily, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='. (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' On the water-wave equations with surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Duke Mathematical Journal,  158(3), 413-499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [4] Alazard, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' & Delort, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Global solutions and asymptotic behavior for two dimensional gravity water waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' ´Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Sup´er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', 48, (2015), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' 5, 1149-1238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [5] Barbosa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', & do Carmo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Stability of hypersurfaces with constant mean curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' In Manfredo P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' do  Carmo–Selected Papers (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' 221-235).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Springer, Berlin, Heidelberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [6] Berti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', & Delort, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Almost global solutions of capillary-gravity water waves equations on the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Springer  International Publishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [7] Berti M, Feola R, Pusateri F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', Birkhoff normal form and long time existence for periodic gravity water waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' arXiv preprint,  arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='11549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' 2018 Oct 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [8] Berti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', & Montalto, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Quasi-periodic standing wave solutions of gravity-capillary water waves (Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' 263, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  1273).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' American mathematical society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [9] Beyer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', & G¨unther, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' On the Cauchy problem for a capillary drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Part I: irrotational motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Mathematical  methods in the applied sciences, 21(12), 1149-1183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [10] Bombieri, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=',& Pila, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The number of integral points on arcs and ovals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Duke Mathematical Journal, 59(2), 337–357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [11] Booker, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', & Sutherland, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' On a question of Mordell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Proceedings of the National Academy of Sciences,  118(11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [12] Burq N, G´erard P, & Tzvetkov N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Strichartz inequalities and the nonlinear Schr¨odinger equation on compact manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  American Journal of Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='126(3):569-605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [13] Christianson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', Hur, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' & Staffilani, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Strichartz estimates for the water-wave problem with surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Communications in Partial Differential Equations, 35(12), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='2195-2252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL NOTES  19  [14] Coutand, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', & Shkoller, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Well-posedness of the free-surface incompressible Euler equations with or without  surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Journal of the American Mathematical Society, 20(3), 829-930.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [15] Deng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', Ionescu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', Pausader, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', & Pusateri, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Global solutions of the gravity-capillary water-wave system  in three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Acta Mathematica, 219(2), 213-402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [16] Delort, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', & Szeftel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Long-time existence for small data nonlinear klein-gordon equations on tori and spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  International Mathematics Research Notices (37), 1897-1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [17] Delort, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', & Imekraz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Long-time existence for the semilinear Klein–Gordon equation on a compact boundary-  less Riemannian manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Communications in Partial Differential Equations, 42(3), 388-416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [18] Germain, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', Masmoudi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', & Shatah, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Global solutions for the gravity water waves equation in dimension 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Annals of Mathematics, 175: 691–754, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [19] Germain, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', Masmoudi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', & Shatah, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Global solutions for capillary waves equation in dimension 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', 68(4): 625–687, 2015  [20] H¨ormander, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Pseudo-differential operaors of type 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Communications in partial differential equations, 13(9),  1085-1111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [21] H¨ormander, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Lectures on nonlinear hyperbolic differential equations (Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' 26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Springer Science & Business Media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [22] Hunter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', Ifrim, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', & Tataru, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Two dimensional water waves in holomorphic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Communications  in Mathematical Physics, 346(2), 483-552.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [23] Ifrim, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', & Tataru, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Two dimensional water waves in holomorphic coordinates II: global solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' arXiv preprint  arXiv:1404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='7583.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [24] Ionescu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', & Pusateri, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Global solutions for the gravity water waves system in 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Inventiones mathematicae,  199(3), 653-804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [25] Ionescu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', & Pusateri, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Recent advances on the global regularity for irrotational water waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Philosophical  Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 376(2111), 20170089.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [26] Ionescu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', & Pusateri, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Long-time existence for multi-dimensional periodic water waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Geometric and  Functional Analysis, 29(3), 811-870.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [27] Lannes, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Well-posedness of the water-waves equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Journal of the American Mathematical Society, 18(3),  605-654.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [28] Lannes, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' The water waves problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Mathematical analysis and asymptotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Mathematical Surveys and Mono-  graphs, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' 188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' American Mathematical Society, Providence, RI, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [29] Magnanini, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', & Poggesi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' On the stability for Alexandrov’s Soap Bubble theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Journal d’Analyse  Math´ematique, 139(1), 179-205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [30] Ming M, & Zhang Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Well-posedness of the water-wave problem with surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Journal de math´ematiques pures et  appliqu´ees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' 2009 Nov 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='92(5):429-55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [31] Ruzhansky, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', & Turunen, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Pseudo-differential operators and symmetries: background analysis and advanced  topics (Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Springer Science & Business Media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [32] Schweizer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' On the three-dimensional Euler equations with a free boundary subject to surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Annales  de l’institut Henri Poincar´e C, Analyse non lin´eaire, 22(6), 753-781.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [33] Sogge, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Concerning the Lp norm of spectral clusters for second-order elliptic operators on compact manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Journal of functional analysis, 77(1), 123-138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [34] Shao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Long Time Behavior of a Quasilinear Hyperbolic System Modelling Elastic Membranes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  org/abs/2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='10663.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [35] Shatah, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (1985) Normal forms and quadratic Klein-Gordon equations, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' 38, 685-696.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [36] Shatah, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=', & Zeng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Geometry and a priori estimates for free boundary problems of the Euler’s equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences,  61(5), 698-744.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [37] Taylor, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Partial differential equations II: Qualitative studies of linear equations (Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' 116).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Springer Science &  Business Media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [38] Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2020) Global regularity for the 3d finite depth capillary water waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Annales Scientifiques de l’´Ecole Normale  Sup´erieure, 53(4): 847–943, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [39] Wu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Well-posedness in Sobolev spaces of the full water wave problem in 2-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Inventiones mathematicae, 130(1),  39-72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [40] Wu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Well-posedness in Sobolev spaces of the full water wave problem in 3-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Journal of the American Mathe-  matical Society, 12(2), 445-495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  20  CHENGYANG SHAO  [41] Wu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Almost global wellposedness of the 2-D full water wave problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Inventiones mathematicae, 177(1), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='45-135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [42] Wu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Global wellposedness of the 3-D full water wave problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Inventiones mathematicae, 184(1), 125-220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content='  [43] Zakharov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' (1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Stability of periodic waves of finite amplitude on the surface of a deep fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
+page_content=' Journal of Applied  Mechanics and Technical Physics, 9(2), 190-194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfT_ca/content/2301.00115v1.pdf'}
diff --git a/LtFJT4oBgHgl3EQfyi0f/content/tmp_files/2301.11638v1.pdf.txt b/LtFJT4oBgHgl3EQfyi0f/content/tmp_files/2301.11638v1.pdf.txt
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+arXiv:2301.11638v1  [math.FA]  27 Jan 2023
+ONE-DIMENSIONAL INTEGRAL RELLICH TYPE INEQUALITIES
+TOHRU OZAWA, PRASUN ROYCHOWDHURY, AND DURVUDKHAN SURAGAN
+Abstract. The motive of this note is twofold. Inspired by the recent develop-
+ment of a new kind of Hardy inequality, here we discuss the corresponding Hardy-
+Rellich and Rellich inequality versions in the integral form. The obtained sharp
+Hardy-Rellich type inequality improves the previously known result. Meanwhile,
+the established sharp Rellich type integral inequality seems new.
+1. Introduction
+In the celebrated paper [8], Godfrey H. Hardy first stated the famous inequality
+which reads as: let 1 < p < ∞ and f be a p-integrable function on (0, ∞), which
+vanishes at zero, then the function r �−→ 1
+r
+� r
+0 f(t) dt is p-integrable over (0, ∞) and
+there holds
+� ∞
+0
+����
+1
+r
+� r
+0
+f(t) dt
+����
+p
+dr ≤
+�
+p
+p − 1
+�p � ∞
+0
+|f(r)|p dr.
+(1.1)
+The constant on the right-hand side of (1.1) is sharp. The development of the famous
+Hardy inequality (1.1) during the period 1906-1928 has its own history and we refer to
+[10] (also, see the preface of [18]). Recent progress by Frank-Laptev-Weidl [9] presents
+a novel one-dimensional inequality with the same sharp constant, which improves the
+classical Hardy inequality (1.1).
+This new version looks as follows: let 1 < p < ∞, then for any f ∈ Lp(0, ∞), which
+vanishes at zero, one has
+� ∞
+0
+sup
+0<s<∞
+���� min
+�1
+r, 1
+s
+� � s
+0
+f(t) dt
+����
+p
+dr ≤
+�
+p
+p − 1
+�p � ∞
+0
+|f(r)|p dr.
+(1.2)
+Certainly, (1.2) gives an improvement of (1.1).
+Recently, the multidimensional
+version in the supercritical case and the discrete version of (1.2) have been established
+in [16] and [15], respectively. In the same spirit, one may ask about the possible
+structure of Hardy-Rellich and Rellich type inequalities. In this short note, we answer
+affirmatively about the possible forms of these two types of inequalities.
+Let us recall the one-dimensional Hardy-Rellich inequality. For f ∈ C2(0, ∞) with
+f ′(0) = 0, there holds
+� ∞
+0
+|f ′(r)|2
+r2
+dr ≤ 4
+� ∞
+0
+|f ′′(r)|2 dr.
+(1.3)
+Date: January 30, 2023.
+2010 Mathematics Subject Classification. 26D10, 35A23.
+Key words and phrases. Hardy inequality, Rellich inequality, Hardy-Rellich inequality, sharp
+constant.
+1
+
+2
+T. OZAWA, P. ROYCHOWDHURY, AND D. SURAGAN
+Starting from it, there have been several articles in which the authors studied many
+improvements of inequality (1.3). Here we mention only a few of them [2, 5, 6, 20, 19]
+and references therein.
+Now let us write (1.3) in the integral form. Note that it can be derived from the
+weighted 1D classical Hardy inequality. This reads as follows: let f ∈ C1(0, ∞), then
+there holds
+� ∞
+0
+|
+� r
+0 f ′(t) dt|2
+r2
+dr ≤ 4
+� ∞
+0
+|f ′(r)|2 dr.
+(1.4)
+Here the constant 4 is sharp.
+We give an improved version of this inequality in
+Theorem 2.1.
+Let us briefly mention another important function inequality the so-called Rellich
+inequality which was first introduced in [14]. It is worth recalling the one-dimensional
+Rellich inequality. The classical (1D-)Rellich inequality states that for f ∈ C2(0, ∞)
+with f(0) = 0 and f ′(0) = 0, there holds
+� ∞
+0
+|f(r)|2
+r4
+dr ≤ 16
+9
+� ∞
+0
+|f ′′(r)|2 dr.
+(1.5)
+Over the past few decades, there has been a constant effort to improve (1.5). Here
+are some references [7, 12, 1, 13, 3, 17, 4]. In this short contribution, we also obtain
+another type of the Rellich inequality (see Theorem 2.2 with p = 2). To the best of our
+knowledge, the most recent progress in this direction was made in [3]. However, a one-
+dimensional study is still missing. Thus, trying to fill this gap is another motivation
+for the present paper. Taking inspiration from there we obtain the following version
+of Rellich inequality, which reads as: for f ∈ L2(0, ∞) there holds
+� ∞
+0
+1
+r4
+� � r
+0
+� τ
+0
+|f(t)| dt dτ
+�2
+dr
+≤
+� ∞
+0
+1
+r4
+� � r
+0
+sup
+0<s<∞ min
+�
+1, τ
+s
+� � s
+0
+|f(t)| dt dτ
+�2
+dr
+≤ 16
+9
+� ∞
+0
+|f(r)|2 dr.
+(1.6)
+Moreover, we will establish the constant 16/9 is a sharp constant. Therefore, (1.6)
+can be compared with (1.5). Note that we have mentioned only the L2(0, ∞) case
+but we will discuss the result for the general Lp(0, ∞) case.
+The plan of this paper is simple. In Section 2 we discuss a few preliminaries and
+then we state the main results. In Section 3 we complete the proofs of the main
+theorems. In the final section, we will briefly mention some more related inequalities.
+2. Preliminaries and main results
+Let us begin this section with basic facts about symmetric decreasing rearrangement.
+For more details, we refer to [11, Chapter 3]. We denote f ∗ the symmetric decreasing
+rearrangement of f. It is well known that f ∗ is a nonnegative, radially symmetric, and
+
+INTEGRAL RELLICH TYPE INEQUALITIES
+3
+nonincreasing function. Irrespective of several properties of f ∗, the useful properties
+in our context is the equimeasurability property, i.e.
+vol({|f| > τ}) = vol({f ∗ > τ}) for all τ ≥ 0.
+By using the layer cake representation and the above property, we have the following
+helpful identity:
+� ∞
+0
+|f(t)|p dt =
+� ∞
+0
+|f ∗(t)|p dt for all p ≥ 1.
+(2.1)
+Also it is clear that for any s > 0 there holds
+� s
+0
+|f(t)| dt ≤
+� s
+0
+f ∗(t) dt.
+(2.2)
+These relations will be very much valuable in the proofs.
+Now, we are ready to state the following important observation.
+Lemma 2.1. Let f be a locally absolutely continuous function on (0, ∞). Then for
+a fixed r > 0 the following identity holds:
+sup
+0<s<∞ min
+�
+1, r
+s
+� � s
+0
+f ∗(t) dt =
+� r
+0
+f ∗(t) dt,
+(2.3)
+where f ∗ is the non-increasing rearrangement of f.
+Proof. We wish to calculate the supremum by using the advantage of the non-increasing
+property of f ∗. For any fixed r > 0, we consider the following two cases:
+Case 1: Let 0 < s ≤ r. Then we obtain
+min
+�
+1, r
+s
+� � s
+0
+f ∗(t) dt =
+� s
+0
+f ∗(t) dt ≤
+� r
+0
+f ∗(t) dt.
+Case 2: Let r ≤ s < ∞. Then we have by change of variable
+min
+�
+1, r
+s
+� � s
+0
+f ∗(t) dt = r
+s
+� s
+0
+f ∗(t) dt ≤ r
+s
+� s
+0
+f ∗(tr/s) dt =
+� r
+0
+f ∗(v) dv.
+In both cases, we get
+min
+�
+1, r
+s
+� � s
+0
+f ∗(t) dt ≤
+� r
+0
+f ∗(t) dt.
+Hence the supremum is attained at s = r and we arrive at
+sup
+0<s<∞ min
+�
+1, r
+s
+� � s
+0
+f ∗(t) dt =
+� r
+0
+f ∗(t) dt.
+□
+Now we are ready to present an improvement of (1.4). That is, this gives a natural
+improvement of the Hardy-Rellich inequality in the integral form.
+Below we will
+describe the corresponding differential form which improves the original Hardy-Rellich
+inequality (1.3) in a simple form.
+
+4
+T. OZAWA, P. ROYCHOWDHURY, AND D. SURAGAN
+Theorem 2.1. Let g ∈ C1(0, ∞), then there holds
+� ∞
+0
+sup
+0<s<∞
+���� min
+�1
+r, 1
+s
+� � r
+0
+g′(t) dt
+����
+2
+dr ≤ 4
+� ∞
+0
+|g′(r)|2 dr.
+(2.4)
+Moreover, the constant is sharp.
+Remark 2.1. Let f ∈ C2(0, ∞), with f ′(0) = 0. Then choosing g(r) = f ′(r) in the
+improved Hardy-Rellich inequality (2.4), we obtain
+� ∞
+0
+sup
+0<s<∞
+���� min
+�1
+r, 1
+s
+�
+f ′(s)
+����
+2
+dr ≤ 4
+� ∞
+0
+|f ′′(r)|2 dr.
+(2.5)
+Moreover, it is straightforward to check the following identity
+sup
+0<s<∞
+���� min
+�1
+r, 1
+s
+�
+f ′(s)
+����
+2
+= max
+�
+sup
+0<s≤r
+|f ′(s)|2
+r2
+,
+sup
+r≤s<∞
+|f ′(s)|2
+s2
+�
+.
+(2.6)
+Then combining (2.5) and (2.6) we deduce
+� ∞
+0
+|f ′(r)|2
+r2
+dr ≤
+� ∞
+0
+max
+�
+sup
+0<s≤r
+|f ′(s)|2
+r2
+,
+sup
+r≤s<∞
+|f ′(s)|2
+s2
+�
+dr ≤ 4
+� ∞
+0
+|f ′′(r)|2 dr.
+(2.7)
+Therefore, (2.7) appears as an immediate improvement of (1.3) in the one-dimension
+differential form. The similar discussion gives the improvement of (1.4) as well (cf.
+[5, Theorem 1.3]).
+Now we are going to discuss the second main result of this note. Before presenting
+the statement first let us recall the classical one-dimensional Lp-Rellich inequality.
+This reads as follows: let p > 1, f ∈ C2(0, ∞) with f(0) = 0 and f ′(0) = 0 there
+holds
+� ∞
+0
+|f(r)|p
+r2p
+dr ≤
+p2p
+(p − 1)p(2p − 1)p
+� ∞
+0
+|f ′′(r)|p dr.
+(2.8)
+Now we are ready to demonstrate the one-dimensional Rellich-type inequality in the
+following integral form.
+Theorem 2.2. Let f ∈ Lp(0, ∞), p > 1. Then we have
+� ∞
+0
+1
+r2p
+� � r
+0
+� τ
+0
+|f(t)| dt dτ
+�p
+dr
+≤
+� ∞
+0
+1
+r2p
+� � r
+0
+sup
+0<s<∞ min
+�
+1, τ
+s
+� � s
+0
+|f(t)| dt dτ
+�p
+dr
+≤
+p2p
+(p − 1)p(2p − 1)p
+� ∞
+0
+|f(r)|p dr.
+(2.9)
+Moreover, the constant is sharp.
+
+INTEGRAL RELLICH TYPE INEQUALITIES
+5
+3. Proofs of Theorems 2.1 and 2.2
+This section is concerned with the proofs of Theorems 2.1 and 2.2. Before going
+further let us recall the following lemma.
+Lemma 3.1. [16, Lemma 3.1] Let 1 < p < ∞. Let g be any nonnegative function
+on (0, ∞). Assume h is a strictly positive non-decreasing function on (0, ∞) such
+that sh(r) ≤ rh(s) for any r, s ∈ (0, ∞) with r ≤ s. Let f be a locally absolutely
+continuous function on (0, ∞). Then we have
+� ∞
+0
+g(r) sup
+0<s<∞
+���� min
+� 1
+h(r),
+1
+h(s)
+� � s
+0
+f(t) dt
+����
+p
+dr ≤
+� ∞
+0
+g(r)
+����
+1
+h(r)
+� r
+0
+f ∗(t) dt
+����
+p
+dr,
+where f ∗ is the non-increasing rearrangement of f.
+Now, as a direct consequence of Lemma (3.1), we derive the proof of Theorem 2.1.
+Proof of Theorem 2.1: Let us consider g(r) = 1 and h(r) = r be functions on
+(0, ∞). Substituting these in Lemma 3.1 with p = 2, then we have
+� ∞
+0
+sup
+0<s<∞
+���� min
+�1
+r, 1
+s
+� � s
+0
+g′(t) dt
+����
+2
+dr ≤
+� ∞
+0
+1
+r2
+����
+� r
+0
+(g′)∗(t) dt
+����
+2
+dr.
+By using the Hardy-Rellich inequality in the form (1.4) for the function (g′)∗, we
+obtain
+� ∞
+0
+sup
+0<s<∞
+���� min
+�1
+r, 1
+s
+� � s
+0
+g′(t) dt
+����
+2
+dr ≤ 4
+� ∞
+0
+|(g′)∗(r)|2 dr
+= 4
+� ∞
+0
+|g′(r)|2 dr.
+In the last step, we have used norm preserving property (2.1). The sharpness follows
+from the optimality of the constant in (1.4). It ends the proof.
+Proof of Theorem 2.2: The first inequality follows from the property of the
+supremum. Now taking the integral of (2.3) from 0 to r we have
+� r
+0
+sup
+0<s<∞ min
+�
+1, τ
+s
+� � s
+0
+f ∗(t) dt dτ =
+� r
+0
+� τ
+0
+f ∗(t) dt dτ.
+(3.1)
+� ∞
+0
+1
+r2p
+� � r
+0
+sup
+0<s<∞
+min
+�
+1, τ
+s
+� � s
+0
+|f(t)| dt dτ
+�p
+dr
+(2.2)
+≤
+� ∞
+0
+1
+r2p
+� � r
+0
+sup
+0<s<∞ min
+�
+1, τ
+s
+� � s
+0
+f ∗(t) dt dτ
+�p
+dr
+(3.1)
+=
+� ∞
+0
+1
+r2p
+� � r
+0
+� τ
+0
+f ∗(t) dt dτ
+�p
+dr
+(2.8)
+≤
+p2p
+(p − 1)p(2p − 1)p
+� ∞
+0
+|f ∗(r)|p dr
+(2.1)
+=
+p2p
+(p − 1)p(2p − 1)p
+� ∞
+0
+|f(r)|p dr.
+
+6
+T. OZAWA, P. ROYCHOWDHURY, AND D. SURAGAN
+This completes the proof of the new type Rellich inequality in the integral form.
+Optimality: We set
+Cp :=
+inf
+f∈Lp(0,∞)\{0}
+� ∞
+0
+1
+r2p
+� � r
+0
+� τ
+0 |f(t)| dt dτ
+�p dr
+� ∞
+0 |f(r)|p dr
+.
+(3.2)
+The validity of (2.9) immediately implies
+Cp ≤
+p2p
+(p − 1)p(2p − 1)p.
+So it remains to show other side and this will be done by giving a proper minimizing
+sequence. We divide the proof in some steps.
+Step 1. Let us start with a cut-off function χ : [0, ∞) → R with the following
+properties:
+1. χ(r) ∈ [0, 1] for all r ∈ [0, ∞) and χ is smooth;
+2. χ satisfies the following
+χ(r) =
+�1
+0 ≤ r ≤ 1,
+0
+2 ≤ r < ∞;
+3. χ is decreasing function, i.e. χ′(r) ≤ 0 for all r ∈ [0, ∞).
+Now for a small ǫ > 0, let us define the minimizing functions {fǫ} as follows:
+fǫ(r) := r
+ǫ−1
+p χ(r).
+Step 2. In this step we will estimate r.h.s. of (2.9). The denominator of (3.2) gives
+� ∞
+0
+|fǫ(r)|p dr =
+� ∞
+0
+rǫ−1χp(r) dr
+=
+� 1
+0
+rǫ−1 dr +
+� 2
+1
+rǫ−1χp(r) dr
+= 1
+ǫ + O(1).
+(3.3)
+Therefore, for a fixed positive ǫ, we have fǫ ∈ Lp(0, ∞).
+Step 3. In this part we will evaluate the numerator of (3.2). Consider the integra-
+tion by parts and we compute
+� ∞
+0
+1
+r2p
+� � r
+0
+� τ
+0
+|fǫ(t)| dt dτ
+�p
+dr
+=
+� ∞
+0
+1
+r2p
+� � r
+0
+� τ
+0
+t
+ǫ−1
+p χ(t) dt dτ
+�p
+dr
+=
+�
+p
+ǫ − 1 + p
+�p � ∞
+0
+1
+r2p
+� � r
+0
+χ(τ)τ
+ǫ−1+p
+p
+dτ −
+� r
+0
+� τ
+0
+t
+ǫ−1+p
+p
+χ′(t) dt dτ
+�p
+dr
+≥
+�
+p
+ǫ − 1 + p
+�p � ∞
+0
+1
+r2p
+� � r
+0
+χ(τ)τ
+ǫ−1+p
+p
+dτ
+�p
+dr
+
+INTEGRAL RELLICH TYPE INEQUALITIES
+7
+=
+�
+p
+ǫ − 1 + p
+�p�
+p
+ǫ − 1 + 2p
+�p � ∞
+0
+1
+r2p
+�
+χ(r)r
+ǫ−1+2p
+p
+−
+� r
+0
+τ
+ǫ−1+2p
+p
+χ′(τ) dτ
+�p
+dr
+≥
+�
+p
+ǫ − 1 + p
+�p�
+p
+ǫ − 1 + 2p
+�p � ∞
+0
+rǫ−1χp(r) dr
+=
+�
+p
+ǫ − 1 + p
+�p�
+p
+ǫ − 1 + 2p
+�p� � 1
+0
+rǫ−1 dr +
+� 2
+1
+rǫ−1χp(r) dr
+�
+= 1
+ǫ
+�
+p
+ǫ − 1 + p
+�p�
+p
+ǫ − 1 + 2p
+�p
++ O(1).
+(3.4)
+In between, exploiting χ′ ≤ 0, we used a simple inequality (a + b)p ≥ ap twice, for
+nonnegative real numbers a and b with p > 1.
+Step 4. Finally, by using (3.3) and (3.4) we deduce the ratio
+� ∞
+0
+1
+r2p
+� � r
+0
+� τ
+0 |f(t)| dt dτ
+�p dr
+� ∞
+0 |f(r)|p dr
+≥
+1
+ǫ
+�
+p
+ǫ−1+p
+�p�
+p
+ǫ−1+2p
+�p + O(1)
+1
+ǫ + O(1)
+→
+p2p
+(p − 1)p(2p − 1)p for ǫ → 0.
+Hence {fǫ} is the required minimizing sequence and, in turn, we have
+Cp =
+p2p
+(p − 1)p(2p − 1)p.
+4. Few more remarks
+Here we shortly revisit some more inequalities. It hints that rescaling above in-
+equalities may generate new inequalities. Let us recall the differential form of the
+inequality (1.2).
+Lemma 4.1. [9, Theorem 1] Let 1 < p < ∞.
+Then, for any locally absolutely
+continuous function f on (0, ∞) with lim infr→0 |f(r)| = 0, there holds
+� ∞
+0
+max
+�
+sup
+0<s≤r
+|f(s)|p
+rp
+, sup
+r≤s<∞
+|f(s)|p
+sp
+�
+dr ≤
+�
+p
+p − 1
+�p � ∞
+0
+|f ′(r)|p dr.
+Now by substituting 1
+r
+� r
+0 f(t) dt and
+� r
+0 f(t) dt instead of f(r) one can have two
+immediate corollaries, respectively.
+Corollary 4.1. Let 1 < p < ∞. Then, for any locally absolutely continuous function
+f on (0, ∞), there holds
+� ∞
+0
+max
+�
+sup
+0<s≤r
+| 1
+s
+� s
+0 f(t) dt|p
+rp
+, sup
+r≤s<∞
+| 1
+s
+� s
+0 f(t) dt|p
+sp
+�
+dr
+≤
+�
+p
+p − 1
+�p
+2p−1
+�� ∞
+0
+|f(r)|p
+rp
+dr +
+� ∞
+0
+1
+r2p
+� � r
+0
+|f(t)|p dt
+�
+dr
+�
+.
+
+8
+T. OZAWA, P. ROYCHOWDHURY, AND D. SURAGAN
+Corollary 4.2. Let 1 < p < ∞. Then, for any locally absolutely continuous function
+f on (0, ∞), there holds
+� ∞
+0
+max
+�
+sup
+0<s≤r
+|
+� s
+0 f(t) dt|p
+rp
+, sup
+r≤s<∞
+|
+� s
+0 f(t) dt|p
+sp
+�
+dr ≤
+�
+p
+p − 1
+�p � ∞
+0
+|f(r)|p dr
+Acknowledgments
+This work was partially supported by the NU program 20122022CRP1601. The
+first author is supported in part by JSPS Kakenhi 18KK0073, 19H00644. The sec-
+ond author is supported in part by National Theoretical Science Research Center
+Operational Plan V-Mathematics Field (2/5) (Project number 111-2124-M-002-014-
+G58-01).
+References
+[1] G. Barbatis, A. Tertikas, On a class of Rellich inequalities, J. Comput. Appl. Math. 194 (2006),
+no. 1, 156–172, MR2230976.
+[2] E. Berchio, D. Ganguly, P. Roychowdhury, Hardy-Rellich and second-order Poincar´e identities
+on the hyperbolic space via Bessel pairs, Calc. Var. Partial Differential Equations 61, no. 4,
+Paper No. 130. (2022), MR4417395.
+[3] N. Bez, S. Machihara, T. Ozawa, Revisiting the Rellich inequality, to appear in Math. Z.,
+(2022), arXiv:2209.02928.
+[4] P. Caldiroli, R. Musina, Rellich inequalities with weights, Calc. Var. Partial Differential Equa-
+tions 45 (2012), 147–164, MR2957654.
+[5] B. Cassano, L. Cossetti, L. Fanelli, Improved Hardy-Rellich inequalities, Commun. Pure Appl.
+Anal. 21 (2022), no. 3, 867–889, MR4389605.
+[6] C. Cazacu, A new proof of the Hardy-Rellich inequality in any dimension, Proc. Roy. Soc.
+Edinburgh Sect. A 150 (2020), no. 6, 2894–2904, MR4190094.
+[7] E. B. Davies, A. M. Hinz, Explicit constants for Rellich inequalities in Lp(Ω), Math. Z. 227
+(1998), no. 3, 511–523, MR1612685.
+[8] G. H. Hardy, Note on a theorem of Hilbert, Math. Z. 6 (1920), no. 3-4, 314–317, MR1544414.
+[9] R. L. Frank, A. Laptev, T. Weidl, An improved one-dimensional Hardy inequality, J. Math.
+Sci. 268 (2022), 323–342, Link.
+[10] A. Kufner, L. Maligranda, L. E. Persson, The prehistory of the Hardy inequality, Amer. Math.
+Monthly 113 (2006), no. 8, 715–732, MR2256532.
+[11] E. H. Lieb, M. Loss, Analysis, Second edition. Graduate Studies in Mathematics, 14. American
+Mathematical Society, Providence, RI, (2001), MR1817225.
+[12] S. Machihara, T. Ozawa, H. Wadade, Remarks on the Rellich inequality, Math. Z. 286 (2017),
+no. 3-4, 1367–1373, MR3671580.
+[13] G. Metafune, L. Negro, M. Sobajima, C. Spina, Rellich inequalites in bounded domains, Math.
+Ann. 379 (2021), no. 1-2, 765–824, MR4211104.
+[14] F. Rellich, Halbbeschr¨ankte Differentialoperatoren h¨oherer Ordnung, (German) Proceedings
+of the International Congress of Mathematicians, 1954, Amsterdam, vol. III, pp. 243-250.
+Erven P. Noordhoff N.V., Groningen; North-Holland Publishing Co., Amsterdam, (1956)
+MR0088624.
+[15] P. Roychowdhury, D. Suragan, Improvement of the discrete Hardy inequality, (2022),
+arXiv:2209.05288.
+[16] P. Roychowdhury, M. Ruzhansky, D. Suragan, Multidimensional Frank-Laptev-Weidl improve-
+ment of the Hardy Inequality, (2022), arXiv:2209.08395.
+
+INTEGRAL RELLICH TYPE INEQUALITIES
+9
+[17] M. Ruzhansky, D. Suragan, Hardy and Rellich inequalities, identities, and sharp remainders
+on homogeneous groups, Adv. Math. 317 (2017), 799–822, MR3682685.
+[18] M. Ruzhansky, D. Suragan, Hardy inequalities on homogeneous groups, 100 years of Hardy
+inequalities. Progress in Mathematics, 327. Birkh¨auser/Springer, Cham, (2019), MR3966452.
+[19] A. Tertikas, N. B. Zographopoulos, Best constants in the Hardy–Rellich inequalities and related
+improvements, Adv. Math. 209 (2007), 407–459, MR2296305.
+[20] D. Yafaev, Sharp constants in the Hardy–Rellich inequalities, J. Funct. Anal. 168 (1999),
+121–144, MR1717839.
+Tohru Ozawa:
+Department of Applied Physics
+Waseda University, Tokyo 169-8555
+Japan
+E-mail address txozawa@waseda.jp
+Prasun Roychowdhury:
+Mathematics Division: National Center for Theoretical Sciences
+NTU, Cosmology Building, No. 1, Sec. 4, Roosevelt Rd., Taipei City 106
+Taiwan
+E-mail address prasunroychowdhury1994@gmail.com
+Durvudkhan Suragan:
+Department of Mathematics
+Nazarbayev University
+Kazakhstan
+E-mail address durvudkhan.suragan@nu.edu.kz
+
diff --git a/LtFJT4oBgHgl3EQfyi0f/content/tmp_files/load_file.txt b/LtFJT4oBgHgl3EQfyi0f/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..df1807ef8d7ba53b5324c8c61b5c425cc29aed6b
--- /dev/null
+++ b/LtFJT4oBgHgl3EQfyi0f/content/tmp_files/load_file.txt
@@ -0,0 +1,515 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf,len=514
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='11638v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='FA]  27 Jan 2023  ONE-DIMENSIONAL INTEGRAL RELLICH TYPE INEQUALITIES  TOHRU OZAWA, PRASUN ROYCHOWDHURY, AND DURVUDKHAN SURAGAN  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' The motive of this note is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Inspired by the recent develop-  ment of a new kind of Hardy inequality, here we discuss the corresponding Hardy-  Rellich and Rellich inequality versions in the integral form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' The obtained sharp  Hardy-Rellich type inequality improves the previously known result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Meanwhile,  the established sharp Rellich type integral inequality seems new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Introduction  In the celebrated paper [8], Godfrey H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Hardy first stated the famous inequality  which reads as: let 1 < p < ∞ and f be a p-integrable function on (0, ∞), which  vanishes at zero, then the function r �−→ 1  r  � r  0 f(t) dt is p-integrable over (0, ∞) and  there holds  � ∞  0  ����  1  r  � r  0  f(t) dt  ����  p  dr ≤  �  p  p − 1  �p � ∞  0  |f(r)|p dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='1)  The constant on the right-hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='1) is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' The development of the famous  Hardy inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='1) during the period 1906-1928 has its own history and we refer to  [10] (also, see the preface of [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Recent progress by Frank-Laptev-Weidl [9] presents  a novel one-dimensional inequality with the same sharp constant, which improves the  classical Hardy inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  This new version looks as follows: let 1 < p < ∞, then for any f ∈ Lp(0, ∞), which  vanishes at zero, one has  � ∞  0  sup  0<s<∞  ���� min  �1  r, 1  s  � � s  0  f(t) dt  ����  p  dr ≤  �  p  p − 1  �p � ∞  0  |f(r)|p dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='2)  Certainly, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='2) gives an improvement of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Recently, the multidimensional  version in the supercritical case and the discrete version of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='2) have been established  in [16] and [15], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' In the same spirit, one may ask about the possible  structure of Hardy-Rellich and Rellich type inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' In this short note, we answer  affirmatively about the possible forms of these two types of inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Let us recall the one-dimensional Hardy-Rellich inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' For f ∈ C2(0, ∞) with  f ′(0) = 0, there holds  � ∞  0  |f ′(r)|2  r2  dr ≤ 4  � ∞  0  |f ′′(r)|2 dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='3)  Date: January 30, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' 26D10, 35A23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Hardy inequality, Rellich inequality, Hardy-Rellich inequality, sharp  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  1  2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' OZAWA, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' ROYCHOWDHURY, AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' SURAGAN  Starting from it, there have been several articles in which the authors studied many  improvements of inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Here we mention only a few of them [2, 5, 6, 20, 19]  and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Now let us write (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='3) in the integral form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Note that it can be derived from the  weighted 1D classical Hardy inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' This reads as follows: let f ∈ C1(0, ∞), then  there holds  � ∞  0  |  � r  0 f ′(t) dt|2  r2  dr ≤ 4  � ∞  0  |f ′(r)|2 dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='4)  Here the constant 4 is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  We give an improved version of this inequality in  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Let us briefly mention another important function inequality the so-called Rellich  inequality which was first introduced in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' It is worth recalling the one-dimensional  Rellich inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' The classical (1D-)Rellich inequality states that for f ∈ C2(0, ∞)  with f(0) = 0 and f ′(0) = 0, there holds  � ∞  0  |f(r)|2  r4  dr ≤ 16  9  � ∞  0  |f ′′(r)|2 dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='5)  Over the past few decades, there has been a constant effort to improve (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Here  are some references [7, 12, 1, 13, 3, 17, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' In this short contribution, we also obtain  another type of the Rellich inequality (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='2 with p = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' To the best of our  knowledge, the most recent progress in this direction was made in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' However, a one-  dimensional study is still missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Thus, trying to fill this gap is another motivation  for the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Taking inspiration from there we obtain the following version  of Rellich inequality, which reads as: for f ∈ L2(0, ∞) there holds  � ∞  0  1  r4  � � r  0  � τ  0  |f(t)| dt dτ  �2  dr  ≤  � ∞  0  1  r4  � � r  0  sup  0<s<∞ min  �  1, τ  s  � � s  0  |f(t)| dt dτ  �2  dr  ≤ 16  9  � ∞  0  |f(r)|2 dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='6)  Moreover, we will establish the constant 16/9 is a sharp constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Therefore, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='6)  can be compared with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Note that we have mentioned only the L2(0, ∞) case  but we will discuss the result for the general Lp(0, ∞) case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  The plan of this paper is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' In Section 2 we discuss a few preliminaries and  then we state the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' In Section 3 we complete the proofs of the main  theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' In the final section, we will briefly mention some more related inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Preliminaries and main results  Let us begin this section with basic facts about symmetric decreasing rearrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  For more details, we refer to [11, Chapter 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' We denote f ∗ the symmetric decreasing  rearrangement of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' It is well known that f ∗ is a nonnegative, radially symmetric, and  INTEGRAL RELLICH TYPE INEQUALITIES  3  nonincreasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Irrespective of several properties of f ∗, the useful properties  in our context is the equimeasurability property, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  vol({|f| > τ}) = vol({f ∗ > τ}) for all τ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  By using the layer cake representation and the above property, we have the following  helpful identity:  � ∞  0  |f(t)|p dt =  � ∞  0  |f ∗(t)|p dt for all p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='1)  Also it is clear that for any s > 0 there holds  � s  0  |f(t)| dt ≤  � s  0  f ∗(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='2)  These relations will be very much valuable in the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Now, we are ready to state the following important observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Let f be a locally absolutely continuous function on (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Then for  a fixed r > 0 the following identity holds:  sup  0<s<∞ min  �  1, r  s  � � s  0  f ∗(t) dt =  � r  0  f ∗(t) dt,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='3)  where f ∗ is the non-increasing rearrangement of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' We wish to calculate the supremum by using the advantage of the non-increasing  property of f ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' For any fixed r > 0, we consider the following two cases:  Case 1: Let 0 < s ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Then we obtain  min  �  1, r  s  � � s  0  f ∗(t) dt =  � s  0  f ∗(t) dt ≤  � r  0  f ∗(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Case 2: Let r ≤ s < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Then we have by change of variable  min  �  1, r  s  � � s  0  f ∗(t) dt = r  s  � s  0  f ∗(t) dt ≤ r  s  � s  0  f ∗(tr/s) dt =  � r  0  f ∗(v) dv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  In both cases, we get  min  �  1, r  s  � � s  0  f ∗(t) dt ≤  � r  0  f ∗(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Hence the supremum is attained at s = r and we arrive at  sup  0<s<∞ min  �  1, r  s  � � s  0  f ∗(t) dt =  � r  0  f ∗(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  □  Now we are ready to present an improvement of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' That is, this gives a natural  improvement of the Hardy-Rellich inequality in the integral form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Below we will  describe the corresponding differential form which improves the original Hardy-Rellich  inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='3) in a simple form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  4  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' OZAWA, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' ROYCHOWDHURY, AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' SURAGAN  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Let g ∈ C1(0, ∞), then there holds  � ∞  0  sup  0<s<∞  ���� min  �1  r, 1  s  � � r  0  g′(t) dt  ����  2  dr ≤ 4  � ∞  0  |g′(r)|2 dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='4)  Moreover, the constant is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Let f ∈ C2(0, ∞), with f ′(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Then choosing g(r) = f ′(r) in the  improved Hardy-Rellich inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='4), we obtain  � ∞  0  sup  0<s<∞  ���� min  �1  r, 1  s  �  f ′(s)  ����  2  dr ≤ 4  � ∞  0  |f ′′(r)|2 dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='5)  Moreover, it is straightforward to check the following identity  sup  0<s<∞  ���� min  �1  r, 1  s  �  f ′(s)  ����  2  = max  �  sup  0<s≤r  |f ′(s)|2  r2  ,  sup  r≤s<∞  |f ′(s)|2  s2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='6)  Then combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='6) we deduce  � ∞  0  |f ′(r)|2  r2  dr ≤  � ∞  0  max  �  sup  0<s≤r  |f ′(s)|2  r2  ,  sup  r≤s<∞  |f ′(s)|2  s2  �  dr ≤ 4  � ∞  0  |f ′′(r)|2 dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='7)  Therefore, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='7) appears as an immediate improvement of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='3) in the one-dimension  differential form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' The similar discussion gives the improvement of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='4) as well (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  [5, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Now we are going to discuss the second main result of this note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Before presenting  the statement first let us recall the classical one-dimensional Lp-Rellich inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  This reads as follows: let p > 1, f ∈ C2(0, ∞) with f(0) = 0 and f ′(0) = 0 there  holds  � ∞  0  |f(r)|p  r2p  dr ≤  p2p  (p − 1)p(2p − 1)p  � ∞  0  |f ′′(r)|p dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='8)  Now we are ready to demonstrate the one-dimensional Rellich-type inequality in the  following integral form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Let f ∈ Lp(0, ∞), p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Then we have  � ∞  0  1  r2p  � � r  0  � τ  0  |f(t)| dt dτ  �p  dr  ≤  � ∞  0  1  r2p  � � r  0  sup  0<s<∞ min  �  1, τ  s  � � s  0  |f(t)| dt dτ  �p  dr  ≤  p2p  (p − 1)p(2p − 1)p  � ∞  0  |f(r)|p dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='9)  Moreover, the constant is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  INTEGRAL RELLICH TYPE INEQUALITIES  5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Proofs of Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='2  This section is concerned with the proofs of Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Before going  further let us recall the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' [16, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='1] Let 1 < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Let g be any nonnegative function  on (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Assume h is a strictly positive non-decreasing function on (0, ∞) such  that sh(r) ≤ rh(s) for any r, s ∈ (0, ∞) with r ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Let f be a locally absolutely  continuous function on (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Then we have  � ∞  0  g(r) sup  0<s<∞  ���� min  � 1  h(r),  1  h(s)  � � s  0  f(t) dt  ����  p  dr ≤  � ∞  0  g(r)  ����  1  h(r)  � r  0  f ∗(t) dt  ����  p  dr,  where f ∗ is the non-increasing rearrangement of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Now, as a direct consequence of Lemma (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='1), we derive the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='1: Let us consider g(r) = 1 and h(r) = r be functions on  (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Substituting these in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='1 with p = 2, then we have  � ∞  0  sup  0<s<∞  ���� min  �1  r, 1  s  � � s  0  g′(t) dt  ����  2  dr ≤  � ∞  0  1  r2  ����  � r  0  (g′)∗(t) dt  ����  2  dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  By using the Hardy-Rellich inequality in the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='4) for the function (g′)∗, we  obtain  � ∞  0  sup  0<s<∞  ���� min  �1  r, 1  s  � � s  0  g′(t) dt  ����  2  dr ≤ 4  � ∞  0  |(g′)∗(r)|2 dr  = 4  � ∞  0  |g′(r)|2 dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  In the last step, we have used norm preserving property (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' The sharpness follows  from the optimality of the constant in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' It ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='2: The first inequality follows from the property of the  supremum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Now taking the integral of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='3) from 0 to r we have  � r  0  sup  0<s<∞ min  �  1, τ  s  � � s  0  f ∗(t) dt dτ =  � r  0  � τ  0  f ∗(t) dt dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='1)  � ∞  0  1  r2p  � � r  0  sup  0<s<∞  min  �  1, τ  s  � � s  0  |f(t)| dt dτ  �p  dr  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='2)  ≤  � ∞  0  1  r2p  � � r  0  sup  0<s<∞ min  �  1, τ  s  � � s  0  f ∗(t) dt dτ  �p  dr  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='1)  =  � ∞  0  1  r2p  � � r  0  � τ  0  f ∗(t) dt dτ  �p  dr  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='8)  ≤  p2p  (p − 1)p(2p − 1)p  � ∞  0  |f ∗(r)|p dr  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='1)  =  p2p  (p − 1)p(2p − 1)p  � ∞  0  |f(r)|p dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  6  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' OZAWA, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' ROYCHOWDHURY, AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' SURAGAN  This completes the proof of the new type Rellich inequality in the integral form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Optimality: We set  Cp :=  inf  f∈Lp(0,∞)\\{0}  � ∞  0  1  r2p  � � r  0  � τ  0 |f(t)| dt dτ  �p dr  � ∞  0 |f(r)|p dr  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='2)  The validity of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='9) immediately implies  Cp ≤  p2p  (p − 1)p(2p − 1)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  So it remains to show other side and this will be done by giving a proper minimizing  sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' We divide the proof in some steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Let us start with a cut-off function χ : [0, ∞) → R with the following  properties:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' χ(r) ∈ [0, 1] for all r ∈ [0, ∞) and χ is smooth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' χ satisfies the following  χ(r) =  �1  0 ≤ r ≤ 1,  0  2 ≤ r < ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' χ is decreasing function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' χ′(r) ≤ 0 for all r ∈ [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Now for a small ǫ > 0, let us define the minimizing functions {fǫ} as follows:  fǫ(r) := r  ǫ−1  p χ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' In this step we will estimate r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' The denominator of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='2) gives  � ∞  0  |fǫ(r)|p dr =  � ∞  0  rǫ−1χp(r) dr  =  � 1  0  rǫ−1 dr +  � 2  1  rǫ−1χp(r) dr  = 1  ǫ + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='3)  Therefore, for a fixed positive ǫ, we have fǫ ∈ Lp(0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' In this part we will evaluate the numerator of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
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+page_content='p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='ǫ − 1 + p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='�p�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='ǫ − 1 + 2p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='�p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='+ O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='4)  In between, exploiting χ′ ≤ 0, we used a simple inequality (a + b)p ≥ ap twice, for  nonnegative real numbers a and b with p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Finally, by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='4) we deduce the ratio  � ∞  0  1  r2p  � � r  0  � τ  0 |f(t)| dt dτ  �p dr  � ∞  0 |f(r)|p dr  ≥  1  ǫ  �  p  ǫ−1+p  �p�  p  ǫ−1+2p  �p + O(1)  1  ǫ + O(1)  →  p2p  (p − 1)p(2p − 1)p for ǫ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Hence {fǫ} is the required minimizing sequence and, in turn, we have  Cp =  p2p  (p − 1)p(2p − 1)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Few more remarks  Here we shortly revisit some more inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' It hints that rescaling above in-  equalities may generate new inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Let us recall the differential form of the  inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' [9, Theorem 1] Let 1 < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Then, for any locally absolutely  continuous function f on (0, ∞) with lim infr→0 |f(r)| = 0, there holds  � ∞  0  max  �  sup  0<s≤r  |f(s)|p  rp  , sup  r≤s<∞  |f(s)|p  sp  �  dr ≤  �  p  p − 1  �p � ∞  0  |f ′(r)|p dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Now by substituting 1  r  � r  0 f(t) dt and  � r  0 f(t) dt instead of f(r) one can have two  immediate corollaries, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Let 1 < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Then, for any locally absolutely continuous function  f on (0, ∞), there holds  � ∞  0  max  �  sup  0<s≤r  | 1  s  � s  0 f(t) dt|p  rp  , sup  r≤s<∞  | 1  s  � s  0 f(t) dt|p  sp  �  dr  ≤  �  p  p − 1  �p  2p−1  �� ∞  0  |f(r)|p  rp  dr +  � ∞  0  1  r2p  � � r  0  |f(t)|p dt  �  dr  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  8  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' OZAWA, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' ROYCHOWDHURY, AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' SURAGAN  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Let 1 < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Then, for any locally absolutely continuous function  f on (0, ∞), there holds  � ∞  0  max  �  sup  0<s≤r  |  � s  0 f(t) dt|p  rp  , sup  r≤s<∞  |  � s  0 f(t) dt|p  sp  �  dr ≤  �  p  p − 1  �p � ∞  0  |f(r)|p dr  Acknowledgments  This work was partially supported by the NU program 20122022CRP1601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' The  first author is supported in part by JSPS Kakenhi 18KK0073, 19H00644.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' The sec-  ond author is supported in part by National Theoretical Science Research Center  Operational Plan V-Mathematics Field (2/5) (Project number 111-2124-M-002-014-  G58-01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  References  [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Barbatis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Tertikas, On a class of Rellich inequalities, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' 194 (2006),  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' 1, 156–172, MR2230976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  [2] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Berchio, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Ganguly, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Roychowdhury, Hardy-Rellich and second-order Poincar´e identities  on the hyperbolic space via Bessel pairs, Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Partial Differential Equations 61, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' 4,  Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' (2022), MR4417395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  [3] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Bez, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Machihara, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Ozawa, Revisiting the Rellich inequality, to appear in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=',  (2022), arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='02928.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  [4] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Caldiroli, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Musina, Rellich inequalities with weights, Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Partial Differential Equa-  tions 45 (2012), 147–164, MR2957654.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  [5] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Cassano, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Cossetti, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Fanelli, Improved Hardy-Rellich inequalities, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' 21 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' 3, 867–889, MR4389605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  [6] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Cazacu, A new proof of the Hardy-Rellich inequality in any dimension, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Edinburgh Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' A 150 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' 6, 2894–2904, MR4190094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  [7] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Davies, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Hinz, Explicit constants for Rellich inequalities in Lp(Ω), Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' 227  (1998), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' 3, 511–523, MR1612685.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  [8] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Hardy, Note on a theorem of Hilbert, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' 6 (1920), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' 3-4, 314–317, MR1544414.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  [9] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Frank, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Laptev, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Weidl, An improved one-dimensional Hardy inequality, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' 268 (2022), 323–342, Link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Kufner, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Maligranda, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Persson, The prehistory of the Hardy inequality, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Monthly 113 (2006), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' 8, 715–732, MR2256532.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  [11] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Lieb, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Loss, Analysis, Second edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Graduate Studies in Mathematics, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' American  Mathematical Society, Providence, RI, (2001), MR1817225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  [12] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Machihara, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Ozawa, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Wadade, Remarks on the Rellich inequality, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' 286 (2017),  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' 3-4, 1367–1373, MR3671580.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  [13] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Metafune, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Negro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Sobajima, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Spina, Rellich inequalites in bounded domains, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' 379 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' 1-2, 765–824, MR4211104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  [14] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Rellich, Halbbeschr¨ankte Differentialoperatoren h¨oherer Ordnung, (German) Proceedings  of the International Congress of Mathematicians, 1954, Amsterdam, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' III, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' 243-250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Erven P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Noordhoff N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=', Groningen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' North-Holland Publishing Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=', Amsterdam, (1956)  MR0088624.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  [15] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Roychowdhury, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Suragan, Improvement of the discrete Hardy inequality, (2022),  arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='05288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  [16] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Roychowdhury, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Ruzhansky, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Suragan, Multidimensional Frank-Laptev-Weidl improve-  ment of the Hardy Inequality, (2022), arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='08395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  INTEGRAL RELLICH TYPE INEQUALITIES  9  [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Ruzhansky, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Suragan, Hardy and Rellich inequalities, identities, and sharp remainders  on homogeneous groups, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' 317 (2017), 799–822, MR3682685.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  [18] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Ruzhansky, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Suragan, Hardy inequalities on homogeneous groups, 100 years of Hardy  inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Progress in Mathematics, 327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Birkh¨auser/Springer, Cham, (2019), MR3966452.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Tertikas, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Zographopoulos, Best constants in the Hardy–Rellich inequalities and related  improvements, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' 209 (2007), 407–459, MR2296305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  [20] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Yafaev, Sharp constants in the Hardy–Rellich inequalities, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' 168 (1999),  121–144, MR1717839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='  Tohru Ozawa:  Department of Applied Physics  Waseda University, Tokyo 169-8555  Japan  E-mail address txozawa@waseda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='jp  Prasun Roychowdhury:  Mathematics Division: National Center for Theoretical Sciences  NTU, Cosmology Building, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' 1, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=' 4, Roosevelt Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content=', Taipei City 106  Taiwan  E-mail address prasunroychowdhury1994@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='com  Durvudkhan Suragan:  Department of Mathematics  Nazarbayev University  Kazakhstan  E-mail address durvudkhan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='suragan@nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
+page_content='kz' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFJT4oBgHgl3EQfyi0f/content/2301.11638v1.pdf'}
diff --git a/MNFRT4oBgHgl3EQfFjeH/content/tmp_files/2301.13480v1.pdf.txt b/MNFRT4oBgHgl3EQfFjeH/content/tmp_files/2301.13480v1.pdf.txt
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+Noname manuscript No.
+(will be inserted by the editor)
+Monitoring fast solar chromospheric activity: the
+MeteoSpace project
+Jean-Marie Malherbe · Thierry Corbard ·
+Ga¨ele Barbary · Fr´ed´eric Morand · Claude
+Collin · Daniel Crussaire · Florence Guitton
+Received: 15 December 2021 / Accepted: date
+Abstract We present in this reference paper an instrumental project dedicated to the
+monitoring of solar activity during solar cycle 25. It concerns the survey of fast evolv-
+ing chromospheric events implied in Space Weather, such as flares, coronal mass
+ejections, filament instabilities and Moreton waves. Coronal waves are produced by
+large flares around the solar maximum and propagate with chromospheric counter-
+parts; they are rare, faint, difficult to observe, and for that reason, challenging. They
+require systematic observations with automatic, fast and multi-channel optical instru-
+ments. MeteoSpace is a high cadence telescope assembly specially designed for that
+purpose. The large amount of data will be freely available to the solar community.
+We describe in details the optical design, the qualification tests and capabilities of
+J.-M. Malherbe
+LESIA, Observatoire de Paris, PSL Research University, CNRS, Meudon, France
+E-mail: Jean-Marie.Malherbe@obspm.fr
+T. Corbard
+Lagrange, Observatoire et Universit´e de la Cˆote d’Azur, CNRS, Nice, France
+E-mail: Thierry.Corbard@oca.eu
+G. Barbary
+LESIA, Observatoire de Paris, PSL Research University, CNRS, Meudon, France
+E-mail: Gaele.Barbary@obspm.fr
+F. Morand
+Calern station, Observatoire et Universit´e de la Cˆote d’Azur, CNRS, Caussols, France
+E-mail: Frederic.Morand@oca.eu
+C. Collin
+LESIA, Observatoire de Paris, PSL Research University, CNRS, Meudon, France
+E-mail: Claude.Collin@obspm.fr
+D. Crussaire
+LESIA, Observatoire de Paris, PSL Research University, CNRS, Meudon, France
+E-mail: Daniel.Crussaire@obspm.fr
+F. Guitton
+Lagrange, Observatoire et Universit´e de la Cˆote d’Azur, CNRS, Nice, France
+E-mail: florence.Guitton@oca.eu
+arXiv:2301.13480v1  [astro-ph.IM]  31 Jan 2023
+
+2
+J.-M. Malherbe et al.
+the telescopes, and show how waves can be detected. MeteoSpace will be installed at
+Calern observatory (Cˆote d’Azur, 1270 m) and will be in full operation in 2023.
+Keywords Sun – Chromosphere – Instrumentation – Imagery – Solar activity –
+Flares – Moreton waves
+1 Introduction
+Solar activity is the primary driver of Space Weather phenomena, such as Coro-
+nal Mass Ejections (CMEs), flares, energetic particles and wind streams which may
+reach and perturb the Earth environment. The solar perspectives of Space Weather
+are reviewed by Schwenn (2006) and more recently by Temmer (2021), while the ter-
+restrial consequences are discussed by Pulkkinen (2007). Solar activity occurs over
+timescales ranging from seconds to minutes in flares and CMEs, which are fast evolv-
+ing and highly dynamic events, originating in the solar atmosphere from non-potential
+magnetic energy. It can be stored in bright faculae (B ≈ 100 G, 1 G = 10−4 T),
+sunspots (B ≈ 1000 G) and large coronal loops above active regions (review by Reale
+(2010)). The monitoring of solar activity requires a systematic survey of the chromo-
+sphere (8000 K), as it is the source of solar events, which propagate to the hot corona
+above (1-2 MK or more). For that purpose, ground-based networks with many sta-
+tions around the world have been organized, such as the Global Hα network (GHN,
+Steinegger et al. (2000)), the Continuous Hα Imaging Network (CHAIN, Ueno et al.
+(2010)) or the GONG Hα network with seven identical telescopes (Harvey et al.,
+2011). In space, observations of various EUV emission lines, formed in the temper-
+ature range 0.1-10 MK of the corona, started with the SOHO/ESA/NASA mission
+(1996) and continue with the improved Solar Dynamics Observatory (SDO/NASA,
+2010). New data are now available from recent spacecrafts, such as Parker Solar
+Probe (NASA, 2018) and Solar Orbiter (ESA, 2020).
+The Space Weather is mainly affected by CMEs (Chen, 2011) and energetic flares
+(Oloketuyi et al., 2019). The integrated X-ray flux of the Sun is permanently moni-
+tored by the GOES/NASA satellites and is a good indicator of solar activity and the
+flare energy. The flux is usually symbolized by letters A, B, C, M and X, respectively
+for 10−8 to 10−4 W m−2 in logarithmic scale, and a multiplicative number. Only C, M
+and X-class flares are significant. Among them, energetic events (M and X) are rela-
+tively rare (respectively 9% and 1%, with only 50 flares between X2.6 and X28 during
+the past 25 years). The largest recent event (4 November 2003, X28, about 1025 J)
+could be compared to the famous and historic Carrington flare (1 September 1859
+seen in white light, maybe X50). Highest energy flares (X-Class) trigger fast coronal
+MHD shock waves (Warmuth, 2015) which propagate at super-magnetosonic speeds
+(500-1000 km s−1) and can produce chromospheric counterparts, discovered much
+earlier by Moreton (1960). They are often described as the signature of the downward
+compression by the moving coronal shock above. Chromospheric Moreton waves are
+fast and rare phenomena, so that study cases are not very numerous (Zhang (2001),
+Narukage et al. (2004), Balasubramaniam et al. (2007), Zhang et al. (2011), Asai et al.
+(2012), Krause et al. (2015), Admiranto et al. (2015), Krause et al. (2018), Cabezas
+et al. (2019)). The usual observing cadence is about 45-60 s, but faster frame rates (5
+
+The MeteoSpace project
+3
+images/minute or more) would be better to investigate with more details the chronol-
+ogy and the physics of such events.
+The MeteoSpace (MTSP) instrument presented in this paper will provide a unique
+opportunity to monitor highly dynamic phenomena around the next solar maximum
+(2025). Our paper is organized as follows. Section 2 summarizes the architecture
+of the MTSP project, while Section 3 presents the CaII K channel. The goals and
+capabilities of the two Hα telescopes are described in Section 4, while Section 5
+discusses the detection of Moreton waves, which is the most challenging purpose of
+the Hα channels. At last, Section 6 describes a possible extension for the future.
+2 Architecture of the MeteoSpace (MTSP) project
+MTSP has an historical background. The chromosphere is observed daily at Meudon
+with Deslandres’s spectroheliograph since 1908. It produces now (x, y, λ) data-cubes
+from which monochromatic images are derived (Malherbe and Dalmasse, 2019). In
+1957, during the International Geophysical Year (IGY), the french astronomers de-
+cided to organize a survey of Hα chromospheric activity at high cadence (60 s) us-
+ing the Lyot filter technology (Lyot, 1944). Two instruments were built for Meudon
+and Haute Provence observatories (Grenat and Laborde (1954), Michard (1965)). In
+1965, a new generation of tunable Lyot filters allowed to observe sequentially the Hα
+line centre and wings, in order to measure Doppler-shifts. This routine, improved in
+1985, was stopped in 2004, due to the filter obsolescence and the retirement of opera-
+tors. About 7 million images have been recorded. In the competitive context of Space
+Weather research and applications, we wish to restart a survey with new automatic
+telescopes and faster cadence (10-15 s), on a sunnier site in order to increase the
+probability of catching rare events, such as large X-class flares and Moreton waves.
+MTSP is a joint project between Paris (OP) and Cˆote d’Azur (OCA) observatories.
+Overall characteristics are summarized in Malherbe et al. (2019). We provide here
+much more instrumental details in order to constitute a reference paper for MTSP.
+The architecture of the MTSP project (Figure 1) is organized around three main
+tasks:
+– Automation procedures (OCA)
+– The real time data processing and archiving procedures (OCA)
+– The telescopes and data acquisition system (OP)
+The assembly of three telescopes (two Hα 6563 ˚A channels for solar activity,
+and a CaII K 3934 ˚A magnetic proxy) is built by OP. A new housing and equato-
+rial mount have been installed at Calern observatory (1270 m, OCA, Figure 2). The
+instruments (Figure 3) will be integrated mid 2022, then tested and commissioned
+in 2023 for systematic and automatic observations. Data will be freely available to
+the international community, without any delay, both for operational and scientific
+purpose. MSTP comes with a specific database (located at Nice computer centre with
+100 TB of disk storage), with possible access through the BASS2000 solar database,
+and will offer later virtual observatory services.
+
+4
+J.-M. Malherbe et al.
+Fig. 1 The MTSP project architecture. The green box is located at Calern station, and the database at Nice
+observatory. Observations will also be available through the BASS2000 data centre with additional Virtual
+Observatory (VO) services. Real time images (JPEG) together with FITS scientific data-sets will be freely
+delivered to the international community.
+2.1 Automation procedures
+MTSP is the first automatic solar telescope assembly available in France (other solar
+instruments are under human control). The telescopes (Figure 3) and housing (Fig-
+ure 2) are under the supervision of a control/command computer (Figure 1). This PC
+is interconnected with other slave machines, such as the data acquisition computer
+(which handles cameras, filters, motor focus, scintillometers, thermal units), the solar
+tracking computer (which drives the equatorial mount) and the House Keeping (HK)
+automation. HK is a complex system for a fully automatic instrument, because many
+environment detectors and cameras must be controlled permanently, concerning the
+telescope status and meteorology (clouds, temperature, humidity, wind). In particular,
+the HK system will decide to start or interrupt observations according to informations
+provided by the environment parameters. Under good observing conditions, the HK
+will roll the housing and open front curtains; the tracking PC will catch and follow
+the Sun, and the instrument PC will start data acquisition after checking the different
+components. In case of danger (such as wind, rain, snow), the HK procedure will
+return an alarm to the central PC which will interrupt observations, order the mount
+
+House
+Mount
+alpha, delta
+motions
+telescopes
+Solar tracking
+PC
+House keeping
+Ethernet
+CCD, Temp. control
+automation
+switch
+Focusing motors
+Filters control
+Data acquisition
+Scintillometers
+CALERN OBSERVATORY
+Control PC
+PC
+1270 m
+Solar database at
+NICE computer
+BASS2000interface
+centre
+and vO services
+Solar activity
+JPEGRealtime
+FITSdatasets
+forecast
+JPEGRealtime
+No level (12 hours)
+images
+N1level(24hours)
+SCIENTIFIC
+USERSThe MeteoSpace project
+5
+Fig. 2 The MTSP housing. Left: the MTSP building at Calern observatory (1270 m, Cˆote d’Azur). It
+moves on two rails and is computer controlled. It is closed by metallic curtains and connected to many
+environment detectors (such as cameras and meteorological devices) for automatic operation. Right: the
+equatorial mount waiting for the instruments.
+to return to the garage position and ask the HK to close the building. SMS messages
+will inform the technical staff of Calern observatory, in particular in case of alarms.
+2.2 Real time data processing and archiving procedures
+Raw 12 bits data are collected by the acquisition PC and transmitted to the control
+computer. There are two data levels, N0 and N1. N0 are raw data written in FITS
+format. For that purpose, FITS keywords with exhaustive data descriptors are simply
+added and N0 images are directed to the Nice archive via the fast network link be-
+tween the observing site and the data centre. N1 is the first level of scientific data.
+For each image, the Sun radius and centre are detected and the solar disk is centred
+in the 35 arcmin FOV. A rotation is then applied in order to present the North up. The
+dark current is subtracted. Corrections of optical distortions are applied, if needed. A
+8 bits JPEG real time image is derived and is immediately uploaded into the database.
+Keywords are added to scientific images to form FITS data-sets, which will be trans-
+ferred after observations (by night), so that the delay for scientific data will be about
+12 hours. The query system of the Nice database will offer, for each channel, individ-
+ual images or large data-sets (zip or tar.gz files) that will be built on line using the time
+interval provided by the requestor. The database is dimensioned to store all observa-
+tions of cycle 25 and will start to operate in 2023. In parallel, the query system will
+be incorporated to the solar BASS2000 data portal. Progressively, it will offer virtual
+observatory services, so that links to other wavelengths will be offered together with
+MTSP data (such as EUV from SDO/AIA or radio from Nanc¸ay Radioheliograh).
+Observations will start with two operating modes: a standard one in the case of low
+activity level (60 s cadence), and a fast mode (10-15 s) in the case of high level. This
+
+House and mount at CALERN observatory (1270 m)6
+J.-M. Malherbe et al.
+mode will be systematically used at the approach of the next solar maximum (2025),
+or will be triggered by indicators such as the real-time GOES/NASA X-ray flux.
+2.3 The telescopes
+Three telescopes, with equivalent focal length 983 mm, forming a solar image of 9.14
+mm diameter, have been developed (Figure 3). The CaII K line channel provides a
+magnetic proxy. It reveals the magnetized areas such as dark spots (1000 G typical)
+and bright facular regions (100 G typical), through broad-band interference filters
+integrating the line core (K3), K2 close wings (±0.2 ˚A) and partially K1 far wings
+(±1.5 ˚A). The two Hα channels monitor chromospheric activity, such as filament
+instabilities, flares, CMEs and eventually Moreton waves. They use narrow bandpass
+filters. In order to have a compact instrument, we selected Fabry-P´erot filters instead
+of Lyot filters previously used at Meudon.
+The telescopes (except filters) have been simulated with the Zemax software. Fig-
+ure 4 shows the Modulation Transfer Functions (MTF) of the aperture, including the
+CCD sampling effect. The cutoff frequencies are 1.36, 0.815 and 1.02 10−3 km−1,
+respectively for the CaII K (80 mm aperture) and the two Hα channels (80 mm and
+100 mm aperture). The Point Spread Functions (PSF) in the image plane are reported.
+The CaII K and Hα telescopes exhibit Strehl ratios of 88% and 93% respectively, and
+the ensquared energy of a point source is 70% in a box of only 2 × 2 pixels for all
+instruments. There is almost no optical distortion. The RMS radius of spot diagrams
+varies from 32% to 65% of the Airy spot radius from the FOV centre to the border,
+in the case of Hα telescopes.
+The optical design and filter tests are detailed in sections 3 and 4, respectively
+for CaII K and Hα telescopes. However, Figure 5 summarizes briefly the overall
+capabilities of MTSP devices. We have two Hα Fabry-P´erot filters, in pupil plane,
+with respectively 0.46 ˚A FWHM (for line wings, filter 1) and 0.34 ˚A FWHM (for
+line centre, filter 2). We also have two interference CaII K filters, in image plane,
+with respectively 1.5 and 1.4 ˚A FWHM (the second one is a spare). The instruments
+are enclosed inside a temperature regulated box at 30±1◦C by active heating and
+passive cooling, but Fabry-P´erot filters have their own and high precision heating
+oven (typical operating values of 65 and 40◦C respectively for filters 1 and 2). CCDs
+are isolated with their own Peltier and air cooling systems; according to the readout
+noise at 8 MHz pixel rate, their dynamic range is about 2700 (12 bits) for a typical
+signal to noise ratio of 100. Exposure times are a few ms only.
+The Hα telescopes, dedicated to fast solar events, can observe the chromosphere
+(8000 K) at high cadence (10-15 s), this is 4 times faster than GONG observations or
+previous Meudon routines, and 3 times faster than SDO/AIA in EUV emission lines
+of the hot corona. The pixel size is about 1 arcsec (2 times better than the previous
+routines at Meudon); it is difficult to achieve a better resolution with small ground
+based instruments observing the Sun.
+
+The MeteoSpace project
+7
+Fig. 3 The MTSP instruments. Top: the mechanical structure and the three telescopes (1, 2 for Hα and
+3 for CaII K). There is room for a fourth telescope, if needed in the future. The instruments are mounted
+on two XT95 optical benches (aluminium 6061) and are enclosed in a thermally regulated and isolated
+box (not shown). The volume is 0.5 × 0.5 × 1.80 m3 and the mass is about 120 kg. Centre: details of the
+complex mechanical interface between the equatorial mount and the two optical benches. Bottom: details
+of the Hα chambers (see also Figure 8 of Section 4).
+3 The CaII K channel
+The CaII K telescope (Figure 6) has an aperture of 80 mm. It is composed of a Taka-
+hashi FS102 objective (820 mm focal length) followed by a magnifying chamber (×
+1.2), a Barr Associates 1.5 ˚A FWHM filter (in image plane) and an interline cooled
+
+Mount mechanical
+Halphacentre(1),Halphawing (2)andCallK(3)telescopes
+interface
+Chambers and
+Mount
+CCD cameras
+Entranceobjectives
+screw
+XT95
+insulation
+optical
+bench
+supports (Al 6061)
+(AI 6061)
+interface with
+mount (Al6061)
+thermal insulation
+(polyamide PA6C)
+lens (L3)
+field corrector (L4)
+motorfocus
+angular adjustment
+Halpha filter
+fine adjustments
+tube
+CCD
+Halphachamber
+(rotation/translation)8
+J.-M. Malherbe et al.
+Fig. 4 The theoretical MTF of the Hα (red) and CaII K (blue) telescopes. Abscissa: spatial fequency in
+km−1 or in arcsec−1. Cutoff frequencies are indicated (bars). The computed PSFs, in the detector plane,
+are reported (the pixels correspond to the CCD sampling). The Hα PSF is for the border of the FOV, while
+the CaII K PSF is for the centre.
+Fig. 5 The MTSP Filter bandwidths. Filters have been controlled with the high dispersion spectrograph of
+Meudon Solar Tower. Left: Hα line without (top) or with (bottom) filter 1. The pupil integrated transmit-
+tance (solid line, top) over the surface filter is drawn (0.46 ˚A FWHM). Middle: Hα line without (top) or
+with (bottom) filter 2. The pupil transmittance is 0.34 ˚A FWHM. Right: CaII K line as seen through the
+two available filters of respectively 1.5 (filter 1, bottom) and 1.4 ˚A FWHM (spare filter 2, top). The trans-
+mittance (solid line) includes K2v, K3 and K2r features, and partially K1v and K1r (v, r letters meaning
+respectively violet and red wings).
+CCD camera (QSI 690, pixel size = 3.69 µm = 0.78 arcsec). The system is protected
+by an UV/IR cutoff filter. The Airy spot size is 5.9 µm or 1.24 arcsec resolution.
+The interference filters were tested with the powerful 14 m spectrograph of Meudon
+Solar Tower at F/75 (R = 300000, 6 m ˚A spectral line sampling in order 15). We found
+that the central wavelength (CWL) is temperature (T) dependant, according to the law
+∆λCWL = C ∆T, where C = 0.07 ˚A/◦C is the measured temperature coefficient. It is
+also function of the incidence angle θ, and follows the law ∆λCWL = K θ 2 with K =
+
+1.0
+Halpha PSF
+0.8
+0.1
+function
+0.01
+0.6
+ transfer
+0.001
+1 pixel = 4.54 microns
+Call K PSF
+Modulation
+0.4
+0.1
+Ha
+Ha 
+Call K
+0.01
+0.2
+1 pixel = 3.69 microns
+arcsec
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.0000
+0.0002
+0.0004
+0.0006
+0.0008
+0.0010
+0.0012
+0.0014
+Spatial frequency (km")3932
+3933
+3934
+3935
+"K1
+<3
+K1r
+K2r.
+$562
+6563
+6564
+656
+6561
+6562
+6563
+-6564
+C9C9
+CallK-PWHM 1.4 A
+3932
+3933
+3934
+3935
+"K1v
+K3
+K1r
+~-K2V
+K2r.
+HalahaFwHMo.3
+K FWHM 1 5 The MeteoSpace project
+9
+Fig. 6 The MTSP CaII K channel. Top: the overall design of the telescope. Bottom: details of the chamber.
+L2/L3 increase the focal length from 820 to 983 mm. Filters are in the image plane at F/12.3. The chamber
+has a motor focus.
+-0.15 ˚A/degree2. As the theory gives K = - 1
+2
+λ0
+n2 (θ in radians), where n is the effective
+index of refraction and λ0 the line wavelength, we derived from the measurements n
+≈ 2.0. With an aperture of F/12.3 (half cone of 2.3◦), the numeric integration over
+the light cone shows that the CWL is blue-shifted (-0.41 ˚A), but this is almost com-
+pensated (+0.49 ˚A) by the operating temperature (+7◦C above the specification of the
+manufacturer), so that the resulting CWL shift is small in comparison to the FWHM
+(the transmittance of the filter centre is displayed in Figure 5, right). This figure also
+reveals that the surface filter is not uniform. We found that that the CWL varies from
+the centre to the border according to the law ∆λCWL = κ x2, where x is the distance
+to the filter centre. We measured κ = -0.006 ˚A/mm2 and +0.003 ˚A/mm2 respectively
+for filters 1 and 2 (the spare filter). It means that the CWL varies between the centre
+of the solar disk and the limb (0 ≤ x ≤ 4.6 mm), numerically ∆λCWL = -0.13 ˚A or
++0.07 ˚A , respectively for filters 1 and 2. This is less than 10% of the FWHM, so that
+the effect does not appear obviously in solar images. Figure 7 shows a comparison
+between the CaII K3 Meudon spectroheliograms (line centre) and MTSP test images
+(formed at lower altitude due to the larger bandwidth): filaments and prominences
+are no more visible, but sunspots appear more contrasted; bright faculae look sim-
+ilar, so that the CaII K MTSP channel produces a good magnetic proxy. It is close
+to the wavelength integration of the spectroheliograph data-cube (x, y, λ with 0.093
+˚A resolution) over the MTSP transmittance.
+4 The two Hα channels
+The two Hα telescopes (Figure 8) have an aperture of 80 mm and 100 mm, respec-
+tively for filters 1 and 2. They are composed of a Takahashi TSA102 objective (816
+mm focal length) followed by an afocal chamber (magnification 1.2), including a
+DayStar Quantum Pro Fabry-P´erot filter in the pupil plane. This system introduces
+field curvature, so that a field corrector (two lenses) forms the final image on an in-
+
+L2
+L3
+Filters
+CCD
+L1
+587.6mm
+144.9
+mm
+f1=820mm
+L2
+CHAMBER
+L3
+Call K filter
+CCD
+e=2mm
+e=2mm
+e=1mm
+e=9.3mm
++935mm
+5.7mm
+144.9
+mm
+98.6mm
+40mm
+m
+5m
+f3=250mm
+UV/IR cut filter
+f2=-220mm10
+J.-M. Malherbe et al.
+Fig. 7 Comparison of Meudon CaII K3 spectroheliograms and MTSP CaII K images, 26 November 2020.
+(a) K3 spectroheliogram (30 s surface scan by the slit, square root to decrease the dynamics). (b) the
+spectroheliograph (x, y, λ) data-cube integrated over the MTSP wavelength transmission curve. (c) the
+MSTP K image, 1.5 ˚A FWHM (2 ms exposure time, the waveband covers K2v, K3 and K2r central
+features of the CaII K line, and partially K1v and K1r wings).
+Fig. 8 The MTSP Hα channels. The filter is located in the pupil plane at F/30 inside the L2/L3 afocal
+system. L4 is a field curvature corrector. The chamber has a motor focus (see also Figure 3). The equivalent
+focal length is 983 mm, as for CaII K. The beam aperture, in the detector plane, is F/12.3 and F/10.0
+respectively for Hα 1 and 2.
+terline cooled CCD camera (QSI 660, pixel size = 4.54 µm = 0.96 arcsec). The Airy
+spot size is 9.8 and 7.9 µm, corresponding to 2.06 and 1.65 arcsec resolution, respec-
+tively for telescopes 1 and 2. The system is protected by an UV/IR cutoff filter in full
+aperture (not drawn). A motor focus is integrated to the chambers.
+The calibration of Fabry-P´erot filters has been done with the spectrograph of
+Meudon Solar Tower, at F/75 (much better than the F/30 filter specification), in or-
+der 9 (10 m ˚A/pixel). Several tests have been performed for both filters. First of all,
+we made a scan of the surface in order to produce a pupil cartography of the CWL
+and FWHM (Figures 9 and 10). For that purpose, the filters (31 mm diameter) were
+translated (-15 mm ≤ x ≤ 15 mm) in front of the spectrograph slit by ∆x = 3 mm
+steps. (λ, y) spectral images, with and without filters, were recorded, for various x-
+positions (top of figures). The wavelength transmittance was derived at several (x,
+y) locations. The results show that the FWHM of both filters is locally in the range
+0.30-0.35 ˚A, but also that filter 2 is better than filter 1 in terms of CWL uniformity.
+By integration on the surface, we computed the resulting bandpass for pupil plane ap-
+plication in afocal systems, leading to 0.46 and 0.34 ˚A, respectively for filters 1 and
+
+FWHM 1.5 A
+(c) Call K MTSP. FWHM 
+al Call
+(b) Call
+A filteDiaphragm
+Image plane, CCD sensor
+Entrance objective L1
+Halpha157mm
+FOV35'
+Halpha269mm
+Imageplane
+D: 10 mm
+Halpha1:80mm
+Mask
+FILTER
+Halpha2:98mm
+Halpha1&2~11mm
+Diaphragm
+Pupil plane
+f1=816mm
+L4 Field
+Halpha134mm
+corrector
+Halpha
+Halpha240mm
+7
+-50/+60mm
+L2
+chamber
+D:32mm
+D:14mm
+~556,6mm
+D:42mm
+f3=250mm
+motor focus
+f2=250mm
+on L4
+284,6mm
+272mm
+243mm
+331,8mm
+218mm
+272mm
+Halpha
+274mm
+telescopes
+58,3mm
+optical design
+~1745mmThe MeteoSpace project
+11
+Fig. 9 Calibration of Hα filter 1. It was translated by 3 mm steps in the spectrograph focus of the Meudon
+Solar Tower (F/75). Top: the Hα line through the filter for 9 x-positions of the slit (in mm). The white
+lines delineate the CWL and FWHM (y-direction). Bottom: the filter cartography. The local FWHM ( ˚A)
+is displayed at left. CWL fluctuations ( ˚A) are not negligible (at right), so that for pupil plane application,
+the integrated bandpass is enlarged to 0.46 ˚A.
+2. The pupil transmittance is Lorentzian shaped in wavelength and drawn in Figure 5
+for both filters.
+Fabry-P´erot filters have secondary lobes, cut by a blocking filter. We found, for
+filter 2, that they appear at ± 25 ˚A from the main lobe with an intensity of only 0.3%
+(Figure 11). For that filter (0.34 ˚A FWHM), the measurements provide the value
+of the finesse (75), the reflection coefficient of the cavity (0.96) and the interference
+order for Hα line (k = 260).
+Meudon spectroheliograph data, which are made of 3D (x, y, λ) data-cubes (0.155
+˚A wavelength resolution) allow to simulate images using various transmittances. Fig-
+ure 12 displays Hα images with different filter characteristics, such as a Lorentzian
+transmittance (0.34 ˚A FWHM) or a 0.50 ˚A FWHM five stage Lyot transmittance.
+It clearly shows that a smaller bandwidth is needed for the Fabry-P´erot to produce
+contrasts similar to the ones provided by Lyot filters. Indeed, the wavelength curve of
+Lyot filters drastically cuts the line wings, contrarily to Lorentzian filters which have
+extended wings. Hence, the photospheric light passes in excess and contaminates the
+contrast of the chromosphere. When 0.50 ˚A Lyot filters are sufficient to select the
+chromosphere, narrower (0.30 ˚A) Fabry-P´erot devices (such as MTSP filter 2) are
+required for the same result.
+The Solar Tower spectrograph allowed us to explore the angular dependance of
+MTSP filters and precise the tilt sensitivity (θ) in terms of CWL and FWHM fluctu-
+ations. This experience also provided an estimate of the effective index of refraction.
+For that purpose, the filter was tilted in the range -3◦ ≤ θ ≤ +3◦ (Figure 13). The
+mean CWL variations are fitted by the law ∆λCWL = K θ 2 with K = -0.38 or -0.37
+
+x=-12mm
+x= -9mm
+x= -6mm
+x= -3rmm
+x=
+Ormim
+x=
+3mm
+x=
+6mm
+X=
+9mm
+×= 12mm
+Slit pasition (x)
+0.12
+0.37
+0.09
+0.36
+0.06
+0.35
+0.34
+0.03
+FWHM (A)
+0.33
+Surface
+0.00
+CWL
+0.32
+filter
+-0.03
+0.31
+-0.06
+cartography
+0.30
+-0.09
+0.29
+-0.12
+0.28
+1512
+J.-M. Malherbe et al.
+Fig. 10 Calibration of Hα filter 2. Top: the Hα line through the filter for 11 x-positions of the spec-
+trograph slit (in mm). The white lines delineate the CWL and FWHM (y-direction). Bottom: the filter
+cartography. The local FWHM ( ˚A) is displayed at left. At right, the CWL ( ˚A) fluctuations are small, so
+that the integrated bandpass for pupil plane application (0.34 ˚A) remains narrow.
+Fig. 11 Secondary transmission peaks of Hα filter 2 in logarithmic intensity scale (wavelength in ab-
+scissa).
+Fig. 12 Simulation of Hα images obtained after filtering the (x, y, λ) spectroheliograph data-cube. (a)
+Spectroheliogram at line centre, 28 October 2021. (b) Same image, through a 0.34 ˚A Lorentzian transmis-
+sion (such as MTSP filter 2). (c) Same image, with a 0.50 ˚A five stage Lyot filter, for comparison.
+
+Transmission (lambda,y), spectral range (in abscissa) 1.65 A
+x=-15mm
+x=-12mm
+x= -9rmm
+x= -6mm
+x= -3mm
+x= 
+Orrim
+X=
+3mm
+x=
+6mm
+x= 
+9mm
+x= 12rmm
+x= 15rmm
+Slit pasition (x)
+0.12
+0.37
+0.09
+0.36
+0.35
+0.06
+0.03
+0.33
+00'0
+0.32
+-0.03
+FWHM (A)
+Surface
+CWL (A)
+0.31
+-0.06
+filter
+0.30
+-0.09
+cartography
+0.29
+-0.12
+0.28
+-0.15 Log scale, Hydrogen lamp
+-25 A
++25 A
+7.15 A
+7.15
+7.15 Ab Hal
+r FWHM 0.34 A
+(c) Halpl
+FWHM 0.50 AThe MeteoSpace project
+13
+Fig. 13 Effect of the tilt angle on the filter transmission curves (abscissa: wavelength in ˚A). Tilt values are
+0◦ (in red), ± 0.25◦, ± 0.75◦, ± 1.25◦, ± 1.75◦, ± 2.25◦ and -2.75◦. The transmission is blue-shifted with
+increasing tilt. Thick line: the Hα profile. Dashed line: the envelope of the blocking filter.
+˚A/degree2, respectively for filters 1 and 2. For the tilt dependance of the mean band-
+pass, we found FWHM = 0.33 + 0.064 θ 2 and 0.35 + 0.052 θ 2 ( ˚A), respectively for
+filters 1 and 2. This means that, for a 1◦ tilt, the bandpass is locally enlarged to 0.40
+˚A and blue-shifted of about -0.40 ˚A. But in the afocal system, at F/30 (half cone
+angle of 1◦), we have to integrate the above formulae over angles, which generates
+a blue-shift of -0.19 ˚A (this value was confirmed by the wavelength line scan made
+with the filter in imagery mode, which produced results of Figure 15). However, the
+F/30 cone angle is not the only one to consider. The solar diameter is 0.53◦, which
+means that the cone incidence θ varies in the range -0.26◦ ≤ θ ≤ +0.26◦. The cor-
+responding blue-shift is, at maximum, -0.024 ˚A, which can be neglected, so that the
+CWL should be almost uniform over the solar image.
+According to the Fabry-P´erot theory, we have K = - 1
+2
+λ0
+n2 (where λ0 is the line
+wavelength). It allows to derive the effective index of refraction n of the overall filter.
+We found n ≈ 1.62. The distance of secondary lobes is 2en
+k2 = 25 ˚A, where k is the
+interference order (260) and e the thickness of the cavity. Hence, we conclude that e
+≈ 7.2 mm.
+The envelope of transmittance curves of Figure 13 corresponds to the blocking
+filter. The estimated FWHM is about 4.5 ˚A; this value, considering a Lorentzian
+shape, is consistent with the intensity (0.3%) of the secondary peaks of Figure 11.
+The CWL depends on the temperature T. We investigated, by the spectroscopic
+means of the Solar Tower, the effect of temperature changes upon filter 2. The trans-
+mittance for temperatures varying from 38 to 48◦C is reported in Figure 14. The re-
+sponse is a red-shift corresponding to the linear law ∆λCWL = κ T - 3.84 ˚A, where T
+is expressed in ◦C and κ is the temperature coefficient equal to 0.0874 ˚A/◦C. Hence,
+it is possible to explore the ± 1.0 ˚A spectral domain centred on the line, but in
+
+1.5×10
+-0.25
++/-0.75.
+unit)
++/-1.25
+(arbitrary
+1.0x10
+transmission
++/-2.25
+filter
+-2.75
+5.0x1
+ Integrated
+1D 
+0
+6559
+6560
+6561
+6562
+6563
+6564
+Wavelength (A)14
+J.-M. Malherbe et al.
+Fig. 14 Effect of the temperature on the filter transmission curves (abscissa: wavelength in ˚A). Tempera-
+ture values are 38, 39, 40, 41, 41.8, 42.8, 43.8, 44.7, 45.6, 46.6 and 47.5◦C. The transmission is red-shifted
+with increasing temperature. Thick line: the Hα profile. Dashed line: the envelope of the blocking filter.
+practice it is a very slow process. We did not notice any FWHM variation with the
+temperature.
+The last qualification test was performed with the filter alone, in image mode,
+without any spectrograph. Filter 1 was mounted in the MTSP afocal system, and we
+explored the wavelength range 6562.0-6563.6 ˚A by temperature variation. It took
+a long time, because the filter needs 10 minutes to stabilize at each wavelength in-
+crement. We recorded also the fluctuations of the solar flux using a scintillometer, in
+order to correct intensities measured by the filter due to atmospheric variations. Then,
+we derived the Hα line profile at the disk centre (Figure 15). We also plotted the line
+profile got by spectroscopic means, and convolved by the Lorentzian transmittance
+of the filter, and found both results in good agreement, after correction of the -0.2
+˚A shift resulting from the F/30 light cone angle.
+Finally, Figure 16 presents the typical Hα observations that are scheduled with
+the above filters. MTSP will provide almost simultaneously two images, the first one
+in the line wing (filter 1, either the blue or the red wing, but not both), and the sec-
+ond one in the line core (filter 2). We discuss in the next section the application to
+the detection of fast evolving Doppler-shifted events, such as Moreton waves, which
+require at least two Hα channels and the high observing cadence of 10-15 s.
+5 Moreton waves detection
+MTSP is particularly well adapted to the detection of highly dynamic events, such
+as Moreton waves originating in energetic flares. As the filter performances are com-
+parable to those of the previous Meudon routine (1985-2004), we have chosen two
+
+1.5×10
+T= 38 39 40 41 42 43 44 45 46 47 48
+unit)
+1.0x10
+transmission
+filter
+5.0×1
+Integrated
+1D
+0
+6560
+6561
+6562
+6563
+6564
+6565
+Wavelength (A)The MeteoSpace project
+15
+Fig. 15 Hα line profiles got in imagery or spectroscopic mode. Black: MTSP measures resulting from
+filter temperature variation. Red: the line observed with the large spectrograph of Meudon Solar Tower,
+and the convolution by the MTSP filter transmittance (dashed). In abscissa: the wavelength ( ˚A) and the
+corresponding LOS velocity (km s−1).
+Fig. 16 Example of MTSP Hα images got during the test campaign of 29 August 2017 with filter 1 (0.46
+˚A FWHM). (a) MTSP, blue wing. (b) MTSP, line centre (0.34 ˚A FWHM filter 2 will be used instead in
+full operation). (c) Meudon spectroheliogram for comparison.
+events observed with this instrument in order to anticipate MTSP capabilities. More-
+ton waves appear in Hα as fronts propagating at typically 500 km s−1. Such veloc-
+ities in the chromosphere (8000 K) are so highly supersonic (Cs ≈ 10 km s−1) and
+superalv´enic (Ca ≈ 10 km s−1 for a 10 G magnetic field) that they are unlikely of
+chromospheric nature. Such phenomena last only a few minutes and are suspected to
+be the chromospheric counterpart of coronal waves propagating in the 200 times hot-
+ter (1.5 MK) and more tenuous corona under the form of fast magnetosonic shocks
+(Cs ≈ 150 km−1, Ca ≈ 200 km s−1 for a 10 G field). The downward compression
+of the chromosphere below the front could be the signature of the coronal shock in
+Hα. Moreton waves occur mainly in X-class flares which are rare events (about one
+event/year above X5, six events above X10 since year 2000). We have chosen the
+
+8x104
+6×104
+H20
+units)
+(arbitrary
+H20
+H20
+4×104
+Intensity
+2×104
+Halpha
+LOs Velocity (km/s)
+60
+-40
+-20
+20
+40
+60
+0
+6561.5
+6562.0
+6562.5
+6563.0
+6563.5
+6564.0
+6564.5
+Wavelength (A)-0.5 A, MTSP 0.46 A FWHM
+(b) Halpha
+MTSP 0.46 A FWHM
+(c) Halpha16
+J.-M. Malherbe et al.
+famous X17.2 flare of 28 October 2003 (Figure 17), after the solar maximum of cy-
+cle 23, and the X1.8 event of 14 October 1999, just before the maximum (Figure 18).
+Largest flares often occur during the descending phase of the solar cycle. In Figure 17
+(b,c,d respectively for Hα centre and ±0.5 ˚A), we have subtracted the reference frame
+just before the event at the same wavelength. Blue/red colours indicate the sign of the
+resulting image (blue, or positive, means brighter; red, or negative, means darker).
+The wave (yellow box) appears as a brightening in Hα core (b). In the red wing sub-
+traction (c), the compression front (R) is negative (or darker) and corresponds to a
+red-shift. It is followed by the relaxation of the chromosphere (B) appearing positive
+(or brighter), because blue-shifted. In the blue wing subtraction (d), this is the con-
+trary: R is positive (or brighter), because red-shifted, while B is negative (or darker)
+with the opposite shift. Figure 15 shows that the order of magnitude of LOS velocities
+for 0.5 ˚A shifts is qualitatively 20 km s−1. This example shows that Moreton waves
+can be detected either by the red or the blue wing. For that reason, MTSP offers only
+one wing.
+Figure 18 presents the X1.8 flare of 14 October 1999, almost 10 times less en-
+ergetic than the previous one, so that the Moreton event is less contrasted. It shows
+exactly what will provide MTSP concerning wave detection with one wing. Most
+events are waited between 2024 and 2028 around the solar maximum (2025) of cycle
+25 (but isolated events, as the X1.0 of 28 October 2021, are possible). The Hα centre
+(a) will be provided by telescope 2, from which a reference frame (just before the
+event) has been subtracted in (b), showing the brightening front. The Moreton event
+appears clearly in the red wing with the red-shifted compression front (R) followed
+by the blue-shifted relaxation front (B). The blue wing could either be chosen. The
+big difference with the 1985-2004 Meudon routine is the observing cadence (4 times
+faster) and the spatial resolution (2 times better). Hence, MTSP will provide much
+more detailed informations to investigate the physics of rare phenomena at the Sun.
+6 A possible extension for MTSP
+We own a NaD1 5896 ˚A Fabry-P´erot filter manufactured by DayStar (0.36 ˚A FWHM),
+for mounting in an afocal design at F/30 similar to the one of the Hα telescopes de-
+scribed above. Magnetograms of the Sun have been successfully produced in this
+line (Land´e factor 1.33) by Mount Wilson (full disk) or by the Narrow-band Filter
+Imager (NFI) onboard HINODE for small regions. NaD1 is a Fraunhofer line formed
+in the low chromosphere above the FeI 6173 ˚A line observed in the photosphere by
+the Helioseismic and Magnetic Imager (HMI) onboard SDO. As DayStar filters are
+linearly polarizing (made of a birefringent material), the incorporation of a Liquid
+Crystal Variable Retarder (LCVR) at the primary focus, providing ± λ
+4 fast modula-
+tion (Malherbe et al., 2007) suffice to produce alternatively I+V and I-V images (I,
+V for Stokes parameters). From the circular polarization rate V
+I and the weak field
+theory (see Stenflo (1994)), it is possible to estimate the LOS magnetic field (BLOS)
+together with the field polarity. Indeed, V
+I is proportional to BLOS and 1
+I
+dI
+dλ . This
+quantity depends on the wavelength and is maximum when measured at the inflexion
+points of the line. Figure 19 displays spectroscopic observations obtained with this
+
+The MeteoSpace project
+17
+Fig. 17 Moreton event of 28 October 2003 at 11:07 UT. Images, in Hα centre, red and blue wings, were
+taken with 60 s time step. (a) Hα centre. (b) Hα centre, minus the reference frame at 11:00 UT (event
+starting time). (c) Hα red wing, minus the reference. (d) Hα blue wing, minus the reference. F, R, B
+indicate respectively the flare location, red and blue shifts of the compression and relaxation fronts of the
+Moreton wave (propagation direction = green arrow).
+Fig. 18 Moreton event of 14 October 1999 at 09:02 UT. Images, in Hα centre and red wing, were taken
+with 60 s time step. (a) Hα centre. (b) Hα centre, minus the reference frame at 09:00 UT (event starting
+time). (c) Hα red wing, minus the reference. F, R, B indicate respectively the flare location, red and blue
+shifts of the propagating front (propagation direction = green arrow).
+
+(a)
+Date.tirme=031028.110712
+Halpha centre
+Date.time=031028.
+Halpha centre
+d
+Dgtetime=031028
+Red wing
+Date time=031028
+Blue wing(a)
+(b)
+Date.time=991014.090247
+Halpha centre
+Date.time=99101
+Halpha centre
+Date.time=g91
+Red wing18
+J.-M. Malherbe et al.
+method at the 8 m spectrograph of the Pic du Midi Turret Dome. We have numeri-
+cally at the inflexion points V
+I ≈ 2.210−4BLOS, where BLOS is expressed in Gauss.
+As the slope of NaD1 wings is steep, Stokes V profiles are sharp. We observed po-
+larization rates up to 0.20 (BLOS ≈ 1000 G) in some sunspots. In order to estimate
+the measurement capabilities in imagery, we have integrated the line profiles over
+the wavelength transmission of the NaD1 filter. The V
+I signal becomes much smaller,
+because the filter FWHM (W1) is much larger than the half width (W2) of the line
+derivative dI
+dλ , which concentrates the polarimetric signal in a very narrow wave-band
+(Figure 19). Hence, the corrective factor to apply to the polarization rate is roughly
+equal to the W1
+W2 ratio and plotted in Figure 20 for various CWL and W1 values. Despite
+of the signal loss due to wavelength integration, it is of interest to consider polarimet-
+ric measurements in imagery. For that purpose, a NaD1 telescope will be tested at
+Meudon in the coming year in the context of a possible extension for MTSP. As the
+polarization rate to measure with the filter has the same order of magnitude than the
+photon noise of a single exposure (1% or 400 G), it is necessary to acquire many
+frames in order to select best images and reduce the noise, as done by Roudier et al.
+(2006). For example, 100 couples (I+V, I-V) will decrease the noise to 40 G. 20 G
+should be achieved with either 2 × 2 binning or with 400 couples. It must be noticed
+that a fast observing cadence is not required for LOS magnetograms, because the
+magnetic field evolves on longer time scales. The method is also, in principle, valid
+for the Hα telescopes, but in practice, Hα is a broad line and the sensitivity to the
+magnetic field is reduced by the factor 6 in comparison to NaD1.
+7 Conclusion
+The MTSP project is dedicated to the survey of fast evolving events in the chromo-
+sphere at the source of solar activity, such as flares and coronal mass ejections. Large
+flares often occur after the solar maximum (2025 for the present cycle) during a few
+years. MTSP, with an outstanding cadence of 10-15 s, has also the major goal to
+investigate Moreton waves, which are extremely fast and rare phenomena associated
+with largest flares. Such events are difficult to detect in the chromosphere, so that only
+a few cases have been studied. With systematic, fast and multi-channel observations
+(two Hα and one CaII K telescopes), MTSP will increase the chances to catch such
+phenomena. Data could be combined to SDO/AIA observations of the low corona
+at slower cadence (45 s) in several EUV channels. MTSP will operate automatically
+at Calern observatory (1270 m) under good seeing and climatic conditions. High ca-
+dence observations will be freely delivered, without any delay, to the international
+community through a dedicated database located at Nice computer centre. MTSP
+will cover cycle 25, from 2023 to, at least, the end of the present decade, where new
+generation solar synoptic networks could start, such as the Next Generation GONG
+project (Hill et al., 2019) or the Solar Physics Research Integrated Network Group
+(SPRING, Gosain et al. (2018)). MSTP will support Solar Orbiter (ESA) and Parker
+Solar Probe (NASA) operations in the coming years.
+
+The MeteoSpace project
+19
+Fig. 19 Simulation of polarization measurement with the 0.36 ˚A FWHM NaD1 filter. Left: spectra of I+V
+and I-V got at Pic du Midi in an active region (wavelength in abscissa, direction of the slit in ordinates,
+spectral pixel 16.8 m ˚A). Middle: the polarization rate V
+I with the quantity 1
+I
+dI
+dλ in the quiet Sun (yellow)
+and the wavelength transmittance of the filter, centred on the blue wing (green, dashed). Right: the polar-
+ization rate V
+I along the slit in the blue wing measured with the spectrograph (solid line) or extrapolated
+for the filter (dotted line); the dashed line is a magnification of the dotted line (ratio given by Figure 20) to
+recover roughly the spectroscopic signal.
+Fig. 20 Simulation of the correction factor to apply to the measurements of the polarization rate with a
+Lorentzian filter, as a function of the FWHM ( ˚A), for various CWL shifts (solid, dashed, dotted lines for
+respectively -0.12, -0.20, -0.30 ˚A CWL shifts). The blue inflexion point of the line is located at -0.12 ˚A.
+The red (dashed) line indicates our filter FWHM; the best CWL shift is -0.20 ˚A.
+
+1.86 A
+0.2
+0.1
+rate
+0.0
+0.1
+-
+-0.2
+0.36 A FWHM filter
+0
+50
+100
+150
+200
+I+V
+I-V
+V/I
+Y-direction (arcsec)15
+10
+-0.30 A
+-0.20 A
+-0.12 A
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+Lorentzian filter FWHM (A)20
+J.-M. Malherbe et al.
+8 Disclosure of potential conflicts of interest
+The authors declare that they have no conflicts of interest.
+9 Data availability statement
+The authors declare that the datasets analysed during the current study are publicly
+available from the corresponding author upon request.
+Acknowledgements We thank the anonymous referee for useful suggestions and comments. We are in-
+debted to Y. Bresson, C. Renaud (OCA), J.-M. Rees, C. Blanchard (OP) and the technical teams of OP
+and OCA for their assistance. We are also grateful for financial support to Paris and Nice observato-
+ries, the Direction G´en´erale de l’Armement, the IDEX Plas@Par, the Programme National Soleil Terre
+(PNST/INSU/CNRS), the DIM ACAV (Ile de France Region) and the IDEX UCA/JEDI.
+References
+Admiranto AG, Priyatikanto R, Yus’an U, Puspitaningrum E (2015) Moreton waves
+and EIT waves related to the flare events of June 3, 2012 and July 6, 2012. In: The
+5th International Conference on Mathematics and Natural Sciences, American In-
+stitute of Physics Conference Series, vol 1677, p 050014, DOI 10.1063/1.4930675,
+1502.04039
+Asai A, Ishii TT, Isobe H, Kitai R, Ichimoto K, UeNo S, Nagata S, Morita S, Nishida
+K, Shiota D, Oi A, Akioka M, Shibata K (2012) First Simultaneous Observation
+of an Hα Moreton Wave, EUV Wave, and Filament/Prominence Oscillations. As-
+trophys. J. Lett.745(2):L18, DOI 10.1088/2041-8205/745/2/L18, 1112.5915
+Balasubramaniam KS, Pevtsov AA, Neidig DF (2007) Are Moreton Waves Coronal
+Phenomena? Astrophys. J.658(2):1372–1379, DOI 10.1086/512001
+Cabezas DP, Asai A, Ichimoto K, Sakaue T, UeNo S, Ishitsuka JK, Shibata K
+(2019) Dynamic Processes of the Moreton Wave on 2014 March 29. Astrophys.
+J.883(1):32, DOI 10.3847/1538-4357/ab3a35, 1908.03534
+Chen PF (2011) Coronal Mass Ejections: Models and Their Observational Basis.
+Living Reviews in Solar Physics 8(1):1, DOI 10.12942/lrsp-2011-1
+Gosain S, Roth M, Hill F, Pevtsov A, Martinez Pillet V, Thompson MJ (2018) Design
+of a next generation synoptic solar observing network: solar physics research inte-
+grated network group (SPRING). In: Evans CJ, Simard L, Takami H (eds) Ground-
+based and Airborne Instrumentation for Astronomy VII, Society of Photo-Optical
+Instrumentation Engineers (SPIE) Conference Series, vol 10702, p 107024H, DOI
+10.1117/12.2306555
+Grenat H, Laborde G (1954) H´eliographe Monochromatique de Lyot. Annales
+d’Astrophysique 17:541
+Harvey JW, Bolding J, Clark R, Hauth D, Hill F, Kroll R, Luis G, Mills N, Purdy
+T, Henney C, Holland D, Winter J (2011) Full-disk Solar H-alpha Images From
+GONG. In: AAS/Solar Physics Division Abstracts #42, AAS/Solar Physics Divi-
+sion Meeting, vol 42, p 17.45
+
+The MeteoSpace project
+21
+Hill F, Hammel H, Martinez-Pillet V, de Wijn A, Gosain S, Burkepile J, Henney CJ,
+McAteer J, Bain HM, Manchester W, Lin H, Roth M, Ichimoto K, Suematsu Y
+(2019) ngGONG: The Next Generation GONG - A New Solar Synoptic Observa-
+tional Network. In: Bulletin of the American Astronomical Society, vol 51, p 74
+Krause G, C´ecere M, Francile C, Costa A, Elaskar S, Schneiter M (2015) Two
+step mechanism for Moreton wave excitations in a blast-wave scenario: the 2006
+December 06 case study. Mon. Not. Roy. Astron. Soc.453(3):2799–2807, DOI
+10.1093/mnras/stv1827, 1505.02723
+Krause G, C´ecere M, Zurbriggen E, Costa A, Francile C, Elaskar S (2018) Are CMEs
+capable of producing Moreton waves? A case study: the 2006 December 6 event.
+Mon. Not. Roy. Astron. Soc.474(1):770–778, DOI 10.1093/mnras/stx2817
+Lyot B (1944) Le Filtre Monochromatique Polarisant et ses Applications en Physique
+Solaire. Annales d’Astrophysique 7:31
+Malherbe JM, Dalmasse K (2019) The New 2018 Version of the Meudon Spectrohe-
+liograph. Solar Phys.294(5):52, DOI 10.1007/s11207-019-1441-7
+Malherbe JM, Roudier T, Moity J, Mein P, Arnaud J, Muller R (2007) Spectro po-
+larimetry with liquid crystals . Mem. Societa Astronomica Italiana.78:203
+Malherbe JM, Corbard T, Dalmasse K, Meteospace Team (2019) Meteospace, a New
+Instrument for Solar Survey at the Calern Observatory. Solar Phys.294(12):177,
+DOI 10.1007/s11207-019-1569-5
+Michard R (1965) Nouvel H´eliographe `a l’Observatoire de Meudon. L’Astronomie
+79:131
+Moreton GE (1960) Hα Observations of Flare-Initiated Disturbances with Velocities
+˜1000 km/sec. Astron. J.65:494, DOI 10.1086/108346
+Narukage N, Eto S, Kadota M, Kitai R, Kurokawa H, Shibata K (2004) Moreton
+waves observed at Hida Observatory. In: Stepanov AV, Benevolenskaya EE, Koso-
+vichev AG (eds) Multi-Wavelength Investigations of Solar Activity, vol 223, pp
+367–370, DOI 10.1017/S1743921304006143
+Oloketuyi J, Liu Y, Zhao M (2019) The Periodic and Temporal Behaviors of Solar
+X-Ray Flares in Solar Cycles 23 and 24. Astrophys. J.874(1):20, DOI 10.3847/
+1538-4357/ab064c
+Pulkkinen T (2007) Space Weather: Terrestrial Perspective. Living Reviews in Solar
+Physics 4(1):1, DOI 10.12942/lrsp-2007-1
+Reale F (2010) Coronal Loops: Observations and Modeling of Confined Plasma. Liv-
+ing Reviews in Solar Physics 7(1):5, DOI 10.12942/lrsp-2010-5, 1010.5927
+Roudier T, Malherbe JM, Moity J, Rondi S, Mein P, Coutard C (2006) Sub arcsec
+evolution of solar magnetic fields. Astron. Astrophys.455(3):1091–1098, DOI 10.
+1051/0004-6361:20064963
+Schwenn R (2006) Space Weather: The Solar Perspective. Living Reviews in Solar
+Physics 3(1):2, DOI 10.12942/lrsp-2006-2
+Steinegger M, Hanslmeier A, Otruba W, Freislich H, Denker C, Goode PR, Marquette
+WM, Varied J, Wang H, Luo G, Chen D, Zhang Q (2000) An Overview of the New
+Global High-Resolution H-alpha Network. Hvar Observatory Bulletin 24(1):179
+Stenflo J (1994) Solar Magnetic Fields: Polarized Radiation Diagnostics, vol 189.
+DOI 10.1007/978-94-015-8246-9
+Temmer M (2021) Space weather: the solar perspective. Living Reviews in Solar
+
+22
+J.-M. Malherbe et al.
+Physics 18(1):4, DOI 10.1007/s41116-021-00030-3, 2104.04261
+Ueno S, Shibata K, Ichimoto K, Kitai R, Nagata S, Kimura G, Nakatani Y (2010)
+Continuous H-alpha Imaging Network Project (CHAIN) with Ground- based Solar
+Telescopes for Space Weather Research. African Skies 14:17
+Warmuth A (2015) Large-scale Globally Propagating Coronal Waves. Living Re-
+views in Solar Physics 12(1):3, DOI 10.1007/lrsp-2015-3
+Zhang H (2001) Moreton wave and its source of disturbances in the X12/3B
+WLF of AR6659 in 1991 June 4. Astron. Astrophys.372:676–685, DOI 10.1051/
+0004-6361:20010379
+Zhang Y, Kitai R, Narukage N, Matsumoto T, Ueno S, Shibata K, Wang J (2011)
+Propagation of Moreton Waves. Pub. Astron. Soc. Japan63:685, DOI 10.1093/
+pasj/63.3.685
+
diff --git a/MNFRT4oBgHgl3EQfFjeH/content/tmp_files/load_file.txt b/MNFRT4oBgHgl3EQfFjeH/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..7439fa9d23e9cecd6333ad0ccae579ba51f13794
--- /dev/null
+++ b/MNFRT4oBgHgl3EQfFjeH/content/tmp_files/load_file.txt
@@ -0,0 +1,795 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf,len=794
+page_content='Noname manuscript No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  (will be inserted by the editor)  Monitoring fast solar chromospheric activity: the  MeteoSpace project  Jean-Marie Malherbe · Thierry Corbard ·  Ga¨ele Barbary · Fr´ed´eric Morand · Claude  Collin · Daniel Crussaire · Florence Guitton  Received: 15 December 2021 / Accepted: date  Abstract We present in this reference paper an instrumental project dedicated to the  monitoring of solar activity during solar cycle 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' It concerns the survey of fast evolv-  ing chromospheric events implied in Space Weather, such as flares, coronal mass  ejections, filament instabilities and Moreton waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Coronal waves are produced by  large flares around the solar maximum and propagate with chromospheric counter-  parts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' they are rare, faint, difficult to observe, and for that reason, challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' They  require systematic observations with automatic, fast and multi-channel optical instru-  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' MeteoSpace is a high cadence telescope assembly specially designed for that  purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The large amount of data will be freely available to the solar community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  We describe in details the optical design, the qualification tests and capabilities of  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Malherbe  LESIA, Observatoire de Paris, PSL Research University, CNRS, Meudon, France  E-mail: Jean-Marie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='Malherbe@obspm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='fr  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Corbard  Lagrange, Observatoire et Universit´e de la Cˆote d’Azur, CNRS, Nice, France  E-mail: Thierry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='Corbard@oca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='eu  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Barbary  LESIA, Observatoire de Paris, PSL Research University, CNRS, Meudon, France  E-mail: Gaele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='Barbary@obspm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='fr  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Morand  Calern station, Observatoire et Universit´e de la Cˆote d’Azur, CNRS, Caussols, France  E-mail: Frederic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='Morand@oca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='eu  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Collin  LESIA, Observatoire de Paris, PSL Research University, CNRS, Meudon, France  E-mail: Claude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='Collin@obspm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='fr  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Crussaire  LESIA, Observatoire de Paris, PSL Research University, CNRS, Meudon, France  E-mail: Daniel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='Crussaire@obspm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='fr  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Guitton  Lagrange, Observatoire et Universit´e de la Cˆote d’Azur, CNRS, Nice, France  E-mail: florence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='Guitton@oca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='eu  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='13480v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='IM]  31 Jan 2023  2  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Malherbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  the telescopes, and show how waves can be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' MeteoSpace will be installed at  Calern observatory (Cˆote d’Azur, 1270 m) and will be in full operation in 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Keywords Sun – Chromosphere – Instrumentation – Imagery – Solar activity –  Flares – Moreton waves  1 Introduction  Solar activity is the primary driver of Space Weather phenomena, such as Coro-  nal Mass Ejections (CMEs), flares, energetic particles and wind streams which may  reach and perturb the Earth environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The solar perspectives of Space Weather  are reviewed by Schwenn (2006) and more recently by Temmer (2021), while the ter-  restrial consequences are discussed by Pulkkinen (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Solar activity occurs over  timescales ranging from seconds to minutes in flares and CMEs, which are fast evolv-  ing and highly dynamic events, originating in the solar atmosphere from non-potential  magnetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' It can be stored in bright faculae (B ≈ 100 G, 1 G = 10−4 T),  sunspots (B ≈ 1000 G) and large coronal loops above active regions (review by Reale  (2010)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The monitoring of solar activity requires a systematic survey of the chromo-  sphere (8000 K), as it is the source of solar events, which propagate to the hot corona  above (1-2 MK or more).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' For that purpose, ground-based networks with many sta-  tions around the world have been organized, such as the Global Hα network (GHN,  Steinegger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' (2000)), the Continuous Hα Imaging Network (CHAIN, Ueno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  (2010)) or the GONG Hα network with seven identical telescopes (Harvey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=',  2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' In space, observations of various EUV emission lines, formed in the temper-  ature range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='1-10 MK of the corona, started with the SOHO/ESA/NASA mission  (1996) and continue with the improved Solar Dynamics Observatory (SDO/NASA,  2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' New data are now available from recent spacecrafts, such as Parker Solar  Probe (NASA, 2018) and Solar Orbiter (ESA, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  The Space Weather is mainly affected by CMEs (Chen, 2011) and energetic flares  (Oloketuyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The integrated X-ray flux of the Sun is permanently moni-  tored by the GOES/NASA satellites and is a good indicator of solar activity and the  flare energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The flux is usually symbolized by letters A, B, C, M and X, respectively  for 10−8 to 10−4 W m−2 in logarithmic scale, and a multiplicative number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Only C, M  and X-class flares are significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Among them, energetic events (M and X) are rela-  tively rare (respectively 9% and 1%, with only 50 flares between X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='6 and X28 during  the past 25 years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The largest recent event (4 November 2003, X28, about 1025 J)  could be compared to the famous and historic Carrington flare (1 September 1859  seen in white light, maybe X50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Highest energy flares (X-Class) trigger fast coronal  MHD shock waves (Warmuth, 2015) which propagate at super-magnetosonic speeds  (500-1000 km s−1) and can produce chromospheric counterparts, discovered much  earlier by Moreton (1960).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' They are often described as the signature of the downward  compression by the moving coronal shock above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Chromospheric Moreton waves are  fast and rare phenomena, so that study cases are not very numerous (Zhang (2001),  Narukage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' (2004), Balasubramaniam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' (2007), Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' (2011), Asai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  (2012), Krause et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' (2015), Admiranto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' (2015), Krause et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' (2018), Cabezas  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The usual observing cadence is about 45-60 s, but faster frame rates (5  The MeteoSpace project  3  images/minute or more) would be better to investigate with more details the chronol-  ogy and the physics of such events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  The MeteoSpace (MTSP) instrument presented in this paper will provide a unique  opportunity to monitor highly dynamic phenomena around the next solar maximum  (2025).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Our paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Section 2 summarizes the architecture  of the MTSP project, while Section 3 presents the CaII K channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The goals and  capabilities of the two Hα telescopes are described in Section 4, while Section 5  discusses the detection of Moreton waves, which is the most challenging purpose of  the Hα channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' At last, Section 6 describes a possible extension for the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  2 Architecture of the MeteoSpace (MTSP) project  MTSP has an historical background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The chromosphere is observed daily at Meudon  with Deslandres’s spectroheliograph since 1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' It produces now (x, y, λ) data-cubes  from which monochromatic images are derived (Malherbe and Dalmasse, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' In  1957, during the International Geophysical Year (IGY), the french astronomers de-  cided to organize a survey of Hα chromospheric activity at high cadence (60 s) us-  ing the Lyot filter technology (Lyot, 1944).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Two instruments were built for Meudon  and Haute Provence observatories (Grenat and Laborde (1954), Michard (1965)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' In  1965, a new generation of tunable Lyot filters allowed to observe sequentially the Hα  line centre and wings, in order to measure Doppler-shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' This routine, improved in  1985, was stopped in 2004, due to the filter obsolescence and the retirement of opera-  tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' About 7 million images have been recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' In the competitive context of Space  Weather research and applications, we wish to restart a survey with new automatic  telescopes and faster cadence (10-15 s), on a sunnier site in order to increase the  probability of catching rare events, such as large X-class flares and Moreton waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  MTSP is a joint project between Paris (OP) and Cˆote d’Azur (OCA) observatories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Overall characteristics are summarized in Malherbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' We provide here  much more instrumental details in order to constitute a reference paper for MTSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  The architecture of the MTSP project (Figure 1) is organized around three main  tasks:  – Automation procedures (OCA)  – The real time data processing and archiving procedures (OCA)  – The telescopes and data acquisition system (OP)  The assembly of three telescopes (two Hα 6563 ˚A channels for solar activity,  and a CaII K 3934 ˚A magnetic proxy) is built by OP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' A new housing and equato-  rial mount have been installed at Calern observatory (1270 m, OCA, Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The  instruments (Figure 3) will be integrated mid 2022, then tested and commissioned  in 2023 for systematic and automatic observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Data will be freely available to  the international community, without any delay, both for operational and scientific  purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' MSTP comes with a specific database (located at Nice computer centre with  100 TB of disk storage), with possible access through the BASS2000 solar database,  and will offer later virtual observatory services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  4  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Malherbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' 1 The MTSP project architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The green box is located at Calern station, and the database at Nice  observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Observations will also be available through the BASS2000 data centre with additional Virtual  Observatory (VO) services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Real time images (JPEG) together with FITS scientific data-sets will be freely  delivered to the international community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='1 Automation procedures  MTSP is the first automatic solar telescope assembly available in France (other solar  instruments are under human control).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The telescopes (Figure 3) and housing (Fig-  ure 2) are under the supervision of a control/command computer (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' This PC  is interconnected with other slave machines, such as the data acquisition computer  (which handles cameras, filters, motor focus, scintillometers, thermal units), the solar  tracking computer (which drives the equatorial mount) and the House Keeping (HK)  automation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' HK is a complex system for a fully automatic instrument, because many  environment detectors and cameras must be controlled permanently, concerning the  telescope status and meteorology (clouds, temperature, humidity, wind).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' In particular,  the HK system will decide to start or interrupt observations according to informations  provided by the environment parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Under good observing conditions, the HK  will roll the housing and open front curtains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' the tracking PC will catch and follow  the Sun, and the instrument PC will start data acquisition after checking the different  components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' In case of danger (such as wind, rain, snow), the HK procedure will  return an alarm to the central PC which will interrupt observations, order the mount  House  Mount  alpha, delta  motions  telescopes  Solar tracking  PC  House keeping  Ethernet  CCD, Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' control  automation  switch  Focusing motors  Filters control  Data acquisition  Scintillometers  CALERN OBSERVATORY  Control PC  PC  1270 m  Solar database at  NICE computer  BASS2000interface  centre  and vO services  Solar activity  JPEGRealtime  FITSdatasets  forecast  JPEGRealtime  No level (12 hours)  images  N1level(24hours)  SCIENTIFIC  USERSThe MeteoSpace project  5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' 2 The MTSP housing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Left: the MTSP building at Calern observatory (1270 m, Cˆote d’Azur).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' It  moves on two rails and is computer controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' It is closed by metallic curtains and connected to many  environment detectors (such as cameras and meteorological devices) for automatic operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Right: the  equatorial mount waiting for the instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  to return to the garage position and ask the HK to close the building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' SMS messages  will inform the technical staff of Calern observatory, in particular in case of alarms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='2 Real time data processing and archiving procedures  Raw 12 bits data are collected by the acquisition PC and transmitted to the control  computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' There are two data levels, N0 and N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' N0 are raw data written in FITS  format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' For that purpose, FITS keywords with exhaustive data descriptors are simply  added and N0 images are directed to the Nice archive via the fast network link be-  tween the observing site and the data centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' N1 is the first level of scientific data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  For each image, the Sun radius and centre are detected and the solar disk is centred  in the 35 arcmin FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' A rotation is then applied in order to present the North up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The  dark current is subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Corrections of optical distortions are applied, if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' A  8 bits JPEG real time image is derived and is immediately uploaded into the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Keywords are added to scientific images to form FITS data-sets, which will be trans-  ferred after observations (by night), so that the delay for scientific data will be about  12 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The query system of the Nice database will offer, for each channel, individ-  ual images or large data-sets (zip or tar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='gz files) that will be built on line using the time  interval provided by the requestor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The database is dimensioned to store all observa-  tions of cycle 25 and will start to operate in 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' In parallel, the query system will  be incorporated to the solar BASS2000 data portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Progressively, it will offer virtual  observatory services, so that links to other wavelengths will be offered together with  MTSP data (such as EUV from SDO/AIA or radio from Nanc¸ay Radioheliograh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Observations will start with two operating modes: a standard one in the case of low  activity level (60 s cadence), and a fast mode (10-15 s) in the case of high level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' This  House and mount at CALERN observatory (1270 m)6  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Malherbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  mode will be systematically used at the approach of the next solar maximum (2025),  or will be triggered by indicators such as the real-time GOES/NASA X-ray flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='3 The telescopes  Three telescopes, with equivalent focal length 983 mm, forming a solar image of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='14  mm diameter, have been developed (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The CaII K line channel provides a  magnetic proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' It reveals the magnetized areas such as dark spots (1000 G typical)  and bright facular regions (100 G typical), through broad-band interference filters  integrating the line core (K3), K2 close wings (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='2 ˚A) and partially K1 far wings  (±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='5 ˚A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The two Hα channels monitor chromospheric activity, such as filament  instabilities, flares, CMEs and eventually Moreton waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' They use narrow bandpass  filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' In order to have a compact instrument, we selected Fabry-P´erot filters instead  of Lyot filters previously used at Meudon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  The telescopes (except filters) have been simulated with the Zemax software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Fig-  ure 4 shows the Modulation Transfer Functions (MTF) of the aperture, including the  CCD sampling effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The cutoff frequencies are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='36, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='815 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='02 10−3 km−1,  respectively for the CaII K (80 mm aperture) and the two Hα channels (80 mm and  100 mm aperture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The Point Spread Functions (PSF) in the image plane are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  The CaII K and Hα telescopes exhibit Strehl ratios of 88% and 93% respectively, and  the ensquared energy of a point source is 70% in a box of only 2 × 2 pixels for all  instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' There is almost no optical distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The RMS radius of spot diagrams  varies from 32% to 65% of the Airy spot radius from the FOV centre to the border,  in the case of Hα telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  The optical design and filter tests are detailed in sections 3 and 4, respectively  for CaII K and Hα telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' However, Figure 5 summarizes briefly the overall  capabilities of MTSP devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' We have two Hα Fabry-P´erot filters, in pupil plane,  with respectively 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='46 ˚A FWHM (for line wings, filter 1) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='34 ˚A FWHM (for  line centre, filter 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' We also have two interference CaII K filters, in image plane,  with respectively 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='4 ˚A FWHM (the second one is a spare).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The instruments  are enclosed inside a temperature regulated box at 30±1◦C by active heating and  passive cooling, but Fabry-P´erot filters have their own and high precision heating  oven (typical operating values of 65 and 40◦C respectively for filters 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' CCDs  are isolated with their own Peltier and air cooling systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' according to the readout  noise at 8 MHz pixel rate, their dynamic range is about 2700 (12 bits) for a typical  signal to noise ratio of 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Exposure times are a few ms only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  The Hα telescopes, dedicated to fast solar events, can observe the chromosphere  (8000 K) at high cadence (10-15 s), this is 4 times faster than GONG observations or  previous Meudon routines, and 3 times faster than SDO/AIA in EUV emission lines  of the hot corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The pixel size is about 1 arcsec (2 times better than the previous  routines at Meudon);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' it is difficult to achieve a better resolution with small ground  based instruments observing the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  The MeteoSpace project  7  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' 3 The MTSP instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Top: the mechanical structure and the three telescopes (1, 2 for Hα and  3 for CaII K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' There is room for a fourth telescope, if needed in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The instruments are mounted  on two XT95 optical benches (aluminium 6061) and are enclosed in a thermally regulated and isolated  box (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The volume is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='5 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='5 × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='80 m3 and the mass is about 120 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Centre: details of the  complex mechanical interface between the equatorial mount and the two optical benches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Bottom: details  of the Hα chambers (see also Figure 8 of Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  3 The CaII K channel  The CaII K telescope (Figure 6) has an aperture of 80 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' It is composed of a Taka-  hashi FS102 objective (820 mm focal length) followed by a magnifying chamber (×  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='2), a Barr Associates 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='5 ˚A FWHM filter (in image plane) and an interline cooled  Mount mechanical  Halphacentre(1),Halphawing (2)andCallK(3)telescopes  interface  Chambers and  Mount  CCD cameras  Entranceobjectives  screw  XT95  insulation  optical  bench  supports (Al 6061)  (AI 6061)  interface with  mount (Al6061)  thermal insulation  (polyamide PA6C)  lens (L3)  field corrector (L4)  motorfocus  angular adjustment  Halpha filter  fine adjustments  tube  CCD  Halphachamber  (rotation/translation)8  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Malherbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' 4 The theoretical MTF of the Hα (red) and CaII K (blue) telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Abscissa: spatial fequency in  km−1 or in arcsec−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Cutoff frequencies are indicated (bars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The computed PSFs, in the detector plane,  are reported (the pixels correspond to the CCD sampling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The Hα PSF is for the border of the FOV, while  the CaII K PSF is for the centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' 5 The MTSP Filter bandwidths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Filters have been controlled with the high dispersion spectrograph of  Meudon Solar Tower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Left: Hα line without (top) or with (bottom) filter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The pupil integrated transmit-  tance (solid line, top) over the surface filter is drawn (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='46 ˚A FWHM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Middle: Hα line without (top) or  with (bottom) filter 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The pupil transmittance is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='34 ˚A FWHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Right: CaII K line as seen through the  two available filters of respectively 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='5 (filter 1, bottom) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='4 ˚A FWHM (spare filter 2, top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The trans-  mittance (solid line) includes K2v, K3 and K2r features, and partially K1v and K1r (v, r letters meaning  respectively violet and red wings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  CCD camera (QSI 690, pixel size = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='69 µm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='78 arcsec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The system is protected  by an UV/IR cutoff filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The Airy spot size is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='9 µm or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='24 arcsec resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  The interference filters were tested with the powerful 14 m spectrograph of Meudon  Solar Tower at F/75 (R = 300000, 6 m ˚A spectral line sampling in order 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' We found  that the central wavelength (CWL) is temperature (T) dependant, according to the law  ∆λCWL = C ∆T, where C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='07 ˚A/◦C is the measured temperature coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' It is  also function of the incidence angle θ, and follows the law ∆λCWL = K θ 2 with K =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='0  Halpha PSF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='1  function  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='6  transfer  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='001  1 pixel = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='54 microns  Call K PSF  Modulation  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='1  Ha  Ha  Call K  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='2  1 pixel = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='69 microns  arcsec  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='0002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='0004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='0006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='0008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='0010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='0012  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='0014  Spatial frequency (km")3932  3933  3934  3935  "K1  <3  K1r  K2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  $562  6563  6564  656  6561  6562  6563  6564  C9C9  CallK-PWHM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='4 A  3932  3933  3934  3935  "K1v  K3  K1r  ~-K2V  K2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  HalahaFwHMo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='3  K FWHM 1 5 The MeteoSpace project  9  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' 6 The MTSP CaII K channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Top: the overall design of the telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Bottom: details of the chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  L2/L3 increase the focal length from 820 to 983 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Filters are in the image plane at F/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The chamber  has a motor focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='15 ˚A/degree2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' As the theory gives K = - 1  2  λ0  n2 (θ in radians), where n is the effective  index of refraction and λ0 the line wavelength, we derived from the measurements n  ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' With an aperture of F/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='3 (half cone of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='3◦), the numeric integration over  the light cone shows that the CWL is blue-shifted (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='41 ˚A), but this is almost com-  pensated (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='49 ˚A) by the operating temperature (+7◦C above the specification of the  manufacturer), so that the resulting CWL shift is small in comparison to the FWHM  (the transmittance of the filter centre is displayed in Figure 5, right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' This figure also  reveals that the surface filter is not uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' We found that that the CWL varies from  the centre to the border according to the law ∆λCWL = κ x2, where x is the distance  to the filter centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' We measured κ = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='006 ˚A/mm2 and +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='003 ˚A/mm2 respectively  for filters 1 and 2 (the spare filter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' It means that the CWL varies between the centre  of the solar disk and the limb (0 ≤ x ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='6 mm), numerically ∆λCWL = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='13 ˚A or  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='07 ˚A , respectively for filters 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' This is less than 10% of the FWHM, so that  the effect does not appear obviously in solar images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Figure 7 shows a comparison  between the CaII K3 Meudon spectroheliograms (line centre) and MTSP test images  (formed at lower altitude due to the larger bandwidth): filaments and prominences  are no more visible, but sunspots appear more contrasted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' bright faculae look sim-  ilar, so that the CaII K MTSP channel produces a good magnetic proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' It is close  to the wavelength integration of the spectroheliograph data-cube (x, y, λ with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='093  ˚A resolution) over the MTSP transmittance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  4 The two Hα channels  The two Hα telescopes (Figure 8) have an aperture of 80 mm and 100 mm, respec-  tively for filters 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' They are composed of a Takahashi TSA102 objective (816  mm focal length) followed by an afocal chamber (magnification 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='2), including a  DayStar Quantum Pro Fabry-P´erot filter in the pupil plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' This system introduces  field curvature, so that a field corrector (two lenses) forms the final image on an in-  L2  L3  Filters  CCD  L1  587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='6mm  144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='9  mm  f1=820mm  L2  CHAMBER  L3  Call K filter  CCD  e=2mm  e=2mm  e=1mm  e=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='3mm  +935mm  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='7mm  144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='9  mm  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='6mm  40mm  m  5m  f3=250mm  UV/IR cut filter  f2=-220mm10  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Malherbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' 7 Comparison of Meudon CaII K3 spectroheliograms and MTSP CaII K images, 26 November 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  (a) K3 spectroheliogram (30 s surface scan by the slit, square root to decrease the dynamics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' (b) the  spectroheliograph (x, y, λ) data-cube integrated over the MTSP wavelength transmission curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' (c) the  MSTP K image, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='5 ˚A FWHM (2 ms exposure time, the waveband covers K2v, K3 and K2r central  features of the CaII K line, and partially K1v and K1r wings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' 8 The MTSP Hα channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The filter is located in the pupil plane at F/30 inside the L2/L3 afocal  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' L4 is a field curvature corrector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The chamber has a motor focus (see also Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The equivalent  focal length is 983 mm, as for CaII K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The beam aperture, in the detector plane, is F/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='3 and F/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='0  respectively for Hα 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  terline cooled CCD camera (QSI 660, pixel size = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='54 µm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='96 arcsec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The Airy  spot size is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='8 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='9 µm, corresponding to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='06 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='65 arcsec resolution, respec-  tively for telescopes 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The system is protected by an UV/IR cutoff filter in full  aperture (not drawn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' A motor focus is integrated to the chambers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  The calibration of Fabry-P´erot filters has been done with the spectrograph of  Meudon Solar Tower, at F/75 (much better than the F/30 filter specification), in or-  der 9 (10 m ˚A/pixel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Several tests have been performed for both filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' First of all,  we made a scan of the surface in order to produce a pupil cartography of the CWL  and FWHM (Figures 9 and 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' For that purpose, the filters (31 mm diameter) were  translated (-15 mm ≤ x ≤ 15 mm) in front of the spectrograph slit by ∆x = 3 mm  steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' (λ, y) spectral images, with and without filters, were recorded, for various x-  positions (top of figures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The wavelength transmittance was derived at several (x,  y) locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The results show that the FWHM of both filters is locally in the range  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='30-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='35 ˚A, but also that filter 2 is better than filter 1 in terms of CWL uniformity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  By integration on the surface, we computed the resulting bandpass for pupil plane ap-  plication in afocal systems, leading to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='46 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='34 ˚A, respectively for filters 1 and  FWHM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='5 A  (c) Call K MTSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' FWHM  al Call  (b) Call  A filteDiaphragm  Image plane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=" CCD sensor  Entrance objective L1  Halpha157mm  FOV35'  Halpha269mm  Imageplane  D: 10 mm  Halpha1:80mm  Mask  FILTER  Halpha2:98mm  Halpha1&2~11mm  Diaphragm  Pupil plane  f1=816mm  L4 Field  Halpha134mm  corrector  Halpha  Halpha240mm  7  50/+60mm  L2  chamber  D:32mm  D:14mm  ~556," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='6mm  D:42mm  f3=250mm  motor focus  f2=250mm  on L4  284,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='6mm  272mm  243mm  331,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='8mm  218mm  272mm  Halpha  274mm  telescopes  58,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='3mm  optical design  ~1745mmThe MeteoSpace project  11  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' 9 Calibration of Hα filter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' It was translated by 3 mm steps in the spectrograph focus of the Meudon  Solar Tower (F/75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Top: the Hα line through the filter for 9 x-positions of the slit (in mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The white  lines delineate the CWL and FWHM (y-direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Bottom: the filter cartography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The local FWHM ( ˚A)  is displayed at left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' CWL fluctuations ( ˚A) are not negligible (at right), so that for pupil plane application,  the integrated bandpass is enlarged to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='46 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The pupil transmittance is Lorentzian shaped in wavelength and drawn in Figure 5  for both filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Fabry-P´erot filters have secondary lobes, cut by a blocking filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' We found, for  filter 2, that they appear at ± 25 ˚A from the main lobe with an intensity of only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='3%  (Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' For that filter (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='34 ˚A FWHM), the measurements provide the value  of the finesse (75), the reflection coefficient of the cavity (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='96) and the interference  order for Hα line (k = 260).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Meudon spectroheliograph data, which are made of 3D (x, y, λ) data-cubes (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='155  ˚A wavelength resolution) allow to simulate images using various transmittances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Fig-  ure 12 displays Hα images with different filter characteristics, such as a Lorentzian  transmittance (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='34 ˚A FWHM) or a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='50 ˚A FWHM five stage Lyot transmittance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  It clearly shows that a smaller bandwidth is needed for the Fabry-P´erot to produce  contrasts similar to the ones provided by Lyot filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Indeed, the wavelength curve of  Lyot filters drastically cuts the line wings, contrarily to Lorentzian filters which have  extended wings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Hence, the photospheric light passes in excess and contaminates the  contrast of the chromosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' When 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='50 ˚A Lyot filters are sufficient to select the  chromosphere, narrower (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='30 ˚A) Fabry-P´erot devices (such as MTSP filter 2) are  required for the same result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  The Solar Tower spectrograph allowed us to explore the angular dependance of  MTSP filters and precise the tilt sensitivity (θ) in terms of CWL and FWHM fluctu-  ations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' This experience also provided an estimate of the effective index of refraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  For that purpose, the filter was tilted in the range -3◦ ≤ θ ≤ +3◦ (Figure 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The  mean CWL variations are fitted by the law ∆λCWL = K θ 2 with K = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='38 or -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='37  x=-12mm  x= -9mm  x= -6mm  x= -3rmm  x=  Ormim  x=  3mm  x=  6mm  X=  9mm  ×= 12mm  Slit pasition (x)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='37  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='36  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='34  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='03  FWHM (A)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='33  Surface  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='00  CWL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='32  filter  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='06  cartography  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='28  1512  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Malherbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' 10 Calibration of Hα filter 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Top: the Hα line through the filter for 11 x-positions of the spec-  trograph slit (in mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The white lines delineate the CWL and FWHM (y-direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Bottom: the filter  cartography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The local FWHM ( ˚A) is displayed at left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' At right, the CWL ( ˚A) fluctuations are small, so  that the integrated bandpass for pupil plane application (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='34 ˚A) remains narrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' 11 Secondary transmission peaks of Hα filter 2 in logarithmic intensity scale (wavelength in ab-  scissa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' 12 Simulation of Hα images obtained after filtering the (x, y, λ) spectroheliograph data-cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' (a)  Spectroheliogram at line centre, 28 October 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' (b) Same image, through a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='34 ˚A Lorentzian transmis-  sion (such as MTSP filter 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' (c) Same image, with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='50 ˚A five stage Lyot filter, for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Transmission (lambda,y), spectral range (in abscissa) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='65 A  x=-15mm  x=-12mm  x= -9rmm  x= -6mm  x= -3mm  x=  Orrim  X=  3mm  x=  6mm  x=  9mm  x= 12rmm  x= 15rmm  Slit pasition (x)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='37  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='36  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content="33  00'0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='03  FWHM (A)  Surface  CWL (A)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='06  filter  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='09  cartography  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='28  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='15 Log scale, Hydrogen lamp  25 A  +25 A  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='15 A  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='15  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='15 Ab Hal  r FWHM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='34 A  (c) Halpl  FWHM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='50 AThe MeteoSpace project  13  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' 13 Effect of the tilt angle on the filter transmission curves (abscissa: wavelength in ˚A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Tilt values are  0◦ (in red), ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='25◦, ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='75◦, ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='25◦, ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='75◦, ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='25◦ and -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='75◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The transmission is blue-shifted with  increasing tilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Thick line: the Hα profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Dashed line: the envelope of the blocking filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  ˚A/degree2, respectively for filters 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' For the tilt dependance of the mean band-  pass, we found FWHM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='33 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='064 θ 2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='35 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='052 θ 2 ( ˚A), respectively for  filters 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' This means that, for a 1◦ tilt, the bandpass is locally enlarged to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='40  ˚A and blue-shifted of about -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='40 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' But in the afocal system, at F/30 (half cone  angle of 1◦), we have to integrate the above formulae over angles, which generates  a blue-shift of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='19 ˚A (this value was confirmed by the wavelength line scan made  with the filter in imagery mode, which produced results of Figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' However, the  F/30 cone angle is not the only one to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The solar diameter is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='53◦, which  means that the cone incidence θ varies in the range -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='26◦ ≤ θ ≤ +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='26◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The cor-  responding blue-shift is, at maximum, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='024 ˚A, which can be neglected, so that the  CWL should be almost uniform over the solar image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  According to the Fabry-P´erot theory, we have K = - 1  2  λ0  n2 (where λ0 is the line  wavelength).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' It allows to derive the effective index of refraction n of the overall filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  We found n ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The distance of secondary lobes is 2en  k2 = 25 ˚A, where k is the  interference order (260) and e the thickness of the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Hence, we conclude that e  ≈ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='2 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  The envelope of transmittance curves of Figure 13 corresponds to the blocking  filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The estimated FWHM is about 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='5 ˚A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' this value, considering a Lorentzian  shape, is consistent with the intensity (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='3%) of the secondary peaks of Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  The CWL depends on the temperature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' We investigated, by the spectroscopic  means of the Solar Tower, the effect of temperature changes upon filter 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The trans-  mittance for temperatures varying from 38 to 48◦C is reported in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The re-  sponse is a red-shift corresponding to the linear law ∆λCWL = κ T - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='84 ˚A, where T  is expressed in ◦C and κ is the temperature coefficient equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='0874 ˚A/◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Hence,  it is possible to explore the ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='0 ˚A spectral domain centred on the line, but in  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='5×10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='25  +/-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  unit)  +/-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='25  (arbitrary  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='0x10  transmission  +/-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='25  filter  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='75  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='0x1  Integrated  1D  0  6559  6560  6561  6562  6563  6564  Wavelength (A)14  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Malherbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' 14 Effect of the temperature on the filter transmission curves (abscissa: wavelength in ˚A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Tempera-  ture values are 38, 39, 40, 41, 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='8, 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='8, 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='8, 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='7, 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='6, 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='6 and 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='5◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The transmission is red-shifted  with increasing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Thick line: the Hα profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Dashed line: the envelope of the blocking filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  practice it is a very slow process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' We did not notice any FWHM variation with the  temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  The last qualification test was performed with the filter alone, in image mode,  without any spectrograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Filter 1 was mounted in the MTSP afocal system, and we  explored the wavelength range 6562.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='0-6563.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='6 ˚A by temperature variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' It took  a long time, because the filter needs 10 minutes to stabilize at each wavelength in-  crement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' We recorded also the fluctuations of the solar flux using a scintillometer, in  order to correct intensities measured by the filter due to atmospheric variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Then,  we derived the Hα line profile at the disk centre (Figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' We also plotted the line  profile got by spectroscopic means, and convolved by the Lorentzian transmittance  of the filter, and found both results in good agreement, after correction of the -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='2  ˚A shift resulting from the F/30 light cone angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Finally, Figure 16 presents the typical Hα observations that are scheduled with  the above filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' MTSP will provide almost simultaneously two images, the first one  in the line wing (filter 1, either the blue or the red wing, but not both), and the sec-  ond one in the line core (filter 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' We discuss in the next section the application to  the detection of fast evolving Doppler-shifted events, such as Moreton waves, which  require at least two Hα channels and the high observing cadence of 10-15 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  5 Moreton waves detection  MTSP is particularly well adapted to the detection of highly dynamic events, such  as Moreton waves originating in energetic flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' As the filter performances are com-  parable to those of the previous Meudon routine (1985-2004), we have chosen two  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='5×10  T= 38 39 40 41 42 43 44 45 46 47 48  unit)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='0x10  transmission  filter  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='0×1  Integrated  1D  0  6560  6561  6562  6563  6564  6565  Wavelength (A)The MeteoSpace project  15  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' 15 Hα line profiles got in imagery or spectroscopic mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Black: MTSP measures resulting from  filter temperature variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Red: the line observed with the large spectrograph of Meudon Solar Tower,  and the convolution by the MTSP filter transmittance (dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' In abscissa: the wavelength ( ˚A) and the  corresponding LOS velocity (km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' 16 Example of MTSP Hα images got during the test campaign of 29 August 2017 with filter 1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='46  ˚A FWHM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' (a) MTSP, blue wing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' (b) MTSP, line centre (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='34 ˚A FWHM filter 2 will be used instead in  full operation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' (c) Meudon spectroheliogram for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  events observed with this instrument in order to anticipate MTSP capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' More-  ton waves appear in Hα as fronts propagating at typically 500 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Such veloc-  ities in the chromosphere (8000 K) are so highly supersonic (Cs ≈ 10 km s−1) and  superalv´enic (Ca ≈ 10 km s−1 for a 10 G magnetic field) that they are unlikely of  chromospheric nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Such phenomena last only a few minutes and are suspected to  be the chromospheric counterpart of coronal waves propagating in the 200 times hot-  ter (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='5 MK) and more tenuous corona under the form of fast magnetosonic shocks  (Cs ≈ 150 km−1, Ca ≈ 200 km s−1 for a 10 G field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The downward compression  of the chromosphere below the front could be the signature of the coronal shock in  Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Moreton waves occur mainly in X-class flares which are rare events (about one  event/year above X5, six events above X10 since year 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' We have chosen the  8x104  6×104  H20  units)  (arbitrary  H20  H20  4×104  Intensity  2×104  Halpha  LOs Velocity (km/s)  60  40  20  20  40  60  0  6561.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='5  6562.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='0  6562.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='5  6563.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='0  6563.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='5  6564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='0  6564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='5  Wavelength (A)-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='5 A, MTSP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='46 A FWHM  (b) Halpha  MTSP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='46 A FWHM  (c) Halpha16  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Malherbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  famous X17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='2 flare of 28 October 2003 (Figure 17), after the solar maximum of cy-  cle 23, and the X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='8 event of 14 October 1999, just before the maximum (Figure 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Largest flares often occur during the descending phase of the solar cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' In Figure 17  (b,c,d respectively for Hα centre and ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='5 ˚A), we have subtracted the reference frame  just before the event at the same wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Blue/red colours indicate the sign of the  resulting image (blue, or positive, means brighter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' red, or negative, means darker).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  The wave (yellow box) appears as a brightening in Hα core (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' In the red wing sub-  traction (c), the compression front (R) is negative (or darker) and corresponds to a  red-shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' It is followed by the relaxation of the chromosphere (B) appearing positive  (or brighter), because blue-shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' In the blue wing subtraction (d), this is the con-  trary: R is positive (or brighter), because red-shifted, while B is negative (or darker)  with the opposite shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Figure 15 shows that the order of magnitude of LOS velocities  for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='5 ˚A shifts is qualitatively 20 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' This example shows that Moreton waves  can be detected either by the red or the blue wing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' For that reason, MTSP offers only  one wing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Figure 18 presents the X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='8 flare of 14 October 1999, almost 10 times less en-  ergetic than the previous one, so that the Moreton event is less contrasted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' It shows  exactly what will provide MTSP concerning wave detection with one wing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Most  events are waited between 2024 and 2028 around the solar maximum (2025) of cycle  25 (but isolated events, as the X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='0 of 28 October 2021, are possible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The Hα centre  (a) will be provided by telescope 2, from which a reference frame (just before the  event) has been subtracted in (b), showing the brightening front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The Moreton event  appears clearly in the red wing with the red-shifted compression front (R) followed  by the blue-shifted relaxation front (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The blue wing could either be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The  big difference with the 1985-2004 Meudon routine is the observing cadence (4 times  faster) and the spatial resolution (2 times better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Hence, MTSP will provide much  more detailed informations to investigate the physics of rare phenomena at the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  6 A possible extension for MTSP  We own a NaD1 5896 ˚A Fabry-P´erot filter manufactured by DayStar (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='36 ˚A FWHM),  for mounting in an afocal design at F/30 similar to the one of the Hα telescopes de-  scribed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Magnetograms of the Sun have been successfully produced in this  line (Land´e factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='33) by Mount Wilson (full disk) or by the Narrow-band Filter  Imager (NFI) onboard HINODE for small regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' NaD1 is a Fraunhofer line formed  in the low chromosphere above the FeI 6173 ˚A line observed in the photosphere by  the Helioseismic and Magnetic Imager (HMI) onboard SDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' As DayStar filters are  linearly polarizing (made of a birefringent material), the incorporation of a Liquid  Crystal Variable Retarder (LCVR) at the primary focus, providing ± λ  4 fast modula-  tion (Malherbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=', 2007) suffice to produce alternatively I+V and I-V images (I,  V for Stokes parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' From the circular polarization rate V  I and the weak field  theory (see Stenflo (1994)), it is possible to estimate the LOS magnetic field (BLOS)  together with the field polarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Indeed, V  I is proportional to BLOS and 1  I  dI  dλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' This  quantity depends on the wavelength and is maximum when measured at the inflexion  points of the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Figure 19 displays spectroscopic observations obtained with this  The MeteoSpace project  17  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' 17 Moreton event of 28 October 2003 at 11:07 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Images, in Hα centre, red and blue wings, were  taken with 60 s time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' (a) Hα centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' (b) Hα centre, minus the reference frame at 11:00 UT (event  starting time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' (c) Hα red wing, minus the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' (d) Hα blue wing, minus the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' F, R, B  indicate respectively the flare location, red and blue shifts of the compression and relaxation fronts of the  Moreton wave (propagation direction = green arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' 18 Moreton event of 14 October 1999 at 09:02 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Images, in Hα centre and red wing, were taken  with 60 s time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' (a) Hα centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' (b) Hα centre, minus the reference frame at 09:00 UT (event starting  time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' (c) Hα red wing, minus the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' F, R, B indicate respectively the flare location, red and blue  shifts of the propagating front (propagation direction = green arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  (a)  Date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='tirme=031028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='110712  Halpha centre  Date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='time=031028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Halpha centre  d  Dgtetime=031028  Red wing  Date time=031028  Blue wing(a)  (b)  Date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='time=991014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='090247  Halpha centre  Date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='time=99101  Halpha centre  Date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='time=g91  Red wing18  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Malherbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  method at the 8 m spectrograph of the Pic du Midi Turret Dome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' We have numeri-  cally at the inflexion points V  I ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='210−4BLOS, where BLOS is expressed in Gauss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  As the slope of NaD1 wings is steep, Stokes V profiles are sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' We observed po-  larization rates up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='20 (BLOS ≈ 1000 G) in some sunspots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' In order to estimate  the measurement capabilities in imagery, we have integrated the line profiles over  the wavelength transmission of the NaD1 filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The V  I signal becomes much smaller,  because the filter FWHM (W1) is much larger than the half width (W2) of the line  derivative dI  dλ , which concentrates the polarimetric signal in a very narrow wave-band  (Figure 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Hence, the corrective factor to apply to the polarization rate is roughly  equal to the W1  W2 ratio and plotted in Figure 20 for various CWL and W1 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Despite  of the signal loss due to wavelength integration, it is of interest to consider polarimet-  ric measurements in imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' For that purpose, a NaD1 telescope will be tested at  Meudon in the coming year in the context of a possible extension for MTSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' As the  polarization rate to measure with the filter has the same order of magnitude than the  photon noise of a single exposure (1% or 400 G), it is necessary to acquire many  frames in order to select best images and reduce the noise, as done by Roudier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' For example, 100 couples (I+V, I-V) will decrease the noise to 40 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' 20 G  should be achieved with either 2 × 2 binning or with 400 couples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' It must be noticed  that a fast observing cadence is not required for LOS magnetograms, because the  magnetic field evolves on longer time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The method is also, in principle, valid  for the Hα telescopes, but in practice, Hα is a broad line and the sensitivity to the  magnetic field is reduced by the factor 6 in comparison to NaD1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  7 Conclusion  The MTSP project is dedicated to the survey of fast evolving events in the chromo-  sphere at the source of solar activity, such as flares and coronal mass ejections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Large  flares often occur after the solar maximum (2025 for the present cycle) during a few  years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' MTSP, with an outstanding cadence of 10-15 s, has also the major goal to  investigate Moreton waves, which are extremely fast and rare phenomena associated  with largest flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Such events are difficult to detect in the chromosphere, so that only  a few cases have been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' With systematic, fast and multi-channel observations  (two Hα and one CaII K telescopes), MTSP will increase the chances to catch such  phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Data could be combined to SDO/AIA observations of the low corona  at slower cadence (45 s) in several EUV channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' MTSP will operate automatically  at Calern observatory (1270 m) under good seeing and climatic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' High ca-  dence observations will be freely delivered, without any delay, to the international  community through a dedicated database located at Nice computer centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' MTSP  will cover cycle 25, from 2023 to, at least, the end of the present decade, where new  generation solar synoptic networks could start, such as the Next Generation GONG  project (Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=', 2019) or the Solar Physics Research Integrated Network Group  (SPRING, Gosain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' MSTP will support Solar Orbiter (ESA) and Parker  Solar Probe (NASA) operations in the coming years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  The MeteoSpace project  19  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' 19 Simulation of polarization measurement with the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='36 ˚A FWHM NaD1 filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Left: spectra of I+V  and I-V got at Pic du Midi in an active region (wavelength in abscissa, direction of the slit in ordinates,  spectral pixel 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='8 m ˚A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Middle: the polarization rate V  I with the quantity 1  I  dI  dλ in the quiet Sun (yellow)  and the wavelength transmittance of the filter, centred on the blue wing (green, dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Right: the polar-  ization rate V  I along the slit in the blue wing measured with the spectrograph (solid line) or extrapolated  for the filter (dotted line);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' the dashed line is a magnification of the dotted line (ratio given by Figure 20) to  recover roughly the spectroscopic signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' 20 Simulation of the correction factor to apply to the measurements of the polarization rate with a  Lorentzian filter, as a function of the FWHM ( ˚A), for various CWL shifts (solid, dashed, dotted lines for  respectively -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='12, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='20, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='30 ˚A CWL shifts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' The blue inflexion point of the line is located at -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='12 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  The red (dashed) line indicates our filter FWHM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' the best CWL shift is -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='20 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='86 A  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='1  rate  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='36 A FWHM filter  0  50  100  150  200  I+V  I-V  V/I  Y-direction (arcsec)15  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='30 A  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='20 A  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='12 A  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='5  Lorentzian filter FWHM (A)20  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Malherbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  8 Disclosure of potential conflicts of interest  The authors declare that they have no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  9 Data availability statement  The authors declare that the datasets analysed during the current study are publicly  available from the corresponding author upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Acknowledgements We thank the anonymous referee for useful suggestions and comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' We are in-  debted to Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Bresson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Renaud (OCA), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Rees, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Blanchard (OP) and the technical teams of OP  and OCA for their assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' We are also grateful for financial support to Paris and Nice observato-  ries, the Direction G´en´erale de l’Armement, the IDEX Plas@Par, the Programme National Soleil Terre  (PNST/INSU/CNRS), the DIM ACAV (Ile de France Region) and the IDEX UCA/JEDI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  References  Admiranto AG, Priyatikanto R, Yus’an U, Puspitaningrum E (2015) Moreton waves  and EIT waves related to the flare events of June 3, 2012 and July 6, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' In: The  5th International Conference on Mathematics and Natural Sciences, American In-  stitute of Physics Conference Series, vol 1677, p 050014, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='4930675,  1502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='04039  Asai A, Ishii TT, Isobe H, Kitai R, Ichimoto K, UeNo S, Nagata S, Morita S, Nishida  K, Shiota D, Oi A, Akioka M, Shibata K (2012) First Simultaneous Observation  of an Hα Moreton Wave, EUV Wave, and Filament/Prominence Oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' As-  trophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='745(2):L18, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='1088/2041-8205/745/2/L18, 1112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='5915  Balasubramaniam KS, Pevtsov AA, Neidig DF (2007) Are Moreton Waves Coronal  Phenomena?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='658(2):1372–1379, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='1086/512001  Cabezas DP, Asai A, Ichimoto K, Sakaue T, UeNo S, Ishitsuka JK, Shibata K  (2019) Dynamic Processes of the Moreton Wave on 2014 March 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='883(1):32, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='3847/1538-4357/ab3a35, 1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='03534  Chen PF (2011) Coronal Mass Ejections: Models and Their Observational Basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Living Reviews in Solar Physics 8(1):1, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='12942/lrsp-2011-1  Gosain S, Roth M, Hill F, Pevtsov A, Martinez Pillet V, Thompson MJ (2018) Design  of a next generation synoptic solar observing network: solar physics research inte-  grated network group (SPRING).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' In: Evans CJ, Simard L, Takami H (eds) Ground-  based and Airborne Instrumentation for Astronomy VII, Society of Photo-Optical  Instrumentation Engineers (SPIE) Conference Series, vol 10702, p 107024H, DOI  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='1117/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='2306555  Grenat H, Laborde G (1954) H´eliographe Monochromatique de Lyot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Annales  d’Astrophysique 17:541  Harvey JW, Bolding J, Clark R, Hauth D, Hill F, Kroll R, Luis G, Mills N, Purdy  T, Henney C, Holland D, Winter J (2011) Full-disk Solar H-alpha Images From  GONG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' In: AAS/Solar Physics Division Abstracts #42, AAS/Solar Physics Divi-  sion Meeting, vol 42, p 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='45  The MeteoSpace project  21  Hill F, Hammel H, Martinez-Pillet V, de Wijn A, Gosain S, Burkepile J, Henney CJ,  McAteer J, Bain HM, Manchester W, Lin H, Roth M, Ichimoto K, Suematsu Y  (2019) ngGONG: The Next Generation GONG - A New Solar Synoptic Observa-  tional Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' In: Bulletin of the American Astronomical Society, vol 51, p 74  Krause G, C´ecere M, Francile C, Costa A, Elaskar S, Schneiter M (2015) Two  step mechanism for Moreton wave excitations in a blast-wave scenario: the 2006  December 06 case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='453(3):2799–2807, DOI  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='1093/mnras/stv1827, 1505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='02723  Krause G, C´ecere M, Zurbriggen E, Costa A, Francile C, Elaskar S (2018) Are CMEs  capable of producing Moreton waves?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' A case study: the 2006 December 6 event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='474(1):770–778, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='1093/mnras/stx2817  Lyot B (1944) Le Filtre Monochromatique Polarisant et ses Applications en Physique  Solaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Annales d’Astrophysique 7:31  Malherbe JM, Dalmasse K (2019) The New 2018 Version of the Meudon Spectrohe-  liograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Solar Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='294(5):52, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='1007/s11207-019-1441-7  Malherbe JM, Roudier T, Moity J, Mein P, Arnaud J, Muller R (2007) Spectro po-  larimetry with liquid crystals .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Societa Astronomica Italiana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='78:203  Malherbe JM, Corbard T, Dalmasse K, Meteospace Team (2019) Meteospace, a New  Instrument for Solar Survey at the Calern Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Solar Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='294(12):177,  DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='1007/s11207-019-1569-5  Michard R (1965) Nouvel H´eliographe `a l’Observatoire de Meudon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' L’Astronomie  79:131  Moreton GE (1960) Hα Observations of Flare-Initiated Disturbances with Velocities  ˜1000 km/sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='65:494, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='1086/108346  Narukage N, Eto S, Kadota M, Kitai R, Kurokawa H, Shibata K (2004) Moreton  waves observed at Hida Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' In: Stepanov AV, Benevolenskaya EE, Koso-  vichev AG (eds) Multi-Wavelength Investigations of Solar Activity, vol 223, pp  367–370, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='1017/S1743921304006143  Oloketuyi J, Liu Y, Zhao M (2019) The Periodic and Temporal Behaviors of Solar  X-Ray Flares in Solar Cycles 23 and 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='874(1):20, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='3847/  1538-4357/ab064c  Pulkkinen T (2007) Space Weather: Terrestrial Perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Living Reviews in Solar  Physics 4(1):1, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='12942/lrsp-2007-1  Reale F (2010) Coronal Loops: Observations and Modeling of Confined Plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Liv-  ing Reviews in Solar Physics 7(1):5, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='12942/lrsp-2010-5, 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='5927  Roudier T, Malherbe JM, Moity J, Rondi S, Mein P, Coutard C (2006) Sub arcsec  evolution of solar magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='455(3):1091–1098, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  1051/0004-6361:20064963  Schwenn R (2006) Space Weather: The Solar Perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Living Reviews in Solar  Physics 3(1):2, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='12942/lrsp-2006-2  Steinegger M, Hanslmeier A, Otruba W, Freislich H, Denker C, Goode PR, Marquette  WM, Varied J, Wang H, Luo G, Chen D, Zhang Q (2000) An Overview of the New  Global High-Resolution H-alpha Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Hvar Observatory Bulletin 24(1):179  Stenflo J (1994) Solar Magnetic Fields: Polarized Radiation Diagnostics, vol 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='1007/978-94-015-8246-9  Temmer M (2021) Space weather: the solar perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Living Reviews in Solar  22  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Malherbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='  Physics 18(1):4, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='1007/s41116-021-00030-3, 2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='04261  Ueno S, Shibata K, Ichimoto K, Kitai R, Nagata S, Kimura G, Nakatani Y (2010)  Continuous H-alpha Imaging Network Project (CHAIN) with Ground- based Solar  Telescopes for Space Weather Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' African Skies 14:17  Warmuth A (2015) Large-scale Globally Propagating Coronal Waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Living Re-  views in Solar Physics 12(1):3, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='1007/lrsp-2015-3  Zhang H (2001) Moreton wave and its source of disturbances in the X12/3B  WLF of AR6659 in 1991 June 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='372:676–685, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='1051/  0004-6361:20010379  Zhang Y, Kitai R, Narukage N, Matsumoto T, Ueno S, Shibata K, Wang J (2011)  Propagation of Moreton Waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Pub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content=' Japan63:685, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='1093/  pasj/63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
+page_content='685' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQfFjeH/content/2301.13480v1.pdf'}
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+arXiv:2301.01304v1  [astro-ph.GA]  3 Jan 2023
+Universality in the Random Walk Structure
+Function of Luminous Quasi-Stellar Objects
+Ji-Jia Tang1, Christian Wolf1,2, and John Tonry3
+1Research School of Astronomy and Astrophysics, Australian National
+University, Cotter Road Weston Creek, ACT 2611, Australia
+2Centre for Gravitational Astrophysics, Australian National University,
+Building 38 Science Road, Acton, ACT 2601, Australia
+3Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive,
+Honolulu, HI 96822-1897, U.S.A.
+draft January 5, 2023
+Abstract
+Rapidly growing black holes are surrounded by accretion disks that make them
+the brightest objects in the Universe. Their brightness is known to be variable, but
+the causes of this are not implied by simple disk models and still debated. Due
+to the small size of accretion disks and their great distance, there are no resolved
+images addressing the puzzle. In this work, we study the dependence of their vari-
+ability on luminosity, wavelength and orbital/thermal timescale. We use over 5,000
+of the most luminous such objects with light curves of almost nightly cadence from
+> 5 years of observations by the NASA/ATLAS project, which provides 2 billion
+magnitude pairs for a structure function analysis. When time is expressed in units
+of orbital or thermal time scale in thin-disk models, we find a universal struc-
+ture function, independent of luminosity and wavelength, supporting the model of
+magneto-rotational instabilities as a main cause. Over a > 1 dex range in time, the
+fractional variability amplitude follows log(A/A0) ≃ 1/2 × log(∆t/tth). De-
+viations from the universality may hold clues on the structure and orientation of
+disks.
+Every massive galaxy in the Universe contains a supermassive black hole at its
+centre; their origin and growth is one of the biggest mysteries in astronomy. Most
+black holes appear dark and are revealed only by their gravity1. During growth spurts,
+however, matter from surrounding areas falls towards the black hole. Its gravitational
+energy is turned via heat into radiation, which highlights the black hole as a ”QSO”
+(Quasi-Stellar Object). Fast-growing black holes are the most luminous objects in the
+Universe. At UV-optical wavelengths, QSOs can outshine 1,000 large galaxies2,3.
+The key for turning gravitational energy into heat is matter flowing through an
+accretion disk4. Matter orbiting the black hole drifts inwards as high friction in the disk
+1
+
+dissipates orbital energy. It is now clear that Keplerian disks with weak magnetic fields
+are prone to magneto-rotational instability (MRI)5. MRI is a fundamental instability
+converting differential rotation into turbulent fluid motions, feeding an energy cascade,
+and explaining the anomalously high viscosity needed for luminous QSO disks6.
+Accretion disks are an elegant explanation for the high luminosity and compactness
+of QSOs. The standard model of optically thick and geometrically thin disks predicts
+their temperature profiles and emission7. As they are too small to be spatially resolved
+by current facilities, we seek clues about their properties from the variability of their
+emission, which is ubiquitous on all timescales. While MRI must be one source of
+short-term fluctuations, another must be the rapidly fluctuating heating from the X-ray
+corona near the black hole8–10. The “lamppost” model exploits the latter and relates
+the sizes of accretion disks to time lags between light curves of different passbands
+probing regions of different temperature and radius in the disk9–15.
+Monitoring of QSO samples has revealed that fluctuations can be described by a
+‘damped random walk’ model16–18: brightness changes tend to be larger when the
+time delay between any two measurements is longer. However, this model provides
+only a behavioural analogy and no physical insight. Mathematically, the light curve of
+an object can be statistically described by a variability ”structure function” (SF): the
+typical brightness difference is expressed as a function of time interval19,20. The SFs of
+QSOs are characteristically different from other types of variable objects, so that QSOs
+can even be identified in a time-domain survey by the SF alone21.
+Measurements of the SF agree on two trends: fluctuations on a fixed timescale get
+stronger in less luminous QSOs and in bluer passbands17,22–26. Thus, the SF can be
+parametrised as
+log A = B0 + γ × log ∆t + BL × log L + Bλ × log λ ,
+(1)
+where amplitudes A of fractional flux changes are in terms of magnitude, and L is the
+measured luminosity of the QSO. Both the time separation between the two measure-
+ments, ∆t, and the wavelength of observed light, λ, are expressed in the QSO rest-
+frame with cosmological redshift and time dilation removed. Outside of the random-
+walk regime, the amplitudes may be suppressed at shortest timescales27–31 for reasons
+that are not yet established. They tend to level off above a damping or decorrelation
+timescale17,31,32, which has been shown to be driven by the thermal timescale of the
+accretion disk32.
+Past works agree that BL < 0 and Bλ < 0 but disagree on the strength of the
+trends. Some complicating factor may not be controlled for in existing studies. The
+SF slope γ = d log A/d log ∆t at fixed (L, λ) is in the range20,22,23,25 of 0.2 to 0.6.
+BL = d log A/d log L at fixed (∆t, λ) describes the dependence on disk luminosity
+(hence size) and is reported in the range17,22,23,25,26 of −0.33 to −0.16. Finally, Bλ =
+d log A/d log λ at fixed (∆t, L) describes the trend with rest-frame wavelength and
+thus location of the fluctuating source within the disk; measurements range17,23,25 from
+−0.75 to −0.48, although observations of non-linear relations also include slope values
+outside this range.
+Here, we present a new analysis of the variability structure function of the ∼ 5, 000
+brightest QSOs in the sky using light curves with unprecedented sampling. The sample
+2
+
+has a median black-hole mass of ∼ 2 billion solar masses according to an analysis of
+spectra from the Sloan Digital Sky Survey (SDSS)33. The light curves are from the
+0.5-metre NASA/ATLAS telescope (Asteroid Terrestrial-impact Last Alert System34),
+which was built to find dangerous asteroids that might threaten the Earth. ATLAS is
+designed to image the entire sky seen from its location in Hawaii four times every night,
+weather permitting. With up to 1,200 nights per QSO per band, we get ∼ 2.1 billion
+magnitude pairs to constrain the QSO SF. This is one of the best data sets for studying
+QSO variability before the new Vera C. Rubin Telescope operates.
+Results
+The new light curves include observations from 2015 to 2021 in two passbands, cyan
+and orange (centred on 533 and 679 nm, respectively). We limit the sample to redshift
+0.5 < z < 3.5 (z < 2.4 in the cyan band), where QSOs are so luminous that their host
+galaxies will not affect the measurements. We reject radio-loud QSOs, where variabil-
+ity may be enhanced by shocks in jets and mask the pure accretion disk behaviour35.
+We also reject gravitationally lensed QSOs and QSOs with close neighbours as seen by
+the ESA Gaia mission36 to avoid blended light curves.
+Previous work typically fitted a global model for the SF, such as Equation 1, to a
+whole data set. The richness of this new data set offers the alternative to split the data
+into subsets without incurring the penalty of noisy statistics. We first split our data into
+twenty bins in ∆t and fit the trends with luminosity and wavelength separately for each
+bin, using the equation
+log(A(∆t)) = B0,∆t + BL,∆t × log L + Bλ,∆t × log λ .
+(2)
+Thus, we get twenty different estimates for BL and Bλ, independently for each ∆t bin.
+As we can now study the trends on different timescales, we may see different regimes
+of behaviour. We use the measured variability on timescales of 1 day to estimate the
+noise in the observations. At the long end, window effects from the finite length of
+the light curves may cause noise in the parameters37. We restrict the wavelength range
+to longer than 130 nm (avoiding flux from the strong Ly-α emission line) and shorter
+than 300 nm (avoiding a previously seen upturn in variability17,22,24, which is currently
+unexplained and discussed further below). We use two types of statistical analysis,
+simple bin averages, and a computationally more intense bootstrap analysis.
+Figure 1 shows the resulting parameters for a range in ∆t from 10 to 337 days. Both
+methods of analysis produce consistent estimates. The estimates for BL and Bλ show
+a strong trend with ∆t. It is immediately clear that a global fit will produce estimates
+for BL and Bλ that depend on the fitting range in ∆t and on the distribution of pairs in
+the range, and thus on subtleties of survey cadence. We suggest this fact explains the
+variety of estimates obtained in the past. We demonstrate how results depend on the
+range of behaviour forced into a global fit in Figure 2, where fiducial global estimates
+for BL and Bλ are obtained by choosing different global fitting ranges for ∆t.
+Finally, we improve the global fit by restricting it to the random-walk regime.
+Given that our sample comprises the most luminous known QSOs, we expect damping
+timescales well in excess of a year, which do not affect our analysis. Having controlled
+3
+
+for ∆t by fitting the trends in each bin separately (Figure 1), we reveal two regimes in
+behaviour: at short ∆t both BL and Bλ drift with ∆t, whereas at longer ∆t they both
+remain constant. Thus, at longer ∆t the choice of fitting range will have little impact.
+Here, we find fit parameters of γ ≈ 1/2, BL ≈ −1/4 and Bλ ≈ −1. Past estimates
+of BL cover values from a lower range17,26 of −0.33 to −0.29, depending on the cho-
+sen parametrisation, to an upper limit23 of −0.16 and include values very close22,25 to
+−1/4.
+Changing the clock for a universal structure function
+So far, BL and Bλ have described how variability amplitudes behave at fixed ∆t. We
+now look at the timescales, on which disks of different L, observed at different λ, show
+a fixed amplitude A and thus potentially a universal structure function if their slopes γ
+are the same as well. We find these timescales to be parametrised as:
+CL = d log ∆t(A)/d log L
+=
+−BL/γ = 0.539 ± 0.004
+(3)
+Cλ = d log ∆t(A)/d log λ
+=
+−Bλ/γ = 2.418 ± 0.023 .
+(4)
+The first study in this direction empirically found CL = 0.47 ± 0.06 and argued
+that the fractional optical variability per disk orbital or thermal timescale is likely con-
+stant38. This argument is based on the well-known fact that the predicted temperature
+profile in thin disks7, T (R) ∝ R−3/4, sets a mean radius for the origin of radiation
+and thus a mean orbital timescale torb of the emitting surface, given λ and a size, and
+thus L, of the disk17. Note, that for a fixed viscosity of the disk material, the thermal
+timescale tth is proportional to the orbital timescale and we get38:
+log torb ∝ log tth ∝ 1/2 log L + 2 log λ .
+(5)
+One later study24 reported CL ≃ 0.41 ± 0.04. Our result shows that the concept of
+a constant fractional variability per orbital or thermal timescale38 is not only consistent
+with measurements of CL but also our Cλ. Thus, we rephrase the whole SF from
+Equation 1 with time in new units, choosing those of thermal timescale given its role
+for the damping timescale32:
+log(A/A0) = γth log(∆t/tth) .
+(6)
+This form is expected for fluctuations arising from MRI, where the equations of motion
+scale with the orbital timescale (and thus the thermal timescale)6,39. It also applies in
+the corona-heated accretion-disk reprocessing (CHAR) model40, which is motivated
+by the concern that a lamppost will not create sufficient amplitude by heating alone.
+The CHAR model proposes that the corona couples directly to the magnetic field in the
+disk and drives stronger fluctuations than expected from a lamppost corona alone.
+We determine tth for each QSO and passband given its (L, λ). Figure 3 shows
+the resulting SF expressed both in natural time and with a tth-scaled clock. Here, we
+increased the time resolution but reduced noise by grouping QSOs into subsamples
+split by (L, λ). The wide (L, λ)-range of our sample causes a wide range of ampli-
+tudes at fixed ∆t (left panel), but at fixed ∆t/tth (right panel) we find indeed a tight
+4
+
+range, supporting the concept of a universal structure function with a constant slope
+and intercept once expressed in suitable units of time. The relation holds over more
+than 1 dex in ∆t/tth, limited by the length of our light curves at long timescales. At
+short timescales, the amplitude suppression relative to the random walk remains. The
+CHAR model actually predicts a suppression with a break timescale proportional to the
+thermal timescale, which would imply a fixed break timescale in Figure 3, right panel.
+However, we see a spread of timescales such that the break depends more strongly on
+luminosity than the thermal timescale, which will be investigated in future work. For
+our final global fit of the SF, we avoid the suppressed regime by using only data at
+log A > −1.4. We find a bootstrap slope with natural clock time of γ = 0.503 ± 0.001
+and with time in units of thermal timescale of γth = 0.510 ± 0.002, extremely consis-
+tent with the γ = 1/2 of a random walk (Table 1 summarises all fit parameters).
+Outlook
+The transition from a regular clock to an orbital or a thermal time-scaled clock has
+standardised the random walk of QSOs as the wide dispersion observed among objects
+with a variety of luminosities and at a variety of restframe wavelengths has disappeared.
+Instead, the SF of QSOs appears now as an intrinsically tight relation.
+A universal SF opens a window to characterise QSOs in more detail by devia-
+tions from the mean relation using future higher-precision measurements. Such devia-
+tions are expected to arise from any effect that varies the orbital and thermal timescale
+estimates. These effects include the dependence of the observed luminosity on disk
+orientation and black-hole spin41, which are not independently known for individual
+objects; the true temperature profile of the accretion disk, which is still debated as it
+may change from T ∝ R−3/4 in an inner part driven by viscosity to T ∝ R−1/2 in an
+outer part dominated by irradiation from an X-ray corona acting as a lamppost39; and
+the actual interior structure of the disk, which may well change from thin to slim disks
+as the accretion rate approaches the Eddington rate39,42,43.
+Deviations from a simple picture are generally evident at wavelengths of λ >
+300 nm. Here, we measure an upturn in A, relative to a trend of Bλ ≈ −1, which has
+been seen before17,22,24,44 (see also Figure 4, right panel). It suggests that fluctuations
+in the outer, cooler parts of QSO accretion disks are to be enhanced by an additional
+mechanism, which has not yet been explained. Also, we excluded λ < 130 nm, as
+we cannot disentangle light from the disk continuum and light from the broad Ly α
+line. The hotter inner disks are harder to study as a wide swath of spectrum in the UV
+domain is absorbed by intervening gas along the line-of-sight to QSOs.
+The SF suppression at short ∆t has been observed across a range in luminosity: ob-
+servations of low-luminosity AGN with the Kepler space mission have revealed breaks
+in the SF on time scales of days to weeks27–29, although it has also been argued that
+there “remain significant levels of spacecraft-induced effects in the standard pipeline
+reduction of the Kepler data”45. Steeper slope changes have been seen in a recent
+study of luminous QSOs on time scales of 10 to 15 days31, while our break time scales
+range from ∼ 15 to ∼ 50 days. The reasons behind these breaks are still under debate:
+either the structure function is intrinsically different such as in the CHAR model40, or
+a ”smearing” effect may be caused by synchronised emission from different parts of
+5
+
+the disk reaching the observer after different time lags30.
+The forthcoming 10-year Legacy Survey in Space and Time (LSST46–48) planned
+at the Vera C. Rubin Telescope for the years 2024-34 will eclipse this data set with
+better data: more passbands will probe a wider temperature range in the accretion disk.
+Deeper photometry will probe QSOs at much fainter luminosity. This, and the reduced
+photometric errors in LSST will provide low-noise structure functions for individual
+objects and reveal how tight and universal the structure function of QSO variability
+really is. LSST should also be a game changer for analysing subtleties in the short-
+term breaks in the SF.
+Conclusion
+We have determined with unprecedented precision the parameters (γ, BL, Bλ) of the
+variability structure function for the UV continuum emission of luminous QSOs. It is
+consistent with a universal random walk once time is expressed in units of the orbital
+or thermal timescales of the emitting material. The mean amplitudes of fluctuations
+can be described by the simple law log(A/A0) = 1/2×log(∆t/tth), where a single fit
+parameter A0 represents the overall QSO population. Note, that the thermal timescale
+has recently been shown to drive the damping timescale of QSO variability as well32,
+although this fact may not have affected the random walk at shorter timescales.
+The result appears consistent with the temperature profiles of thin-disk models7, at
+least at wavelengths from 130 to 300 nm. Deviations observed at longer wavelengths
+that show the cooler, outer parts of disks17,22,24,44 still need to be explained. It also
+appears consistent with an origin of the fluctuations in magneto-rotational instabilities
+(MRI) in differentially rotating accretion disks that are made of magnetised plasma;
+there, the equations of motion scale with the orbital timescale6,39. MRI is suggested to
+produce variability with a random walk phenomenology49 and γ ≈ 0.4 to 0.5. Disks
+could thus be ordered in a 1-parameter family with the mean thermal timescale at a
+chosen wavelength as an ordering parameter. The structure function of QSOs may then
+be a universal relation apart from the short-term suppression. It appears intrinsically
+tight and would be broadened in observation by disk orientation and black hole spin,
+as well as host galaxy dust extinction affecting the inferred luminosity. It would also
+be broadened by intrinsic variety in vertical disk structure driven by accretion rates.
+The role of the lamppost for driving QSO variability is likely confined to a contri-
+bution of modest amplitude on shorter timescales. If the CHAR model also described
+the physics of disk variability correctly, then it would be hard to disentangle genuine
+lamppost reverberation from outwards-propagating magnetic fields; and if the veloc-
+ity of the latter dropped below the speed of light40, the sizes of disks inferred by a
+reverberation analysis could be overestimated.
+Future data from the 10-year LSST project will shed light on these questions and
+more subtle properties with its extremely precise measurements for a nearly all-sky
+QSO sample of unprecedented size, depth, variety and completeness.
+6
+
+Online Methods
+QSO sample and data cleaning
+We combined the Million Quasars Catalog50 (MILLIQUAS v7.1 update) with the Gaia
+eDR3 catalogue36 and select 6,163 spectroscopically confirmed, non-lensed QSOs
+with Gaia magnitude Rp < 17.5, redshift 0.5 < z < 3.5, declination δ > −45◦,
+Galactic foreground reddening51 of E(B − V )SFD < 0.15 and Gaia BpRp Excess
+Factor < 1.3 (indicative of single, unblended, point sources). We also required that the
+MILLIQUAS and Gaia positions are coincident within 0.3′′, which excludes blended
+objects of similar SED such as the multiple images of lensed QSOs, and that sources
+have no Gaia neighbours within 15′′ to keep the ATLAS photometry unaffected. We
+confirmed with recent lists of lensed QSOs52,53 that none of these are included in our
+sample. Our sample has 6,115 sources in common with NASA/ATLAS. The ATLAS
+photometric system is consistent with other reliable sources of photometry such as
+Pan-STARRS and Gaia54.
+The selection is based on Gaia eDR3 mean photometry, which is arguably the best-
+calibrated among large astronomical datasets55; the Gaia observations extend from
+mid-2014 to mid-2017 and overlap with the period of the ATLAS lightcurves (LCs).
+Thus, there will be no noticeable selection bias in favour of QSOs that might be brighter
+during an early selection epoch but fainter when the LCs are observed56.
+We further remove six QSOs from the sample, which have large flux errors through-
+out their LCs (90-percentile flux errors in the orange band of σfν,o > 150µJy or in the
+cyan band LC of σfν,c > 85µJy). On individual images, the faintest objects in the
+sample have signal-to-noise ratios (SNR) of > 11 in the orange band and > 13 in the
+cyan band. Up to four images are available per band and night.
+We reject radio-loud QSOs by cross-matching the sample with catalogues from the
+Faint Images of the Radio Sky at Twenty-cm57 (FIRST) Survey, the NRAO VLA Sky
+Survey58 (NVSS), and the Sydney University Molonglo Sky Survey59 (SUMSS). We
+use matching radii between the MILLIQUAS and FIRST, NVSS, and SUMSS coordi-
+nates of 3”, 12”, and 11”, respectively. 1,242 QSOs are matched with at least one radio
+catalogue. If a QSO is matched to multiple radio sources in NVSS or SUMSS, the ra-
+dio detection with the closest separation is used. We apply a radio loudness criterion of
+0.4×(mo −tNVSS/SUMSS) > 1.3 for matches with NVSS and SUMSS. Radio sources
+below this threshold are labelled radio-intermediate. QSOs with matches in FIRST but
+not in NVSS turned out to be all radio-intermediate. The sample contains 775 radio-
+loud and 467 radio-intermediate objects, and 4,848 radio non-detections. Thus, our
+final sample contains 5,315 QSOs, of which 3,607 have single-epoch virial black hole
+mass measurements derived from SDSS spectra33.
+We clean the ATLAS LCs first by excluding all observations with large errors of
+log(σfν,o) > −3.94−0.12×(mo−16.5) and log(σfν,c) > −4.17−0.10×(mc−16.5),
+which constitute about 5 percent of all observations; and further by comparing each
+measurement with other observations within ±7 days and removing outliers with a 2σ-
+clipping to reject spurious variability, e.g., due to temporary chance blending with faint
+asteroids. If there is only one observation within ±7 days, it is retained.
+7
+
+Luminosity
+We estimate QSO luminosities at restframe 3,000 ˚A (L3000) using a linear interpolation
+of the Gaia Bp and Rp magnitudes. At redshifts of 0.5 < z < 0.7 and z > 1.56, the
+restframe 3,000 ˚A point is extrapolated beyond the pivotal wavelengths of the Gaia
+filters. We correct for foreground dust extinction in the Milky Way51 using bandpass
+coefficients60 of RBp = 3.378 and RRp = 2.035. Luminosity distances are derived
+in a flat ΛCDM cosmology with Ωm = 0.3 and a Hubble-Lemaˆıtre constant of H0 =
+70 km sec−1 Mpc−1. Luminosity errors from this procedure are expected to be < 20%
+and thus much smaller than those introduced by host galaxy dust and the orientation
+of the disk. Luminosity estimates from broadband photometry does not distinguish
+between contributions from the accretion disk continuum and those from line emission,
+especially from the Fe complex that is strong in the vicinity of 3,000 ˚A . Comparing our
+L3000 estimates with those obtained by spectral fitting on the available subsample33,
+we find an average offset of 0.07 dex and reduce our L3000 values to correct for line
+contributions. Bolometric luminosities, as used by some authors24, are derived with
+log(Lbol/L3000) = log(fBC × λ) = log(5.15 × 3000 ˚A) = 4.189, using fBC = 5.15
+as the bolometric correction factor61.
+Thermal timescale
+For each QSO and passband, we estimate the characteristic (defined in the following)
+thermal timescale of the emitting material in the accretion disk. We combine the equa-
+tion for tth 39 and an equation for the scale length of the disk given L, λ values62, and
+get:
+tth
+days = 2.89
+1.8 ×
+1
+86400 × 10−13 × (45GM⊙fBCλ3000L3000λ4
+rf
+16π6α2ηhpc4 cos i
+)0.5
+= 7.40 × 10−27 × ( L3000
+erg/s/ ˚A)0.5 × (λrf
+˚A )2 ,
+(7)
+where hp is the Planck constant, α = 0.163 is the assumed viscosity parameter, and i =
+45◦ is the mean inclination. We assume a radiative efficiency of η = Lbol/( ˙Mc2) =
+0.1, where ˙M is the mass accretion rate; changes in η due to variations in black-hole
+spin affect the inner cutoff of the accretion disk. For a given
+˙M, such a variation
+has little effect on the outer disk, the overall L3000, and the thermal timescales at the
+wavelengths observed here. Instead, it mostly affects the emission of higher-energy
+photons and Lbol.
+Variability structure function
+We adopt the following definition of the variability structure function (SF)64:
+A =
+�π
+2 < ∆m >2 − < σ2 >, with ∆m = |mi − mj|,
+(8)
+8
+
+where mi and mj are any two observations and σ is the magnitude error due to noise.
+We find that the magnitude errors quoted by ATLAS underestimate the noise and in-
+stead derive σ2 from the apparent intra-day variability, which we expect to be negligi-
+ble for radio-quiet QSOs. We determine < σ2 > as a function of observed magnitude,
+mobs, such that the SF is anchored at A ≈ 0 for ∆t < 1 day. The required noise model
+is shown in Figure 5 and given by
+log(< σ2 >) = n0 + n1mobs ,
+(9)
+with (n0, n1) = (−12.411, 0.585) and (−12.428, 0.573) for the orange and cyan pass-
+band, respectively (pure Poisson noise suggests n1 = 0.4).
+Informed by the pair sampling distribution of the LCs, we reduce window effects
+from the finite extent of the LCs by ignoring pairs with ∆tobs ≥ 0.47max(∆tobs).
+This cut is applied in the observed frame, while the rest of the paper uses the rest-
+frame ∆t = ∆tobs/(1 + z), corrected for cosmological time dilation.
+Binning the time axis
+We bin the time axis in two different ways depending on purpose. A first case is used
+for the relation log A(∆t) in Equation 2 and Figures 1, 2 and 4. A second case is used
+for fits over the final fitting range and for log A(∆t/tth) in Equation 6, Figure 3 and
+Table 1.
+In the first case, we split the ∆t axis into 20 bins. The first ∆t bin contains all pairs
+with ∆t < 1 day, which is used only for noise analysis. Further bins are chosen to
+balance the number of pairs among the bins. The average number of pairs per ∆t bin
+are ∼103 million and ∼8 million for the orange and cyan passband, respectively. We
+split the z axis into 9 bins, thus grouping QSOs with observations of similar restframe
+wavelengths. In the cyan passband, we exclude QSOs at 2.4 < z < 3.5 to avoid
+contamination from the Ly-α line. In the orange passband, we use this z range as a
+single, highest-z, bin. The remaining eight bins cover the range of 0.5 < z < 2.4
+with roughly balanced QSO numbers. In each z bin, QSOs are further split into ten
+L3000 bins, each with almost equal QSOs numbers. In total, there are 1,800 bins in
+(z, L3000, ∆t). In each bin, the mean value of z and L3000 and the mid-point of ∆t is
+used for the analysis and figures.
+In the second case, we split the ∆t axis into 100 bins. The shortest ∆t < 1 day
+bin is exactly the same as in the first binning method, and further bins are balancing
+the number of pairs among the bins for both passbands. The QSOs are grouped into
+16 bins along the tth axis with an equal number of QSOs per bin, which is 2,950 and
+4,134 in orange and cyan, respectively. In the ∆t/tth axis, we split the observed pairs
+into 100 bins of constant number, independently for each tth group in each passband
+after excluding pairs with ∆t < 1 and ∆t > 337 days. Combining both passbands,
+there are in total 3200 bins for (tth, ∆t) and separately 3200 bins for (tth, ∆t/tth). In
+each bin, the mid-point of ∆t and ∆t/tth is used for the analysis and figures.
+9
+
+Simple bin average structure function
+For each individual QSO, we calculate its SF with Equation 8 from all available pairs
+in each ∆t or ∆t/tth bin. We use a 3σ-clipped mean of magnitude differences as
+< ∆m >. Then, for each (z, L3000, ∆t) or (tth, ∆t) or (tth, ∆t/tth) bin we calculate
+the median squared amplitude, A2, and the standard error of the median (SEMED),
+which ensures that QSOs with negative noise-corrected amplitudes A2, due to an over-
+subtraction by the ⟨σ2⟩ noise model, are still included in the statistics.
+Bootstrapping the structure function
+Using bootstrapping, we can account better for the contribution of each QSO. Here,
+we select random pairs in each of the (z, L3000, ∆t) or (tth, ∆t) or (tth, ∆t/tth) bins.
+We randomly pick a QSO N times from all the QSOs in that bin, where N is the total
+number of QSOs in that bin. For each picked QSO, we randomly select 300 distinct
+pairs in the orange and 30 distinct pairs in the cyan passband, and then calculate the SF
+using Equation 8. This process is repeated 100 times to get the bootstrap distribution,
+from which we calculate the mean and standard deviation for each bin.
+SF fitting
+We fit a structure function of the form
+log(A(∆t)) = B0,∆t + BL,∆t log(
+L3000
+1042.7erg/s/ ˚A) + Bλ,∆t log(
+λrf
+2200 ˚A)
+(10)
+using a Levenberg-Marquardt least-squares fit65 to the (z, L3000) bins for each ∆t
+bin separately. Here, log A and its error may be results of bin averages or from the
+bootstrap method. λrf is the pivotal wavelength of a passband divided by 1 + z. The
+normalisation factors for λrf and L3000 are chosen to be close to the median of the
+sample. Data with λrf > 3000 ˚A are excluded from the fit as it deviates from a linear
+relation. In Figure 4, we show an example for one ∆t bin: the left and middle panels
+show the two passbands. The right panel shows the intercepts of the fits at log(L3000)−
+42.7 for different λrf. We also compare the upturn at longer wavelengths with results
+from the literature24. The B0,∆t, BL,∆t, and Bλ,∆t from each ∆t bin are shown in
+Figure 1.
+Global parameter estimates using broad ∆t ranges
+We fit a global model of the form
+log(A) = B0 + BL log(
+L3000
+1042.7erg/s/ ˚A) + Bλ log(
+λrf
+2200 ˚A) + γ log( ∆t
+216d)
+(11)
+to the SF data in the (z, L3000, ∆t) bins with λrf < 3000 ˚A ˙We test the robustness
+of global parameter averages by varying the time span of data that is included in the
+global fit; we vary the lower edge ∆t0, while leaving the upper edge at ∆t=337 days
+unchanged. Figure 2 shows the parameter estimates as f(∆t0).
+10
+
+The structure function as f(∆t) and f(∆t/tth)
+For Figure 3 we split the sample for each passband separately into 16 groups ranked
+by tth. We then fit Equation 11 to the points with log A > −1.4 using the errors for
+weighting (see Table 1 and left panel of Figure 3 for results). Similarly, we fit the bins
+with log A > −1.4 to Equation 6 (see Table 1 and right panel of Figure 3 for results).
+For reference, we find log A0 = −1.185 when fixing γ = 1/2 in the bootstrap method.
+Data availability
+The QSO samples are selected from the publicly available MILLIQUAS database com-
+plemented with publicly available Gaia data. Data from NASA/ATLAS are publicly
+available at https://fallingstar-data.com/forcedphot/. The QSO light curves used in this
+study are available from the corresponding author on request.
+Code availability
+The main analysis routines have been written by the corresponding author in IDL and
+are available on request.
+Acknowledgements
+JJT was supported by the Taiwan Australian National University PhD scholarship and
+the Australian Research Council (ARC) through Discovery Project DP190100252 and
+also acknowledges support by the Institute of Astronomy and Astrophysics, Academia
+Sinica (ASIAA). JT has been funded in part by the Stromlo Distinguished Visitor Pro-
+gram at RSAA. We thank Mark Krumholz for comments on the manuscript and Stefan
+Wagner for helpful discussions.
+This work uses data from the University of Hawaii’s ATLAS project, funded through
+NASA grants NN12AR55G, 80NSSC18K0284, and 80NSSC18K1575, with contri-
+butions from the Queen’s University Belfast, STScI, the South African Astronomical
+Observatory, and the Millennium Institute of Astrophysics, Chile.
+This work has made use of SDSS spectroscopic data. Funding for the Sloan Dig-
+ital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S.
+Department of Energy Office of Science, and the Participating Institutions. SDSS-IV
+acknowledges support and resources from the Center for High Performance Comput-
+ing at the University of Utah. The SDSS website is http://www.sdss.org. SDSS-IV is
+managed by the Astrophysical Research Consortium for the Participating Institutions
+of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie
+Institution for Science, Carnegie Mellon University, Center for Astrophysics — Har-
+vard & Smithsonian, the Chilean Participation Group, the French Participation Group,
+Instituto de Astrof´ısica de Canarias, The Johns Hopkins University, Kavli Institute
+for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the
+11
+
+Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Insti-
+tut f¨ur Astrophysik Potsdam (AIP), Max-Planck-Institut f¨ur Astronomie (MPIA Hei-
+delberg), Max-Planck-Institut f¨ur Astrophysik (MPA Garching), Max-Planck-Institut
+f¨ur Extraterrestrische Physik (MPE), National Astronomical Observatories of China,
+New Mexico State University, New York University, University of Notre Dame, Ob-
+servat´ario Nacional / MCTI, The Ohio State University, Pennsylvania State University,
+Shanghai Astronomical Observatory, United Kingdom Participation Group, Universi-
+dad Nacional Aut´onoma de M´exico, University of Arizona, University of Colorado
+Boulder, University of Oxford, University of Portsmouth, University of Utah, Univer-
+sity of Virginia, University of Washington, University of Wisconsin, Vanderbilt Uni-
+versity, and Yale University.
+This research has made use of data obtained from LAMOST quasar survey. LAM-
+OST is a National Major Scientific Project built by the Chinese Academy of Sciences.
+Funding for the project has been provided by the National Development and Reform
+Commission. LAMOST is operated and managed by the National Astronomical Ob-
+servatories, Chinese Academy of Sciences.
+This work has made use of data from the European Space Agency (ESA) mission
+Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
+Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium).
+Funding for the DPAC has been provided by national institutions, in particular the
+institutions participating in the Gaia Multilateral Agreement.
+Author Contributions Statement
+JJT contributed the data analysis and drafting and editing of the article. CW contributed
+the conception and design of the work, the supervision of JJT, as well as drafting and
+editing of the article. JT contributed the observation and data collection.
+Competing Interests Statement
+The authors declare no competing interests.
+12
+
+Table 1: Coefficients for final fit results.
+method
+bootstrap
+bin averages
+B0
+−0.944 ± 0.001
+−0.985 ± 0.001
+BL
+−0.271 ± 0.002
+−0.277 ± 0.004
+Bλ
+−1.216 ± 0.011
+−1.265 ± 0.021
+γ
++0.503 ± 0.002
++0.447 ± 0.003
+CL
++0.539 ± 0.004
++0.620 ± 0.010
+Cλ
++2.418 ± 0.023
++2.831 ± 0.052
+log A0
+−1.187 ± 0.001
+−1.210 ± 0.001
+γth
++0.510 ± 0.002
++0.463 ± 0.003
+13
+
+Figure 1: Coefficients of the structure function from Equation 10 in narrow time in-
+tervals, comparing bootstrap results and simple bin averages. This figure contains the
+result of 5315 QSOs. The bootstrap error bars are the standard deviation of the dis-
+tribution while the error bars of the simple bin averages are the standard error of the
+median (SEMED). Data points and error bars are based on ∼ 2 billion magnitude pairs
+from 5315 QSOs.
+14
+
+Figure 2: Global coefficient estimates from Equation 11 (bootstrap case). ∆t0 is the
+short end of the fitting window, while the long end of the window (∆tend) is fixed at
+337 days. The crosses and their error bars are the means and the standard deviations of
+the bootstrap distributions. When short timescales are included, the estimates drift, as
+non-constant behaviour is included into the fit. Restricting the window to the longest
+timescales, shrinks the data set and makes the fits noisier (data points and error bars are
+based on ∼ 2 billion magnitude pairs from 5315 QSOs).
+15
+
+Figure 3: Variability amplitudes log A vs. log ∆t (left) and vs. log(∆t/tth) (right).
+The colour encodes the thermal timescale of the measurement from blue for the shortest
+to red for the longest tth. The best-fit lines to the data at log A > −1.4 have slopes of
+γ = 0.503 (left) and γth = 0.510 (right), consistent with a random walk. Solid lines
+show the fit range while the dotted lines are extrapolated. At short ∆t, the variability
+appears suppressed and this effect reaches to longer ∆t in larger (here: redder) disks.
+16
+
+Figure 4: One example time separation bin, ∆trest = [125; 141] days. The 5,315
+individual QSOs are points, while large symbols and their error bars are bootstrap
+mean and standard deviation of the variability amplitudes in bins of luminosity L3000
+and redshift z, lines are fits to bootstrap mean values (left: orange passband; centre:
+cyan passband). Right: intercepts and errors of the fits at fixed L3000 for different
+rest-frame wavelengths, where either passband is seen to sample the same underlying
+behaviour. Solid lines are fit ranges, dashed lines are extrapolations shown for clarity.
+Note an upwards deviation at λ > 300 nm, which is consistent with the dotted line
+from one past work24 and also seen in other works22,44.
+Figure 5: The variability amplitude, π
+2 < ∆m >2, at ∆t < 1 day as a function of
+observed magnitude. Gray symbols are individual QSOs. Dotted lines show the 3σ-
+clipped mean within fine magnitude bins. Solid lines show the fit of Equation 9.
+17
+
+References
+[1] Kormendy, J. & Ho, L. C. Coevolution (or not) of supermassive black holes and
+host galaxies. ARA&A 51, 511–653 (2013).
+[2] Richards, G. T. et al. The sloan digital sky survey quasar survey: Quasar lumi-
+nosity function from data release 3. AJ 131, 2766–2787 (2006).
+[3] Wolf, C. et al. Discovery of the most ultra-luminous qso using gaia, skymapper,
+and wise. PASA 35, e024 (2018).
+[4] Pringle, J. E. & Rees, M. J. Accretion disc models for compact x-ray sources.
+A&A 21, 1 (1972).
+[5] Balbus, S. A. & Hawley, J. F. A powerful local shear instability in weakly mag-
+netized disks. i. linear analysis. ApJ 376, 214 (1991).
+[6] Balbus, S. A. Enhanced angular momentum transport in accretion disks. ARA&A
+41, 555–597 (2003).
+[7] Shakura, N. I. & Sunyaev, R. A. Reprint of 1973a&amp;a....24..337s. black holes
+in binary systems. observational appearance. A&A 500, 33–51 (1973).
+[8] Clavel, J. et al. Correlated hard x-ray and ultraviolet variability in ngc 5548. ApJ
+393, 113 (1992).
+[9] Peterson, B. M. Reverberation mapping of active galactic nuclei. PASP 105, 247
+(1993).
+[10] Cackett, E. M., Horne, K. & Winkler, H. Testing thermal reprocessing in active
+galactic nuclei accretion discs. MNRAS 380, 669–682 (2007).
+[11] Shappee, B. J. et al. The man behind the curtain: X-rays drive the uv through nir
+variability in the 2013 active galactic nucleus outburst in ngc 2617. ApJ 788, 48
+(2014).
+[12] Fausnaugh, M. M. et al.
+Space telescope and optical reverberation mapping
+project. iii. optical continuum emission and broadband time delays in ngc 5548.
+ApJ 821, 56 (2016).
+[13] Jiang, Y.-F. et al. Detection of time lags between quasar continuum emission
+bands based on pan-starrs light curves. ApJ 836, 186 (2017).
+[14] Fausnaugh, M. M. et al. Continuum reverberation mapping of the accretion disks
+in two seyfert 1 galaxies. ApJ 854, 107 (2018).
+[15] Yu, Z. et al. Quasar accretion disk sizes from continuum reverberation mapping
+in the des standard-star fields. ApJS 246, 16 (2020).
+[16] Kelly, B. C., Bechtold, J. & Siemiginowska, A. Are the variations in quasar
+optical flux driven by thermal fluctuations? ApJ 698, 895–910 (2009).
+18
+
+[17] MacLeod, C. L. et al. Modeling the time variability of sdss stripe 82 quasars as a
+damped random walk. ApJ 721, 1014–1033 (2010).
+[18] Wang, H. & Shi, Y. The deviation of optical variability of radio-quiet quasars
+from damped random walk. Ap&SS 364, 27 (2019).
+[19] Hughes, P. A., Aller, H. D. & Aller, M. F. The university of michigan radio
+astronomy data base. i. structure function analysis and the relation between bl
+lacertae objects and quasi-stellar objects. ApJ 396, 469 (1992).
+[20] Kozłowski, S. Revisiting stochastic variability of agns with structure functions.
+ApJ 826, 118 (2016).
+[21] Palanque-Delabrouille, N. et al. Variability selected high-redshift quasars on sdss
+stripe 82. A&A 530, A122 (2011).
+[22] Vanden Berk, D. E. et al. The ensemble photometric variability of ˜25,000 quasars
+in the sloan digital sky survey. ApJ 601, 692–714 (2004).
+[23] Morganson, E. et al. Measuring quasar variability with pan-starrs1 and sdss. ApJ
+784, 92 (2014).
+[24] Caplar, N., Lilly, S. J. & Trakhtenbrot, B. Optical variability of agns in the ptf/iptf
+survey. ApJ 834, 111 (2017).
+[25] Li, Z., McGreer, I. D., Wu, X.-B., Fan, X. & Yang, Q. The ensemble photometric
+variability of over 105 quasars in the dark energy camera legacy survey and the
+sloan digital sky survey. ApJ 861, 6 (2018).
+[26] Suberlak, K. L., Ivezi´c, ˇZ. & MacLeod, C. Improving damped random walk
+parameters for sdss stripe 82 quasars with pan-starrs1. ApJ 907, 96 (2021).
+[27] Mushotzky, R. F., Edelson, R., Baumgartner, W. & Gandhi, P. Kepler observations
+of rapid optical variability in active galactic nuclei. ApJ 743, L12 (2011).
+[28] Kasliwal, V. P., Vogeley, M. S. & Richards, G. T. Are the variability properties
+of the kepler agn light curves consistent with a damped random walk? MNRAS
+451, 4328–4345 (2015).
+[29] Smith, K. L. et al. The kepler light curves of agn: A detailed analysis. ApJ 857,
+141 (2018).
+[30] Tachibana, Y. et al. Deep modeling of quasar variability. ApJ 903, 54 (2020).
+[31] Stone, Z. et al.
+Optical variability of quasars with 20-year photometric light
+curves. MNRAS (2022).
+[32] Burke, C. J. et al. A characteristic optical variability time scale in astrophysical
+accretion disks. Science 373, 789–792 (2021).
+[33] Rakshit, S., Stalin, C. S. & Kotilainen, J. Spectral properties of quasars from
+sloan digital sky survey data release 14: The catalog. ApJS 249, 17 (2020).
+19
+
+[34] Tonry, J. L. et al. Atlas: A high-cadence all-sky survey system. PASP 130,
+064505 (2018).
+[35] Heckman, T. M. Long-time-scale optical variability in quasars. PASP 88, 844–
+848 (1976).
+[36] Collaboration, G. et al. Gaia early data release 3. summary of the contents and
+survey properties. A&A 649, A1 (2021).
+[37] Kozłowski, S. Limitations on the recovery of the true agn variability parameters
+using damped random walk modeling. A&A 597, A128 (2017).
+[38] Kelly, B. C., Treu, T., Malkan, M., Pancoast, A. & Woo, J.-H. Active galactic
+nucleus black hole mass estimates in the era of time domain astronomy. ApJ 779,
+187 (2013).
+[39] Frank, J., King, A. & Raine, D. J. Accretion Power in Astrophysics: Third Edition
+(Cambridge University Press, 2002).
+[40] Sun, M. et al. Corona-heated accretion-disk reprocessing: A physical model to
+decipher the melody of agn uv/optical twinkling. ApJ 891, 178 (2020).
+[41] Campitiello, S., Ghisellini, G., Sbarrato, T. & Calderone, G. How to constrain
+mass and spin of supermassive black holes through their disk emission. A&A
+612, A59 (2018).
+[42] Abramowicz, M. A., Czerny, B., Lasota, J. P. & Szuszkiewicz, E. Slim accretion
+disks. ApJ 332, 646 (1988).
+[43] Sadowski, A. et al. Relativistic slim disks with vertical structure. A&A 527, A17
+(2011).
+[44] Yu, W., Richards, G. T., Vogeley, M. S., Moreno, J. & Graham, M. J. Examining
+agn uv/optical variability beyond the simple damped random walk. arXiv e-prints
+arXiv:2201.08943 (2022).
+[45] Kasliwal, V. P., Vogeley, M. S., Richards, G. T., Williams, J. & Carini, M. T. Do
+the kepler agn light curves need reprocessing? MNRAS 453, 2075–2081 (2015).
+[46] Brandt, W. N. et al.
+Active galaxy science in the lsst deep-drilling fields:
+Footprints, cadence requirements, and total-depth requirements. arXiv e-prints
+arXiv:1811.06542 (2018).
+[47] Ivezi´c, ˇZ. et al. Lsst: From science drivers to reference design and anticipated
+data products. ApJ 873, 111 (2019).
+[48] Chan, J. H. H., Millon, M., Bonvin, V. & Courbin, F. Twisted quasar light curves:
+implications for continuum reverberation mapping of accretion disks. A&A 636,
+A52 (2020).
+20
+
+[49] Kawaguchi, T., Mineshige, S., Umemura, M. & Turner, E. L. Optical variability
+in active galactic nuclei: Starbursts or disk instabilities?
+ApJ 504, 671–679
+(1998).
+[50] Flesch, E. W. The half million quasars (hmq) catalogue. PASA 32, e010 (2015).
+[51] Schlegel, D. J., Finkbeiner, D. P. & Davis, M. Maps of dust infrared emission
+for use in estimation of reddening and cosmic microwave background radiation
+foregrounds. ApJ 500, 525–553 (1998).
+[52] Gravitationally lensed quasar database. https://research.ast.cam.ac.uk/lensedquasars/.
+Accessed: 2022-06-03.
+[53] Lemon, C. et al. Gravitationally lensed quasars in gaia – iv. 150 new lenses,
+quasar pairs, and projected quasars. arXiv e-prints arXiv:2206.07714 (2022).
+[54] Tonry, J. L. et al. The atlas all-sky stellar reference catalog. ApJ 867, 105 (2018).
+[55] Riello, M. et al. Gaia early data release 3. photometric content and validation.
+A&A 649, A3 (2021).
+[56] Shen, Y. & Burke, C. J. A sample bias in quasar variability studies. ApJ 918, L19
+(2021).
+[57] Becker, R. H., White, R. L. & Helfand, D. J. The first survey: Faint images of the
+radio sky at twenty centimeters. ApJ 450, 559 (1995).
+[58] Condon, J. J. et al. The nrao vla sky survey. AJ 115, 1693–1716 (1998).
+[59] Mauch, T. et al. Sumss: a wide-field radio imaging survey of the southern sky -
+ii. the source catalogue. MNRAS 342, 1117–1130 (2003).
+[60] Casagrande, L. & VandenBerg, D. A. On the use of gaia magnitudes and new
+tables of bolometric corrections. MNRAS 479, L102–L107 (2018).
+[61] Richards, G. T. et al. Spectral energy distributions and multiwavelength selection
+of type 1 quasars. ApJS 166, 470–497 (2006).
+[62] Morgan, C. W., Kochanek, C. S., Morgan, N. D. & Falco, E. E. The quasar
+accretion disk size-black hole mass relation. ApJ 712, 1129–1136 (2010).
+[63] King, A. R., Pringle, J. E. & Livio, M. Accretion disc viscosity: how big is alpha?
+MNRAS 376, 1740–1746 (2007).
+[64] di Clemente, A., Giallongo, E., Natali, G., Trevese, D. & Vagnetti, F. The vari-
+ability of quasars. ii. frequency dependence. ApJ 463, 466 (1996).
+[65] Markwardt, C. B. Non-linear least-squares fitting in idl with mpfit. In Bohlender,
+D. A., Durand, D. & Dowler, P. (eds.) Astronomical Data Analysis Software and
+Systems XVIII, vol. 411 of Astronomical Society of the Pacific Conference Series,
+251 (2009).
+21
+
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+page_content=' Christian Wolf1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
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+page_content=' Australian National  University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
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+page_content=' Australian National University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Building 38 Science Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
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+page_content=' University of Hawaii,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' 2680 Woodlawn Drive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Honolulu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' HI 96822-1897,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  draft January 5, 2023  Abstract  Rapidly growing black holes are surrounded by accretion disks that make them  the brightest objects in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Their brightness is known to be variable, but  the causes of this are not implied by simple disk models and still debated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Due  to the small size of accretion disks and their great distance, there are no resolved  images addressing the puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' In this work, we study the dependence of their vari-  ability on luminosity, wavelength and orbital/thermal timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' We use over 5,000  of the most luminous such objects with light curves of almost nightly cadence from  > 5 years of observations by the NASA/ATLAS project, which provides 2 billion  magnitude pairs for a structure function analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' When time is expressed in units  of orbital or thermal time scale in thin-disk models, we find a universal struc-  ture function, independent of luminosity and wavelength, supporting the model of  magneto-rotational instabilities as a main cause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Over a > 1 dex range in time, the  fractional variability amplitude follows log(A/A0) ≃ 1/2 × log(∆t/tth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' De-  viations from the universality may hold clues on the structure and orientation of  disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Every massive galaxy in the Universe contains a supermassive black hole at its  centre;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' their origin and growth is one of the biggest mysteries in astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Most  black holes appear dark and are revealed only by their gravity1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' During growth spurts,  however, matter from surrounding areas falls towards the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Its gravitational  energy is turned via heat into radiation, which highlights the black hole as a ”QSO”  (Quasi-Stellar Object).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Fast-growing black holes are the most luminous objects in the  Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' At UV-optical wavelengths, QSOs can outshine 1,000 large galaxies2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  The key for turning gravitational energy into heat is matter flowing through an  accretion disk4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Matter orbiting the black hole drifts inwards as high friction in the disk  1  dissipates orbital energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' It is now clear that Keplerian disks with weak magnetic fields  are prone to magneto-rotational instability (MRI)5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' MRI is a fundamental instability  converting differential rotation into turbulent fluid motions, feeding an energy cascade,  and explaining the anomalously high viscosity needed for luminous QSO disks6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Accretion disks are an elegant explanation for the high luminosity and compactness  of QSOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The standard model of optically thick and geometrically thin disks predicts  their temperature profiles and emission7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' As they are too small to be spatially resolved  by current facilities, we seek clues about their properties from the variability of their  emission, which is ubiquitous on all timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' While MRI must be one source of  short-term fluctuations, another must be the rapidly fluctuating heating from the X-ray  corona near the black hole8–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The “lamppost” model exploits the latter and relates  the sizes of accretion disks to time lags between light curves of different passbands  probing regions of different temperature and radius in the disk9–15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Monitoring of QSO samples has revealed that fluctuations can be described by a  ‘damped random walk’ model16–18: brightness changes tend to be larger when the  time delay between any two measurements is longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' However, this model provides  only a behavioural analogy and no physical insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Mathematically, the light curve of  an object can be statistically described by a variability ”structure function” (SF): the  typical brightness difference is expressed as a function of time interval19,20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The SFs of  QSOs are characteristically different from other types of variable objects, so that QSOs  can even be identified in a time-domain survey by the SF alone21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Measurements of the SF agree on two trends: fluctuations on a fixed timescale get  stronger in less luminous QSOs and in bluer passbands17,22–26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Thus, the SF can be  parametrised as  log A = B0 + γ × log ∆t + BL × log L + Bλ × log λ ,  (1)  where amplitudes A of fractional flux changes are in terms of magnitude, and L is the  measured luminosity of the QSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Both the time separation between the two measure-  ments, ∆t, and the wavelength of observed light, λ, are expressed in the QSO rest-  frame with cosmological redshift and time dilation removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Outside of the random-  walk regime, the amplitudes may be suppressed at shortest timescales27–31 for reasons  that are not yet established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' They tend to level off above a damping or decorrelation  timescale17,31,32, which has been shown to be driven by the thermal timescale of the  accretion disk32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Past works agree that BL < 0 and Bλ < 0 but disagree on the strength of the  trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Some complicating factor may not be controlled for in existing studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The  SF slope γ = d log A/d log ∆t at fixed (L, λ) is in the range20,22,23,25 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='2 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  BL = d log A/d log L at fixed (∆t, λ) describes the dependence on disk luminosity  (hence size) and is reported in the range17,22,23,25,26 of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='33 to −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Finally, Bλ =  d log A/d log λ at fixed (∆t, L) describes the trend with rest-frame wavelength and  thus location of the fluctuating source within the disk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' measurements range17,23,25 from  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='75 to −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='48, although observations of non-linear relations also include slope values  outside this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Here, we present a new analysis of the variability structure function of the ∼ 5, 000  brightest QSOs in the sky using light curves with unprecedented sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The sample  2  has a median black-hole mass of ∼ 2 billion solar masses according to an analysis of  spectra from the Sloan Digital Sky Survey (SDSS)33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The light curves are from the  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='5-metre NASA/ATLAS telescope (Asteroid Terrestrial-impact Last Alert System34),  which was built to find dangerous asteroids that might threaten the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ATLAS is  designed to image the entire sky seen from its location in Hawaii four times every night,  weather permitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' With up to 1,200 nights per QSO per band, we get ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='1 billion  magnitude pairs to constrain the QSO SF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' This is one of the best data sets for studying  QSO variability before the new Vera C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Rubin Telescope operates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Results  The new light curves include observations from 2015 to 2021 in two passbands, cyan  and orange (centred on 533 and 679 nm, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' We limit the sample to redshift  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='5 < z < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='5 (z < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='4 in the cyan band), where QSOs are so luminous that their host  galaxies will not affect the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' We reject radio-loud QSOs, where variabil-  ity may be enhanced by shocks in jets and mask the pure accretion disk behaviour35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  We also reject gravitationally lensed QSOs and QSOs with close neighbours as seen by  the ESA Gaia mission36 to avoid blended light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Previous work typically fitted a global model for the SF, such as Equation 1, to a  whole data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The richness of this new data set offers the alternative to split the data  into subsets without incurring the penalty of noisy statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' We first split our data into  twenty bins in ∆t and fit the trends with luminosity and wavelength separately for each  bin, using the equation  log(A(∆t)) = B0,∆t + BL,∆t × log L + Bλ,∆t × log λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  (2)  Thus, we get twenty different estimates for BL and Bλ, independently for each ∆t bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  As we can now study the trends on different timescales, we may see different regimes  of behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' We use the measured variability on timescales of 1 day to estimate the  noise in the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' At the long end, window effects from the finite length of  the light curves may cause noise in the parameters37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' We restrict the wavelength range  to longer than 130 nm (avoiding flux from the strong Ly-α emission line) and shorter  than 300 nm (avoiding a previously seen upturn in variability17,22,24, which is currently  unexplained and discussed further below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' We use two types of statistical analysis,  simple bin averages, and a computationally more intense bootstrap analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Figure 1 shows the resulting parameters for a range in ∆t from 10 to 337 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Both  methods of analysis produce consistent estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The estimates for BL and Bλ show  a strong trend with ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' It is immediately clear that a global fit will produce estimates  for BL and Bλ that depend on the fitting range in ∆t and on the distribution of pairs in  the range, and thus on subtleties of survey cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' We suggest this fact explains the  variety of estimates obtained in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' We demonstrate how results depend on the  range of behaviour forced into a global fit in Figure 2, where fiducial global estimates  for BL and Bλ are obtained by choosing different global fitting ranges for ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Finally, we improve the global fit by restricting it to the random-walk regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Given that our sample comprises the most luminous known QSOs, we expect damping  timescales well in excess of a year, which do not affect our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Having controlled  3  for ∆t by fitting the trends in each bin separately (Figure 1), we reveal two regimes in  behaviour: at short ∆t both BL and Bλ drift with ∆t, whereas at longer ∆t they both  remain constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Thus, at longer ∆t the choice of fitting range will have little impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Here, we find fit parameters of γ ≈ 1/2, BL ≈ −1/4 and Bλ ≈ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Past estimates  of BL cover values from a lower range17,26 of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='33 to −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='29, depending on the cho-  sen parametrisation, to an upper limit23 of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='16 and include values very close22,25 to  −1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Changing the clock for a universal structure function  So far, BL and Bλ have described how variability amplitudes behave at fixed ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' We  now look at the timescales, on which disks of different L, observed at different λ, show  a fixed amplitude A and thus potentially a universal structure function if their slopes γ  are the same as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' We find these timescales to be parametrised as:  CL = d log ∆t(A)/d log L  =  −BL/γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='539 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='004  (3)  Cλ = d log ∆t(A)/d log λ  =  −Bλ/γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='418 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='023 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  (4)  The first study in this direction empirically found CL = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='06 and argued  that the fractional optical variability per disk orbital or thermal timescale is likely con-  stant38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' This argument is based on the well-known fact that the predicted temperature  profile in thin disks7, T (R) ∝ R−3/4, sets a mean radius for the origin of radiation  and thus a mean orbital timescale torb of the emitting surface, given λ and a size, and  thus L, of the disk17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Note, that for a fixed viscosity of the disk material, the thermal  timescale tth is proportional to the orbital timescale and we get38:  log torb ∝ log tth ∝ 1/2 log L + 2 log λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  (5)  One later study24 reported CL ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Our result shows that the concept of  a constant fractional variability per orbital or thermal timescale38 is not only consistent  with measurements of CL but also our Cλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Thus, we rephrase the whole SF from  Equation 1 with time in new units, choosing those of thermal timescale given its role  for the damping timescale32:  log(A/A0) = γth log(∆t/tth) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  (6)  This form is expected for fluctuations arising from MRI, where the equations of motion  scale with the orbital timescale (and thus the thermal timescale)6,39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' It also applies in  the corona-heated accretion-disk reprocessing (CHAR) model40, which is motivated  by the concern that a lamppost will not create sufficient amplitude by heating alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  The CHAR model proposes that the corona couples directly to the magnetic field in the  disk and drives stronger fluctuations than expected from a lamppost corona alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  We determine tth for each QSO and passband given its (L, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Figure 3 shows  the resulting SF expressed both in natural time and with a tth-scaled clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Here, we  increased the time resolution but reduced noise by grouping QSOs into subsamples  split by (L, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The wide (L, λ)-range of our sample causes a wide range of ampli-  tudes at fixed ∆t (left panel), but at fixed ∆t/tth (right panel) we find indeed a tight  4  range, supporting the concept of a universal structure function with a constant slope  and intercept once expressed in suitable units of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The relation holds over more  than 1 dex in ∆t/tth, limited by the length of our light curves at long timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' At  short timescales, the amplitude suppression relative to the random walk remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The  CHAR model actually predicts a suppression with a break timescale proportional to the  thermal timescale, which would imply a fixed break timescale in Figure 3, right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  However, we see a spread of timescales such that the break depends more strongly on  luminosity than the thermal timescale, which will be investigated in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' For  our final global fit of the SF, we avoid the suppressed regime by using only data at  log A > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' We find a bootstrap slope with natural clock time of γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='503 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='001  and with time in units of thermal timescale of γth = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='510 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='002, extremely consis-  tent with the γ = 1/2 of a random walk (Table 1 summarises all fit parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Outlook  The transition from a regular clock to an orbital or a thermal time-scaled clock has  standardised the random walk of QSOs as the wide dispersion observed among objects  with a variety of luminosities and at a variety of restframe wavelengths has disappeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Instead, the SF of QSOs appears now as an intrinsically tight relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  A universal SF opens a window to characterise QSOs in more detail by devia-  tions from the mean relation using future higher-precision measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Such devia-  tions are expected to arise from any effect that varies the orbital and thermal timescale  estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' These effects include the dependence of the observed luminosity on disk  orientation and black-hole spin41, which are not independently known for individual  objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' the true temperature profile of the accretion disk, which is still debated as it  may change from T ∝ R−3/4 in an inner part driven by viscosity to T ∝ R−1/2 in an  outer part dominated by irradiation from an X-ray corona acting as a lamppost39;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' and  the actual interior structure of the disk, which may well change from thin to slim disks  as the accretion rate approaches the Eddington rate39,42,43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Deviations from a simple picture are generally evident at wavelengths of λ >  300 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Here, we measure an upturn in A, relative to a trend of Bλ ≈ −1, which has  been seen before17,22,24,44 (see also Figure 4, right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' It suggests that fluctuations  in the outer, cooler parts of QSO accretion disks are to be enhanced by an additional  mechanism, which has not yet been explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Also, we excluded λ < 130 nm, as  we cannot disentangle light from the disk continuum and light from the broad Ly α  line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The hotter inner disks are harder to study as a wide swath of spectrum in the UV  domain is absorbed by intervening gas along the line-of-sight to QSOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  The SF suppression at short ∆t has been observed across a range in luminosity: ob-  servations of low-luminosity AGN with the Kepler space mission have revealed breaks  in the SF on time scales of days to weeks27–29, although it has also been argued that  there “remain significant levels of spacecraft-induced effects in the standard pipeline  reduction of the Kepler data”45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Steeper slope changes have been seen in a recent  study of luminous QSOs on time scales of 10 to 15 days31, while our break time scales  range from ∼ 15 to ∼ 50 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The reasons behind these breaks are still under debate:  either the structure function is intrinsically different such as in the CHAR model40, or  a ”smearing” effect may be caused by synchronised emission from different parts of  5  the disk reaching the observer after different time lags30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  The forthcoming 10-year Legacy Survey in Space and Time (LSST46–48) planned  at the Vera C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Rubin Telescope for the years 2024-34 will eclipse this data set with  better data: more passbands will probe a wider temperature range in the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Deeper photometry will probe QSOs at much fainter luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' This, and the reduced  photometric errors in LSST will provide low-noise structure functions for individual  objects and reveal how tight and universal the structure function of QSO variability  really is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' LSST should also be a game changer for analysing subtleties in the short-  term breaks in the SF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Conclusion  We have determined with unprecedented precision the parameters (γ, BL, Bλ) of the  variability structure function for the UV continuum emission of luminous QSOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' It is  consistent with a universal random walk once time is expressed in units of the orbital  or thermal timescales of the emitting material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The mean amplitudes of fluctuations  can be described by the simple law log(A/A0) = 1/2×log(∆t/tth), where a single fit  parameter A0 represents the overall QSO population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Note, that the thermal timescale  has recently been shown to drive the damping timescale of QSO variability as well32,  although this fact may not have affected the random walk at shorter timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  The result appears consistent with the temperature profiles of thin-disk models7, at  least at wavelengths from 130 to 300 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Deviations observed at longer wavelengths  that show the cooler, outer parts of disks17,22,24,44 still need to be explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' It also  appears consistent with an origin of the fluctuations in magneto-rotational instabilities  (MRI) in differentially rotating accretion disks that are made of magnetised plasma;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  there, the equations of motion scale with the orbital timescale6,39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' MRI is suggested to  produce variability with a random walk phenomenology49 and γ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='4 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Disks  could thus be ordered in a 1-parameter family with the mean thermal timescale at a  chosen wavelength as an ordering parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The structure function of QSOs may then  be a universal relation apart from the short-term suppression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' It appears intrinsically  tight and would be broadened in observation by disk orientation and black hole spin,  as well as host galaxy dust extinction affecting the inferred luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' It would also  be broadened by intrinsic variety in vertical disk structure driven by accretion rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  The role of the lamppost for driving QSO variability is likely confined to a contri-  bution of modest amplitude on shorter timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' If the CHAR model also described  the physics of disk variability correctly, then it would be hard to disentangle genuine  lamppost reverberation from outwards-propagating magnetic fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' and if the veloc-  ity of the latter dropped below the speed of light40, the sizes of disks inferred by a  reverberation analysis could be overestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Future data from the 10-year LSST project will shed light on these questions and  more subtle properties with its extremely precise measurements for a nearly all-sky  QSO sample of unprecedented size, depth, variety and completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  6  Online Methods  QSO sample and data cleaning  We combined the Million Quasars Catalog50 (MILLIQUAS v7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='1 update) with the Gaia  eDR3 catalogue36 and select 6,163 spectroscopically confirmed, non-lensed QSOs  with Gaia magnitude Rp < 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='5, redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='5 < z < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='5, declination δ > −45◦,  Galactic foreground reddening51 of E(B − V )SFD < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='15 and Gaia BpRp Excess  Factor < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='3 (indicative of single, unblended, point sources).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' We also required that the  MILLIQUAS and Gaia positions are coincident within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='3′′, which excludes blended  objects of similar SED such as the multiple images of lensed QSOs, and that sources  have no Gaia neighbours within 15′′ to keep the ATLAS photometry unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' We  confirmed with recent lists of lensed QSOs52,53 that none of these are included in our  sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Our sample has 6,115 sources in common with NASA/ATLAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The ATLAS  photometric system is consistent with other reliable sources of photometry such as  Pan-STARRS and Gaia54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  The selection is based on Gaia eDR3 mean photometry, which is arguably the best-  calibrated among large astronomical datasets55;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' the Gaia observations extend from  mid-2014 to mid-2017 and overlap with the period of the ATLAS lightcurves (LCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Thus, there will be no noticeable selection bias in favour of QSOs that might be brighter  during an early selection epoch but fainter when the LCs are observed56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  We further remove six QSOs from the sample, which have large flux errors through-  out their LCs (90-percentile flux errors in the orange band of σfν,o > 150µJy or in the  cyan band LC of σfν,c > 85µJy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' On individual images, the faintest objects in the  sample have signal-to-noise ratios (SNR) of > 11 in the orange band and > 13 in the  cyan band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Up to four images are available per band and night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  We reject radio-loud QSOs by cross-matching the sample with catalogues from the  Faint Images of the Radio Sky at Twenty-cm57 (FIRST) Survey, the NRAO VLA Sky  Survey58 (NVSS), and the Sydney University Molonglo Sky Survey59 (SUMSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' We  use matching radii between the MILLIQUAS and FIRST, NVSS, and SUMSS coordi-  nates of 3”, 12”, and 11”, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' 1,242 QSOs are matched with at least one radio  catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' If a QSO is matched to multiple radio sources in NVSS or SUMSS, the ra-  dio detection with the closest separation is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' We apply a radio loudness criterion of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='4×(mo −tNVSS/SUMSS) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='3 for matches with NVSS and SUMSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Radio sources  below this threshold are labelled radio-intermediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' QSOs with matches in FIRST but  not in NVSS turned out to be all radio-intermediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The sample contains 775 radio-  loud and 467 radio-intermediate objects, and 4,848 radio non-detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Thus, our  final sample contains 5,315 QSOs, of which 3,607 have single-epoch virial black hole  mass measurements derived from SDSS spectra33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  We clean the ATLAS LCs first by excluding all observations with large errors of  log(σfν,o) > −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='94−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='12×(mo−16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='5) and log(σfν,c) > −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='17−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='10×(mc−16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='5),  which constitute about 5 percent of all observations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' and further by comparing each  measurement with other observations within ±7 days and removing outliers with a 2σ-  clipping to reject spurious variability, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', due to temporary chance blending with faint  asteroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' If there is only one observation within ±7 days, it is retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  7  Luminosity  We estimate QSO luminosities at restframe 3,000 ˚A (L3000) using a linear interpolation  of the Gaia Bp and Rp magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' At redshifts of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='5 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='7 and z > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='56, the  restframe 3,000 ˚A point is extrapolated beyond the pivotal wavelengths of the Gaia  filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' We correct for foreground dust extinction in the Milky Way51 using bandpass  coefficients60 of RBp = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='378 and RRp = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Luminosity distances are derived  in a flat ΛCDM cosmology with Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='3 and a Hubble-Lemaˆıtre constant of H0 =  70 km sec−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Luminosity errors from this procedure are expected to be < 20%  and thus much smaller than those introduced by host galaxy dust and the orientation  of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Luminosity estimates from broadband photometry does not distinguish  between contributions from the accretion disk continuum and those from line emission,  especially from the Fe complex that is strong in the vicinity of 3,000 ˚A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Comparing our  L3000 estimates with those obtained by spectral fitting on the available subsample33,  we find an average offset of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='07 dex and reduce our L3000 values to correct for line  contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Bolometric luminosities, as used by some authors24, are derived with  log(Lbol/L3000) = log(fBC × λ) = log(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='15 × 3000 ˚A) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='189, using fBC = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='15  as the bolometric correction factor61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Thermal timescale  For each QSO and passband, we estimate the characteristic (defined in the following)  thermal timescale of the emitting material in the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' We combine the equa-  tion for tth 39 and an equation for the scale length of the disk given L, λ values62, and  get:  tth  days = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='89  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='8 ×  1  86400 × 10−13 × (45GM⊙fBCλ3000L3000λ4  rf  16π6α2ηhpc4 cos i  )0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='5  = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='40 × 10−27 × ( L3000  erg/s/ ˚A)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='5 × (λrf  ˚A )2 ,  (7)  where hp is the Planck constant, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='163 is the assumed viscosity parameter, and i =  45◦ is the mean inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' We assume a radiative efficiency of η = Lbol/( ˙Mc2) =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='1, where ˙M is the mass accretion rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' changes in η due to variations in black-hole  spin affect the inner cutoff of the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' For a given  ˙M, such a variation  has little effect on the outer disk, the overall L3000, and the thermal timescales at the  wavelengths observed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Instead, it mostly affects the emission of higher-energy  photons and Lbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Variability structure function  We adopt the following definition of the variability structure function (SF)64:  A =  �π  2 < ∆m >2 − < σ2 >, with ∆m = |mi − mj|,  (8)  8  where mi and mj are any two observations and σ is the magnitude error due to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  We find that the magnitude errors quoted by ATLAS underestimate the noise and in-  stead derive σ2 from the apparent intra-day variability, which we expect to be negligi-  ble for radio-quiet QSOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' We determine < σ2 > as a function of observed magnitude,  mobs, such that the SF is anchored at A ≈ 0 for ∆t < 1 day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The required noise model  is shown in Figure 5 and given by  log(< σ2 >) = n0 + n1mobs ,  (9)  with (n0, n1) = (−12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='411, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='585) and (−12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='428, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='573) for the orange and cyan pass-  band, respectively (pure Poisson noise suggests n1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Informed by the pair sampling distribution of the LCs, we reduce window effects  from the finite extent of the LCs by ignoring pairs with ∆tobs ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='47max(∆tobs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  This cut is applied in the observed frame, while the rest of the paper uses the rest-  frame ∆t = ∆tobs/(1 + z), corrected for cosmological time dilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Binning the time axis  We bin the time axis in two different ways depending on purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' A first case is used  for the relation log A(∆t) in Equation 2 and Figures 1, 2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' A second case is used  for fits over the final fitting range and for log A(∆t/tth) in Equation 6, Figure 3 and  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  In the first case, we split the ∆t axis into 20 bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The first ∆t bin contains all pairs  with ∆t < 1 day, which is used only for noise analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Further bins are chosen to  balance the number of pairs among the bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The average number of pairs per ∆t bin  are ∼103 million and ∼8 million for the orange and cyan passband, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' We  split the z axis into 9 bins, thus grouping QSOs with observations of similar restframe  wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' In the cyan passband, we exclude QSOs at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='4 < z < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='5 to avoid  contamination from the Ly-α line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' In the orange passband, we use this z range as a  single, highest-z, bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The remaining eight bins cover the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='5 < z < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='4  with roughly balanced QSO numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' In each z bin, QSOs are further split into ten  L3000 bins, each with almost equal QSOs numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' In total, there are 1,800 bins in  (z, L3000, ∆t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' In each bin, the mean value of z and L3000 and the mid-point of ∆t is  used for the analysis and figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  In the second case, we split the ∆t axis into 100 bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The shortest ∆t < 1 day  bin is exactly the same as in the first binning method, and further bins are balancing  the number of pairs among the bins for both passbands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The QSOs are grouped into  16 bins along the tth axis with an equal number of QSOs per bin, which is 2,950 and  4,134 in orange and cyan, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' In the ∆t/tth axis, we split the observed pairs  into 100 bins of constant number, independently for each tth group in each passband  after excluding pairs with ∆t < 1 and ∆t > 337 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Combining both passbands,  there are in total 3200 bins for (tth, ∆t) and separately 3200 bins for (tth, ∆t/tth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' In  each bin, the mid-point of ∆t and ∆t/tth is used for the analysis and figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  9  Simple bin average structure function  For each individual QSO, we calculate its SF with Equation 8 from all available pairs  in each ∆t or ∆t/tth bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' We use a 3σ-clipped mean of magnitude differences as  < ∆m >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Then, for each (z, L3000, ∆t) or (tth, ∆t) or (tth, ∆t/tth) bin we calculate  the median squared amplitude, A2, and the standard error of the median (SEMED),  which ensures that QSOs with negative noise-corrected amplitudes A2, due to an over-  subtraction by the ⟨σ2⟩ noise model, are still included in the statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Bootstrapping the structure function  Using bootstrapping, we can account better for the contribution of each QSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Here,  we select random pairs in each of the (z, L3000, ∆t) or (tth, ∆t) or (tth, ∆t/tth) bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  We randomly pick a QSO N times from all the QSOs in that bin, where N is the total  number of QSOs in that bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' For each picked QSO, we randomly select 300 distinct  pairs in the orange and 30 distinct pairs in the cyan passband, and then calculate the SF  using Equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' This process is repeated 100 times to get the bootstrap distribution,  from which we calculate the mean and standard deviation for each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  SF fitting  We fit a structure function of the form  log(A(∆t)) = B0,∆t + BL,∆t log(  L3000  1042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='7erg/s/ ˚A) + Bλ,∆t log(  λrf  2200 ˚A)  (10)  using a Levenberg-Marquardt least-squares fit65 to the (z, L3000) bins for each ∆t  bin separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Here, log A and its error may be results of bin averages or from the  bootstrap method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' λrf is the pivotal wavelength of a passband divided by 1 + z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The  normalisation factors for λrf and L3000 are chosen to be close to the median of the  sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Data with λrf > 3000 ˚A are excluded from the fit as it deviates from a linear  relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' In Figure 4, we show an example for one ∆t bin: the left and middle panels  show the two passbands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The right panel shows the intercepts of the fits at log(L3000)−  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='7 for different λrf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' We also compare the upturn at longer wavelengths with results  from the literature24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The B0,∆t, BL,∆t, and Bλ,∆t from each ∆t bin are shown in  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Global parameter estimates using broad ∆t ranges  We fit a global model of the form  log(A) = B0 + BL log(  L3000  1042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='7erg/s/ ˚A) + Bλ log(  λrf  2200 ˚A) + γ log( ∆t  216d)  (11)  to the SF data in the (z, L3000, ∆t) bins with λrf < 3000 ˚A ˙We test the robustness  of global parameter averages by varying the time span of data that is included in the  global fit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' we vary the lower edge ∆t0, while leaving the upper edge at ∆t=337 days  unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Figure 2 shows the parameter estimates as f(∆t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  10  The structure function as f(∆t) and f(∆t/tth)  For Figure 3 we split the sample for each passband separately into 16 groups ranked  by tth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' We then fit Equation 11 to the points with log A > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='4 using the errors for  weighting (see Table 1 and left panel of Figure 3 for results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Similarly, we fit the bins  with log A > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='4 to Equation 6 (see Table 1 and right panel of Figure 3 for results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  For reference, we find log A0 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='185 when fixing γ = 1/2 in the bootstrap method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Data availability  The QSO samples are selected from the publicly available MILLIQUAS database com-  plemented with publicly available Gaia data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Data from NASA/ATLAS are publicly  available at https://fallingstar-data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='com/forcedphot/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The QSO light curves used in this  study are available from the corresponding author on request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Code availability  The main analysis routines have been written by the corresponding author in IDL and  are available on request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Acknowledgements  JJT was supported by the Taiwan Australian National University PhD scholarship and  the Australian Research Council (ARC) through Discovery Project DP190100252 and  also acknowledges support by the Institute of Astronomy and Astrophysics, Academia  Sinica (ASIAA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' JT has been funded in part by the Stromlo Distinguished Visitor Pro-  gram at RSAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' We thank Mark Krumholz for comments on the manuscript and Stefan  Wagner for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  This work uses data from the University of Hawaii’s ATLAS project, funded through  NASA grants NN12AR55G, 80NSSC18K0284, and 80NSSC18K1575, with contri-  butions from the Queen’s University Belfast, STScI, the South African Astronomical  Observatory, and the Millennium Institute of Astrophysics, Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  This work has made use of SDSS spectroscopic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Funding for the Sloan Dig-  ital Sky Survey IV has been provided by the Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Sloan Foundation, the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Department of Energy Office of Science, and the Participating Institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' SDSS-IV  acknowledges support and resources from the Center for High Performance Comput-  ing at the University of Utah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The SDSS website is http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='sdss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' SDSS-IV is  managed by the Astrophysical Research Consortium for the Participating Institutions  of the SDSS Collaboration including the Brazilian Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' the Carnegie  Institution for Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Carnegie Mellon University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Center for Astrophysics — Har-  vard & Smithsonian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' the Chilean Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' the French Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Instituto de Astrof´ısica de Canarias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The Johns Hopkins University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Kavli Institute  for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' the  11  Korean Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Lawrence Berkeley National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Leibniz Insti-  tut f¨ur Astrophysik Potsdam (AIP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Max-Planck-Institut f¨ur Astronomie (MPIA Hei-  delberg),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Max-Planck-Institut f¨ur Astrophysik (MPA Garching),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Max-Planck-Institut  f¨ur Extraterrestrische Physik (MPE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' National Astronomical Observatories of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  New Mexico State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' New York University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' University of Notre Dame,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Ob-  servat´ario Nacional / MCTI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The Ohio State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Pennsylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Shanghai Astronomical Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' United Kingdom Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Universi-  dad Nacional Aut´onoma de M´exico,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' University of Arizona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' University of Colorado  Boulder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' University of Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' University of Portsmouth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' University of Utah,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Univer-  sity of Virginia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' University of Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' University of Wisconsin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Vanderbilt Uni-  versity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' and Yale University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  This research has made use of data obtained from LAMOST quasar survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' LAM-  OST is a National Major Scientific Project built by the Chinese Academy of Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Funding for the project has been provided by the National Development and Reform  Commission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' LAMOST is operated and managed by the National Astronomical Ob-  servatories, Chinese Academy of Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  This work has made use of data from the European Space Agency (ESA) mission  Gaia (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='int/gaia), processed by the Gaia Data Processing and  Analysis Consortium (DPAC, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='int/web/gaia/dpac/consortium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Funding for the DPAC has been provided by national institutions, in particular the  institutions participating in the Gaia Multilateral Agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Author Contributions Statement  JJT contributed the data analysis and drafting and editing of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' CW contributed  the conception and design of the work, the supervision of JJT, as well as drafting and  editing of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' JT contributed the observation and data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Competing Interests Statement  The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  12  Table 1: Coefficients for final fit results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  method  bootstrap  bin averages  B0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='944 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='001  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='985 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='001  BL  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='271 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='002  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='277 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='004  Bλ  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='216 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='011  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='265 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='021  γ  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='503 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='002  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='447 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='003  CL  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='539 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='004  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='620 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='010  Cλ  +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='418 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='023  +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='831 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='052  log A0  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='187 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='001  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='210 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='001  γth  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='510 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='002  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='463 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='003  13  Figure 1: Coefficients of the structure function from Equation 10 in narrow time in-  tervals, comparing bootstrap results and simple bin averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' This figure contains the  result of 5315 QSOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The bootstrap error bars are the standard deviation of the dis-  tribution while the error bars of the simple bin averages are the standard error of the  median (SEMED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Data points and error bars are based on ∼ 2 billion magnitude pairs  from 5315 QSOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  14  Figure 2: Global coefficient estimates from Equation 11 (bootstrap case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ∆t0 is the  short end of the fitting window, while the long end of the window (∆tend) is fixed at  337 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The crosses and their error bars are the means and the standard deviations of  the bootstrap distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' When short timescales are included, the estimates drift, as  non-constant behaviour is included into the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Restricting the window to the longest  timescales, shrinks the data set and makes the fits noisier (data points and error bars are  based on ∼ 2 billion magnitude pairs from 5315 QSOs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  15  Figure 3: Variability amplitudes log A vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' log ∆t (left) and vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' log(∆t/tth) (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  The colour encodes the thermal timescale of the measurement from blue for the shortest  to red for the longest tth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The best-fit lines to the data at log A > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='4 have slopes of  γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='503 (left) and γth = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='510 (right), consistent with a random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Solid lines  show the fit range while the dotted lines are extrapolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' At short ∆t, the variability  appears suppressed and this effect reaches to longer ∆t in larger (here: redder) disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  16  Figure 4: One example time separation bin, ∆trest = [125;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' 141] days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The 5,315  individual QSOs are points, while large symbols and their error bars are bootstrap  mean and standard deviation of the variability amplitudes in bins of luminosity L3000  and redshift z, lines are fits to bootstrap mean values (left: orange passband;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' centre:  cyan passband).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Right: intercepts and errors of the fits at fixed L3000 for different  rest-frame wavelengths, where either passband is seen to sample the same underlying  behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Solid lines are fit ranges, dashed lines are extrapolations shown for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Note an upwards deviation at λ > 300 nm, which is consistent with the dotted line  from one past work24 and also seen in other works22,44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Figure 5: The variability amplitude, π  2 < ∆m >2, at ∆t < 1 day as a function of  observed magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Gray symbols are individual QSOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Dotted lines show the 3σ-  clipped mean within fine magnitude bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Solid lines show the fit of Equation 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  17  References  [1] Kormendy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Ho, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Coevolution (or not) of supermassive black holes and  host galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ARA&A 51, 511–653 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [2] Richards, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The sloan digital sky survey quasar survey: Quasar lumi-  nosity function from data release 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' AJ 131, 2766–2787 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [3] Wolf, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Discovery of the most ultra-luminous qso using gaia, skymapper,  and wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' PASA 35, e024 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [4] Pringle, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Rees, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Accretion disc models for compact x-ray sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  A&A 21, 1 (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [5] Balbus, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Hawley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' A powerful local shear instability in weakly mag-  netized disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' linear analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJ 376, 214 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [6] Balbus, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Enhanced angular momentum transport in accretion disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ARA&A  41, 555–597 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [7] Shakura, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Sunyaev, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Reprint of 1973a&amp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='.24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='.337s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' black holes  in binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' observational appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' A&A 500, 33–51 (1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [8] Clavel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Correlated hard x-ray and ultraviolet variability in ngc 5548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJ  393, 113 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [9] Peterson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Reverberation mapping of active galactic nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' PASP 105, 247  (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [10] Cackett, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Horne, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Winkler, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Testing thermal reprocessing in active  galactic nuclei accretion discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' MNRAS 380, 669–682 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [11] Shappee, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The man behind the curtain: X-rays drive the uv through nir  variability in the 2013 active galactic nucleus outburst in ngc 2617.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJ 788, 48  (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [12] Fausnaugh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Space telescope and optical reverberation mapping  project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' optical continuum emission and broadband time delays in ngc 5548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  ApJ 821, 56 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [13] Jiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Detection of time lags between quasar continuum emission  bands based on pan-starrs light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJ 836, 186 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [14] Fausnaugh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Continuum reverberation mapping of the accretion disks  in two seyfert 1 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJ 854, 107 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [15] Yu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Quasar accretion disk sizes from continuum reverberation mapping  in the des standard-star fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJS 246, 16 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [16] Kelly, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Bechtold, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Siemiginowska, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Are the variations in quasar  optical flux driven by thermal fluctuations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJ 698, 895–910 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  18  [17] MacLeod, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Modeling the time variability of sdss stripe 82 quasars as a  damped random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJ 721, 1014–1033 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [18] Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Shi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The deviation of optical variability of radio-quiet quasars  from damped random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Ap&SS 364, 27 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [19] Hughes, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Aller, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Aller, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The university of michigan radio  astronomy data base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' structure function analysis and the relation between bl  lacertae objects and quasi-stellar objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJ 396, 469 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [20] Kozłowski, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Revisiting stochastic variability of agns with structure functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  ApJ 826, 118 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [21] Palanque-Delabrouille, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Variability selected high-redshift quasars on sdss  stripe 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' A&A 530, A122 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [22] Vanden Berk, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The ensemble photometric variability of ˜25,000 quasars  in the sloan digital sky survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJ 601, 692–714 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [23] Morganson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Measuring quasar variability with pan-starrs1 and sdss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJ  784, 92 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [24] Caplar, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Lilly, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Trakhtenbrot, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Optical variability of agns in the ptf/iptf  survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJ 834, 111 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [25] Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', McGreer, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Wu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Fan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Yang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The ensemble photometric  variability of over 105 quasars in the dark energy camera legacy survey and the  sloan digital sky survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJ 861, 6 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [26] Suberlak, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Ivezi´c, ˇZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & MacLeod, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Improving damped random walk  parameters for sdss stripe 82 quasars with pan-starrs1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJ 907, 96 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [27] Mushotzky, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Edelson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Baumgartner, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Gandhi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Kepler observations  of rapid optical variability in active galactic nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJ 743, L12 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [28] Kasliwal, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Vogeley, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Richards, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Are the variability properties  of the kepler agn light curves consistent with a damped random walk?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' MNRAS  451, 4328–4345 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [29] Smith, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The kepler light curves of agn: A detailed analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJ 857,  141 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [30] Tachibana, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Deep modeling of quasar variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJ 903, 54 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [31] Stone, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Optical variability of quasars with 20-year photometric light  curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' MNRAS (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [32] Burke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' A characteristic optical variability time scale in astrophysical  accretion disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Science 373, 789–792 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [33] Rakshit, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Stalin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Kotilainen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Spectral properties of quasars from  sloan digital sky survey data release 14: The catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJS 249, 17 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  19  [34] Tonry, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Atlas: A high-cadence all-sky survey system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' PASP 130,  064505 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [35] Heckman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Long-time-scale optical variability in quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' PASP 88, 844–  848 (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [36] Collaboration, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Gaia early data release 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' summary of the contents and  survey properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' A&A 649, A1 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [37] Kozłowski, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Limitations on the recovery of the true agn variability parameters  using damped random walk modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' A&A 597, A128 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [38] Kelly, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Treu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Malkan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Pancoast, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Woo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Active galactic  nucleus black hole mass estimates in the era of time domain astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJ 779,  187 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [39] Frank, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', King, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Raine, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Accretion Power in Astrophysics: Third Edition  (Cambridge University Press, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [40] Sun, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Corona-heated accretion-disk reprocessing: A physical model to  decipher the melody of agn uv/optical twinkling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJ 891, 178 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [41] Campitiello, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Ghisellini, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Sbarrato, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Calderone, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' How to constrain  mass and spin of supermassive black holes through their disk emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' A&A  612, A59 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [42] Abramowicz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Czerny, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Lasota, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Szuszkiewicz, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Slim accretion  disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJ 332, 646 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [43] Sadowski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Relativistic slim disks with vertical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' A&A 527, A17  (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [44] Yu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Richards, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Vogeley, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Moreno, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Graham, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Examining  agn uv/optical variability beyond the simple damped random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' arXiv e-prints  arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='08943 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [45] Kasliwal, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Vogeley, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Richards, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Williams, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Carini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Do  the kepler agn light curves need reprocessing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' MNRAS 453, 2075–2081 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [46] Brandt, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Active galaxy science in the lsst deep-drilling fields:  Footprints, cadence requirements, and total-depth requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' arXiv e-prints  arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='06542 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [47] Ivezi´c, ˇZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Lsst: From science drivers to reference design and anticipated  data products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJ 873, 111 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [48] Chan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Millon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Bonvin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Courbin, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Twisted quasar light curves:  implications for continuum reverberation mapping of accretion disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' A&A 636,  A52 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  20  [49] Kawaguchi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Mineshige, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Umemura, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Turner, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Optical variability  in active galactic nuclei: Starbursts or disk instabilities?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  ApJ 504, 671–679  (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [50] Flesch, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The half million quasars (hmq) catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' PASA 32, e010 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [51] Schlegel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Finkbeiner, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Davis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Maps of dust infrared emission  for use in estimation of reddening and cosmic microwave background radiation  foregrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJ 500, 525–553 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [52] Gravitationally lensed quasar database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' https://research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='ast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='uk/lensedquasars/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  Accessed: 2022-06-03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [53] Lemon, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Gravitationally lensed quasars in gaia – iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' 150 new lenses,  quasar pairs, and projected quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' arXiv e-prints arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='07714 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [54] Tonry, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The atlas all-sky stellar reference catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJ 867, 105 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [55] Riello, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Gaia early data release 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' photometric content and validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  A&A 649, A3 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [56] Shen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Burke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' A sample bias in quasar variability studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJ 918, L19  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [57] Becker, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', White, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Helfand, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The first survey: Faint images of the  radio sky at twenty centimeters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJ 450, 559 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [58] Condon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The nrao vla sky survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' AJ 115, 1693–1716 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [59] Mauch, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Sumss: a wide-field radio imaging survey of the southern sky -  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' the source catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' MNRAS 342, 1117–1130 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [60] Casagrande, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & VandenBerg, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' On the use of gaia magnitudes and new  tables of bolometric corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' MNRAS 479, L102–L107 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [61] Richards, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Spectral energy distributions and multiwavelength selection  of type 1 quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJS 166, 470–497 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [62] Morgan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Kochanek, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Morgan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Falco, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The quasar  accretion disk size-black hole mass relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJ 712, 1129–1136 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [63] King, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Pringle, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Livio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Accretion disc viscosity: how big is alpha?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  MNRAS 376, 1740–1746 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [64] di Clemente, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Giallongo, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Natali, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Trevese, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Vagnetti, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' The vari-  ability of quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' frequency dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' ApJ 463, 466 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  [65] Markwardt, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' Non-linear least-squares fitting in idl with mpfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' In Bohlender,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=', Durand, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' & Dowler, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=') Astronomical Data Analysis Software and  Systems XVIII, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content=' 411 of Astronomical Society of the Pacific Conference Series,  251 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
+page_content='  21' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfWfzu/content/2301.01304v1.pdf'}
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+LINKING NUMBER OF MODULAR KNOTS
+JAMES RICKARDS
+Abstract. We compute the linking number of two modular knots in the space PSL(2, Z)\ PSL(2, R) with
+the trefoil filled in, which answers a question posed by Ghys in 2007. This computation is realized through
+the correspondence between modular links and Lorenz links, and can be thought of as an intersection number
+involving Conway topographs. We compare this to a second formula for the linking number of Lorenz links,
+which was proven by Stephen F. Kennedy in 1994.
+1. Introduction
+The manifold X = PSL(2, Z)\ PSL(2, R) is diffeomorphic to S3 minus a trefoil (for a proof, see [Mil71]).
+The modular flow on X is given by right multiplication by φ(t) =
+�
+et
+0
+0 e−t
+�
+, and periodic orbits of the modular
+flow are called modular knots. These knots are parametrized by conjugacy classes of primitive hyperbolic
+matrices, as described in the following definition.
+Definition 1.1. Let A ∈ PSL(2, Z) be a primitive hyperbolic matrix with largest eigenvalue λ > 1. Then
+there exists a matrix M ∈ PSL(2, R) with M −1AM =
+� λ
+0
+0 λ−1
+�
+. Define the knot kA to be:
+kA : [0, log(λ)) → X
+t → Mφ(t).
+The path kA is a knot since Mφ(log(λ)) = AM ∼ M = Mφ(0) in X. Furthermore, it does not depend on
+the choice of M, and is constant across the PSL(2, Z) conjugacy class of A.
+As part of his 2006 ICM address, Ghys studied these knots ([Ghy07] and [GL06]). He proved that the
+linking number of kA with the removed trefoil can be given by R(A), the Rademacher function of A (see the
+paper by Atiyah, [Ati87], for various equivalent definitions). He then asked: what is the linking number of
+kA and kB?
+One difficulty of this question is the presence of the missing trefoil: to compute the linking number of our
+knots, you need to “fill the trefoil in”. However, if you don’t leave the space PSL(2, Z)\ PSL(2, R), there is
+no obvious way in which to do that. One partial fix is due to Duke, Imamo¯glu, and T´oth in 2017 ([DIT17]).
+In their paper, they used weight 2 cocycles to generalize the Rademacher function, and proved that this
+generalization gives the linking number of kA +kA−1 with kB +kB−1. The link kA +kA−1 is null-homologous,
+which enabled them to do their computations without leaving PSL(2, Z)\ PSL(2, R).
+As part of their paper, they noted that link(kA + kA−1, kB + kB−1) could be interpreted as an intersection
+number of closed geodesics on the modular curve. This topic was studied further in [Ric21a], and then
+Date: January 5, 2023.
+2020 Mathematics Subject Classification. Primary 37E15; Secondary 11E16, 11F23, 37C27, 57K10, 57K31.
+Key words and phrases. Modular knot, Lorenz knot, linking number, topograph, quadratic form.
+I thank Vishal P. Patil; a conversation about knot theory with him inspired me to have a look at this problem again. This
+work was partially supported by NSF-CAREER CNS-1652238 (PI Katherine E. Stange).
+1
+arXiv:2301.01334v1  [math.DS]  3 Jan 2023
+
+generalized to the case of geodesics on a Shimura curve in [Ric21b]. Since these results all deal with the links
+kA + kA−1, they do not provide a full solution to Ghys’s original question. By harnessing the connection to
+Lorenz links, in Theorem 2.3 we give a combinatorial computation of link(kA, kB), settling the question. In
+Section 7, we compare this finer invariant to the symmetrized version of Duke et al., and give some data to
+illustrate the difference.
+We should note that the linking number of Lorenz knots has been computed before. In [Ken94], Stephen
+F. Kennedy gives formulas for the linking numbers of Lorenz knots, as well as horseshoe knots. This formula
+involves the alphabetization of words, and the fact that the two formulas coincide does not appear to be
+trivial. For sake of comparison, we record his main result in Section 6.
+2. Main result
+This paper will take us into the world of dynamical systems and Lorenz equations, so we will adapt their
+terminology.
+Definition 2.1. A Lorenz word is any finite aperiodic word in the letters L, R, with length at least 2. A
+single shift of a Lorenz word W is the word obtained by taking the first letter and moving it to the end, and
+is denoted s(W). A cyclic shift is the result of any number of single shifts. Two Lorenz words are said to be
+equivalent if they differ by a cyclic shift. A Lorenz sequence is a doubly infinite periodic sequence, formed by
+repeating a fixed Lorenz word in both directions. Denote the Lorenz sequence associated to a Lorenz word
+W by S(W).
+Given a Lorenz word W, we can substitute in the following matrices for L and R and multiply out to
+produce a primitive hyperbolic matrix:
+L :=
+�
+1
+1
+0
+1
+�
+,
+R :=
+�
+1
+0
+1
+1
+�
+.
+Denote the matrix formed by mat(W). Given a primitive hyperbolic matrix A, define L (A) to be the set of
+Lorenz words W for which A is conjugate to mat(W) over PSL(2, Z). The following lemma is classical (and
+can be easily proven with the material in Section 3).
+Lemma 2.2. The set L (A) is non-empty, and consists of a single equivalence class of Lorenz words.
+The set L (A) is clearly constant across an conjugacy class of primitive hyperbolic matrices. Since A is
+hyperbolic, the equation Ax = x has two real solutions, called the roots of A (where A acts by M¨obius
+transformation). As we will describe in Remark 3.3, L (A) can be computed directly from the continued
+fraction representation of one of the roots of A.
+Theorem 2.3. Let A, B ∈ PSL(2, Z) be primitive hyperbolic matrices that are not conjugate to each other.
+Let WA ∈ L (A) and WB ∈ L (B), and write WA = a1a2 · · · am and WB = b1b2 · · · bn. Then − link(kA, kB)
+is equal to the number of triples of integers (i, j, x) such that:
+• 1 ≤ i ≤ m, 1 ≤ j ≤ n, and x ≥ 0;
+• ai = L and bj = R;
+• ai+k = bj+k for all integers 1 ≤ k ≤ x − 1;
+• ai+x = R and bj+x = L;
+2
+
+where the indices are taken modulo the periods (m and n respective to a and b). In particular, link(kA, kB)
+is always a negative integer.
+Another interpretation of the combinatorial computation in Theorem 2.3 is we are computing (modulo the
+periods) occurrences of (possibly empty) words W ′ such that LW ′R appears in S(WA) and RW ′L appears
+in S(WB).
+For an example of Theorem 2.3, consider the matrices A = ( 3 5
+7 12 ) and B = ( 2 1
+1 1 ), which correspond to
+words WA = RRLLRL and WB = LR. In Figure 1, we depict the knots kA and kB in R3 with the trefoil
+filled in, and can numerically compute that their linking number is −3 (using the standard convention that
+an overcrossing from left to right has sign +1).
+Figure 1. kA is in red and kB is in blue. Image created with Matplotlib, [Hun07].
+In view of Theorem 2.3, we pick up the three triples (i, j, x) = (3, 2, 4), (4, 2, 0), (6, 2, 0), corresponding to
+. . . RR LLRLRR LLRL . . .
+. . . L RLRLRL R . . .
+. . . RRL LR LRRLLRL . . .
+. . . L RL R . . .
+. . . RRLLR LR RLLRL . . .
+. . . L RL R . . .
+The strategy to prove Theorem 2.3 is to use Ghys’s result that modular links are isotopic to Lorenz links
+(top of page 272 of [Ghy07]). We then compute the linking number of Lorenz links corresponding to the
+given Lorenz words using Birman–Williams’ template theory, [BW83].
+Remark 2.4. Theorem 4.1 of [BW83] proves that the linking number of Lorenz links in non-zero, which is
+a corollary of Theorem 2.3.
+3
+
+Remark 2.5. The work of Birman-Williams (along with most other Lorenz knot papers) uses the opposite
+sign convention for the linking number, so that it is always positive. Since we framed the question in terms of
+the number theoretic picture, we will instead follow the convention used by Ghys and Duke-Imamo¯glu-T´oth
+in [Ghy07] and [DIT17].
+3. Connection to Conway’s topograph and quadratic forms
+Despite the strong connection to Lorenz links, motivation for the formula of Theorem 2.3 came from
+the aforementioned works [DIT17] and [Ric21a]. Given a primitive hyperbolic A ∈ PSL(2, Z), denote the
+geodesic connecting the two roots of A by ℓA. This descends to a closed geodesic ˜ℓA on the modular curve,
+PSL(2, Z)\H. The unsigned intersection number of two closed geodesics ℓ1 and ℓ2 , denoted Int(ℓ1, ℓ2), counts
+the number of transverse intersections of the (unoriented) curves.
+Theorem 3.1 (Follows from Theorem 6.3 and Lemma 6.7 of [DIT17]). Let A, B ∈ PSL(2, Z) be primitive,
+hyperbolic, and not conjugate to each other or each other’s inverse. Then
+link(kA + kA−1, kB + kB−1) = − Int(˜ℓA, ˜ℓB).
+While it’s not obvious how this intersection number can break up into a sum of four linking numbers,
+a natural decomposition appears when we consider the Conway topographs of the corresponding quadratic
+forms.
+Given a primitive hyperbolic matrix A =
+� a b
+c d
+�
+∈ PSL(2, Z), the equation Ax = x translates to cx2 +(d−
+a)x − b = 0. Let g = gcd(c, d − a, b), and the quadratic form associated to A is
+qA(x, y) := 1
+g
+�
+cx2 + (d − a)xy − by2�
+:= 1
+g [c, d − a, −b].
+This is a primitive integral indefinite binary quadratic form, and conjugacy classes of matrices correspond
+to PSL(2, Z)-equivalence classes of quadratic forms. Furthermore, this is a bijection, with the inverse map
+realized by taking the automorph of qA (for more details, see Proposition 1.4 of [Sar82]).
+The Conway topograph is a combinatorial object associated to a quadratic form (for a more comprehensive
+study of the topograph, see Chapter 4 of [Hat22], or Section 3 of [Ric21a]). The base object of the topograph
+is an infinite connected 3-regular graph embedded in R2. In particular, if we add an orientation to an edge,
+there is a well-defined notion of “left” and “right”. A path can therefore be represented by a word in L and
+R, denoting left and right respectively.
+The topograph for q(x, y) divides R2 into regions, and numbers can be placed in the regions and on the
+edges such that:
+• Numbers in regions represent values properly represented by q(x, y) (i.e. q(x, y) when x, y ∈ Z are
+coprime);
+• If an edge contains the number b and is adjacent to regions with numbers a, c, then [a, ±b, c] is a
+quadratic form similar to q. In fact, the entire equivalence class of forms similar to q arises in this
+fashion.
+• By assigning a “positive direction” to each edge, we can determine if we need to take +b or −b in
+the form.
+See Figure 2 for part of the topograph of q(x, y) = 7x2 + 9xy − 5y2.
+4
+
+7
+11
+41
+37
+31
+5
+−5
+−11
+−7
+7
+−31
+9
+23
+37
+45
+21
+1
+11
+29
+1
+9
+21
+31
+43
+23
+5
+43
+9
+19
+19
+31
+33
+45
+33
+37
+29
+−5
+−41
+−37
+Figure 2. Topograph of [7, 9, −5]. Red numbers are in regions, and black numbers are on
+edges. The direction of the arrow determines the sign of the edge numbers.
+When q is indefinite, there is a doubly infinite path (called the river) that separates the positive and
+negative numbers placed in regions. The river is periodic with minimal period of length at least 2, and
+therefore can be represented as a Lorenz sequence.
+Definition 3.2. Let q(x, y) be a primitive integral indefinite binary quadratic form. A river word for q is a
+Lorenz word that represents the minimal period of the path taken by the river on the topograph of q. It has
+length at least 2, and is unique up to cyclic shift. Let Riv(q) denote the set of river words of q
+Remark 3.3 (Remark 3.5 of [Ric21a]). Let q = [a, b, c] have discriminant D, and let the continued fraction
+of −b+
+√
+D
+2a
+(the first root of q) be
+[a0, a1, . . .] = [a0, a1, . . . , as, as+1, . . . , as+p],
+where s is the smallest integer such that the continued fraction is periodic after index s, and p is the smallest
+even integer such that the sequence has period p. Consider the sequence of 0’s and 1’s formed by:
+• s + 1 (mod 2) repeated as+1 times;
+• s + 2 (mod 2) repeated as+2 times;
+• · · ·
+• s + p (mod 2) repeated as+p times.
+If we replace the 0s with Ls and 1s with Rs, then the word formed is in Riv(q). For example, the first root of
+[7, 9, −5] is −9+
+√
+221
+14
+, which has continued fraction [0, 2, 2, 1, 1]. This gives the river word RRLLRL, which
+agrees with Figure 2: start on the left hand side at [7, 9, −5], walk via RRLLRL, and you end up at another
+river edge representing [7, 9, −5].
+When drawing the topograph of an indefinite form, the river is normally “flattened” with the positive
+numbers above the river.
+Remark 3.4. Let q = [a, b, c] be a “positive river form”, i.e. a > 0 and c < 0, and let W be the river word
+formed by starting at q. Then mat(W) is the automorph of q, i.e. mat(W) ◦ q = q.
+5
+
+The topograph machinery can also be used to give a nice proof of Lemma 2.2. If A is a primitive hyperbolic
+matrix, then Riv(qA) = L (A).
+To connect this back to Theorem 3.1, let A, B ∈ PSL(2, Z) be primitive and hyperbolic. Let TA be the
+topograph associated to qA, and let TB be the topograph associated to qB. By picking an oriented edge of
+each topograph, we can superimpose one on top of the other, so that a region has an associated ordered
+pair: the number from TA, followed by the number from TB. There are four possibilities for the signs of the
+numbers in this pair: (+, +), (+, −), (−, +), (−, −), and it turns out that we get an intersection between ˜ℓA
+and ˜ℓB if all four combinations appear (see [Ric21a] for more details). This is equivalent to the rivers of the
+topgraphs starting off disjoint, meeting, and then crossing each other.
+In particular, if we have “intersecting topographs”, then you can translate either topograph by the corre-
+sponding river word to get another pair of intersecting topographs. This equivalence gives rise to the same
+intersection point on the modular curve, and is the only way to produce the same point. Theorem 3.5 follows.
+Theorem 3.5 (Theorem 5 of [Ric21a]). Let A, B ∈ PSL(2, Z) be primitive and hyperbolic, corresponding to
+respective topographs TA and TB. Then Int(˜ℓA, ˜ℓB) is equal to the number of ways to superimpose TB on top
+of TA so that the rivers meet and cross, modulo the periods of the rivers.
+Consider the combinatorics of Theorem 3.5: start with TA, flattened so that the river RA is horizontal.
+In order to meet and cross, the river RB for TB can meet RA from either the left or the right hand side
+(i.e. top/bottom in terms of the picture). It can then flow in the same or the opposite direction, leading to
+4 possibilities.
+Definition 3.6. Let IntRS(A, B) denote the topograph intersection number where RB joins RA from the
+right hand side, and the rivers flow in the same direction. Call this the RS-intersection number.
+The RS-intersection number is exactly the quantity we want.
+Theorem 3.7. The topograph RS-intersection number, IntRS(A, B), coincides with − link(kA, kB).
+Before giving the proof, consider the example from Figure 1, i.e. A = ( 3 5
+7 12 ) and B = ( 2 1
+1 1 ), corresponding
+to river words RRLLRL and LR. The topograph for A was displayed in Figure 2, and all 3 possible RS-
+intersections are shown in Figure 3.
+· · ·
+· · ·
+Figure 3. RS-intersection of RRLLRL and LR. One and a half periods of the topo-
+graph of RRLLRL are shown in solid black, and the three RS-intersections with LR are in
+blue/red/green and dotted/dashed.
+Proof of Theorem 3.7. Consider how to combinatorially compute IntRS. Let RA correspond to the word
+a1a2 · · · am, and let RB correspond to the word b1b2 · · · bn. For RB to join RA from the right hand side, we
+6
+
+must have a pair of indices (i, j) with 1 ≤ i ≤ m and 1 ≤ j ≤ n such that ai = L and bj = R. Then the
+rivers flow in the same direction for x ∈ Z≥0 steps, corresponding to ai+k = bj+k for 1 ≤ k ≤ x − 1. Since
+RB crosses RA, we must exit with ai+x = R and bj+x = L. In particular, this is exactly the same description
+as provided in Theorem 2.3, so it follows from the proof in Section 5.
+□
+Remark 3.8. On the topograph side, IntRS(A, B) is still defined when A and B are conjugate. On the
+modular curve side of things, this corresponds to transverse self-intersections of the geodesic. However, it is
+not entirely clear what the interpretation should be in terms of knot theory. If A = ( 2 1
+1 1 ), then IntRS(A, A) =
+1, whereas kA is the unknot. Perhaps there is a natural framing of the modular knots so that IntRS(A, A)
+becomes the self-linking number.
+4. Lorenz knots
+In an attempt to model atmospheric convection, meteorologist Edward Norton Lorenz came up with the
+following system of differential equations (where t represents time):
+dx
+dt = 10(y − x),
+dy
+dy = 28x − y − xz,
+dz
+dt = xy − 8
+3z.
+By picking an initial starting point, the ODE follows a path to determine a flow. See Figure 4 for an example.
+Figure 4. A sample solution to the Lorenz system. Image created with Mathematica,
+[Inc22].
+7
+
+In [Lor63], Lorenz proved that while paths must all eventually enter and remain in a bounded region,
+they are very susceptible to miniscule changes in the initial inputs. This observation led to the beginning of
+chaos theory.
+We will study a different aspect of this theory, namely the knots formed as solutions to Lorenz’s equations.
+A Lorenz knot is defined to be a closed periodic orbit of this ODE, and a Lorenz link is a set of Lorenz knots.
+Lorenz knots/links can be studied with template theory, introduced by Birman and Williams in [BW83].
+As seen in Figure 4, the solutions seem to form a “butterfly”: looping around one of two circles. This can
+be made precise with the Lorenz template, which is a branched surface in R3 with a semi-flow that describes
+the behaviour of Lorenz knots. See Figure 5 for a depiction of the Lorenz template.
+L
+R
+Figure 5. The Lorenz template.
+Remark 4.1. The basic tools to create the Lorenz template came from the work of Guckenheimer and
+Williams in [GW79], [Wil79]. The proof that the template actually corresponds to Lorenz’s original equations
+is due to Tucker in [Tuc02].
+Let the branch (the piece connecting the left and right circles) be [0, 1]. A path starting at x ∈ (0, 1)−{1/2}
+will wind around the left (if x < 1/2) or the right (if x > 1/2) loop, ending back up at f(x) ∈ (0, 1).
+Furthermore, we have the following properties:
+• f(x) ̸= x for all x ∈ (0, 1) − {1/2};
+• f restricted to (0, 1/2) is a continuous increasing bijection with (0, 1);
+• f restricted to (1/2, 1) is a continuous increasing bijection with (0, 1);
+• flows corresponding to x and x′ with 0 < x < x′ < 1/2 (or 1/2 < x < x′ < 1) do not cross;
+• flows coming from the left loop meet the branch above flows coming from the right loop (denoted
+by dotted lines in the figure).
+In fact (see Section 2.4 of [BW83]), the function f(x) = 2x (mod 1) will model the Lorenz template.
+Remark 4.2. In Figure 4, the right hand side of the orbits come back to the branch in front of the left
+hand side orbits. This is opposite to what is done in the Lorenz template, Figure 5. I’m not certain of why
+this is happens, but it is the convention used in [BW83] and [Ghy07], so we will follow it. In any case, the
+only side affect is a potential difference in sign.
+Since a Lorenz knot is a periodic closed flow, we can associate a word to it via the sequence of L’s and
+R’s it follows on the template.
+8
+
+Definition 4.3. The Lorenz word of a Lorenz knot is the sequence of L’s and R’s that the knot visits as it
+travels the template over one period.
+It is clear that the Lorenz word is only defined up to cyclic shifts. Note the similarity between the Lorenz
+word of a Lorenz knot and the Lorenz word of a primitive hyperbolic matrix! By the remarkable work of
+Ghys in [Ghy07], these two worlds are the same.
+Theorem 4.4 (Ghys, 2006). Consider a modular link corresponding to distinct Lorenz words W1, W2, . . . , Wn.
+This coincides (knot-theoretically) with the Lorenz link corresponding to Lorenz words W1, W2, . . . , Wn.
+In particular, to prove Theorem 2.3, it suffices to prove it for Lorenz links.
+5. Proof of the main result
+To compute link(kA, kB), by Theorem 4.4, it suffices to compute the linking number of the corresponding
+knots on the Lorenz template. Let these knots be k′
+1 and k′
+2 respectively. The linking number is then the
+number of times k′
+2 crosses under k′
+1, and accounting for sign: +1 from right to left, and −1 from left to
+right.
+Let W1 = a1a2 · · · am and W2 = b1b2 · · · bn be the distinct Lorenz words corresponding to A and B
+respectively. For each 1 ≤ i ≤ m, let Ki be the part of the k′
+1 corresponding to ai on the template, and
+say that it leaves from xi ∈ (0, 1) (and therefore flows to xi+1, with indices taken modulo m). Similarly, for
+each 1 ≤ j ≤ n, let Lj be the part of k′
+2 on the template corresponding to bj, and say that it leaves from
+yj ∈ (0, 1). To compute link(W1, W2), it suffices to compute link(Ki, Lj) for each pair (i, j) with 1 ≤ i ≤ m
+and 1 ≤ j ≤ n and add them up.
+Case 1: ai = bj. In this case, both knots flow around the same side of the template, not intersecting
+each other, and return to the branch in the same order as they started. See Figure 6 for a depiction of the
+situation. This contributes nothing to the linking number.
+L
+R
+L
+R
+Figure 6. Flowing around the same side of the template.
+Case 2: ai = L and bj = R. The knots flow around opposite loops, and may or may not cross as they
+return.
+• If they do not cross, then we must have xi+1 < yi+1. Therefore we must have either ai+1 = bj+1,
+or ai+1 = L and bj+1 = R. In the first case, the loops will flow around the corresponding side, and
+again retain their ordering. Therefore we again have ai+2 = bj+2 or ai+2 = L and bj+2 = R. This
+process continues, until eventually the knots take opposite sides (after at most mn iterations), which
+must be L for ai+k and R for bj+k.
+9
+
+To sum up this sub-case, if they do not cross, then there is some word X such that the Lorenz words
+are LXL and RXR starting at (ai, bj).
+• If they do cross, then we have yi+1 < xi+1. The sign of crossing is also −1, since the left branch
+comes in on top of the right branch. The analysis of the previous case holds, except now when the
+knots separate, we must have ai+k = R and bi+k = L. In other words, there is some X such that the
+Lorenz words are LXR and RXL starting at (ai, bj).
+Since the conditions found in the two sub-cases are disjoint and cover every possibility, they are if and
+only if. In particular, occurrences of LXR and RXL contribute −1 to the linking.
+Case 3: ai = R and bj = L. This is identical to case 2, except now a crossing corresponds to k′
+1 coming
+in underneath k′
+2, which we do not count.
+Theorem 2.3 follows.
+6. Comparison to Kennedy’s formula
+In [Ken94], Kennedy provides another computation of link(W1, W2), where W1 and W2 are two Lorenz
+words (that are not equivalent under cyclic shift).
+Definition 6.1. Let σ be a permutation of {1, 2, . . . , n}. Define the crossing count of σ to be �n
+i=1 |σ(i)−i|.
+For i = 1, 2, let Wi have period ni, and write out the following words in order (where s represents a single
+shift):
+W1, s(W1), s2(W1), . . . , sn1−1(W1), W2, s(W2), . . . , sn2−1(W2).
+Let σ1 denote the permutation that alphabetizes the first n1 words, let σ2 denote the permutation that
+alphabetizes the second n2 words, and let σ3 denote the permutation that alphabetizes all n1 + n2 words.
+Theorem 6.2 (Theorem 1 of [Ken94]). The linking number of the Lorenz knots with words W1 and W2 is
+1
+4 (C(σ1) + C(σ2) − C(σ3)) .
+For example, consider our example of W1 = RRLLRL and W2 = LR, as examined in Figures 1 and 3.
+The alphabetizations are in Table 1.
+Table 1. Alphabetization of RRLLRL and LR.
+n
+Word
+Joint order
+Individual order
+1
+RRLLRL
+8
+6
+2
+RLLRLR
+6
+4
+3
+LLRLRR
+1
+1
+4
+LRLRRL
+3
+2
+5
+RLRRLL
+7
+5
+6
+LRRLLR
+4
+3
+7
+LR
+2
+1
+8
+RL
+5
+2
+10
+
+Therefore σ1 = (6, 4, 1, 2, 5, 3), σ2 = (1, 2), and σ3 = (8, 6, 1, 3, 7, 4, 2, 5). This gives
+C(σ1) = |6 − 1| + |4 − 2| + |1 − 3| + |2 − 4| + |5 − 5| + |3 − 6| =14
+C(σ2) = |1 − 1| + |2 − 2| =0
+C(σ3) = |8 − 1| + |6 − 2| + |1 − 3| + |3 − 4| + |7 − 5| + |4 − 6| + |2 − 7| + |5 − 8| =26,
+hence
+link(W1, W2) = 1
+4 (14 + 0 − 26) = −3,
+which agrees with our computation. A direct combinatorial proof that the counts in Theorems 6.2 and 2.3
+are equal does not appear to be obvious.
+7. Comparison to Duke-Imamo¯glu-T´oth’s symmetrized linking number
+In certain cases, link(kA, kB) can be deduced from link(kA + kA−1, kB + kB−1), which was computed in
+[DIT17] and [Ric21a].
+Definition 7.1. A primitive hyperbolic matrix A ∈ PSL(2, R) is called reciprocal if A is conjugate to A−1.
+The matrix A is reciprocal if and only if the corresponding quadratic form qA is reciprocal, i.e. qA is
+PSL(2, Z)-equivalent to −qA. Furthermore, if W is the Lorenz word associated to A, then A is reciprocal if
+and only if when you write W backwards and swap L’s and R’s, you end up with a cyclic shift of W. For
+example, A = ( 3 5
+7 12 ) (from the example in Figure 1) is reciprocal, since it corresponds to the Lorenz word
+W = RRLLRL. Writing this word backwards and swapping R’s and L’s gives RLRRLL, which is a cyclic
+shift of W by 4 places.
+Using Theorem 2.3 (see Proposition 3.8 of [Ric21a] and the surrounding commentary for more details) we
+can prove the following result.
+Proposition 7.2. Let A or B be reciprocal. Then
+link(kA, kB) = 1
+4 link(kA + kA−1, kB + kB−1).
+In particular, if either A or B is reciprocal, then link(kA, kB) can be deduced from the symmetrized linking
+number. If both A and B are not reciprocal, then this is no longer true, and we obtain new information.
+For example, consider the non-reciprocal matrices
+A =
+�
+3
+2
+1
+1
+�
+,
+B =
+�
+14
+9
+3
+2
+�
+,
+C =
+�
+19
+7
+8
+3
+�
+,
+D =
+�
+28
+9
+3
+1
+�
+,
+which correspond to the Lorenz words
+WA = LLR,
+WB = LLLLRLR,
+WC = LLRRLRR,
+WD = LLLLLLLLLRRR.
+Table 2 gives the corresponding linking numbers, and it can be observed that Proposition 7.2 does not
+necessarily hold when neither matrix is reciprocal.
+11
+
+Table 2. Linking number comparison.
+X, Y
+link(kX, kY )
+link(kX, kY −1)
+link(kX + kX−1, kY + kY −1)
+A, B
+4
+2
+12
+A, C
+3
+4
+14
+A, D
+3
+3
+12
+B, C
+6
+8
+28
+B, D
+7
+5
+24
+C, D
+7
+7
+28
+References
+[Ati87]
+Michael Atiyah. The logarithm of the Dedekind η-function. Math. Ann., 278(1-4):335–380, 1987.
+[BW83]
+Joan S. Birman and R. F. Williams. Knotted periodic orbits in dynamical systems. I. Lorenz’s equations. Topology,
+22(1):47–82, 1983.
+[DIT17] W. Duke, ¨O. Imamo¯glu, and ´A. T´oth. Modular cocycles and linking numbers. Duke Math. J., 166(6):1179–1210, 2017.
+[Ghy07]
+´Etienne Ghys. Knots and dynamics. In International Congress of Mathematicians. Vol. I, pages 247–277. Eur. Math.
+Soc., Z¨urich, 2007.
+[GL06]
+´Etienne Ghys and Jos Leys. Lorenz and modular flows: a visual introduction. http://www.ams.org/publicoutreach/
+feature-column/fcarc-lorenz, 2006. AMS feature column. Accessed 16 December 2022.
+[GW79] John Guckenheimer and R. F. Williams. Structural stability of Lorenz attractors. Inst. Hautes ´Etudes Sci. Publ.
+Math., (50):59–72, 1979.
+[Hat22]
+Allen Hatcher. Topology of Numbers. American Mathematical Society, Providence, RI, 2022.
+[Hun07] J. D. Hunter. Matplotlib: A 2d graphics environment. Computing in Science & Engineering, 9(3):90–95, 2007.
+[Inc22]
+Wolfram Research, Inc. Mathematica, Version 13.1, 2022.
+[Ken94] Stephen F. Kennedy. Algorithms for the linking numbers of Lorenz and horseshoe knots. Houston J. Math., 20(4):705–
+712, 1994.
+[Lor63]
+Edward N. Lorenz. Deterministic nonperiodic flow. J. Atmospheric Sci., 20(2):130–141, 1963.
+[Mil71]
+John Milnor. Introduction to algebraic K-theory. Princeton University Press, Princeton, N.J.; University of Tokyo
+Press, Tokyo, 1971. Annals of Mathematics Studies, No. 72.
+[Ric21a] James Rickards. Computing intersections of closed geodesics on the modular curve. J. Number Theory, 225:374–408,
+2021.
+[Ric21b] James Rickards. Counting intersection numbers of closed geodesics on Shimura curves. https://arxiv.org/abs/2104.
+01968, 2021.
+[Sar82]
+Peter Sarnak. Class numbers of indefinite binary quadratic forms. J. Number Theory, 15(2):229–247, 1982.
+[Tuc02]
+Warwick Tucker. A rigorous ODE solver and Smale’s 14th problem. Found. Comput. Math., 2(1):53–117, 2002.
+[Wil79]
+R. F. Williams. The structure of Lorenz attractors. Inst. Hautes ´Etudes Sci. Publ. Math., (50):73–99, 1979.
+University of Colorado Boulder, Boulder, Colorado, USA
+Email address: james.rickards@colorado.edu
+URL: https://math.colorado.edu/~jari2770/
+12
+
diff --git a/WdAzT4oBgHgl3EQfYfwQ/content/tmp_files/load_file.txt b/WdAzT4oBgHgl3EQfYfwQ/content/tmp_files/load_file.txt
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@@ -0,0 +1,512 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf,len=511
+page_content='LINKING NUMBER OF MODULAR KNOTS  JAMES RICKARDS  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' We compute the linking number of two modular knots in the space PSL(2, Z)\\ PSL(2, R) with  the trefoil filled in, which answers a question posed by Ghys in 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' This computation is realized through  the correspondence between modular links and Lorenz links, and can be thought of as an intersection number  involving Conway topographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' We compare this to a second formula for the linking number of Lorenz links,  which was proven by Stephen F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Kennedy in 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Introduction  The manifold X = PSL(2, Z)\\ PSL(2, R) is diffeomorphic to S3 minus a trefoil (for a proof, see [Mil71]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  The modular flow on X is given by right multiplication by φ(t) =  �  et  0  0 e−t  �  , and periodic orbits of the modular  flow are called modular knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' These knots are parametrized by conjugacy classes of primitive hyperbolic  matrices, as described in the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Let A ∈ PSL(2, Z) be a primitive hyperbolic matrix with largest eigenvalue λ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Then  there exists a matrix M ∈ PSL(2, R) with M −1AM =  � λ  0  0 λ−1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Define the knot kA to be:  kA : [0, log(λ)) → X  t → Mφ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  The path kA is a knot since Mφ(log(λ)) = AM ∼ M = Mφ(0) in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Furthermore, it does not depend on  the choice of M, and is constant across the PSL(2, Z) conjugacy class of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  As part of his 2006 ICM address, Ghys studied these knots ([Ghy07] and [GL06]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' He proved that the  linking number of kA with the removed trefoil can be given by R(A), the Rademacher function of A (see the  paper by Atiyah, [Ati87], for various equivalent definitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' He then asked: what is the linking number of  kA and kB?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  One difficulty of this question is the presence of the missing trefoil: to compute the linking number of our  knots, you need to “fill the trefoil in”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' However, if you don’t leave the space PSL(2, Z)\\ PSL(2, R), there is  no obvious way in which to do that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' One partial fix is due to Duke, Imamo¯glu, and T´oth in 2017 ([DIT17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  In their paper, they used weight 2 cocycles to generalize the Rademacher function, and proved that this  generalization gives the linking number of kA +kA−1 with kB +kB−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' The link kA +kA−1 is null-homologous,  which enabled them to do their computations without leaving PSL(2, Z)\\ PSL(2, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  As part of their paper, they noted that link(kA + kA−1, kB + kB−1) could be interpreted as an intersection  number of closed geodesics on the modular curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' This topic was studied further in [Ric21a], and then  Date: January 5, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Primary 37E15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Secondary 11E16, 11F23, 37C27, 57K10, 57K31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Modular knot, Lorenz knot, linking number, topograph, quadratic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  I thank Vishal P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Patil;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' a conversation about knot theory with him inspired me to have a look at this problem again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' This  work was partially supported by NSF-CAREER CNS-1652238 (PI Katherine E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Stange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='01334v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='DS]  3 Jan 2023  generalized to the case of geodesics on a Shimura curve in [Ric21b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Since these results all deal with the links  kA + kA−1, they do not provide a full solution to Ghys’s original question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' By harnessing the connection to  Lorenz links, in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='3 we give a combinatorial computation of link(kA, kB), settling the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' In  Section 7, we compare this finer invariant to the symmetrized version of Duke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=', and give some data to  illustrate the difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  We should note that the linking number of Lorenz knots has been computed before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' In [Ken94], Stephen  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Kennedy gives formulas for the linking numbers of Lorenz knots, as well as horseshoe knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' This formula  involves the alphabetization of words, and the fact that the two formulas coincide does not appear to be  trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' For sake of comparison, we record his main result in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Main result  This paper will take us into the world of dynamical systems and Lorenz equations, so we will adapt their  terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' A Lorenz word is any finite aperiodic word in the letters L, R, with length at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' A  single shift of a Lorenz word W is the word obtained by taking the first letter and moving it to the end, and  is denoted s(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' A cyclic shift is the result of any number of single shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Two Lorenz words are said to be  equivalent if they differ by a cyclic shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' A Lorenz sequence is a doubly infinite periodic sequence, formed by  repeating a fixed Lorenz word in both directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Denote the Lorenz sequence associated to a Lorenz word  W by S(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Given a Lorenz word W, we can substitute in the following matrices for L and R and multiply out to  produce a primitive hyperbolic matrix:  L :=  �  1  1  0  1  �  ,  R :=  �  1  0  1  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Denote the matrix formed by mat(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Given a primitive hyperbolic matrix A, define L (A) to be the set of  Lorenz words W for which A is conjugate to mat(W) over PSL(2, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' The following lemma is classical (and  can be easily proven with the material in Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' The set L (A) is non-empty, and consists of a single equivalence class of Lorenz words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  The set L (A) is clearly constant across an conjugacy class of primitive hyperbolic matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Since A is  hyperbolic, the equation Ax = x has two real solutions, called the roots of A (where A acts by M¨obius  transformation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' As we will describe in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='3, L (A) can be computed directly from the continued  fraction representation of one of the roots of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Let A, B ∈ PSL(2, Z) be primitive hyperbolic matrices that are not conjugate to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Let WA ∈ L (A) and WB ∈ L (B), and write WA = a1a2 · · · am and WB = b1b2 · · · bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Then − link(kA, kB)  is equal to the number of triples of integers (i, j, x) such that:  1 ≤ i ≤ m, 1 ≤ j ≤ n, and x ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  ai = L and bj = R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  ai+k = bj+k for all integers 1 ≤ k ≤ x − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  ai+x = R and bj+x = L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  2  where the indices are taken modulo the periods (m and n respective to a and b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' In particular, link(kA, kB)  is always a negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Another interpretation of the combinatorial computation in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='3 is we are computing (modulo the  periods) occurrences of (possibly empty) words W ′ such that LW ′R appears in S(WA) and RW ′L appears  in S(WB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  For an example of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='3, consider the matrices A = ( 3 5  7 12 ) and B = ( 2 1  1 1 ), which correspond to  words WA = RRLLRL and WB = LR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' In Figure 1, we depict the knots kA and kB in R3 with the trefoil  filled in, and can numerically compute that their linking number is −3 (using the standard convention that  an overcrossing from left to right has sign +1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' kA is in red and kB is in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Image created with Matplotlib, [Hun07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  In view of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='3, we pick up the three triples (i, j, x) = (3, 2, 4), (4, 2, 0), (6, 2, 0), corresponding to  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' RR LLRLRR LLRL .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' L RLRLRL R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' RRL LR LRRLLRL .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' L RL R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' RRLLR LR RLLRL .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' L RL R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  The strategy to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='3 is to use Ghys’s result that modular links are isotopic to Lorenz links  (top of page 272 of [Ghy07]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' We then compute the linking number of Lorenz links corresponding to the  given Lorenz words using Birman–Williams’ template theory, [BW83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='1 of [BW83] proves that the linking number of Lorenz links in non-zero, which is  a corollary of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  3  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' The work of Birman-Williams (along with most other Lorenz knot papers) uses the opposite  sign convention for the linking number, so that it is always positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Since we framed the question in terms of  the number theoretic picture, we will instead follow the convention used by Ghys and Duke-Imamo¯glu-T´oth  in [Ghy07] and [DIT17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Connection to Conway’s topograph and quadratic forms  Despite the strong connection to Lorenz links, motivation for the formula of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='3 came from  the aforementioned works [DIT17] and [Ric21a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Given a primitive hyperbolic A ∈ PSL(2, Z), denote the  geodesic connecting the two roots of A by ℓA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' This descends to a closed geodesic ˜ℓA on the modular curve,  PSL(2, Z)\\H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' The unsigned intersection number of two closed geodesics ℓ1 and ℓ2 , denoted Int(ℓ1, ℓ2), counts  the number of transverse intersections of the (unoriented) curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='1 (Follows from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='3 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='7 of [DIT17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Let A, B ∈ PSL(2, Z) be primitive,  hyperbolic, and not conjugate to each other or each other’s inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Then  link(kA + kA−1, kB + kB−1) = − Int(˜ℓA, ˜ℓB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  While it’s not obvious how this intersection number can break up into a sum of four linking numbers,  a natural decomposition appears when we consider the Conway topographs of the corresponding quadratic  forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Given a primitive hyperbolic matrix A =  � a b  c d  �  ∈ PSL(2, Z), the equation Ax = x translates to cx2 +(d−  a)x − b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Let g = gcd(c, d − a, b), and the quadratic form associated to A is  qA(x, y) := 1  g  �  cx2 + (d − a)xy − by2�  := 1  g [c, d − a, −b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  This is a primitive integral indefinite binary quadratic form, and conjugacy classes of matrices correspond  to PSL(2, Z)-equivalence classes of quadratic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Furthermore, this is a bijection, with the inverse map  realized by taking the automorph of qA (for more details, see Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='4 of [Sar82]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  The Conway topograph is a combinatorial object associated to a quadratic form (for a more comprehensive  study of the topograph, see Chapter 4 of [Hat22], or Section 3 of [Ric21a]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' The base object of the topograph  is an infinite connected 3-regular graph embedded in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' In particular, if we add an orientation to an edge,  there is a well-defined notion of “left” and “right”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' A path can therefore be represented by a word in L and  R, denoting left and right respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  The topograph for q(x, y) divides R2 into regions, and numbers can be placed in the regions and on the  edges such that:  Numbers in regions represent values properly represented by q(x, y) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' q(x, y) when x, y ∈ Z are  coprime);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  If an edge contains the number b and is adjacent to regions with numbers a, c, then [a, ±b, c] is a  quadratic form similar to q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' In fact, the entire equivalence class of forms similar to q arises in this  fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  By assigning a “positive direction” to each edge, we can determine if we need to take +b or −b in  the form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  See Figure 2 for part of the topograph of q(x, y) = 7x2 + 9xy − 5y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  4  7  11  41  37  31  5  −5  −11  −7  7  −31  9  23  37  45  21  1  11  29  1  9  21  31  43  23  5  43  9  19  19  31  33  45  33  37  29  −5  −41  −37  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Topograph of [7, 9, −5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Red numbers are in regions, and black numbers are on  edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' The direction of the arrow determines the sign of the edge numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  When q is indefinite, there is a doubly infinite path (called the river) that separates the positive and  negative numbers placed in regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' The river is periodic with minimal period of length at least 2, and  therefore can be represented as a Lorenz sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Let q(x, y) be a primitive integral indefinite binary quadratic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' A river word for q is a  Lorenz word that represents the minimal period of the path taken by the river on the topograph of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' It has  length at least 2, and is unique up to cyclic shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Let Riv(q) denote the set of river words of q  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='3 (Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='5 of [Ric21a]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Let q = [a, b, c] have discriminant D, and let the continued fraction  of −b+  √  D  2a  (the first root of q) be  [a0, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='] = [a0, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' , as, as+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' , as+p],  where s is the smallest integer such that the continued fraction is periodic after index s, and p is the smallest  even integer such that the sequence has period p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Consider the sequence of 0’s and 1’s formed by:  s + 1 (mod 2) repeated as+1 times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  s + 2 (mod 2) repeated as+2 times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  · · ·  s + p (mod 2) repeated as+p times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  If we replace the 0s with Ls and 1s with Rs, then the word formed is in Riv(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' For example, the first root of  [7, 9, −5] is −9+  √  221  14  , which has continued fraction [0, 2, 2, 1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' This gives the river word RRLLRL, which  agrees with Figure 2: start on the left hand side at [7, 9, −5], walk via RRLLRL, and you end up at another  river edge representing [7, 9, −5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  When drawing the topograph of an indefinite form, the river is normally “flattened” with the positive  numbers above the river.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Let q = [a, b, c] be a “positive river form”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' a > 0 and c < 0, and let W be the river word  formed by starting at q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Then mat(W) is the automorph of q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' mat(W) ◦ q = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  5  The topograph machinery can also be used to give a nice proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' If A is a primitive hyperbolic  matrix, then Riv(qA) = L (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  To connect this back to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='1, let A, B ∈ PSL(2, Z) be primitive and hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Let TA be the  topograph associated to qA, and let TB be the topograph associated to qB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' By picking an oriented edge of  each topograph, we can superimpose one on top of the other, so that a region has an associated ordered  pair: the number from TA, followed by the number from TB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' There are four possibilities for the signs of the  numbers in this pair: (+, +), (+, −), (−, +), (−, −), and it turns out that we get an intersection between ˜ℓA  and ˜ℓB if all four combinations appear (see [Ric21a] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' This is equivalent to the rivers of the  topgraphs starting off disjoint, meeting, and then crossing each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  In particular, if we have “intersecting topographs”, then you can translate either topograph by the corre-  sponding river word to get another pair of intersecting topographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' This equivalence gives rise to the same  intersection point on the modular curve, and is the only way to produce the same point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='5 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='5 (Theorem 5 of [Ric21a]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Let A, B ∈ PSL(2, Z) be primitive and hyperbolic, corresponding to  respective topographs TA and TB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Then Int(˜ℓA, ˜ℓB) is equal to the number of ways to superimpose TB on top  of TA so that the rivers meet and cross, modulo the periods of the rivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Consider the combinatorics of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='5: start with TA, flattened so that the river RA is horizontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  In order to meet and cross, the river RB for TB can meet RA from either the left or the right hand side  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' top/bottom in terms of the picture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' It can then flow in the same or the opposite direction, leading to  4 possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Let IntRS(A, B) denote the topograph intersection number where RB joins RA from the  right hand side, and the rivers flow in the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Call this the RS-intersection number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  The RS-intersection number is exactly the quantity we want.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' The topograph RS-intersection number, IntRS(A, B), coincides with − link(kA, kB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Before giving the proof, consider the example from Figure 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' A = ( 3 5  7 12 ) and B = ( 2 1  1 1 ), corresponding  to river words RRLLRL and LR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' The topograph for A was displayed in Figure 2, and all 3 possible RS-  intersections are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  · ·  · ·  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' RS-intersection of RRLLRL and LR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' One and a half periods of the topo-  graph of RRLLRL are shown in solid black, and the three RS-intersections with LR are in  blue/red/green and dotted/dashed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Consider how to combinatorially compute IntRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Let RA correspond to the word  a1a2 · · · am, and let RB correspond to the word b1b2 · · · bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' For RB to join RA from the right hand side, we  6  must have a pair of indices (i, j) with 1 ≤ i ≤ m and 1 ≤ j ≤ n such that ai = L and bj = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Then the  rivers flow in the same direction for x ∈ Z≥0 steps, corresponding to ai+k = bj+k for 1 ≤ k ≤ x − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Since  RB crosses RA, we must exit with ai+x = R and bj+x = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' In particular, this is exactly the same description  as provided in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='3, so it follows from the proof in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' On the topograph side, IntRS(A, B) is still defined when A and B are conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' On the  modular curve side of things, this corresponds to transverse self-intersections of the geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' However, it is  not entirely clear what the interpretation should be in terms of knot theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' If A = ( 2 1  1 1 ), then IntRS(A, A) =  1, whereas kA is the unknot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Perhaps there is a natural framing of the modular knots so that IntRS(A, A)  becomes the self-linking number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Lorenz knots  In an attempt to model atmospheric convection, meteorologist Edward Norton Lorenz came up with the  following system of differential equations (where t represents time):  dx  dt = 10(y − x),  dy  dy = 28x − y − xz,  dz  dt = xy − 8  3z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  By picking an initial starting point, the ODE follows a path to determine a flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' See Figure 4 for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' A sample solution to the Lorenz system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Image created with Mathematica,  [Inc22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  7  In [Lor63], Lorenz proved that while paths must all eventually enter and remain in a bounded region,  they are very susceptible to miniscule changes in the initial inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' This observation led to the beginning of  chaos theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  We will study a different aspect of this theory, namely the knots formed as solutions to Lorenz’s equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  A Lorenz knot is defined to be a closed periodic orbit of this ODE, and a Lorenz link is a set of Lorenz knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Lorenz knots/links can be studied with template theory, introduced by Birman and Williams in [BW83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  As seen in Figure 4, the solutions seem to form a “butterfly”: looping around one of two circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' This can  be made precise with the Lorenz template, which is a branched surface in R3 with a semi-flow that describes  the behaviour of Lorenz knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' See Figure 5 for a depiction of the Lorenz template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  L  R  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' The Lorenz template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' The basic tools to create the Lorenz template came from the work of Guckenheimer and  Williams in [GW79], [Wil79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' The proof that the template actually corresponds to Lorenz’s original equations  is due to Tucker in [Tuc02].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Let the branch (the piece connecting the left and right circles) be [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' A path starting at x ∈ (0, 1)−{1/2}  will wind around the left (if x < 1/2) or the right (if x > 1/2) loop, ending back up at f(x) ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Furthermore, we have the following properties:  f(x) ̸= x for all x ∈ (0, 1) − {1/2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  f restricted to (0, 1/2) is a continuous increasing bijection with (0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  f restricted to (1/2, 1) is a continuous increasing bijection with (0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  flows corresponding to x and x′ with 0 < x < x′ < 1/2 (or 1/2 < x < x′ < 1) do not cross;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  flows coming from the left loop meet the branch above flows coming from the right loop (denoted  by dotted lines in the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  In fact (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='4 of [BW83]), the function f(x) = 2x (mod 1) will model the Lorenz template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' In Figure 4, the right hand side of the orbits come back to the branch in front of the left  hand side orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' This is opposite to what is done in the Lorenz template, Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' I’m not certain of why  this is happens, but it is the convention used in [BW83] and [Ghy07], so we will follow it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' In any case, the  only side affect is a potential difference in sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Since a Lorenz knot is a periodic closed flow, we can associate a word to it via the sequence of L’s and  R’s it follows on the template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  8  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' The Lorenz word of a Lorenz knot is the sequence of L’s and R’s that the knot visits as it  travels the template over one period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  It is clear that the Lorenz word is only defined up to cyclic shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Note the similarity between the Lorenz  word of a Lorenz knot and the Lorenz word of a primitive hyperbolic matrix!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' By the remarkable work of  Ghys in [Ghy07], these two worlds are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='4 (Ghys, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Consider a modular link corresponding to distinct Lorenz words W1, W2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' , Wn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  This coincides (knot-theoretically) with the Lorenz link corresponding to Lorenz words W1, W2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' , Wn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  In particular, to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='3, it suffices to prove it for Lorenz links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Proof of the main result  To compute link(kA, kB), by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='4, it suffices to compute the linking number of the corresponding  knots on the Lorenz template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Let these knots be k′  1 and k′  2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' The linking number is then the  number of times k′  2 crosses under k′  1, and accounting for sign: +1 from right to left, and −1 from left to  right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Let W1 = a1a2 · · · am and W2 = b1b2 · · · bn be the distinct Lorenz words corresponding to A and B  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' For each 1 ≤ i ≤ m, let Ki be the part of the k′  1 corresponding to ai on the template, and  say that it leaves from xi ∈ (0, 1) (and therefore flows to xi+1, with indices taken modulo m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Similarly, for  each 1 ≤ j ≤ n, let Lj be the part of k′  2 on the template corresponding to bj, and say that it leaves from  yj ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' To compute link(W1, W2), it suffices to compute link(Ki, Lj) for each pair (i, j) with 1 ≤ i ≤ m  and 1 ≤ j ≤ n and add them up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Case 1: ai = bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' In this case, both knots flow around the same side of the template, not intersecting  each other, and return to the branch in the same order as they started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' See Figure 6 for a depiction of the  situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' This contributes nothing to the linking number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  L  R  L  R  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Flowing around the same side of the template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Case 2: ai = L and bj = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' The knots flow around opposite loops, and may or may not cross as they  return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  If they do not cross, then we must have xi+1 < yi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Therefore we must have either ai+1 = bj+1,  or ai+1 = L and bj+1 = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' In the first case, the loops will flow around the corresponding side, and  again retain their ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Therefore we again have ai+2 = bj+2 or ai+2 = L and bj+2 = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' This  process continues, until eventually the knots take opposite sides (after at most mn iterations), which  must be L for ai+k and R for bj+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  9  To sum up this sub-case, if they do not cross, then there is some word X such that the Lorenz words  are LXL and RXR starting at (ai, bj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  If they do cross, then we have yi+1 < xi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' The sign of crossing is also −1, since the left branch  comes in on top of the right branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' The analysis of the previous case holds, except now when the  knots separate, we must have ai+k = R and bi+k = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' In other words, there is some X such that the  Lorenz words are LXR and RXL starting at (ai, bj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Since the conditions found in the two sub-cases are disjoint and cover every possibility, they are if and  only if.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' In particular, occurrences of LXR and RXL contribute −1 to the linking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Case 3: ai = R and bj = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' This is identical to case 2, except now a crossing corresponds to k′  1 coming  in underneath k′  2, which we do not count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='3 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Comparison to Kennedy’s formula  In [Ken94], Kennedy provides another computation of link(W1, W2), where W1 and W2 are two Lorenz  words (that are not equivalent under cyclic shift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Let σ be a permutation of {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Define the crossing count of σ to be �n  i=1 |σ(i)−i|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  For i = 1, 2, let Wi have period ni, and write out the following words in order (where s represents a single  shift):  W1, s(W1), s2(W1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' , sn1−1(W1), W2, s(W2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' , sn2−1(W2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Let σ1 denote the permutation that alphabetizes the first n1 words, let σ2 denote the permutation that  alphabetizes the second n2 words, and let σ3 denote the permutation that alphabetizes all n1 + n2 words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='2 (Theorem 1 of [Ken94]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' The linking number of the Lorenz knots with words W1 and W2 is  1  4 (C(σ1) + C(σ2) − C(σ3)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  For example, consider our example of W1 = RRLLRL and W2 = LR, as examined in Figures 1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  The alphabetizations are in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Alphabetization of RRLLRL and LR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  n  Word  Joint order  Individual order  1  RRLLRL  8  6  2  RLLRLR  6  4  3  LLRLRR  1  1  4  LRLRRL  3  2  5  RLRRLL  7  5  6  LRRLLR  4  3  7  LR  2  1  8  RL  5  2  10  Therefore σ1 = (6, 4, 1, 2, 5, 3), σ2 = (1, 2), and σ3 = (8, 6, 1, 3, 7, 4, 2, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' This gives  C(σ1) = |6 − 1| + |4 − 2| + |1 − 3| + |2 − 4| + |5 − 5| + |3 − 6| =14  C(σ2) = |1 − 1| + |2 − 2| =0  C(σ3) = |8 − 1| + |6 − 2| + |1 − 3| + |3 − 4| + |7 − 5| + |4 − 6| + |2 − 7| + |5 − 8| =26,  hence  link(W1, W2) = 1  4 (14 + 0 − 26) = −3,  which agrees with our computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' A direct combinatorial proof that the counts in Theorems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='3  are equal does not appear to be obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Comparison to Duke-Imamo¯glu-T´oth’s symmetrized linking number  In certain cases, link(kA, kB) can be deduced from link(kA + kA−1, kB + kB−1), which was computed in  [DIT17] and [Ric21a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' A primitive hyperbolic matrix A ∈ PSL(2, R) is called reciprocal if A is conjugate to A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  The matrix A is reciprocal if and only if the corresponding quadratic form qA is reciprocal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' qA is  PSL(2, Z)-equivalent to −qA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Furthermore, if W is the Lorenz word associated to A, then A is reciprocal if  and only if when you write W backwards and swap L’s and R’s, you end up with a cyclic shift of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' For  example, A = ( 3 5  7 12 ) (from the example in Figure 1) is reciprocal, since it corresponds to the Lorenz word  W = RRLLRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Writing this word backwards and swapping R’s and L’s gives RLRRLL, which is a cyclic  shift of W by 4 places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='3 (see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='8 of [Ric21a] and the surrounding commentary for more details) we  can prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Let A or B be reciprocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Then  link(kA, kB) = 1  4 link(kA + kA−1, kB + kB−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  In particular, if either A or B is reciprocal, then link(kA, kB) can be deduced from the symmetrized linking  number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' If both A and B are not reciprocal, then this is no longer true, and we obtain new information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  For example, consider the non-reciprocal matrices  A =  �  3  2  1  1  �  ,  B =  �  14  9  3  2  �  ,  C =  �  19  7  8  3  �  ,  D =  �  28  9  3  1  �  ,  which correspond to the Lorenz words  WA = LLR,  WB = LLLLRLR,  WC = LLRRLRR,  WD = LLLLLLLLLRRR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Table 2 gives the corresponding linking numbers, and it can be observed that Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='2 does not  necessarily hold when neither matrix is reciprocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  11  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Linking number comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  X, Y  link(kX, kY )  link(kX, kY −1)  link(kX + kX−1, kY + kY −1)  A, B  4  2  12  A, C  3  4  14  A, D  3  3  12  B, C  6  8  28  B, D  7  5  24  C, D  7  7  28  References  [Ati87]  Michael Atiyah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' The logarithm of the Dedekind η-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=', 278(1-4):335–380, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  [BW83]  Joan S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Birman and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Williams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Knotted periodic orbits in dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Lorenz’s equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Topology,  22(1):47–82, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  [DIT17] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Duke, ¨O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Imamo¯glu, and ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' T´oth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Modular cocycles and linking numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Duke Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=', 166(6):1179–1210, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  [Ghy07]  ´Etienne Ghys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Knots and dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' In International Congress of Mathematicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' I, pages 247–277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=', Z¨urich, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  [GL06]  ´Etienne Ghys and Jos Leys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Lorenz and modular flows: a visual introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='ams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='org/publicoutreach/  feature-column/fcarc-lorenz, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' AMS feature column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Accessed 16 December 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  [GW79] John Guckenheimer and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Williams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Structural stability of Lorenz attractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Hautes ´Etudes Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=', (50):59–72, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  [Hat22]  Allen Hatcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Topology of Numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' American Mathematical Society, Providence, RI, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  [Hun07] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Hunter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Matplotlib: A 2d graphics environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Computing in Science & Engineering, 9(3):90–95, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  [Inc22]  Wolfram Research, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Mathematica, Version 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='1, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  [Ken94] Stephen F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Kennedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Algorithms for the linking numbers of Lorenz and horseshoe knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Houston J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=', 20(4):705–  712, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  [Lor63]  Edward N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Lorenz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Deterministic nonperiodic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Atmospheric Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=', 20(2):130–141, 1963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  [Mil71]  John Milnor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Introduction to algebraic K-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Princeton University Press, Princeton, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' University of Tokyo  Press, Tokyo, 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Annals of Mathematics Studies, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  [Ric21a] James Rickards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Computing intersections of closed geodesics on the modular curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Number Theory, 225:374–408,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  [Ric21b] James Rickards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Counting intersection numbers of closed geodesics on Shimura curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='org/abs/2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  01968, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  [Sar82]  Peter Sarnak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Class numbers of indefinite binary quadratic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Number Theory, 15(2):229–247, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  [Tuc02]  Warwick Tucker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' A rigorous ODE solver and Smale’s 14th problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=', 2(1):53–117, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  [Wil79]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Williams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' The structure of Lorenz attractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Hautes ´Etudes Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content=', (50):73–99, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='  University of Colorado Boulder, Boulder, Colorado, USA  Email address: james.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='rickards@colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='edu  URL: https://math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
+page_content='edu/~jari2770/  12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAzT4oBgHgl3EQfYfwQ/content/2301.01334v1.pdf'}
diff --git a/WtAyT4oBgHgl3EQfh_ga/content/tmp_files/2301.00385v1.pdf.txt b/WtAyT4oBgHgl3EQfh_ga/content/tmp_files/2301.00385v1.pdf.txt
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@@ -0,0 +1,1707 @@
+arXiv:2301.00385v1  [math.CA]  1 Jan 2023
+Inner Riesz pseudo-balayage and its applications to minimum
+energy problems with external fields
+Natalia Zorii
+Dedicated to Professor Stephen J. Gardiner on the occasion of his 65th birthday
+Abstract. For the Riesz kernel κα(x, y) := |x − y|α−n of order 0 < α < n on Rn, n ⩾ 2, we
+introduce the so-called inner pseudo-balayage ˆωA of a (Radon) measure ω on Rn to a set A ⊂ Rn
+as the (unique) measure minimizing the Gauss functional
+ˆ
+κα(x, y) d(µ ⊗ µ)(x, y) − 2
+ˆ
+κα(x, y) d(ω ⊗ µ)(x, y)
+over the class E+(A) of all positive measures µ of finite energy, concentrated on A.
+For quite
+general signed ω (not necessarily of finite energy) and A (not necessarily closed), such ˆωA does
+exist, and it maintains the basic features of inner balayage for positive measures (defined when
+α ⩽ 2), except for those implied by the domination principle. (To illustrate the latter, we point out
+that, in contrast to what occurs for the balayage, the inner pseudo-balayage of a positive measure
+may increase its total mass.) The inner pseudo-balayage ˆωA is further shown to be a powerful tool in
+the problem of minimizing the Gauss functional over all µ ∈ E+(A) with µ(Rn) = 1, which enables
+us to improve substantially many recent results on this topic, by strengthening their formulations
+and/or by extending the areas of their applications. For instance, if A is a quasiclosed set of nonzero
+inner capacity c∗(A), and if ω is a signed measure, compactly supported in Rn \ ClRnA, then the
+problem in question is solvable if and only if either c∗(A) < ∞, or ˆωA(Rn) ⩾ 1. In particular,
+if c∗(A) = ∞, then the problem has no solution whenever ω+(Rn) < 1/Cn,α, where Cn,α := 1 if
+α ⩽ 2, and Cn,α := 2n−α otherwise; whereas ω−(Rn), the total amount of the negative charge, has
+no influence on this phenomenon. The results obtained are illustrated by some examples.
+1. Inner pseudo-balayage: a motivation and a model case
+This paper deals with the theory of potentials with respect to the α-Riesz kernels
+κα(x, y) := |x − y|α−n of order 0 < α < n on Rn, n ⩾ 2, |x − y| being the Euclidean
+distance in Rn. Our main goal is to proceed further with the study of minimum
+α-Riesz energy problems in the presence of external fields f, a point of interest for
+many researchers (see e.g. the monographs [2, 20] and references therein, [14, 18],
+[21]–[25], as well as [1, 8, 32, 33], some of the most recent papers on this topic).
+In the current work we improve substantially many recent results in this field, by
+strengthening their formulations and/or by extending the areas of their applications
+(see Section 6 for the results obtained). This has become possible due to the devel-
+opment of a new tool, the inner pseudo-balayage (see Section 3, cf. also the present
+section for a motivation of the proposed definition as well as for a model case).
+It is well known that the α-Riesz balayage (sweeping out) serves as an efficient
+tool in the problems in question (see e.g. [8, 23, 25, 33]). However, its application is
+only limited to the case of α ranging over (0, 2], and to external fields f of the form
+f(x) := −Uω(x) := −
+ˆ
+κα(x, y) dω(y),
+(1.1)
+2010 Mathematics Subject Classification: Primary 31C15.
+Key words: Minimum Riesz energy problems with external fields, inner Riesz balayage, inner
+Riesz pseudo-balayage.
+
+2
+Natalia Zorii
+where ω is a suitable positive Radon measure.
+To extend the area of application of such a tool to arbitrary α ∈ (0, n) and/or to
+external fields f given by (1.1), but now with signed ω involved, we generalize the
+standard concept of inner balayage of positive measures (defined for α ∈ (0, 2]) to the
+so-called inner pseudo-balayage of signed measures, by maintaining the basic features
+of the former concept — except for those implied by the domination principle.
+Being crucial to our study of minimum energy problems with external fields, the
+concept of inner pseudo-balayage is also of independent interest, looking promising
+for further generalizations and other applications.
+Before introducing it, we first
+review some basic facts of the theory of α-Riesz potentials.
+We denote by M the linear space of all (real-valued Radon) measures µ on Rn,
+equipped with the vague topology of pointwise convergence on the class C0(Rn) of
+all continuous functions ϕ : Rn → R of compact support, and by M+ the cone of all
+positive µ ∈ M, where µ is positive if and only if µ(ϕ) ⩾ 0 for all positive ϕ ∈ C0(Rn).
+Given µ, ν ∈ M, the potential Uµ and the mutual energy I(µ, ν) are introduced by
+Uµ(x) :=
+ˆ
+κα(x, y) dµ(y),
+x ∈ Rn,
+I(µ, ν) :=
+ˆ
+κα(x, y) d(µ ⊗ ν)(x, y),
+respectively, provided that the integral on the right is well defined (as a finite number
+or ±∞). For µ = ν, I(µ, ν) defines the energy I(µ) := I(µ, µ) of µ ∈ M.
+The following property of strict positive definiteness for the α-Riesz kernels, dis-
+covered by M. Riesz [19, Chapter I, Eq. (13)] (cf. also [17, Theorem 1.15]), is crucial
+to the current study: I(µ) ⩾ 0 for any (signed) µ ∈ M, and I(µ) = 0 ⇐⇒ µ = 0.
+This implies that all (signed) µ ∈ M with I(µ) < ∞ form a pre-Hilbert space E with
+the inner product ⟨µ, ν⟩ := I(µ, ν) and the energy norm ∥µ∥ :=
+�
+I(µ), see e.g. [10,
+Lemma 3.1.2]. The topology on E defined by means of this norm, is said to be strong.
+Another fact decisive to this paper is that the cone E+ := E ∩ M+ is strongly
+complete, and that the strong topology on E+ is finer than the (induced) vague
+topology on E+ (see J. Deny [6]; for α = 2, cf. also H. Cartan [4]). Thus any strong
+Cauchy sequence (net) (µj) ⊂ E+ converges both strongly and vaguely to the same
+unique limit µ0 ∈ E+, the strong topology on E as well as the vague topology on M
+being Hausdorff. (Following B. Fuglede [10], such a kernel is said to be perfect.)
+1.1. A model case. As a model case for introducing the concept of inner α-Riesz
+pseudo-balayage, consider first a closed set F ⊂ Rn and a (signed) measure ω ∈ M
+of finite energy. Since the class M+(F) of all µ ∈ M+ with the support S(µ) ⊂ F is
+vaguely closed [3, Section III.2, Proposition 6], the convex cone E+(F) := M+(F)∩E
+is strongly closed, and hence strongly complete, the α-Riesz kernel being perfect. By
+applying [9] (Theorem 1.12.3 and Proposition 1.12.4(2)), we therefore conclude that
+for the given ω ∈ E, there exists the unique Pω ∈ E+(F) such that
+∥ω − Pω∥ =
+min
+µ∈E+(F ) ∥ω − µ∥,
+(1.2)
+and the same Pω is uniquely characterized within E+(F) by the two relations
+⟨Pω − ω, µ⟩ ⩾ 0 for all µ ∈ E+(F),
+(1.3)
+⟨Pω − ω, Pω⟩ = 0.
+(1.4)
+This Pω is said to be the orthogonal projection of ω ∈ E onto E+(F).
+
+Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields
+3
+By a slight modification of [26, Proof of Theorem 3.1] we infer from the above
+that Pω is the only measure in E+(F) having the two properties
+UP ω ⩾ Uω n.e. on F,
+(1.5)
+UP ω = Uω Pω-a.e.,
+(1.6)
+where the abbreviation n.e. (nearly everywhere) means that the inequality holds true
+everywhere on F except for a subset N ⊂ F of inner capacity zero: c∗(N) = 0.1
+Assume for a moment that α ⩽ 2, and that the above ω is positive, i.e. ω ∈ E+.
+By use of the complete maximum principle [17, Theorems 1.27, 1.29], we derive from
+(1.5) and (1.6) that Pω is then uniquely characterized within E+(F) by the equality
+UP ω = Uω n.e. on F,
+(1.7)
+see [26, Theorem 3.1], and hence Pω is actually the balayage ωF of ω ∈ E+ onto F:
+Pω = ωF.
+¶ However, if either α > 2, or if the above ω is signed, then relations (1.2)–(1.6)
+still hold, but they no longer result in (1.7).
+Motivated by this observation, we introduce the following definition.
+Definition 1.1. The pseudo-balayage ˆωF of ω ∈ E onto a closed set F ⊂ Rn
+with respect to the α-Riesz kernel of arbitrary order α ∈ (0, n) is defined as the only
+measure in E+(F) satisfying (1.2) (with ˆωF in place of Pω); or equivalently, as the
+unique measure in E+(F) having properties (1.3)–(1.6) (with ˆωF in place of Pω).
+Remark 1.2. Assume for a moment that ω is positive. It follows from the above
+that the pseudo-balayage ˆωF coincides with the balayage ωF whenever α ⩽ 2; while
+otherwise, the former concept presents a natural extension of the latter, the problem
+of balayage for α > 2 being unsolvable.2 But if now ω is signed, then for α ⩽ 2, both
+ˆωF and ωF still exist and are unique, whereas, in general,
+ˆωF ̸= ωF,
+the balayage of signed ω being defined by linearity (for more details see Remark 3.3).
+Remark 1.3. In Section 3 below, we shall extend the above definition of the
+pseudo-balayage ˆωF, given in the model case of ω ∈ E and closed F ⊂ Rn, to:
+• ω ∈ M that are not necessarily of finite energy.
+• F ⊂ Rn that are not necessarily closed.
+For the former goal, we observe that problem (1.2) is equivalent to that of minimizing
+the Gauss functional ∥µ∥2 − 2
+´
+Uω dµ, which makes sense not only for ω ∈ E.
+We complete this section with some general conventions, used in what follows.
+From now on, when speaking of a (signed) measure µ ∈ M, we understand that
+its potential Uµ is well defined and finite almost everywhere with respect to the
+Lebesgue measure on Rn; or equivalently (cf. [17, Section I.3.7]) that
+ˆ
+|y|>1
+d|µ|(y)
+|y|n−α < ∞,
+(1.8)
+1For closed F, the set N of all x ∈ F where (1.5) fails is Borel, hence capacitable, and so (1.5)
+actually holds true even quasi-everywhere (q.e.) on F, namely everywhere on F except for N of outer
+capacity zero: c∗(N) = 0. (For the concepts of inner and outer capacities, see [17, Section II.2.6].)
+2See e.g. [17, Section IV.5.20]; this is caused by the fact that for α > 2, the maximum principle
+fails to hold. Therefore, when speaking of α-Riesz balayage, we understand that α ∈ (0, 2].
+
+4
+Natalia Zorii
+where |µ| := µ+ + µ−, µ+ and µ− being the positive and negative parts of µ in the
+Hahn–Jordan decomposition. Actually, then (and only then) Uµ is finite q.e. on Rn,
+cf. [17, Section III.1.1]. This would necessarily hold if µ were required to be bounded
+(that is, with |µ|(Rn) < ∞), or of finite energy, cf. [10, Corollary to Lemma 3.2.3].
+A measure µ ∈ M+ is said to be concentrated on a set A ⊂ Rn if Ac := Rn \ A
+is µ-negligible, or equivalently if A is µ-measurable and µ = µ|A, µ|A being the
+restriction of µ to A. Denoting by M+(A) the cone of all µ ∈ M+ concentrated on
+A, we further write E+(A) := M+(A) ∩ E, and let E′(A) stand for the closure of
+E+(A) in the strong topology on E+. We emphasize that E′(A) is strongly complete,
+being a strongly closed subcone of the strongly complete cone E+.
+Given A ⊂ Rn, denote by CA the upward directed set of all compact subsets K
+of A, where K1 ⩽ K2 if and only if K1 ⊂ K2. If a net (xK)K∈CA ⊂ Y converges to
+x0 ∈ Y , Y being a topological space, then we shall indicate this fact by writing
+xK → x0 in Y as K ↑ A.
+2. On the inner Riesz balayage
+Before proceeding with an extension of the concept of pseudo-balayage announced
+in Remark 1.3, we first recall some basic facts of the theory of inner α-Riesz balayage.
+Such a theory, generalizing Cartan’s pioneering work [5] on the inner Newtonian
+balayage (α = 2) to any α ∈ (0, 2], was initiated in the author’s recent papers [26, 27],
+and it was further developed in [28]–[30].3 Throughout this section, 0 < α ⩽ 2.
+Definition 2.1 ([26, Sections 3, 4]). The inner balayage ωA of a measure ω ∈ M+
+to a set A ⊂ Rn is defined as the measure of minimum potential in the class ΓA,ω,
+ΓA,ω :=
+�
+µ ∈ M+ : Uµ ⩾ Uω n.e. on A
+�
+.
+That is, ωA ∈ ΓA,ω and
+UωA = min
+µ∈ΓA,ω Uµ on Rn.
+(2.1)
+Theorem 2.2 ([26, Sections 3, 4]). Given arbitrary ω ∈ M+ and A ⊂ Rn, the
+inner balayage ωA, introduced by Definition 2.1, exists and is unique. Furthermore,4
+UωA = Uω n.e. on A,
+(2.2)
+UωA ⩽ Uω on Rn.
+The same ωA can alternatively be characterized by means of either of the following
+(equivalent) assertions:
+(a) ωA is the unique measure in M+ satisfying the symmetry relation
+I(ωA, σ) = I(σA, ω) for all σ ∈ E+,
+where σA denotes the only measure in E′(A) with UσA = Uσ n.e. on A.5
+3See also [31] for an application of this theory to Deny’s principle of positivity of mass.
+4As pointed out in [26, Remark 3.12], (2.2) no longer characterizes ωA uniquely (as it does for
+closed A and ω ∈ E+). For more details see footnote 5, Corollary 2.4, Remark 2.5, and Theorem 2.6.
+5For any σ ∈ E+ and any A ⊂ Rn, the measure σA ∈ E′(A) having the property U σA = U σ n.e.
+on A, exists and is unique. It is, in fact, the orthogonal projection of σ in the pre-Hilbert space E
+onto the convex, strongly complete cone E′(A); that is (compare with (1.2)),
+∥σ − σA∥ =
+min
+µ∈E′(A) ∥σ − µ∥.
+The same σA is uniquely characterized within M+ by the extremal property (2.1) with µ := σ.
+
+Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields
+5
+(b) ωA is the unique measure in M+ satisfying either of the two limit relations
+ωA
+j → ωA vaguely in M+ as j → ∞,
+UωA
+j ↑ UωA pointwise on Rn as j → ∞,
+where (ωj) ⊂ E+ is an arbitrary sequence having the property6
+Uωj ↑ Uω pointwise on Rn as j → ∞,
+whereas ωA
+j denotes the only measure in E′(A) with UωA
+j = Uωj n.e. on A.
+Remark 2.3. For signed ω ∈ M, we define the inner balayage ωA by linearity:
+ωA := (ω+)A − (ω−)A.
+(2.3)
+If moreover the mutual energy I(ω, σ) is well defined for all σ ∈ E, then this ωA is
+uniquely characterized by the symmetry relation
+I(ωA, σ) = I(σA, ω) for all σ ∈ E,
+which actually only needs to be verified for certain countably many σ ∈ E, indepen-
+dent of the choice of ω (cf. [30], Theorem 1.4 and Remark 1.4).
+• In the rest of this paper, we shall always require A ⊂ Rn to have the property7
+(P1) E+(A) is strongly closed.
+Then (and only then)
+E′(A) = E+(A),
+and hence Theorem 2.2 remains valid with E′(A) replaced throughout by E+(A). In
+particular, the following useful corollary holds true.
+Corollary 2.4. For this A and for any ω ∈ E+, the inner balayage ωA is, in fact,
+the orthogonal projection of ω onto the (convex, strongly complete) cone E+(A):
+∥ω − ωA∥ =
+min
+µ∈E+(A) ∥ω − µ∥.
+The same ωA is uniquely characterized within E+(A) by UωA = Uω n.e. on A.
+Remark 2.5. Assumption (P1) is important for the validity of Corollary 2.4.
+Indeed, if α = 2 and A := Br, r ∈ (0, ∞), then for any ω ∈ M+(Bc
+r) with ω(B
+c
+r) > 0,
+we have S(ωBr) = Sr [27, Theorems 4.1, 5.1], and hence the inner balayage ωBr is
+not concentrated on the set Br itself. (Actually, S(ωBr) ∩ Br = ∅.)8
+The following generalization of Corollary 2.4 will be useful in the sequel.
+Theorem 2.6. Given ω ∈ M+, assume that ωA is of finite energy.9 Then ωA
+is concentrated on A, that is, ωA ∈ E+(A), and it is the unique solution to the
+6Such ωj ∈ E+, j ∈ N, do exist; they can be defined, for instance, by means of the formula
+U ωj := min
+�
+U ω, jU λ�
+,
+λ ∈ E+ being fixed (see e.g. [17, p. 272] or [5, p. 257, footnote]). Here we have used the fact that
+for any µ1, µ2 ∈ M+, there is µ0 ∈ M+ such that U µ0 := min {U µ1, U µ2} [17, Theorem 1.31].
+7As shown in [33, Theorem 3.9], (P1) is fulfilled, for instance, if A is quasiclosed (quasicompact),
+that is, if A can be approximated in outer capacity by closed (compact) sets, see Fuglede [11].
+8Here and in the sequel we use the notations Br := {|x| < r}, Br := {|x| ⩽ r}, Sr := {|x| = r}.
+9This occurs e.g. if ω itself is of finite energy, or if ω is bounded and meets the separation condition
+inf
+(x,y)∈S(ω)×A |x − y| > 0.
+
+6
+Natalia Zorii
+problem of minimizing the Gauss functional ∥µ∥2 −2
+´
+Uω dµ, µ ranging over E+(A).
+Alternatively, ωA is uniquely characterized within E+(A) by UωA = Uω n.e. on A.
+Proof. Since ωA = (ωA)A (see [26, Corollary 4.2]), Corollary 2.4 applied to ωA ∈ E+
+shows that, indeed, ωA ∈ E+(A), and hence ωA is the orthogonal projection of itself
+onto E+(A); or equivalently, it is the (unique) solution to the problem of minimizing
+the functional ∥µ∥2−2
+´
+UωA dµ, µ ranging over E+(A). This implies the former part
+of the claim by noting that
+´
+UωA dµ =
+´
+Uω dµ for all µ ∈ E+(A), which, in turn,
+is derived from (2.2) by use of the fact, to be often used in what follows, that any
+µ-measurable subset of A with c∗(·) = 0 is µ-negligible for any µ ∈ E+(A).
+For the latter part, assume (2.2) holds for some µ0 ∈ E+(A) in place of ωA. By
+the strengthened version of countable subadditivity for inner capacity (Lemma 2.7),
+Uµ0 = Uω = UωA n.e. on A,
+whence µ0 = (ωA)A, again by Corollary 2.4 applied to ωA ∈ E+. Combining this with
+(ωA)A = ωA (see above) gives µ0 = ωA, thereby completing the whole proof.
+□
+Lemma 2.7. For arbitrary Q ⊂ Rn and Borel Uj ⊂ Rn,
+c∗
+��
+j∈N
+Q ∩ Uj
+�
+⩽
+�
+j∈N
+c∗(Q ∩ Uj).
+Proof. See [10, pp. 157–158] (for α = 2, cf. [5, p. 253]); compare with [17, p. 144].
+□
+3. An extension of the concept of pseudo-balayage
+In what follows, we assume that
+c∗(A) > 0.
+(3.1)
+Then (and only then) the class E+(A) is not reduced to {0}, see [10, Lemma 2.3.1],
+and the problems in question become nontrivial.
+Fix a (signed) measure ω ∈ M (not necessarily of finite energy), and define the
+external field f : Rn → [−∞, ∞] by means of the formula
+f := −Uω.
+Being the difference between two lower semicontinuous (l.s.c.) functions, f is Borel
+measurable, and, due to (1.8), f is finite q.e. on Rn.
+Let E+
+f (A) stand for the convex cone of all µ ∈ E+(A) such that f is µ-integrable;
+then for every µ ∈ E+
+f (A), the Gauss functional10
+If(µ) := ∥µ∥2 + 2
+ˆ
+f dµ = ∥µ∥2 − 2
+ˆ
+Uω dµ
+is finite. Denoting
+ˆwf(A) :=
+inf
+µ∈E+
+f (A) If(µ),
+(3.2)
+we have
+− ∞ ⩽ ˆwf(A) ⩽ 0,
+(3.3)
+the upper estimate being caused by the fact that 0 ∈ E+
+f (A) while If(0) = 0.
+Definition 3.1. A measure ˆωA ∈ E+
+f (A) with If(ˆωA) = ˆwf(A) is said to be the
+inner α-Riesz pseudo-balayage of ω onto A.
+10In constructive function theory, the Gauss functional is also referred to as the f-weighted energy.
+
+Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields
+7
+Lemma 3.2. The inner pseudo-balayage ˆωA is unique (if it exists).
+Proof. This follows by standard methods based on the convexity of the class E+
+f (A)
+and the parallelogram identity in the pre-Hilbert space E, by use of the strict positive
+definiteness of the Riesz kernel. (See e.g. [21, Proof of Lemma 6]. Note that this proof
+requires ˆwf(A) to be finite, which however necessarily holds whenever ˆωA exists.)
+□
+Remark 3.3. If ω is positive, then, as seen from Theorem 2.6, the concept of
+inner pseudo-balayage ˆωA extends that of inner balayage ωA (introduced for α ⩽ 2)
+to arbitrary α ∈ (0, n). See the present section as well as Section 4 for details.
+¶ But if ω is signed, then the inner balayage ωA, defined for α ⩽ 2 by means of
+(2.3), may not coincide with the inner pseudo-balayage ˆωA. (Thus, in case α ⩽ 2,
+the theory of inner pseudo-balayage may be thought of as an alternative theory of
+inner balayage for signed measures, which is not equivalent to the standard one.)
+Indeed, take ω ∈ M such that ω = −ω− ̸= 0. Then ωA = −(ω−)A ̸= 0, whereas
+ˆωA, minimizing ∥µ∥2 + 2
+´
+Uω− dµ ⩾ 0 over E+
+f (A), is obviously 0, and so ˆωA ̸= ωA.
+Remark 3.4. It follows easily from Definition 3.1 that, if ˆωA exists, then so does
+(�
+cω)A for any c ∈ [0, ∞), and moreover
+(�
+cω)A = cˆωA.
+(3.4)
+However, this fails to hold if c < 0, cf. Remark 3.3.
+3.1. On the existence of the inner pseudo-balayage. Recall that we are working
+under the permanent requirements (P1) and (3.1). In the rest of this paper, we also
+assume that ω ∈ M satisfies either of the following properties:11
+(P2) ω ∈ E.
+(P3) ω+ is bounded, Uω is upper semicontinuous on A := ClRnA, and
+MA := sup
+x∈A
+U|ω|(x) < ∞.
+(3.5)
+Note that in case (P2), the class E+
+f (A) actually coincides with the whole of
+E+(A), cf. (3.13), while in case (P3), it necessarily contains all bounded µ ∈ E+(A).
+Theorem 3.5. The inner pseudo-balayage ˆωA, introduced by Definition 3.1, does
+exist. Hence,
+− ∞ < ˆwf(A) ⩽ 0.
+(3.6)
+The same ˆωA can alternatively be characterized by either of the following (a) or (b):
+(a) ˆωA is the only measure in E+
+f (A) having the two properties
+ˆ
+U ˆωA−ω dµ ⩾ 0 for all µ ∈ E+
+f (A),
+(3.7)
+ˆ
+U ˆωA−ω dˆωA = 0.
+(3.8)
+(b) ˆωA is the only measure in E+(A) having the two properties12
+U ˆωA ⩾ Uω n.e. on A,
+(3.9)
+U ˆωA = Uω
+ˆωA-a.e.
+(3.10)
+11(P3) is certainly fulfilled if ω ∈ M is compactly supported in A
+c.
+12Due to (3.9), U ˆωA ⩾ U ω holds true ˆωA-a.e.; hence, (3.10) is equivalent to the apparently weaker
+relation U ˆωA ⩽ U ω ˆωA-a.e. Similarly, (3.8) can be replaced by
+´
+U ˆωA−ω dˆωA ⩽ 0.
+
+8
+Natalia Zorii
+Proof. Fixing ν ∈ E+
+f (A), we shall first show that (3.9) and (3.10) hold true for ν in
+place of ˆωA if and only if so do (3.7) and (3.8). It is enough to verify only the "if"
+part of this claim, the opposite being obvious from the fact that any µ-measurable
+subset of A with c∗(·) = 0 is µ-negligible for any µ ∈ E+(A).
+Assuming, therefore, that (3.7) and (3.8) hold true (for ν in place of ˆωA), suppose
+to the contrary that (3.9) fails. But then there is compact K ⊂ A such that Uν < Uω
+on K while c(K) > 0,13 hence
+´
+Uν−ω dτ < 0 for any τ ∈ E+(K), τ ̸= 0, which
+contradicts (3.7),
+´
+f dτ being obviously finite. Thus (3.9) does indeed hold, whence
+Uν ⩾ Uω ν-a.e.
+(3.11)
+Further, assuming to the contrary that (3.10) fails to hold, we infer from (3.11)
+that there exists compact Q ⊂ A such that ν(Q) > 0 while Uν > Uω on Q. Together
+with (3.11), this yields
+´
+Uν−ω dν > 0, which however contradicts (3.8).
+The equivalence thereby verified enables us to prove the statement on the unique-
+ness in each of assertions (a) and (b). Indeed, suppose that (3.9) and (3.10) are ful-
+filled by some ν, ν′ ∈ E+(A) in place of ˆωA. Noting from (3.10) that then necessarily
+ν, ν′ ∈ E+
+f (A), we conclude by applying (3.7) and (3.8) to each of ν and ν′ that
+⟨ν, ν′⟩ ⩾
+ˆ
+Uω dν′ = ∥ν′∥2,
+⟨ν′, ν⟩ ⩾
+ˆ
+Uω dν = ∥ν∥2.
+Therefore,
+0 ⩽ ∥ν − ν′∥2 =
+�
+∥ν∥2 − ⟨ν′, ν⟩
+�
++
+�
+∥ν′∥2 − ⟨ν, ν′⟩
+�
+⩽ 0,
+whence ν = ν′, by virtue of the strict positive definiteness of the α-Riesz kernel.
+Case (P2). Assume first that (P2) holds; then the Gauss functional has the form
+If(µ) = ∥ω − µ∥2 − ∥ω∥2 for all µ ∈ E+(A),
+(3.12)
+whence
+E+
+f (A) = E+(A).
+(3.13)
+Thus the problem on the existence of the inner pseudo-balayage ˆωA is reduced to
+that on the existence of the orthogonal projection of ω ∈ E onto E+(A), i.e.
+ˆωA ∈ E+(A) and ∥ω − ˆωA∥ =
+min
+µ∈E+(A) ∥ω − µ∥.
+Since the convex cone E+(A) is strongly closed by (P1), hence strongly complete,
+E+ being strongly complete by the perfectness of the α-Riesz kernel, applying [9]
+(Theorem 1.12.3 and Proposition 1.12.4(2)) shows that such an orthogonal projection
+does exist, and it is uniquely characterized within E+(A) by both (3.7) and (3.8). In
+view of the equivalence of (a) and (b) proved above, this implies the theorem.
+Case (P3). The remaining case (P3) will be treated in four steps.
+Step 1. The purpose of this step is to show that the inner pseudo-balayage ˆωA
+exists if and only if there exists the (unique) measure µ0 ∈ E+
+f (A) satisfying (a)
+(equivalently, (b)), and then necessarily µ0 = ˆωA.
+Assume first that ˆωA exists. To verify (3.9), suppose to the contrary that there is
+a compact set K ⊂ A with c(K) > 0, such that U ˆωA < Uω on K. A straightforward
+verification then shows that for any τ ∈ E+(K), τ ̸= 0, and any t ∈ (0, ∞),
+If(ˆωA + tτ) − If(ˆωA) = 2t
+ˆ �
+U ˆωA − Uω�
+dτ + t2∥τ∥2.
+(3.14)
+13If A is capacitable (e.g. Borel), we write c(A) := c∗(A) = c∗(A).
+
+Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields
+9
+As ∥τ∥ < ∞, the value on the right in (3.14) (hence, also that on the left) is < 0 when
+t > 0 is small enough, which however contradicts Definition 3.1, for ˆωA+tτ ∈ E+
+f (A).
+Having thus established (3.9), we obtain
+U ˆωA ⩾ Uω
+ˆωA-a.e.
+(3.15)
+Suppose now that (3.10) fails to hold; then there exists a compact set Q ⊂ A
+with ˆωA(Q) > 0, such that U ˆωA > Uω on Q, cf. (3.15). Denoting υ := ˆωA|Q, we have
+ˆωA − tυ ∈ E+
+f (A) for all t ∈ (0, 1), hence
+If(ˆωA − tυ) − If(ˆωA) = −2t
+ˆ �
+U ˆωA − Uω�
+dυ + t2∥υ∥2,
+which again contradicts Definition 3.1 when t is small enough. (Note that, by (3.10),
+U ˆωA ⩽ Uω on S(ˆωA),
+(3.16)
+because Uω is upper semicontinuous (u.s.c.) on A by (P3), while U ˆωA is l.s.c. on Rn.)
+For the "if" part of the claim, assume that (a) holds true for some (unique)
+µ0 ∈ E+
+f (A). To show that then necessarily µ0 = ˆωA, we only need to verify that
+If(µ) − If(µ0) ⩾ 0 for any µ ∈ E+
+f (A).
+(3.17)
+But obviously
+If(µ) − If(µ0) = ∥µ − µ0 + µ0∥2 − 2
+ˆ
+Uω dµ − ∥µ0∥2 + 2
+ˆ
+Uω dµ0
+= ∥µ − µ0∥2 + 2
+ˆ
+Uµ0−ω d(µ − µ0),
+and relations (3.7) and (3.8) (with µ0 in place of ˆωA) immediately lead to (3.17).
+Step 2. To complete the proof of the theorem, it thus remains to establish the
+existence of the inner pseudo-balayage ˆωA. To this end, suppose throughout this step
+that A = K is compact. As c(K) > 0, see (3.1), we may restrict ourselves to nonzero
+measures µ ∈ E+(K). For each of those µ, there are t ∈ (0, ∞) and τ ∈ E+(K) with
+τ(K) = 1 such that µ = tτ. The potential U|ω| being bounded on K by (P3),
+If(µ) = t2∥τ∥2 − 2t
+ˆ
+Uω dτ ⩾ t2∥τ∥2 − 2t
+ˆ
+Uω+ dτ
+⩾ t2c(K)−1 − 2tMK = t2�
+c(K)−1 − 2MKt−1�
+,
+(3.18)
+MK ∈ [0, ∞) being introduced by (3.5) (with A := K). Hence, by virtue of (3.18),
+If(µ) > 0 for all µ ∈ E+(K) having the property
+µ(K) > 2MKc(K) =: LK ∈ [0, ∞).
+On the other hand, ˆwf(K) ⩽ 0, cf. (3.3). In view of the above, ˆwf(K) would
+therefore be the same if E+
+f (K) in (3.2) were replaced by
+E+
+LK(K) :=
+�
+µ ∈ E+(K) : µ(K) ⩽ LK
+�
+,
+(3.19)
+that is,
+ˆwf(K) =
+inf
+µ∈E+
+LK (K) If(µ) =: ˆwf,LK(K).
+(3.20)
+Hence,
+−∞ < −2MKLK ⩽ ˆwf(K) ⩽ 0.
+
+10
+Natalia Zorii
+Choose a (minimizing) sequence (µj) ⊂ E+
+LK(K) such that
+lim
+j→∞ If(µj) = ˆwf,LK(K).
+Being vaguely bounded, cf. (3.19), the sequence (µj) is vaguely relatively compact [3,
+Section III.1, Proposition 15], and so there is a subsequence (µjk) converging vaguely
+to some µ0 ∈ M+(K). (Here we have used the first countability of the vague topology
+on M, see [30, Lemma 4.4], as well as the fact that M+(K) is vaguely closed, see [3,
+Section III.2, Proposition 6].) By the principle of descent [17, Eq. (1.4.5)],
+∥µ0∥2 ⩽ lim inf
+k→∞ ∥µjk∥2.
+Furthermore, by [3, Section IV.4.4, Corollary 3],
+ˆ
+Uω dµ0 ⩾ lim sup
+k→∞
+ˆ
+Uω dµjk ∈ (−∞, ∞),
+Uω being bounded and u.s.c. on the (compact) set K. This altogether gives
+ˆwf(K) ⩽ If(µ0) ⩽ lim inf
+k→∞ If(µjk) = ˆwf,LK(K),
+which combined with (3.20) shows that µ0 serves as the pseudo-balayage ˆωK.
+Step 3. Our next aim is to show that the constant LK, satisfying (3.20), can be
+defined to be independent of K ∈ CA large enough. (To be exact, here and in the
+sequel K ∈ CA is chosen to follow some K0 with c(K0) > 0.)
+By (3.9) and (3.16), U ˆωK = Uω n.e. on S := S(ˆωK), hence γS-a.e., where γS
+denotes the capacitary measure on S (see [10, Theorem 2.5]). Since UγS ⩾ 1 holds
+true n.e. on S, hence ˆωK-a.e., we get, by Fubini’s theorem,
+ˆωK(Rn) =
+ˆ
+1 dˆωK ⩽
+ˆ
+UγS dˆωK =
+ˆ
+U ˆωK dγS
+=
+ˆ
+Uω dγS =
+ˆ
+UγS dω ⩽
+ˆ
+UγS dω+.
+As UγS ⩽ 1 on S(γS), applying the Frostman maximum principle if α ⩽ 2 (see [17,
+Theorem 1.10]), or [17, Theorem 1.5] otherwise, gives
+ˆωK(Rn) ⩽ Cn,αω+(Rn) =: L for all compact K ⊂ A,
+(3.21)
+where
+Cn,α :=
+�
+1
+if α ⩽ 2,
+2n−α
+otherwise.
+(3.22)
+We are thus led to the following conclusion, crucial to our proof.
+¶ We have
+ˆwf(K) = ˆwf,L(K) ∈ [−2MAL, 0] for all K ∈ CA,
+(3.23)
+MA ∈ [0, ∞) and L ∈ [0, ∞) being introduced by (3.5) and (3.21), respectively.
+The infimum ˆwf,L(K) is an actual minimum with the minimizer ˆωK.
+Step 4. To establish the existence of ˆωA for noncompact A, we first note that the
+net
+�
+ˆwf,L(K)
+�
+K∈CA decreases, and moreover, by (3.23),
+∞ < lim
+K↑A ˆwf,L(K) ⩽ 0.
+(3.24)
+For any compact K, K′ ⊂ A such that K ⊂ K′ and c(K) > 0,
+(ˆωK + ˆωK′)/2 ∈ E+
+L (K′),
+
+Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields
+11
+whence
+∥ˆωK + ˆωK′∥2 − 4
+ˆ
+Uω d(ˆωK + ˆωK′) ⩾ 4 ˆwf,L(K′) = 4If(ˆωK′).
+Applying the parallelogram identity to ˆωK, ˆωK′ ∈ E+ we therefore get
+∥ˆωK − ˆωK′∥2 ⩽ 2If(ˆωK) − 2If(ˆωK′).
+(3.25)
+Noting from (3.24) that the net
+�
+If(ˆωK)
+�
+K⩾K0 is Cauchy in R, we infer from (3.25)
+that the net (ˆωK)K⩾K0 is strong Cauchy in E+(A). The cone E+(A) being strongly
+closed (hence strongly complete) by (P1), there exists ζ ∈ E+(A) such that
+ˆωK → ζ strongly (hence vaguely) in E+(A) as K ↑ A.
+(3.26)
+Moreover, ζ ∈ E+
+L (A), the mapping µ �→ µ(Rn) being vaguely l.s.c. on M+.14
+We claim that this ζ serves as the inner pseudo-balayage ˆωA. As shown above
+(Step 1), this will follow once we verify (3.9) and (3.10) for ζ in place of ˆωA.
+To verify (3.9) (for ζ in place of ˆωA), it is enough to do this for any given compact
+K∗ ⊂ A. The strong topology on E+ being first-countable, in view of (3.26) there is
+a subsequence (ˆωKj)j∈N of the net (ˆωK)K∈CA such that Kj ⊃ K∗ for all j, and
+ˆωKj → ζ strongly (hence vaguely) in E+ as j → ∞.
+(3.27)
+Passing if necessary to a subsequence and changing the notations, we conclude from
+(3.27), by use of [10, p. 166, Remark], that
+Uζ = lim
+j→∞ U ˆωKj
+n.e. on Rn.
+(3.28)
+Applying now (3.9) to each ˆωKj, and then letting j → ∞, on account of the countable
+subadditivity of inner capacity on Borel sets [17, p. 144] we infer from (3.28) that
+(3.9) (with ζ in place of ˆωA) does indeed hold n.e. on K∗, whence n.e. on A.
+To establish (3.10) for ζ in place of ˆωA, we note from (3.16) applied to Kj that
+U ˆωKj ⩽ Uω on S(ˆωKj),
+(3.29)
+(Kj) being the sequence chosen above. Since (ˆωKj) converges to ζ vaguely, see (3.27),
+for every x ∈ S(ζ) there exist a subsequence (Kjk) of (Kj) and points xjk ∈ S(ˆωKjk)
+such that xjk, k ∈ N, approach x as k → ∞. Thus, by (3.29),
+U ˆω
+Kjk (xjk) ⩽ Uω(xjk) for all k ∈ N.
+Letting here k → ∞, in view of the upper semicontinuity of Uω on A and the lower
+semicontinuity of the mapping (x, µ) �→ Uµ(x) on Rn × M+, where M+ is equipped
+with the vague topology [10, Lemma 2.2.1(b)], we obtain (3.10) for ζ in place of ˆωA.
+This implies that
+ζ = ˆωA,
+(3.30)
+thereby completing the proof of the whole theorem.
+□
+14See [3, Section IV.1, Proposition 4] applied to the (positive, l.s.c.) function 1 on Rn. It is
+useful to point out that in the case where the set in question is compact, the mapping µ �→
+´
+g dµ
+remains vaguely l.s.c. on M+(K) for any l.s.c. function g (not necessarily positive). This follows
+by replacing g by g′ := g + c ⩾ 0, where c ∈ (0, ∞), a l.s.c. function on a compact set being lower
+bounded, and then by making use of the vague continuity of the mapping µ �→ µ(K) on M+(K).
+
+12
+Natalia Zorii
+Remark 3.6. Assume for a moment that ω is positive, and that A is quasiclosed
+and Borel. If moreover either ω ∈ E+, or Uω|A is bounded while c∗(A) < ∞,15 then
+Theorem 3.5 can be deduced from Fuglede’s result [12, Theorem 4.10] on the outer
+pseudo-balayage with respect to a perfect kernel on a locally compact (Hausdorff)
+space. The methods developed in the above proof are essentially different from those
+in [12], which enabled us to establish the existence of the inner Riesz pseudo-balayage
+ˆωA for pretty general ω and A, see (P1) and (P2), or (P1) and (P3). In this regard, it
+is also worth noting that the above proof seems to admit a generalization to suitable
+perfect kernels on locally compact spaces, which we plan to pursue in future work.
+3.2. Further properties of the inner pseudo-balayage. The following theorem
+justifies the term "inner" pseudo-balayage.
+Theorem 3.7. ˆωK → ˆωA strongly and vaguely in E+ as K ↑ A.
+Proof. In case (P3), this follows by substituting (3.30) into (3.26).
+It is thus left to consider case (P2). Then ω ∈ E, and therefore ˆωK, resp. ˆωA,
+is the orthogonal projection of ω onto the (convex, strongly complete) cone E+(K),
+resp. E+(A). A slight modification of the proof of (3.25) shows that
+∥ˆωK − ˆωK′∥2 ⩽ 2If(ˆωK) − 2If(ˆωK′) whenever K ⊂ K′
+(K, K′ ∈ CA).
+Noting that the net
+�
+ˆwf(K)
+�
+K∈CA is decreasing and, by (3.12), bounded:
+−∥ω∥2 ⩽ ˆwf(K) ⩽ 0 for all K ∈ CA,
+we conclude from the above that the net (ˆωK)K∈CA ⊂ E+(A) is strong Cauchy, and
+hence converges strongly and vaguely to some (unique) µ0 ∈ E+(A). This implies
+ˆwf(A) ⩽ If(µ0) = lim
+K↑A If(ˆωK) = lim
+K↑A ˆwf(K),
+the former equality being derived from the strong convergence of (ˆωK) to µ0 by use
+of (3.12). To verify that this µ0 actually equals ˆωA, it thus remains to show that
+lim
+K↑A ˆwf(K) ⩽ ˆwf(A).
+(3.31)
+But for every µ ∈ E+(A),
+If(µ) = lim
+K↑A If(µ|K) ⩾ lim
+K↑A ˆwf(K),
+where the equality follows by applying [10, Lemma 1.2.2] to each of the positive, l.s.c.,
+µ-integrable functions κα, Uω+, and Uω−, the set A being µ-measurable. Letting now
+µ range over E+(A) we get (3.31), thereby completing the proof of the theorem.
+□
+Corollary 3.8. If Uω is u.s.c. on A (which holds in particular in case (P3)), then
+ˆωA(Rn) ⩽ Cn,αω+(Rn),
+(3.32)
+Cn,α being introduced by (3.22).
+Proof. In view of the upper semicontinuity of Uω on A, the proof of (3.21), provided
+in case (P3), remains valid in case (P2) as well. Hence, in both cases (P2) and (P3),
+ˆωK(Rn) ⩽ Cn,αω+(Rn) for all K ∈ CA,
+which results in (3.32) since the net (ˆωK)K∈CA converges vaguely to ˆωA (Theorem 3.7)
+while the mapping µ �→ µ(Rn) is vaguely l.s.c. on M+ (cf. footnote 14).
+□
+15A quasiclosed set of finite outer capacity is actually quasicompact [12, Lemma 3.14].
+
+Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields
+13
+4. The comparison of the concepts of pseudo-balayage and balayage
+The aim of Examples 4.1–4.4 below is to demonstrate that usual nice properties
+of the inner balayage may fail to hold when dealing with the inner pseudo-balayage,
+and this occurs even in the simplest case of the Dirac measure and a sphere.
+¶ The first fact illustrating the difference between these two concepts is that the
+inner pseudo-balayage may increase the total mass of a positive measure (see e.g.
+(4.5), pertaining to α > 2). Recall that, if α ∈ (0, 2], then, by [26, Corollary 4.9],
+µA(Rn) ⩽ µ(Rn) for any µ ∈ M+ and A ⊂ Rn.
+(4.1)
+¶ The second one is that, for α > 2, there exist a set A ⊂ Rn which is not inner
+α-thin at infinity16 and a measure µ ∈ M+ such that
+ˆµA(Rn) > µ(Rn),
+see e.g. (4.5). In contrast to that, not being inner α-thin at infinity is necessary and
+sufficient for equality to prevail in (4.1) for all µ ∈ M+, see [27, Corollary 5.3].
+Example 4.1. Let D ⊂ Rn be a bounded (connected, open) domain, ω := εx0,
+where εx0 is the unit Dirac measure at x0 ∈ D, and let A be the inverse of D \ {x0}
+with respect to the sphere Sx0,1 := {|x − x0| = 1}. For these A and ω, (P1) and (P3)
+are fulfilled, and hence the pseudo-balayage ˆεA
+x0 exists and is unique (Theorem 3.5).
+Assume first that α ∈ (0, 2]. Since the balayage εA
+x0 of εx0 onto A is obviously of
+finite energy, Theorem 2.6 yields
+ˆεA
+x0 = εA
+x0.
+(4.2)
+Applying [17, Section IV.5.20] we therefore infer that the pseudo-balayage ˆεA
+x0 is
+actually the Kelvin transform γ∗
+D of the capacitary measure γD on D with respect to
+the sphere Sx0,1. Hence, in particular,
+S(ˆεA
+x0) =
+�
+∂A
+if α = 2,
+A
+if α < 2,
+(4.3)
+where ∂A := ∂RnA. The set A not being α-thin at infinity, we also get, in consequence
+of (4.2) and [13, Theorem 3.22],
+ˆεA
+x0(Rn) = εx0(Rn) = 1.
+(4.4)
+Let now α ∈ (2, n). We aim to show that then, in contrast to (4.4),17
+1 = εx0(Rn) < ˆεA
+x0(Rn) ⩽ 2n−α,
+(4.5)
+the latter inequality being valid by virtue of (3.32).
+As is known, the capacitary measure γD on D is the (unique) measure in E+(D)
+of (finite) total mass c(D), and such that (see [17, Sections II.1.3, II.3.13])
+UγD ⩾ 1 n.e. on D,
+(4.6)
+UγD = 1 n.e. on S(γD),
+(4.7)
+UγD > 1 on D,
+(4.8)
+16By [16, Definition 3.1], A ⊂ Rn is said to be inner α-thin at infinity if
+�
+j∈N
+c∗(Aj)
+qj(n−α) < ∞,
+where q ∈ (1, ∞) and Aj := A ∩ {x ∈ Rn :
+qj ⩽ |x| < qj+1}.
+For α ∈ (0, 2], see also [27,
+Definition 2.1], while for α = 2 and Borel A, see [7, pp. 175–176].
+17Compare with (6.11).
+
+14
+Natalia Zorii
+(4.8) being caused by the fact that for α ∈ (2, n), the α-Riesz potential of a positive
+measure is superharmonic on Rn [17, Theorem 1.4].
+For the Kelvin transform γ∗
+D of γD, applying [17, Eqs. (4.5.2)–(4.5.4)] yields
+Uγ∗
+D(x) = |x − x0|α−nUγD(x∗),
+I(γ∗
+D) = I(γD),
+γ∗
+D(Rn) = UγD(x0),
+(4.9)
+x∗ being the inverse of x with respect to Sx0,1. Combined with (4.6) and (4.7), this
+shows that γ∗
+D is a measure of the class E+(A) having the properties
+Uγ∗
+D ⩾ Uεx0 n.e. on A,
+Uγ∗
+D = Uεx0 n.e. on S(γ∗
+D),
+whence (3.9) and (3.10) with ω := εx0 and ˆωA := γ∗
+D. Thus, by virtue of Theorem 3.5,
+ˆεA
+x0 = γ∗
+D,
+which together with (4.8) and (4.9) establishes (4.5). Also note that
+S(ˆεA
+x0) ⊂ ∂A
+(compare with (4.3)).
+Example 4.2. A slight modification of arguments in Example 4.1 shows that if
+G ⊂ Rn is an open, relatively compact set, then for any α ∈ (2, n) and any x0 ∈ G,
+1 < ˆεGc
+x0 (Rn) ⩽ 2n−α.
+Example 4.3. Let α ∈ (2, n), A := Bc
+R := {|x| ⩾ R}, R ∈ (0, ∞), and let ω :=
+ε0, where ε0 denotes the unit Dirac measure at x = 0. As follows from Example 4.1,
+ˆε
+Bc
+R
+0
+= γ∗
+Br,
+where γBr is the capacitary measure on the ball Br, r := 1/R, and γ∗
+Br is the Kelvin
+transform of γBr with respect to the unit sphere S1. Therefore, by symmetry reasons
+applied to γBr, ˆε
+Bc
+R
+0
+is uniformly distributed over the sphere SR, and such that
+1 < ˆε
+Bc
+R
+0 (Rn) ⩽ 2n−α.
+Example 4.4. Let α ∈ (2, n), A := SR, R ∈ (0, ∞), and let ω := ε0. As shown
+in Example 4.3,
+S(ˆε
+Bc
+R
+0 ) = SR,
+which in view of Definition 3.1 gives
+ˆεSR
+0
+= ˆε
+Bc
+R
+0 .
+Thus ˆεSR
+0
+is uniformly distributed over the sphere SR, and such that
+1 < ˆεSR
+0 (Rn) ⩽ 2n−α.
+Also note that ˆεSR
+0
+is, in fact, the Kelvin transform of the capacitary measure γSr
+( = γBr), where r := 1/R, with respect to the unit sphere S1.
+
+Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields
+15
+5. The inner Gauss variational problem
+As before, consider a set A ⊂ Rn such that (3.1) and (P1) are fulfilled, a (signed)
+measure ω ∈ M satisfying either (P2) or (P3), and the external field f given by
+f := −Uω.
+According to Theorem 3.5, the problem of minimizing the Gauss functional If(µ),
+If(µ) := ∥µ∥2 + 2
+ˆ
+f dµ = ∥µ∥2 − 2
+ˆ
+Uω dµ,
+over the class E+
+f (A) of all µ ∈ E+(A) with finite If(µ) is uniquely solvable, and its
+solution ˆωA, called the inner pseudo-balayage of ω onto A, is uniquely characterized
+within E+
+f (A) by both (3.7) and (3.8) — or, equivalently, by both (3.9) and (3.10).
+The rest of the paper is to show that the concept of inner pseudo-balayage serves
+as a powerful tool in the inner Gauss variational problem, which reads as follows.
+Problem 5.1. Does there exist λA,f minimizing If(µ) within ˘E+(A)? Here,
+˘E+(A) :=
+�
+µ ∈ E+(A) : µ(Rn) = 1}.
+Recent results on Problem 5.1, which was originated by C.F. Gauss [14], are
+reviewed in the monographs [2, 20] (see also numerous references therein); for some
+of the latest researches on this topic, see [8, 32, 33].
+Remark 5.2. If A = K is compact while f is l.s.c. on K, then the existence of
+the minimizer λK,f can easily be verified, by use of the fact that the class ˘E+(K) is
+vaguely compact, cf. [3, Section III.1.9, Corollary 3], while the Gauss functional If(·)
+is vaguely l.s.c. on M+(K), the latter being obvious from the principle of descent and
+the vague lower semicontinuity of the mapping µ �→
+´
+f dµ on M+(K) (footnote 14).
+However, such a proof, based on the vague topology only, is no longer applicable if
+either A is noncompact, or f is not l.s.c.
+To investigate Problem 5.1 in the general case where A is noncompact and/or
+f is not l.s.c., we have recently developed an approach based on the systematic use
+of both the strong and vague topologies on the pre-Hilbert space E, which utilized
+essentially the perfectness of the Riesz kernels, see [33]. However, if c∗(A) = ∞, then
+the analysis performed in [33] was only limited to α ⩽ 2 and ω ⩾ 0, being mainly
+based on the theory of inner balayage for positive measures.
+Motivated by this observation, we generalize the approach, suggested in [33], to
+arbitrary α ∈ (0, n) and signed ω, by use of the theory of inner pseudo-balayage,
+developed in Sections 3, 4 above. The results thereby established are formulated in
+Section 6, and are proved in Sections 7–9. It is worth emphasizing that those results
+improve substantially many recent ones from [8, 33] (see Section 6.3 for some details).
+5.1. Preliminary results. To begin with, observe that under either of assumptions
+(P2) or (P3), the Gauss functional If(µ) is finite for all µ ∈ ˘E+(A). Thus
+˘E+(A) ⊂ E+
+f (A),
+(5.1)
+and therefore
+min
+µ∈E+
+f (A) If(µ) =: ˆwf(A) ⩽ wf(A) :=
+inf
+µ∈ ˘E+(A) If(µ).
+(5.2)
+Lemma 5.3. −∞ < wf(A) < ∞.
+
+16
+Natalia Zorii
+Proof. According to [21, Lemma 5], wf(A) < ∞ is equivalent to the inequality
+c∗
+�
+{x ∈ A : |f|(x) < ∞}
+�
+> 0,
+which indeed holds true by virtue of (3.1) and the fact that f is finite n.e. on Rn.
+(Here the strengthened version of countable subadditivity for inner capacity has been
+utilized, see Lemma 2.7.) Finally, combining (5.2) and (3.6) gives wf(A) > −∞.
+□
+The solution λA,f to Problem 5.1 is unique (if it exists), which can be proved by
+use of the convexity of the class ˘E+(A) and the parallelogram identity in the pre-Hil-
+bert space E. Such λA,f is said to be the inner f-weighted equilibrium measure.
+The following theorem, providing characteristic properties of λA,f, can be derived
+from the author’s earlier paper [21] (see Theorems 1, 2 and Proposition 1 therein).
+Theorem 5.4. For λ ∈ ˘E+(A) to be the (unique) solution λA,f to Problem 5.1,
+it is necessary and sufficient that either of the following two inequalities be fulfilled:
+Uλ
+f ⩾
+ˆ
+Uλ
+f dλ n.e. on A,
+(5.3)
+Uλ
+f ⩽ wf(A) −
+ˆ
+f dλ λ-a.e. on A,
+(5.4)
+where
+Uλ
+f := Uλ + f
+is said to be the f-weighted potential of λ. If (5.3) or (5.4) holds true, then actually
+ˆ
+Uλ
+f dλ = wf(A) −
+ˆ
+f dλ =: cA,f ∈ (−∞, ∞),
+(5.5)
+cA,f being referred to as the inner f-weighted equilibrium constant.18
+Remark 5.5. If f is l.s.c. on A (which occurs e.g. in case (P3)), then, by (5.4),
+U
+λA,f
+f
+⩽ cA,f on S(λA,f),
+which combined with (5.3) gives
+U
+λA,f
+f
+= cA,f n.e. on A ∩ S(λA,f).
+6. On the existence of λA,f and its properties
+In all that follows, except for Corollary 6.8, we assume A and f to satisfy the
+permanent requirements, reminded at the beginning of Section 5.
+6.1. On the solvability of Problem 5.1. Sufficient and/or necessary conditions
+for the existence of the (unique) solution λA,f to Problem 5.1 are established in
+Theorems 6.1, 6.3, 6.4, 6.7 and Corollaries 6.2, 6.5, 6.6, 6.8 below.
+Theorem 6.1. For λA,f to exist, it is sufficient that19
+c∗(A) < ∞.
+(6.1)
+Corollary 6.2. λA,f does exist whenever A is quasicompact.
+Proof. This is obvious since for quasicompact A, both (P1) and (6.1) hold true, by
+virtue of [33, Theorem 3.9] and [11, Definition 2.1], respectively.
+□
+18Similarly to [20, p. 27], cA,f might also be referred to as the inner modified Robin constant.
+19See Remark 8.2 below for an extension of Theorem 6.1 and Corollary 6.2.
+
+Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields
+17
+Theorem 6.3. For λA,f to exist, it is sufficient that
+ˆωA(Rn) = 1,
+(6.2)
+ˆωA being the inner pseudo-balayage of ω onto A. Furthermore, then
+λA,f = ˆωA,
+wf(A) = ˆwf(A),
+cA,f = 0,
+cA,f being the inner f-weighted equilibrium constant.
+• On account of Theorem 6.1, in the rest of this section we assume that
+c∗(A) = ∞.
+(6.3)
+• Unless (P2) holds, assume additionally that
+lim
+|x|→∞, x∈A Uω−(x) = 0.
+(6.4)
+Theorem 6.4. Problem 5.1 is unsolvable whenever
+ˆωA(Rn) < 1.
+(6.5)
+Corollary 6.5. If ω+ = 0, then Problem 5.1 is always unsolvable.
+Proof. Indeed, if ω = −ω−, then (6.5) is fulfilled, for ˆωA = 0 (see Remark 3.3).
+□
+Corollary 6.6. If Uω is u.s.c. on A, then Problem 5.1 is unsolvable whenever
+ω+(Rn) < 1/Cn,α,
+Cn,α being introduced by (3.22).
+Proof. This follows from Theorem 6.4 by use of (3.32).
+□
+Theorem 6.7. Unless (P2) holds, assume Uω is continuous20 on A and
+lim
+|x|→∞, x∈A Uω±(x) = 0.
+(6.6)
+Then
+λA,f exists ⇐⇒ ˆωA(Rn) ⩾ 1.
+If moreover
+ˆωA(Rn) > 1,
+(6.7)
+then21
+λA,f ̸= ˆωA,
+wf(A) ̸= ˆwf(A),
+cA,f ̸= 0.
+(6.8)
+Corollary 6.8. Dropping assumption (6.3) as well as all those imposed on ω,
+consider signed ω ∈ M, compactly supported in A
+c. Then λA,f exists if and only
+if either c∗(A) < ∞, or ˆωA(Rn) ⩾ 1.
+In particular, λA,f does not exist if both
+c∗(A) = ∞ and ω+(Rn) < 1/Cn,α are fulfilled, Cn,α being introduced by (3.22).
+Proof. This follows by combining Theorems 6.1, 6.7 and Corollary 6.6.
+□
+Remark 6.9. Thus, if ω ∈ M is compactly supported in A
+c while c∗(A) = ∞,
+then, according to Corollary 6.8, Problem 5.1 is unsolvable whenever ω+(Rn) is small
+enough, whereas ω−(Rn), the total amount of the negative charge, has no influence
+on this phenomenon. (Note that, when appealing to the electrostatic interpretation
+of the problem, the fact just observed agrees with our physical intuition.)
+20When speaking of a continuous function, we understand that the values are finite numbers.
+21Compare with Theorem 6.3 as well as with Remark 8.3.
+
+18
+Natalia Zorii
+6.2. On the description of the support S(λA,f). The following Theorem 6.10
+establishes sufficient conditions for the minimizer λA,f to be of compact support,
+whereas Examples 6.11 and 6.12 analyze their sharpness.
+Theorem 6.10. Under the hypotheses of Theorem 6.7, assume moreover that
+A is not inner α-thin at infinity. Then S(λA,f) is compact whenever (6.7) is fulfilled.
+Examples 6.11 and 6.12 provide explicit formulae for S(λA,f) for some specific
+A and f. The latter equality in (6.9) as well as both equalities in (6.13) show that
+Theorem 6.10 would fail in general if ˆωA(Rn) > 1 were replaced by ˆωA(Rn) = 1.
+Example 6.11. For A := Bc
+R := {|x| ⩾ R}, R ∈ (0, ∞), and for the unit Dirac
+measure ε0 at x = 0, define
+ω := ε0/q,
+where q := ˆε
+Bc
+R
+0 (Rn).
+As shown in Example 4.1, q = 1 if α ⩽ 2, and q > 1 otherwise. By (3.4),
+ˆωBc
+R = ˆε
+Bc
+R
+0 /q,
+whence
+ˆωBc
+R(Rn) = ˆε
+Bc
+R
+0 (Rn)/q = 1.
+Therefore, by virtue of Theorem 6.3, the solution λBc
+R,f to Problem 5.1 with f := −Uω
+does exist, and moreover
+λBc
+R,f = ˆωBc
+R = ˆε
+Bc
+R
+0 /q.
+In view of what was pointed out in Examples 4.1 and 4.3, this gives
+S(λBc
+R,f) =
+�
+SR
+if α ⩾ 2,
+Bc
+R
+otherwise.
+(6.9)
+Example 6.12. Let Bz,r be the closed ball of radius r centered at the point
+z := (0, . . . , 0, −r), and let A0 denote the inverse of Bz,r \ {0} with respect to the
+unit sphere S1 centered at the origin 0. Then A0 is a closed subset of Rn, not α-thin
+at infinity, such that 0 ̸∈ A0, and whose boundary ∂A0 is unbounded.
+In a manner similar to that in Example 4.1, we see that
+ˆεA0
+0
+= γ∗
+Bz,r,
+(6.10)
+γ∗
+Bz,r being the Kelvin transform of γBz,r, the capacitary measure on Bz,r, with respect
+to S1. Noting that γ∗
+Bz,r(Rn) = UγBz,r(0) (cf. (4.9)), whereas UγBz,r(0) = 1, we obtain
+ˆεA0
+0 (Rn) = 1 for any α ∈ (0, n)
+(6.11)
+(compare with (4.5)). Applying Theorem 6.3 we therefore infer that the solution
+λA0,f to Problem 5.1 with A := A0 and f := −Uε0 does exist, and moreover
+λA0,f = ˆεA0
+0 .
+(6.12)
+By virtue of the description of S(γBz,r) [17, Section II.3.13], (6.10) and (6.12) yield
+S(λA0,f) =
+�
+∂A0
+if α ⩾ 2,
+A0
+otherwise.
+(6.13)
+
+Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields
+19
+6.3. Remark. The results thereby obtained improve substantially many recent ones
+from [8, 33], by strengthening their formulations and/or by extending the areas of
+their applications. For instance, [8, Corollary 2.6] only deals with closed sets A that
+are not thin at infinity, and with external fields f of the form −Uω, where ω := cεx0,
+c ∈ (0, ∞), εx0 being the unit Dirac measure at x0 ̸∈ A. However, even for these
+very particular A and ω, all the assertions in [8, Corollary 2.6] are in general weaker
+than the relevant ones, established above. This is caused, in particular, by the fact
+that those assertions from [8] are given in terms of ω(Rn), whereas ours — in terms
+of ˆωA(Rn) (see Section 4 for the relations between these two values).
+Regarding the advantages of our current approach in comparison with that sug-
+gested in [33], see Remark 5.2 above.
+7. Proof of Theorem 6.3
+Due to condition (6.2), the inner pseudo-balayage ˆωA, minimizing If(µ) over the
+class E+
+f (A), actually belongs to its proper subclass ˘E+(A), see (5.1). Therefore,
+wf(A) ⩽ If(ˆωA) = ˆwf(A),
+which combined with (5.2) gives
+If(ˆωA) = wf(A) = ˆwf(A).
+Thus ˆωA serves as the (unique) solution to Problem 5.1, i.e. ˆωA = λA,f. Substituting
+this equality into (3.8), we get
+ˆ
+U
+λA,f
+f
+dλA,f =
+ˆ �
+U ˆωA − Uω�
+dˆωA = 0,
+which according to Theorem 5.4 establishes the remaining relation cA,f = 0.
+8. Proofs of Theorems 6.1 and 6.4
+8.1. Extremal measures. Let Mf(A) stand for the (nonempty) set of all nets
+(µs)s∈S ⊂ ˘E+(A) having the property
+lim
+s∈S If(µs) = wf(A);
+(8.1)
+those nets (µs)s∈S are said to be minimizing (in Problem 5.1). Using the finiteness
+of wf(A) (Lemma 5.3), the convexity of ˘E+(A), and the perfectness of the α-Riesz
+kernel, one can see with the aid of arguments similar to those in [33, Lemma 4.2]
+that there exists the unique ξA,f ∈ E+ such that, for every (µs)s∈S ∈ Mf(A),
+µs → ξA,f strongly and vaguely in E+ (as s ranges through S).
+(8.2)
+This ξ := ξA,f will be referred to as the extremal measure (in Problem 5.1).
+Due to (8.2), we have
+ξA,f ∈ E+(A),
+(8.3)
+E+(A) being strongly closed by (P1), and moreover
+ξA,f(Rn) ⩽ 1,
+(8.4)
+the mapping µ �→ µ(Rn) being vaguely l.s.c. on M+ [3, Section IV.1, Proposition 4].
+The following simple observation is crucial to the proofs given below.
+
+20
+Natalia Zorii
+Lemma 8.1. Problem 5.1 is solvable if and only if
+ξA,f(Rn) = 1 and If(ξA,f) = wf(A),
+(8.5)
+and in the affirmative case
+λA,f = ξA,f.
+(8.6)
+Proof. The "if" part is evident by (8.3), whereas the opposite is implied by the fact
+that the trivial net (λA,f) is obviously minimizing, and hence converges strongly to
+both λA,f and ξA,f. Since the strong topology on E is Hausdorff, (8.6) follows.
+□
+8.2. Proof of Theorem 6.1. Fix a minimizing sequence (µj) ∈ Mf(A); by (8.2),
+µj → ξ strongly and vaguely in E+ (as j → ∞),
+ξ := ξA,f being the (unique) extremal measure in Problem 5.1, whence
+sup
+j∈N
+∥µj∥ < ∞.
+(8.7)
+To show that this ξ serves as the solution to Problem 5.1, it is enough to verify (8.5).
+Since µj → ξ vaguely, applying [3, Section IV.1, Proposition 4] gives
+ξ(Rn) ⩽ lim inf
+j→∞
+µj(Rn) = 1,
+(8.8)
+whereas [3, Section IV.4.4, Corollary 3] yields
+ˆ
+1K dξ ⩾ lim sup
+j→∞
+ˆ
+1K dµj for every compact K ⊂ Rn,
+(8.9)
+the indicator function 1K of K being bounded, of compact support, and u.s.c. on Rn.
+Combining (8.8) and (8.9) with
+ξ(Rn) = lim
+K↑Rn
+ˆ
+1K dξ,
+we get
+1 ⩾ ξ(Rn) ⩾ lim sup
+(j,K)∈N×C
+ˆ
+1K dµj = 1 − lim inf
+(j,K)∈N×C
+ˆ
+1A\K dµj,
+N × C being the directed product of the directed sets N and C := CRn [15, p. 68].
+The former relation in (8.5) will therefore follow once we establish the equality
+lim inf
+(j,K)∈N×C
+ˆ
+1A\K dµj = 0.
+(8.10)
+By [17, Theorem 2.6] applied to A \ K, K ∈ C being arbitrarily chosen, there
+exists the (unique) inner capacitary measure γA\K, minimizing the energy ∥µ∥2 over
+the (convex) set ΓA\K consisting of all µ ∈ E+ with
+Uµ ⩾ 1 n.e. on A \ K.
+For any K′ ∈ C such that K ⊂ K′, we have ΓA\K ⊂ ΓA\K′, and [17, Lemma 2.2]
+therefore gives
+∥γA\K − γA\K′∥2 ⩽ ∥γA\K∥2 − ∥γA\K′∥2.
+(8.11)
+
+Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields
+21
+Since ∥γA\K∥2 = c∗(A\K) [17, Theorem 2.6], ∥γA\K∥2 decreases as K ranges through
+C, which together with (8.11) implies that the net (γA\K)K∈C ⊂ E+ is Cauchy in the
+strong topology on E+. Noting that (γA\K)K∈C converges vaguely to zero,22 we get
+γA\K → 0 strongly in E+ as K ↑ Rn,
+(8.12)
+the α-Riesz kernel being perfect.
+It follows from the above that
+UγA\K ⩾ 1A\K n.e. on A \ K,
+(8.13)
+and, therefore, µj-a.e. for all j ∈ N. Integrating (8.13) with respect to µj we obtain,
+by the Cauchy–Schwarz (Bunyakovski) inequality,
+ˆ
+1A\K dµj ⩽
+ˆ
+UγA\K dµj ⩽ ∥γA\K∥ · ∥µj∥ for all K ∈ C and j ∈ N.
+Combined with (8.7) and (8.12), this gives (8.10), hence ξ ∈ ˘E+(A), and consequently
+wf(A) ⩽ If(ξ).
+(8.14)
+To complete the proof of the theorem, it remains to verify the latter relation in
+(8.5), which in view of (8.1) and (8.14) is reduced to the inequality
+If(ξ) ⩽ lim
+j→∞ If(µj).
+(8.15)
+In case (P2), (8.15) follows at once from the strong convergence of (µj) to ξ, by
+applying (3.12) to each of µj and ξ. Otherwise, case (P3) holds, and hence f = −Uω
+is l.s.c. and bounded on A. Thus there is c ∈ (0, ∞) such that f ′ := f + c ⩾ 0 on A,
+and [3, Section IV.1, Proposition 4] applied to f ′ gives
+ˆ
+f dξ + c =
+ˆ
+f ′ dξ ⩽ lim inf
+j→∞
+ˆ
+f ′ dµj = lim inf
+j→∞
+ˆ
+f dµj + c,
+the first and last equalities being valid by virtue of ξ(Rn) = µj(Rn) = 1. Therefore,
+ˆ
+f dξ ⩽ lim
+j→∞
+ˆ
+f dµj.
+Multiplied by 2, and then added to
+lim
+j→∞ ∥µj∥2 = ∥ξ∥2,
+this results in (8.15), thereby completing the whole proof.
+Remark 8.2. A slight generalization of the above proof shows that Theorem 6.1
+and, hence, Corollary 6.2 remain valid for an external field f represented as the sum
+f := u + Uϑ,
+where ϑ ∈ E, while u : A → (−∞, ∞] is l.s.c., bounded from below, and such that
+c∗({x ∈ A : u(x) < ∞}) > 0.
+22Indeed, for any given ϕ ∈ C0(Rn), there exists an open, relatively compact set G ⊂ Rn such
+that ϕ(x) = 0 for all x ̸∈ G. Hence, γA\K(ϕ) = 0 for all K ∈ C with K ⊃ G, and the claim follows.
+
+22
+Natalia Zorii
+8.3. Proof of Theorem 6.4. As c∗(A) = ∞ by (6.3), there are mutually noninter-
+secting, compact sets Kj ⊂ A, j ∈ N, such that |x| ⩾ j for all x ∈ Kj and c(Kj) ⩾ j.
+If λj := γKj/c(Kj) ∈ ˘E+(Kj) denotes the normalized capacitary measure on Kj, then
+∥λj∥ → 0 as j → ∞,
+(8.16)
+λj → 0 vaguely in E+ as j → ∞,
+(8.17)
+the latter being implied by the fact that for any compact subset K of Rn, we have
+K ∩ S(λj) = ∅ for all j large enough. Define
+µj := ˆωA + cjλj,
+where cj := 1 − ˆωA(Rn).
+(8.18)
+Noting from (6.5) that
+0 < cj ⩽ 1 for all j,
+(8.19)
+we get µj ∈ ˘E+(A) for all j, whence
+wf(A) ⩽ lim inf
+j→∞
+If(µj).
+(8.20)
+On the other hand, If(ˆωA) = ˆwf(A) (see Definition 3.1 and Theorem 3.5). By
+means of a straightforward verification, we derive from (8.16), (8.18), and (8.19) that
+lim sup
+j→∞
+If(µj) ⩽ lim sup
+j→∞
+�
+If(ˆωA) + 2cj
+ˆ
+Uω− dλj
+�
+⩽ ˆwf(A) + 2L∞,
+(8.21)
+where
+L∞ := lim sup
+j→∞
+ˆ
+Uω− dλj.
+Let first (P2) take place. Applying the Cauchy–Schwarz inequality to the mea-
+sures ω−, λj ∈ E+, and then letting j → ∞, we infer from (8.16) that
+L∞ = 0.
+(8.22)
+Otherwise, (P3) and hence (6.4) must be fulfilled, which again results in (8.22), λj
+being the unit measure supported by A ∩ {|x| ⩾ j}. Substituting (8.22) into (8.21),
+and then combining the inequality thus obtained with (8.20) and (5.2), we get
+lim
+j→∞ If(µj) = ˆwf(A) = wf(A),
+(8.23)
+which shows that the sequence (µj) is, in fact, minimizing in Problem 5.1, and hence
+converges both strongly and vaguely to the extremal measure ξA,f:
+µj → ξA,f strongly and vaguely in E+ as j → ∞.
+On account of (8.17)–(8.19), this gives
+ˆωA = ξA,f,
+the vague topology on M being Hausdorff. Therefore, by (6.5),
+ξA,f(Rn) = ˆωA(Rn) < 1,
+and an application of Lemma 8.1 shows that Problem 5.1 is indeed unsolvable.
+Remark 8.3. As shown in (8.23), under the assumptions of Theorem 6.4, equal-
+ity prevails in (5.2), i.e.
+ˆwf(A) = wf(A)
+(compare with Theorems 6.3 and 6.7).
+
+Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields
+23
+9. Proofs of Theorems 6.7 and 6.10
+9.1. Auxiliary results. According to Corollary 6.2, for every K ∈ CA such that
+K ⩾ K0, where c(K0) > 0, there is the (unique) solution λK,f to Problem 5.1 with
+A := K, whereas by virtue of Lemma 9.1 below, those λK,f form a minimizing net:
+(λK,f)K⩾K0 ∈ Mf(A).
+(9.1)
+Lemma 9.1. wf(K) ↓ wf(A) as K ↑ A.
+Proof. For any µ ∈ ˘E+(A), µ(K) ↑ 1 as K ↑ A. Applying [10, Lemma 1.2.2] to each
+of the (positive, l.s.c., µ-integrable) functions κα, Uω+, and Uω−, we therefore get
+If(µ) = lim
+K↑A If(µ|K) = lim
+K↑A If(νK) ⩾ lim
+K↑A wf(K),
+where
+νK := µ|K/µ(K) ∈ ˘E+(K),
+K ∈ CA being large enough. Letting now µ range over ˘E+(A) gives
+wf(A) ⩾ lim
+K↑A wf(K),
+whence the lemma, the opposite being obvious by the monotonicity.
+□
+• Unless case (P2) takes place, assume in what follows that the external field
+f = −Uω is continuous on A, and that (6.6) is fulfilled.
+Lemma 9.2. For the extremal measure ξA,f, we have
+If(ξA,f) = wf(A).
+(9.2)
+Proof. By virtue of (8.2) and (9.1),
+λK,f → ξA,f strongly and vaguely in E+ as K ↑ A,
+hence
+lim
+K↑A ∥λK,f∥2 = ∥ξA,f∥2.
+(9.3)
+If case (P2) takes place, then the strong convergence of (λK,f)K⩾K0 to ξA,f yields, by
+applying (3.12) to each of λK,f and ξA,f,
+lim
+K↑A If(λK,f) = If(ξA,f),
+(9.4)
+whence, by Lemma 9.1,
+If(ξA,f) = lim
+K↑A wf(K) = wf(A).
+In the remaining case (P3), for any t > 0 choose r so that
+|f| < t/2 on A ∩ B
+c
+r,
+which is possible in view of (6.6). On account of (8.4),
+����
+ˆ
+B
+c
+r
+f d(λK,f − ξA,f)
+���� < t for all K ⩾ K0.
+(9.5)
+The above r can certainly be chosen so that
+ξA,f(Sr) = 0,
+the measure ξA,f being bounded. Then, according to [17, Theorem 0.5′],
+λK,f|Br → ξA,f|Br vaguely in M+ as K ↑ A,
+
+24
+Natalia Zorii
+whence, by the continuity of f on A,
+lim
+K↑A
+ˆ
+f dλK,f|Br =
+ˆ
+f dξA,f|Br,
+which combined with (9.5), taken for t > 0 arbitrarily small, results in23
+lim
+K↑A
+ˆ
+f dλK,f =
+ˆ
+f dξA,f.
+(9.6)
+Multiplied by 2, and then added to (9.3), this yields (9.4), whence (9.2), again by
+making use of Lemma 9.1.
+□
+Corollary 9.3. λA,f exists if and only if ξA,f(Rn) = 1, and in the affirmative
+case λA,f = ξA,f.
+Proof. This follows by combining Lemmas 8.1 and 9.2.
+□
+Lemma 9.4. For the extremal measure ξ = ξA,f, we have
+Uξ
+f ⩾ Cξ n.e. on A,
+(9.7)
+Uξ
+f ⩽ Cξ on S(ξ),
+(9.8)
+where
+Cξ :=
+ˆ
+Uξ
+f dξ ∈ (−∞, ∞).
+(9.9)
+Proof. In case (P3), the finiteness of
+´
+Uξ
+f dξ follows from the boundedness of Uω± on
+A, the extremal measure ξ being bounded by (8.4); while in case (P2), it is obvious.
+By Theorem 5.4 applied to each K ∈ CA large enough,
+U
+λK,f
+f
+⩾ cK,f n.e. on K,
+(9.10)
+U
+λK,f
+f
+⩽ cK,f on S(λK,f),
+(9.11)
+where
+cK,f =
+ˆ
+U
+λK,f
+f
+dλK,f.
+Furthermore, by combining (9.3) with (9.6),
+lim
+K↑A cK,f = Cξ,
+(9.12)
+Cξ being the (finite) constant appearing in (9.9).
+Fix K∗ ∈ CA. The strong topology on E+ being first-countable, one can choose
+a subsequence (λKj,f)j∈N of the net (λK,f)K∈CA such that
+λKj,f → ξ strongly (hence vaguely) in E+ as j → ∞.
+(9.13)
+There is certainly no loss of generality in assuming that
+K∗ ⊂ Kj for all j,
+for if not, we replace Kj by K′
+j := Kj ∪ K∗; then, by the monotonicity of
+�
+wf(K)
+�
+,
+the sequence (λK′
+j,f)j∈N remains minimizing, and hence also converges strongly to ξ.
+Due to the arbitrary choice of K∗ ∈ CA, (9.7) will follow once we show that
+Uξ
+f ⩾ Cξ n.e. on K∗.
+(9.14)
+23In case (P2), (9.6) holds true as well, which is obtained by subtracting (9.3) from (9.4). Alter-
+natively, it can be derived from the strong convergence of the net (λK,f)K⩾K0 to ξA,f, by applying
+the Cauchy–Schwarz inequality to the measures ω, λK,f − ξA,f ∈ E.
+
+Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields
+25
+Passing if necessary to a subsequence and changing the notations, we conclude from
+(9.13), by virtue of [10, p. 166, Remark], that
+Uξ = lim
+j→∞ UλKj ,f n.e. on Rn.
+(9.15)
+Applying now (9.10) to each Kj, and then letting j → ∞, on account of (9.12) and
+(9.15) we arrive at (9.14). (Here the countable subadditivity of inner capacity on
+Borel sets has been utilized.)
+Since (λKj,f) converges to ξ vaguely, see (9.13), for every x ∈ S(ξ) there exist a
+subsequence (Kjk) of the sequence (Kj) and points xjk ∈ S(λKjk,f), k ∈ N, such that
+xjk approach x as k → ∞. Thus, according to (9.11),
+U
+λKjk ,f
+f
+(xjk) ⩽
+ˆ
+U
+λKjk ,f
+f
+dλKjk,f for all k ∈ N.
+Letting here k → ∞, and applying (9.12), the continuity of f on A, and the lower
+semicontinuity of the mapping (x, µ) �→ Uµ(x) on Rn×M+, M+ being equipped with
+the vague topology [10, Lemma 2.2.1(b)], we get the remaining inequality (9.8).
+□
+9.2. Proof of Theorem 6.7. We first remark from Theorems 6.3 and 6.4 that it is
+actually enough to consider the case when (6.7) is fulfilled.
+For the extremal measure ξ = ξA,f, we infer from (9.7) and (9.8) that
+Uξ
+f = Cξ n.e. on S(ξ) ∩ A,
+whence
+Uξ
+f = Cξ ξ-a.e.,
+(9.16)
+the measure ξ being of the class E+(A). We claim that then necessarily
+Cξ ̸= 0.
+(9.17)
+Indeed, if this were not true, then (9.7) and (9.16) would imply, by virtue of Theo-
+rem 3.5, that ξ = ˆωA, which however contradicts (6.7), for ξ(Rn) ⩽ 1 by (8.4).
+Integrating (9.16) with respect to ξ we obtain
+ˆ
+Uξ
+f dξ = Cξ · ξ(Rn),
+whence, by (9.9) and (9.17),
+ξ(Rn) = 1.
+Applying Corollary 9.3 we see that under assumption (6.7), λA,f does indeed exist,
+and moreover λA,f = ξ.
+The equality λA,f = ξ implies, by use of (5.5), (9.9), and (9.17), that
+cA,f =
+ˆ
+U
+λA,f
+f
+dλA,f =
+ˆ
+Uξ
+f dξ = Cξ ̸= 0,
+which proves the third relation in (6.8). The first is obvious since, by (6.7),
+λA,f(Rn) = 1 < ˆωA(Rn).
+Finally, the first relation implies the second, for if not, then wf(A) = ˆwf(A), whence
+λA,f = ˆωA, by the uniqueness of ˆωA and the inclusion ˘E+(A) ⊂ E+
+f (A), cf. (5.1).
+
+26
+Natalia Zorii
+9.3. Proof of Theorem 6.10. According to Theorem 6.7, under the stated assump-
+tions there exists the solution λA,f to Problem 5.1, and moreover, by (6.8),
+cA,f ̸= 0.
+(9.18)
+Assume to the contrary that S(λA,f) is noncompact. As seen from Remark 5.5, then
+there exists a sequence (xj) ⊂ A such that |xj| → ∞ as j → ∞, and
+U
+λA,f
+f
+(xj) = cA,f for all j ∈ N.
+On account of (6.6), this yields
+lim inf
+j→∞
+UλA,f(xj) = cA,f,
+whence cA,f ⩾ 0, which in view of (9.18) shows that, actually,
+cA,f > 0.
+(9.19)
+But, by (5.3), U
+λA,f
+f
+⩾ cA,f n.e. on A, which together with (6.6) and (9.19) gives
+lim inf
+|x|→∞, x∈A UλA,f (x) > 0.
+However, this is impossible in consequence of [16, Remark 4.12(i)], the set A not
+being inner α-thin at infinity.
+References
+[1] Benko, D., Dragnev, P.D., Orive, R.: On point-mass Riesz external fields on the real axis. J.
+Math. Anal. Appl. 491, 124299 (2020)
+[2] Borodachov, S.V., Hardin, D.P., Saff, E.B.: Discrete Energy on Rectifiable Sets. Springer,
+Berlin (2019)
+[3] Bourbaki, N.: Integration. Chapters 1–6. Springer, Berlin (2004)
+[4] Cartan, H.: Théorie du potentiel newtonien: énergie, capacité, suites de potentiels. Bull. Soc.
+Math. France 73, 74–106 (1945)
+[5] Cartan, H.: Théorie générale du balayage en potentiel newtonien. Ann. Univ. Fourier Grenoble
+22, 221–280 (1946)
+[6] Deny, J.: Les potentiels d’énergie finie. Acta Math. 82, 107–183 (1950)
+[7] Doob, J.L.: Classical Potential Theory and Its Probabilistic Counterpart. Springer, Berlin
+(1984)
+[8] Dragnev, P.D., Orive, R., Saff, E.B., Wielonsky F.: Riesz energy problems with external fields
+and related theory. Constr. Approx. (2022). https://doi.org/10.1007/s00365-022-09588-z
+[9] Edwards, R.E.: Functional Analysis. Theory and Applications. Holt, Rinehart and Winston,
+New York (1965)
+[10] Fuglede, B.: On the theory of potentials in locally compact spaces. Acta Math. 103, 139–215
+(1960)
+[11] Fuglede, B.: The quasi topology associated with a countably subadditive set function. Ann.
+Inst. Fourier Grenoble 21, 123–169 (1971)
+[12] Fuglede, B.: Symmetric function kernels and sweeping of measures. Anal. Math. 42, 225–259
+(2016)
+[13] Fuglede, B., Zorii, N.: Green kernels associated with Riesz kernels. Ann. Acad. Sci. Fenn. Math.
+43, 121–145 (2018)
+[14] Gauss, C.F.:
+Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnisse des
+Quadrats der Entfernung wirkenden Anziehungs– und Abstoßungs–Kräfte (1839). Werke 5,
+197–244 (1867)
+
+Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields
+27
+[15] Kelley, J.L.: General Topology. Princeton, New York (1957)
+[16] Kurokawa, T., Mizuta, Y.: On the order at infinity of Riesz potentials. Hiroshima Math. J. 9,
+533–545 (1979)
+[17] Landkof, N.S.: Foundations of Modern Potential Theory. Springer, Berlin (1972)
+[18] Ohtsuka, M.: On potentials in locally compact spaces. J. Sci. Hiroshima Univ. Ser. A-I 25,
+135–352 (1961)
+[19] Riesz, M.: Intégrales de Riemann–Liouville et potentiels. Acta Szeged 9, 1–42 (1938)
+[20] Saff, E.B., Totik, V: Logarithmic Potentials with External Fields. Springer, Berlin (1997)
+[21] Zorii, N.V.: Equilibrium potentials with external fields. Ukrainian Math. J. 55, 1423–1444
+(2003)
+[22] Zorii, N.V.: Equilibrium problems for potentials with external fields. Ukrainian Math. J. 55,
+1588–1618 (2003)
+[23] Zorii, N.: Constrained energy problems with external fields for vector measures. Math. Nachr.
+285, 1144–1165 (2012)
+[24] Zorii, N.: Equilibrium problems for infinite dimensional vector potentials with external fields.
+Potential Anal. 38, 397–432 (2013)
+[25] Zorii, N.: Necessary and sufficient conditions for the solvability of the Gauss variational problem
+for infinite dimensional vector measures. Potential Anal. 41, 81–115 (2014)
+[26] Zorii, N.: A theory of inner Riesz balayage and its applications. Bull. Pol. Acad. Sci. Math.
+68, 41–67 (2020)
+[27] Zorii, N.: Harmonic measure, equilibrium measure, and thinness at infinity in the theory of
+Riesz potentials. Potential Anal. 57, 447–472 (2022)
+[28] Zorii, N.: Balayage of measures on a locally compact space. Anal. Math. 48, 249–277 (2022)
+[29] Zorii, N.: On the theory of capacities on locally compact spaces and its interaction with the
+theory of balayage. Potential Anal. (2022). https://doi.org/10.1007/s11118-022-10010-3
+[30] Zorii, N.:
+On the theory of balayage on locally compact spaces. Potential Anal. (2022).
+https://doi.org/10.1007/s11118-022-10024-x
+[31] Zorii, N.: On the role of the point at infinity in Deny’s principle of positivity of mass for Riesz
+potentials. arXiv:2202.12418 (2022)
+[32] Zorii, N.:
+Minimum energy problems with external fields on locally compact spaces.
+arXiv:2207.14342 (2022)
+[33] Zorii, N.: Minimum Riesz energy problems with external fields. arXiv:2209.05891 (2022)
+Institute of Mathematics, Academy of Sciences of Ukraine, Tereshchenkivska 3,
+01601, Kyiv-4, Ukraine, natalia.zorii@gmail.com
+
diff --git a/WtAyT4oBgHgl3EQfh_ga/content/tmp_files/load_file.txt b/WtAyT4oBgHgl3EQfh_ga/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,1528 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf,len=1527
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='00385v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='CA]  1 Jan 2023  Inner Riesz pseudo-balayage and its applications to minimum  energy problems with external fields  Natalia Zorii  Dedicated to Professor Stephen J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Gardiner on the occasion of his 65th birthday  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' For the Riesz kernel κα(x, y) := |x − y|α−n of order 0 < α < n on Rn, n ⩾ 2, we  introduce the so-called inner pseudo-balayage ˆωA of a (Radon) measure ω on Rn to a set A ⊂ Rn  as the (unique) measure minimizing the Gauss functional  ˆ  κα(x, y) d(µ ⊗ µ)(x, y) − 2  ˆ  κα(x, y) d(ω ⊗ µ)(x, y)  over the class E+(A) of all positive measures µ of finite energy, concentrated on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  For quite  general signed ω (not necessarily of finite energy) and A (not necessarily closed), such ˆωA does  exist, and it maintains the basic features of inner balayage for positive measures (defined when  α ⩽ 2), except for those implied by the domination principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' (To illustrate the latter, we point out  that, in contrast to what occurs for the balayage, the inner pseudo-balayage of a positive measure  may increase its total mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=') The inner pseudo-balayage ˆωA is further shown to be a powerful tool in  the problem of minimizing the Gauss functional over all µ ∈ E+(A) with µ(Rn) = 1, which enables  us to improve substantially many recent results on this topic, by strengthening their formulations  and/or by extending the areas of their applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' For instance, if A is a quasiclosed set of nonzero  inner capacity c∗(A), and if ω is a signed measure, compactly supported in Rn \\ ClRnA, then the  problem in question is solvable if and only if either c∗(A) < ∞, or ˆωA(Rn) ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' In particular,  if c∗(A) = ∞, then the problem has no solution whenever ω+(Rn) < 1/Cn,α, where Cn,α := 1 if  α ⩽ 2, and Cn,α := 2n−α otherwise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' whereas ω−(Rn), the total amount of the negative charge, has  no influence on this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' The results obtained are illustrated by some examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Inner pseudo-balayage: a motivation and a model case  This paper deals with the theory of potentials with respect to the α-Riesz kernels  κα(x, y) := |x − y|α−n of order 0 < α < n on Rn, n ⩾ 2, |x − y| being the Euclidean  distance in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Our main goal is to proceed further with the study of minimum  α-Riesz energy problems in the presence of external fields f, a point of interest for  many researchers (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' the monographs [2, 20] and references therein, [14, 18],  [21]–[25], as well as [1, 8, 32, 33], some of the most recent papers on this topic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  In the current work we improve substantially many recent results in this field, by  strengthening their formulations and/or by extending the areas of their applications  (see Section 6 for the results obtained).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' This has become possible due to the devel-  opment of a new tool, the inner pseudo-balayage (see Section 3, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' also the present  section for a motivation of the proposed definition as well as for a model case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  It is well known that the α-Riesz balayage (sweeping out) serves as an efficient  tool in the problems in question (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' [8, 23, 25, 33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' However, its application is  only limited to the case of α ranging over (0, 2], and to external fields f of the form  f(x) := −Uω(x) := −  ˆ  κα(x, y) dω(y),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1)  2010 Mathematics Subject Classification: Primary 31C15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Key words: Minimum Riesz energy problems with external fields, inner Riesz balayage, inner  Riesz pseudo-balayage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  2  Natalia Zorii  where ω is a suitable positive Radon measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  To extend the area of application of such a tool to arbitrary α ∈ (0, n) and/or to  external fields f given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1), but now with signed ω involved, we generalize the  standard concept of inner balayage of positive measures (defined for α ∈ (0, 2]) to the  so-called inner pseudo-balayage of signed measures, by maintaining the basic features  of the former concept — except for those implied by the domination principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Being crucial to our study of minimum energy problems with external fields, the  concept of inner pseudo-balayage is also of independent interest, looking promising  for further generalizations and other applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Before introducing it, we first  review some basic facts of the theory of α-Riesz potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  We denote by M the linear space of all (real-valued Radon) measures µ on Rn,  equipped with the vague topology of pointwise convergence on the class C0(Rn) of  all continuous functions ϕ : Rn → R of compact support, and by M+ the cone of all  positive µ ∈ M, where µ is positive if and only if µ(ϕ) ⩾ 0 for all positive ϕ ∈ C0(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Given µ, ν ∈ M, the potential Uµ and the mutual energy I(µ, ν) are introduced by  Uµ(x) :=  ˆ  κα(x, y) dµ(y),  x ∈ Rn,  I(µ, ν) :=  ˆ  κα(x, y) d(µ ⊗ ν)(x, y),  respectively, provided that the integral on the right is well defined (as a finite number  or ±∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' For µ = ν, I(µ, ν) defines the energy I(µ) := I(µ, µ) of µ ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  The following property of strict positive definiteness for the α-Riesz kernels, dis-  covered by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Riesz [19, Chapter I, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' (13)] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' also [17, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='15]), is crucial  to the current study: I(µ) ⩾ 0 for any (signed) µ ∈ M, and I(µ) = 0 ⇐⇒ µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  This implies that all (signed) µ ∈ M with I(µ) < ∞ form a pre-Hilbert space E with  the inner product ⟨µ, ν⟩ := I(µ, ν) and the energy norm ∥µ∥ :=  �  I(µ), see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' [10,  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' The topology on E defined by means of this norm, is said to be strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Another fact decisive to this paper is that the cone E+ := E ∩ M+ is strongly  complete, and that the strong topology on E+ is finer than the (induced) vague  topology on E+ (see J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Deny [6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' for α = 2, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' also H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Cartan [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Thus any strong  Cauchy sequence (net) (µj) ⊂ E+ converges both strongly and vaguely to the same  unique limit µ0 ∈ E+, the strong topology on E as well as the vague topology on M  being Hausdorff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' (Following B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Fuglede [10], such a kernel is said to be perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=')  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' A model case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' As a model case for introducing the concept of inner α-Riesz  pseudo-balayage, consider first a closed set F ⊂ Rn and a (signed) measure ω ∈ M  of finite energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Since the class M+(F) of all µ ∈ M+ with the support S(µ) ⊂ F is  vaguely closed [3, Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2, Proposition 6], the convex cone E+(F) := M+(F)∩E  is strongly closed, and hence strongly complete, the α-Riesz kernel being perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' By  applying [9] (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3 and Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4(2)), we therefore conclude that  for the given ω ∈ E, there exists the unique Pω ∈ E+(F) such that  ∥ω − Pω∥ =  min  µ∈E+(F ) ∥ω − µ∥,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2)  and the same Pω is uniquely characterized within E+(F) by the two relations  ⟨Pω − ω, µ⟩ ⩾ 0 for all µ ∈ E+(F),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3)  ⟨Pω − ω, Pω⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4)  This Pω is said to be the orthogonal projection of ω ∈ E onto E+(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields  3  By a slight modification of [26, Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1] we infer from the above  that Pω is the only measure in E+(F) having the two properties  UP ω ⩾ Uω n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on F,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5)  UP ω = Uω Pω-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=',  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6)  where the abbreviation n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' (nearly everywhere) means that the inequality holds true  everywhere on F except for a subset N ⊂ F of inner capacity zero: c∗(N) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1  Assume for a moment that α ⩽ 2, and that the above ω is positive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' ω ∈ E+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  By use of the complete maximum principle [17, Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='27, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='29], we derive from  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6) that Pω is then uniquely characterized within E+(F) by the equality  UP ω = Uω n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on F,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7)  see [26, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1], and hence Pω is actually the balayage ωF of ω ∈ E+ onto F:  Pω = ωF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  ¶ However, if either α > 2, or if the above ω is signed, then relations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6)  still hold, but they no longer result in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Motivated by this observation, we introduce the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' The pseudo-balayage ˆωF of ω ∈ E onto a closed set F ⊂ Rn  with respect to the α-Riesz kernel of arbitrary order α ∈ (0, n) is defined as the only  measure in E+(F) satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2) (with ˆωF in place of Pω);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' or equivalently, as the  unique measure in E+(F) having properties (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6) (with ˆωF in place of Pω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Assume for a moment that ω is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' It follows from the above  that the pseudo-balayage ˆωF coincides with the balayage ωF whenever α ⩽ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' while  otherwise, the former concept presents a natural extension of the latter, the problem  of balayage for α > 2 being unsolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2 But if now ω is signed, then for α ⩽ 2, both  ˆωF and ωF still exist and are unique, whereas, in general,  ˆωF ̸= ωF,  the balayage of signed ω being defined by linearity (for more details see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' In Section 3 below, we shall extend the above definition of the  pseudo-balayage ˆωF, given in the model case of ω ∈ E and closed F ⊂ Rn, to:  ω ∈ M that are not necessarily of finite energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  F ⊂ Rn that are not necessarily closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  For the former goal, we observe that problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2) is equivalent to that of minimizing  the Gauss functional ∥µ∥2 − 2  ´  Uω dµ, which makes sense not only for ω ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  We complete this section with some general conventions, used in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  From now on, when speaking of a (signed) measure µ ∈ M, we understand that  its potential Uµ is well defined and finite almost everywhere with respect to the  Lebesgue measure on Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' or equivalently (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' [17, Section I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7]) that  ˆ  |y|>1  d|µ|(y)  |y|n−α < ∞,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8)  1For closed F, the set N of all x ∈ F where (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5) fails is Borel, hence capacitable, and so (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5)  actually holds true even quasi-everywhere (q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=') on F, namely everywhere on F except for N of outer  capacity zero: c∗(N) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' (For the concepts of inner and outer capacities, see [17, Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=')  2See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' [17, Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='20];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' this is caused by the fact that for α > 2, the maximum principle  fails to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Therefore, when speaking of α-Riesz balayage, we understand that α ∈ (0, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  4  Natalia Zorii  where |µ| := µ+ + µ−, µ+ and µ− being the positive and negative parts of µ in the  Hahn–Jordan decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Actually, then (and only then) Uµ is finite q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on Rn,  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' [17, Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' This would necessarily hold if µ were required to be bounded  (that is, with |µ|(Rn) < ∞), or of finite energy, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' [10, Corollary to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  A measure µ ∈ M+ is said to be concentrated on a set A ⊂ Rn if Ac := Rn \\ A  is µ-negligible, or equivalently if A is µ-measurable and µ = µ|A, µ|A being the  restriction of µ to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Denoting by M+(A) the cone of all µ ∈ M+ concentrated on  A, we further write E+(A) := M+(A) ∩ E, and let E′(A) stand for the closure of  E+(A) in the strong topology on E+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' We emphasize that E′(A) is strongly complete,  being a strongly closed subcone of the strongly complete cone E+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Given A ⊂ Rn, denote by CA the upward directed set of all compact subsets K  of A, where K1 ⩽ K2 if and only if K1 ⊂ K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' If a net (xK)K∈CA ⊂ Y converges to  x0 ∈ Y , Y being a topological space, then we shall indicate this fact by writing  xK → x0 in Y as K ↑ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' On the inner Riesz balayage  Before proceeding with an extension of the concept of pseudo-balayage announced  in Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3, we first recall some basic facts of the theory of inner α-Riesz balayage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Such a theory, generalizing Cartan’s pioneering work [5] on the inner Newtonian  balayage (α = 2) to any α ∈ (0, 2], was initiated in the author’s recent papers [26, 27],  and it was further developed in [28]–[30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3 Throughout this section, 0 < α ⩽ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1 ([26, Sections 3, 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' The inner balayage ωA of a measure ω ∈ M+  to a set A ⊂ Rn is defined as the measure of minimum potential in the class ΓA,ω,  ΓA,ω :=  �  µ ∈ M+ : Uµ ⩾ Uω n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on A  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  That is, ωA ∈ ΓA,ω and  UωA = min  µ∈ΓA,ω Uµ on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1)  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2 ([26, Sections 3, 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Given arbitrary ω ∈ M+ and A ⊂ Rn, the  inner balayage ωA, introduced by Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1, exists and is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Furthermore,4  UωA = Uω n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on A,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2)  UωA ⩽ Uω on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  The same ωA can alternatively be characterized by means of either of the following  (equivalent) assertions:  (a) ωA is the unique measure in M+ satisfying the symmetry relation  I(ωA, σ) = I(σA, ω) for all σ ∈ E+,  where σA denotes the only measure in E′(A) with UσA = Uσ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5  3See also [31] for an application of this theory to Deny’s principle of positivity of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  4As pointed out in [26, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='12], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2) no longer characterizes ωA uniquely (as it does for  closed A and ω ∈ E+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' For more details see footnote 5, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5, and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  5For any σ ∈ E+ and any A ⊂ Rn, the measure σA ∈ E′(A) having the property U σA = U σ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  on A, exists and is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' It is, in fact, the orthogonal projection of σ in the pre-Hilbert space E  onto the convex, strongly complete cone E′(A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' that is (compare with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2)),  ∥σ − σA∥ =  min  µ∈E′(A) ∥σ − µ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  The same σA is uniquely characterized within M+ by the extremal property (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1) with µ := σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields  5  (b) ωA is the unique measure in M+ satisfying either of the two limit relations  ωA  j → ωA vaguely in M+ as j → ∞,  UωA  j ↑ UωA pointwise on Rn as j → ∞,  where (ωj) ⊂ E+ is an arbitrary sequence having the property6  Uωj ↑ Uω pointwise on Rn as j → ∞,  whereas ωA  j denotes the only measure in E′(A) with UωA  j = Uωj n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' For signed ω ∈ M, we define the inner balayage ωA by linearity:  ωA := (ω+)A − (ω−)A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3)  If moreover the mutual energy I(ω, σ) is well defined for all σ ∈ E, then this ωA is  uniquely characterized by the symmetry relation  I(ωA, σ) = I(σA, ω) for all σ ∈ E,  which actually only needs to be verified for certain countably many σ ∈ E, indepen-  dent of the choice of ω (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' [30], Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4 and Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  In the rest of this paper, we shall always require A ⊂ Rn to have the property7  (P1) E+(A) is strongly closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Then (and only then)  E′(A) = E+(A),  and hence Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2 remains valid with E′(A) replaced throughout by E+(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' In  particular, the following useful corollary holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' For this A and for any ω ∈ E+, the inner balayage ωA is, in fact,  the orthogonal projection of ω onto the (convex, strongly complete) cone E+(A):  ∥ω − ωA∥ =  min  µ∈E+(A) ∥ω − µ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  The same ωA is uniquely characterized within E+(A) by UωA = Uω n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Assumption (P1) is important for the validity of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Indeed, if α = 2 and A := Br, r ∈ (0, ∞), then for any ω ∈ M+(Bc  r) with ω(B  c  r) > 0,  we have S(ωBr) = Sr [27, Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1], and hence the inner balayage ωBr is  not concentrated on the set Br itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' (Actually, S(ωBr) ∩ Br = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' )8  The following generalization of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4 will be useful in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Given ω ∈ M+, assume that ωA is of finite energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9 Then ωA  is concentrated on A, that is, ωA ∈ E+(A), and it is the unique solution to the  6Such ωj ∈ E+, j ∈ N, do exist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' they can be defined, for instance, by means of the formula  U ωj := min  �  U ω, jU λ�  ,  λ ∈ E+ being fixed (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' [17, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' 272] or [5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' 257, footnote]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Here we have used the fact that  for any µ1, µ2 ∈ M+, there is µ0 ∈ M+ such that U µ0 := min {U µ1, U µ2} [17, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  7As shown in [33, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9], (P1) is fulfilled, for instance, if A is quasiclosed (quasicompact),  that is, if A can be approximated in outer capacity by closed (compact) sets, see Fuglede [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  8Here and in the sequel we use the notations Br := {|x| < r}, Br := {|x| ⩽ r}, Sr := {|x| = r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  9This occurs e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' if ω itself is of finite energy, or if ω is bounded and meets the separation condition  inf  (x,y)∈S(ω)×A |x − y| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  6  Natalia Zorii  problem of minimizing the Gauss functional ∥µ∥2 −2  ´  Uω dµ, µ ranging over E+(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Alternatively, ωA is uniquely characterized within E+(A) by UωA = Uω n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Since ωA = (ωA)A (see [26, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2]), Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4 applied to ωA ∈ E+  shows that, indeed, ωA ∈ E+(A), and hence ωA is the orthogonal projection of itself  onto E+(A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' or equivalently, it is the (unique) solution to the problem of minimizing  the functional ∥µ∥2−2  ´  UωA dµ, µ ranging over E+(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' This implies the former part  of the claim by noting that  ´  UωA dµ =  ´  Uω dµ for all µ ∈ E+(A), which, in turn,  is derived from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2) by use of the fact, to be often used in what follows, that any  µ-measurable subset of A with c∗(·) = 0 is µ-negligible for any µ ∈ E+(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  For the latter part, assume (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2) holds for some µ0 ∈ E+(A) in place of ωA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' By  the strengthened version of countable subadditivity for inner capacity (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7),  Uµ0 = Uω = UωA n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on A,  whence µ0 = (ωA)A, again by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4 applied to ωA ∈ E+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Combining this with  (ωA)A = ωA (see above) gives µ0 = ωA, thereby completing the whole proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  □  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' For arbitrary Q ⊂ Rn and Borel Uj ⊂ Rn,  c∗  ��  j∈N  Q ∩ Uj  �  ⩽  �  j∈N  c∗(Q ∩ Uj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' See [10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' 157–158] (for α = 2, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' [5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' 253]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' compare with [17, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' 144].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' An extension of the concept of pseudo-balayage  In what follows, we assume that  c∗(A) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1)  Then (and only then) the class E+(A) is not reduced to {0}, see [10, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1],  and the problems in question become nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Fix a (signed) measure ω ∈ M (not necessarily of finite energy), and define the  external field f : Rn → [−∞, ∞] by means of the formula  f := −Uω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Being the difference between two lower semicontinuous (l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=') functions, f is Borel  measurable, and, due to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8), f is finite q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Let E+  f (A) stand for the convex cone of all µ ∈ E+(A) such that f is µ-integrable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  then for every µ ∈ E+  f (A), the Gauss functional10  If(µ) := ∥µ∥2 + 2  ˆ  f dµ = ∥µ∥2 − 2  ˆ  Uω dµ  is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Denoting  ˆwf(A) :=  inf  µ∈E+  f (A) If(µ),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2)  we have  − ∞ ⩽ ˆwf(A) ⩽ 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3)  the upper estimate being caused by the fact that 0 ∈ E+  f (A) while If(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' A measure ˆωA ∈ E+  f (A) with If(ˆωA) = ˆwf(A) is said to be the  inner α-Riesz pseudo-balayage of ω onto A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  10In constructive function theory, the Gauss functional is also referred to as the f-weighted energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields  7  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' The inner pseudo-balayage ˆωA is unique (if it exists).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' This follows by standard methods based on the convexity of the class E+  f (A)  and the parallelogram identity in the pre-Hilbert space E, by use of the strict positive  definiteness of the Riesz kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' (See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' [21, Proof of Lemma 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Note that this proof  requires ˆwf(A) to be finite, which however necessarily holds whenever ˆωA exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=')  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' If ω is positive, then, as seen from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6, the concept of  inner pseudo-balayage ˆωA extends that of inner balayage ωA (introduced for α ⩽ 2)  to arbitrary α ∈ (0, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' See the present section as well as Section 4 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  ¶ But if ω is signed, then the inner balayage ωA, defined for α ⩽ 2 by means of  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3), may not coincide with the inner pseudo-balayage ˆωA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' (Thus, in case α ⩽ 2,  the theory of inner pseudo-balayage may be thought of as an alternative theory of  inner balayage for signed measures, which is not equivalent to the standard one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=')  Indeed, take ω ∈ M such that ω = −ω− ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Then ωA = −(ω−)A ̸= 0, whereas  ˆωA, minimizing ∥µ∥2 + 2  ´  Uω− dµ ⩾ 0 over E+  f (A), is obviously 0, and so ˆωA ̸= ωA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' It follows easily from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1 that, if ˆωA exists, then so does  (�  cω)A for any c ∈ [0, ∞), and moreover  (�  cω)A = cˆωA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4)  However, this fails to hold if c < 0, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' On the existence of the inner pseudo-balayage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Recall that we are working  under the permanent requirements (P1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' In the rest of this paper, we also  assume that ω ∈ M satisfies either of the following properties:11  (P2) ω ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (P3) ω+ is bounded, Uω is upper semicontinuous on A := ClRnA, and  MA := sup  x∈A  U|ω|(x) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5)  Note that in case (P2), the class E+  f (A) actually coincides with the whole of  E+(A), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='13), while in case (P3), it necessarily contains all bounded µ ∈ E+(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' The inner pseudo-balayage ˆωA, introduced by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1, does  exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Hence,  − ∞ < ˆwf(A) ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6)  The same ˆωA can alternatively be characterized by either of the following (a) or (b):  (a) ˆωA is the only measure in E+  f (A) having the two properties  ˆ  U ˆωA−ω dµ ⩾ 0 for all µ ∈ E+  f (A),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7)  ˆ  U ˆωA−ω dˆωA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8)  (b) ˆωA is the only measure in E+(A) having the two properties12  U ˆωA ⩾ Uω n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on A,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9)  U ˆωA = Uω  ˆωA-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='10)  11(P3) is certainly fulfilled if ω ∈ M is compactly supported in A  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  12Due to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9), U ˆωA ⩾ U ω holds true ˆωA-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' hence, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='10) is equivalent to the apparently weaker  relation U ˆωA ⩽ U ω ˆωA-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Similarly, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8) can be replaced by  ´  U ˆωA−ω dˆωA ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  8  Natalia Zorii  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Fixing ν ∈ E+  f (A), we shall first show that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='10) hold true for ν in  place of ˆωA if and only if so do (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' It is enough to verify only the "if"  part of this claim, the opposite being obvious from the fact that any µ-measurable  subset of A with c∗(·) = 0 is µ-negligible for any µ ∈ E+(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Assuming, therefore, that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8) hold true (for ν in place of ˆωA), suppose  to the contrary that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9) fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' But then there is compact K ⊂ A such that Uν < Uω  on K while c(K) > 0,13 hence  ´  Uν−ω dτ < 0 for any τ ∈ E+(K), τ ̸= 0, which  contradicts (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7),  ´  f dτ being obviously finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Thus (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9) does indeed hold, whence  Uν ⩾ Uω ν-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='11)  Further, assuming to the contrary that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='10) fails to hold, we infer from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='11)  that there exists compact Q ⊂ A such that ν(Q) > 0 while Uν > Uω on Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Together  with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='11), this yields  ´  Uν−ω dν > 0, which however contradicts (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  The equivalence thereby verified enables us to prove the statement on the unique-  ness in each of assertions (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Indeed, suppose that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='10) are ful-  filled by some ν, ν′ ∈ E+(A) in place of ˆωA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Noting from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='10) that then necessarily  ν, ν′ ∈ E+  f (A), we conclude by applying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8) to each of ν and ν′ that  ⟨ν, ν′⟩ ⩾  ˆ  Uω dν′ = ∥ν′∥2,  ⟨ν′, ν⟩ ⩾  ˆ  Uω dν = ∥ν∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Therefore,  0 ⩽ ∥ν − ν′∥2 =  �  ∥ν∥2 − ⟨ν′, ν⟩  �  +  �  ∥ν′∥2 − ⟨ν, ν′⟩  �  ⩽ 0,  whence ν = ν′, by virtue of the strict positive definiteness of the α-Riesz kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Case (P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Assume first that (P2) holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' then the Gauss functional has the form  If(µ) = ∥ω − µ∥2 − ∥ω∥2 for all µ ∈ E+(A),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='12)  whence  E+  f (A) = E+(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='13)  Thus the problem on the existence of the inner pseudo-balayage ˆωA is reduced to  that on the existence of the orthogonal projection of ω ∈ E onto E+(A), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  ˆωA ∈ E+(A) and ∥ω − ˆωA∥ =  min  µ∈E+(A) ∥ω − µ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Since the convex cone E+(A) is strongly closed by (P1), hence strongly complete,  E+ being strongly complete by the perfectness of the α-Riesz kernel, applying [9]  (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3 and Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4(2)) shows that such an orthogonal projection  does exist, and it is uniquely characterized within E+(A) by both (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' In  view of the equivalence of (a) and (b) proved above, this implies the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Case (P3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' The remaining case (P3) will be treated in four steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' The purpose of this step is to show that the inner pseudo-balayage ˆωA  exists if and only if there exists the (unique) measure µ0 ∈ E+  f (A) satisfying (a)  (equivalently, (b)), and then necessarily µ0 = ˆωA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Assume first that ˆωA exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' To verify (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9), suppose to the contrary that there is  a compact set K ⊂ A with c(K) > 0, such that U ˆωA < Uω on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' A straightforward  verification then shows that for any τ ∈ E+(K), τ ̸= 0, and any t ∈ (0, ∞),  If(ˆωA + tτ) − If(ˆωA) = 2t  ˆ �  U ˆωA − Uω�  dτ + t2∥τ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='14)  13If A is capacitable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Borel), we write c(A) := c∗(A) = c∗(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields  9  As ∥τ∥ < ∞, the value on the right in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='14) (hence, also that on the left) is < 0 when  t > 0 is small enough, which however contradicts Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1, for ˆωA+tτ ∈ E+  f (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Having thus established (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9), we obtain  U ˆωA ⩾ Uω  ˆωA-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='15)  Suppose now that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='10) fails to hold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' then there exists a compact set Q ⊂ A  with ˆωA(Q) > 0, such that U ˆωA > Uω on Q, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Denoting υ := ˆωA|Q, we have  ˆωA − tυ ∈ E+  f (A) for all t ∈ (0, 1), hence  If(ˆωA − tυ) − If(ˆωA) = −2t  ˆ �  U ˆωA − Uω�  dυ + t2∥υ∥2,  which again contradicts Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1 when t is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' (Note that, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='10),  U ˆωA ⩽ Uω on S(ˆωA),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='16)  because Uω is upper semicontinuous (u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=') on A by (P3), while U ˆωA is l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=')  For the "if" part of the claim, assume that (a) holds true for some (unique)  µ0 ∈ E+  f (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' To show that then necessarily µ0 = ˆωA, we only need to verify that  If(µ) − If(µ0) ⩾ 0 for any µ ∈ E+  f (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='17)  But obviously  If(µ) − If(µ0) = ∥µ − µ0 + µ0∥2 − 2  ˆ  Uω dµ − ∥µ0∥2 + 2  ˆ  Uω dµ0  = ∥µ − µ0∥2 + 2  ˆ  Uµ0−ω d(µ − µ0),  and relations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8) (with µ0 in place of ˆωA) immediately lead to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' To complete the proof of the theorem, it thus remains to establish the  existence of the inner pseudo-balayage ˆωA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' To this end, suppose throughout this step  that A = K is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' As c(K) > 0, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1), we may restrict ourselves to nonzero  measures µ ∈ E+(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' For each of those µ, there are t ∈ (0, ∞) and τ ∈ E+(K) with  τ(K) = 1 such that µ = tτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' The potential U|ω| being bounded on K by (P3),  If(µ) = t2∥τ∥2 − 2t  ˆ  Uω dτ ⩾ t2∥τ∥2 − 2t  ˆ  Uω+ dτ  ⩾ t2c(K)−1 − 2tMK = t2�  c(K)−1 − 2MKt−1�  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='18)  MK ∈ [0, ∞) being introduced by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5) (with A := K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Hence, by virtue of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='18),  If(µ) > 0 for all µ ∈ E+(K) having the property  µ(K) > 2MKc(K) =: LK ∈ [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  On the other hand, ˆwf(K) ⩽ 0, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' In view of the above, ˆwf(K) would  therefore be the same if E+  f (K) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2) were replaced by  E+  LK(K) :=  �  µ ∈ E+(K) : µ(K) ⩽ LK  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='19)  that is,  ˆwf(K) =  inf  µ∈E+  LK (K) If(µ) =: ˆwf,LK(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='20)  Hence,  −∞ < −2MKLK ⩽ ˆwf(K) ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  10  Natalia Zorii  Choose a (minimizing) sequence (µj) ⊂ E+  LK(K) such that  lim  j→∞ If(µj) = ˆwf,LK(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Being vaguely bounded, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='19), the sequence (µj) is vaguely relatively compact [3,  Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1, Proposition 15], and so there is a subsequence (µjk) converging vaguely  to some µ0 ∈ M+(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' (Here we have used the first countability of the vague topology  on M, see [30, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4], as well as the fact that M+(K) is vaguely closed, see [3,  Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2, Proposition 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=') By the principle of descent [17, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5)],  ∥µ0∥2 ⩽ lim inf  k→∞ ∥µjk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Furthermore, by [3, Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4, Corollary 3],  ˆ  Uω dµ0 ⩾ lim sup  k→∞  ˆ  Uω dµjk ∈ (−∞, ∞),  Uω being bounded and u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on the (compact) set K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' This altogether gives  ˆwf(K) ⩽ If(µ0) ⩽ lim inf  k→∞ If(µjk) = ˆwf,LK(K),  which combined with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='20) shows that µ0 serves as the pseudo-balayage ˆωK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Our next aim is to show that the constant LK, satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='20), can be  defined to be independent of K ∈ CA large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' (To be exact, here and in the  sequel K ∈ CA is chosen to follow some K0 with c(K0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=')  By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='16), U ˆωK = Uω n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on S := S(ˆωK), hence γS-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=', where γS  denotes the capacitary measure on S (see [10, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Since UγS ⩾ 1 holds  true n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on S, hence ˆωK-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=', we get, by Fubini’s theorem,  ˆωK(Rn) =  ˆ  1 dˆωK ⩽  ˆ  UγS dˆωK =  ˆ  U ˆωK dγS  =  ˆ  Uω dγS =  ˆ  UγS dω ⩽  ˆ  UγS dω+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  As UγS ⩽ 1 on S(γS), applying the Frostman maximum principle if α ⩽ 2 (see [17,  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='10]), or [17, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5] otherwise, gives  ˆωK(Rn) ⩽ Cn,αω+(Rn) =: L for all compact K ⊂ A,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='21)  where  Cn,α :=  �  1  if α ⩽ 2,  2n−α  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='22)  We are thus led to the following conclusion, crucial to our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  ¶ We have  ˆwf(K) = ˆwf,L(K) ∈ [−2MAL, 0] for all K ∈ CA,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='23)  MA ∈ [0, ∞) and L ∈ [0, ∞) being introduced by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='21), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  The infimum ˆwf,L(K) is an actual minimum with the minimizer ˆωK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' To establish the existence of ˆωA for noncompact A, we first note that the  net  �  ˆwf,L(K)  �  K∈CA decreases, and moreover, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='23),  ∞ < lim  K↑A ˆwf,L(K) ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='24)  For any compact K, K′ ⊂ A such that K ⊂ K′ and c(K) > 0,  (ˆωK + ˆωK′)/2 ∈ E+  L (K′),  Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields  11  whence  ∥ˆωK + ˆωK′∥2 − 4  ˆ  Uω d(ˆωK + ˆωK′) ⩾ 4 ˆwf,L(K′) = 4If(ˆωK′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Applying the parallelogram identity to ˆωK, ˆωK′ ∈ E+ we therefore get  ∥ˆωK − ˆωK′∥2 ⩽ 2If(ˆωK) − 2If(ˆωK′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='25)  Noting from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='24) that the net  �  If(ˆωK)  �  K⩾K0 is Cauchy in R, we infer from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='25)  that the net (ˆωK)K⩾K0 is strong Cauchy in E+(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' The cone E+(A) being strongly  closed (hence strongly complete) by (P1), there exists ζ ∈ E+(A) such that  ˆωK → ζ strongly (hence vaguely) in E+(A) as K ↑ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='26)  Moreover, ζ ∈ E+  L (A), the mapping µ �→ µ(Rn) being vaguely l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on M+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='14  We claim that this ζ serves as the inner pseudo-balayage ˆωA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' As shown above  (Step 1), this will follow once we verify (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='10) for ζ in place of ˆωA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  To verify (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9) (for ζ in place of ˆωA), it is enough to do this for any given compact  K∗ ⊂ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' The strong topology on E+ being first-countable, in view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='26) there is  a subsequence (ˆωKj)j∈N of the net (ˆωK)K∈CA such that Kj ⊃ K∗ for all j, and  ˆωKj → ζ strongly (hence vaguely) in E+ as j → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='27)  Passing if necessary to a subsequence and changing the notations, we conclude from  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='27), by use of [10, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' 166, Remark], that  Uζ = lim  j→∞ U ˆωKj  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='28)  Applying now (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9) to each ˆωKj, and then letting j → ∞, on account of the countable  subadditivity of inner capacity on Borel sets [17, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' 144] we infer from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='28) that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9) (with ζ in place of ˆωA) does indeed hold n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on K∗, whence n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  To establish (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='10) for ζ in place of ˆωA, we note from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='16) applied to Kj that  U ˆωKj ⩽ Uω on S(ˆωKj),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='29)  (Kj) being the sequence chosen above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Since (ˆωKj) converges to ζ vaguely, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='27),  for every x ∈ S(ζ) there exist a subsequence (Kjk) of (Kj) and points xjk ∈ S(ˆωKjk)  such that xjk, k ∈ N, approach x as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Thus, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='29),  U ˆω  Kjk (xjk) ⩽ Uω(xjk) for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Letting here k → ∞, in view of the upper semicontinuity of Uω on A and the lower  semicontinuity of the mapping (x, µ) �→ Uµ(x) on Rn × M+, where M+ is equipped  with the vague topology [10, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1(b)], we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='10) for ζ in place of ˆωA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  This implies that  ζ = ˆωA,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='30)  thereby completing the proof of the whole theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  □  14See [3, Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1, Proposition 4] applied to the (positive, l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=') function 1 on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' It is  useful to point out that in the case where the set in question is compact, the mapping µ �→  ´  g dµ  remains vaguely l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on M+(K) for any l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' function g (not necessarily positive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' This follows  by replacing g by g′ := g + c ⩾ 0, where c ∈ (0, ∞), a l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' function on a compact set being lower  bounded, and then by making use of the vague continuity of the mapping µ �→ µ(K) on M+(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  12  Natalia Zorii  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Assume for a moment that ω is positive, and that A is quasiclosed  and Borel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' If moreover either ω ∈ E+, or Uω|A is bounded while c∗(A) < ∞,15 then  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5 can be deduced from Fuglede’s result [12, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='10] on the outer  pseudo-balayage with respect to a perfect kernel on a locally compact (Hausdorff)  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' The methods developed in the above proof are essentially different from those  in [12], which enabled us to establish the existence of the inner Riesz pseudo-balayage  ˆωA for pretty general ω and A, see (P1) and (P2), or (P1) and (P3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' In this regard, it  is also worth noting that the above proof seems to admit a generalization to suitable  perfect kernels on locally compact spaces, which we plan to pursue in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Further properties of the inner pseudo-balayage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' The following theorem  justifies the term "inner" pseudo-balayage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' ˆωK → ˆωA strongly and vaguely in E+ as K ↑ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' In case (P3), this follows by substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='30) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  It is thus left to consider case (P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Then ω ∈ E, and therefore ˆωK, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' ˆωA,  is the orthogonal projection of ω onto the (convex, strongly complete) cone E+(K),  resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' E+(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' A slight modification of the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='25) shows that  ∥ˆωK − ˆωK′∥2 ⩽ 2If(ˆωK) − 2If(ˆωK′) whenever K ⊂ K′  (K, K′ ∈ CA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Noting that the net  �  ˆwf(K)  �  K∈CA is decreasing and, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='12), bounded:  −∥ω∥2 ⩽ ˆwf(K) ⩽ 0 for all K ∈ CA,  we conclude from the above that the net (ˆωK)K∈CA ⊂ E+(A) is strong Cauchy, and  hence converges strongly and vaguely to some (unique) µ0 ∈ E+(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' This implies  ˆwf(A) ⩽ If(µ0) = lim  K↑A If(ˆωK) = lim  K↑A ˆwf(K),  the former equality being derived from the strong convergence of (ˆωK) to µ0 by use  of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' To verify that this µ0 actually equals ˆωA, it thus remains to show that  lim  K↑A ˆwf(K) ⩽ ˆwf(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='31)  But for every µ ∈ E+(A),  If(µ) = lim  K↑A If(µ|K) ⩾ lim  K↑A ˆwf(K),  where the equality follows by applying [10, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2] to each of the positive, l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=',  µ-integrable functions κα, Uω+, and Uω−, the set A being µ-measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Letting now  µ range over E+(A) we get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='31), thereby completing the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  □  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' If Uω is u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on A (which holds in particular in case (P3)), then  ˆωA(Rn) ⩽ Cn,αω+(Rn),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='32)  Cn,α being introduced by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' In view of the upper semicontinuity of Uω on A, the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='21), provided  in case (P3), remains valid in case (P2) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Hence, in both cases (P2) and (P3),  ˆωK(Rn) ⩽ Cn,αω+(Rn) for all K ∈ CA,  which results in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='32) since the net (ˆωK)K∈CA converges vaguely to ˆωA (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7)  while the mapping µ �→ µ(Rn) is vaguely l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on M+ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' footnote 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  □  15A quasiclosed set of finite outer capacity is actually quasicompact [12, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields  13  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' The comparison of the concepts of pseudo-balayage and balayage  The aim of Examples 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4 below is to demonstrate that usual nice properties  of the inner balayage may fail to hold when dealing with the inner pseudo-balayage,  and this occurs even in the simplest case of the Dirac measure and a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  ¶ The first fact illustrating the difference between these two concepts is that the  inner pseudo-balayage may increase the total mass of a positive measure (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5), pertaining to α > 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Recall that, if α ∈ (0, 2], then, by [26, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9],  µA(Rn) ⩽ µ(Rn) for any µ ∈ M+ and A ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1)  ¶ The second one is that, for α > 2, there exist a set A ⊂ Rn which is not inner  α-thin at infinity16 and a measure µ ∈ M+ such that  ˆµA(Rn) > µ(Rn),  see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' In contrast to that, not being inner α-thin at infinity is necessary and  sufficient for equality to prevail in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1) for all µ ∈ M+, see [27, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Let D ⊂ Rn be a bounded (connected, open) domain, ω := εx0,  where εx0 is the unit Dirac measure at x0 ∈ D, and let A be the inverse of D \\ {x0}  with respect to the sphere Sx0,1 := {|x − x0| = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' For these A and ω, (P1) and (P3)  are fulfilled, and hence the pseudo-balayage ˆεA  x0 exists and is unique (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Assume first that α ∈ (0, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Since the balayage εA  x0 of εx0 onto A is obviously of  finite energy, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6 yields  ˆεA  x0 = εA  x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2)  Applying [17, Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='20] we therefore infer that the pseudo-balayage ˆεA  x0 is  actually the Kelvin transform γ∗  D of the capacitary measure γD on D with respect to  the sphere Sx0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Hence, in particular,  S(ˆεA  x0) =  �  ∂A  if α = 2,  A  if α < 2,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3)  where ∂A := ∂RnA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' The set A not being α-thin at infinity, we also get, in consequence  of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2) and [13, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='22],  ˆεA  x0(Rn) = εx0(Rn) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4)  Let now α ∈ (2, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' We aim to show that then, in contrast to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4),17  1 = εx0(Rn) < ˆεA  x0(Rn) ⩽ 2n−α,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5)  the latter inequality being valid by virtue of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  As is known, the capacitary measure γD on D is the (unique) measure in E+(D)  of (finite) total mass c(D), and such that (see [17, Sections II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3, II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='13])  UγD ⩾ 1 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on D,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6)  UγD = 1 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on S(γD),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7)  UγD > 1 on D,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8)  16By [16, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1], A ⊂ Rn is said to be inner α-thin at infinity if  �  j∈N  c∗(Aj)  qj(n−α) < ∞,  where q ∈ (1, ∞) and Aj := A ∩ {x ∈ Rn :  qj ⩽ |x| < qj+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  For α ∈ (0, 2], see also [27,  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1], while for α = 2 and Borel A, see [7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' 175–176].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  17Compare with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  14  Natalia Zorii  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8) being caused by the fact that for α ∈ (2, n), the α-Riesz potential of a positive  measure is superharmonic on Rn [17, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  For the Kelvin transform γ∗  D of γD, applying [17, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4)] yields  Uγ∗  D(x) = |x − x0|α−nUγD(x∗),  I(γ∗  D) = I(γD),  γ∗  D(Rn) = UγD(x0),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9)  x∗ being the inverse of x with respect to Sx0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Combined with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7), this  shows that γ∗  D is a measure of the class E+(A) having the properties  Uγ∗  D ⩾ Uεx0 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on A,  Uγ∗  D = Uεx0 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on S(γ∗  D),  whence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='10) with ω := εx0 and ˆωA := γ∗  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Thus, by virtue of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5,  ˆεA  x0 = γ∗  D,  which together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9) establishes (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Also note that  S(ˆεA  x0) ⊂ ∂A  (compare with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' A slight modification of arguments in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1 shows that if  G ⊂ Rn is an open, relatively compact set, then for any α ∈ (2, n) and any x0 ∈ G,  1 < ˆεGc  x0 (Rn) ⩽ 2n−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Let α ∈ (2, n), A := Bc  R := {|x| ⩾ R}, R ∈ (0, ∞), and let ω :=  ε0, where ε0 denotes the unit Dirac measure at x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' As follows from Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1,  ˆε  Bc  R  0  = γ∗  Br,  where γBr is the capacitary measure on the ball Br, r := 1/R, and γ∗  Br is the Kelvin  transform of γBr with respect to the unit sphere S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Therefore, by symmetry reasons  applied to γBr, ˆε  Bc  R  0  is uniformly distributed over the sphere SR, and such that  1 < ˆε  Bc  R  0 (Rn) ⩽ 2n−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Let α ∈ (2, n), A := SR, R ∈ (0, ∞), and let ω := ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' As shown  in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3,  S(ˆε  Bc  R  0 ) = SR,  which in view of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1 gives  ˆεSR  0  = ˆε  Bc  R  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Thus ˆεSR  0  is uniformly distributed over the sphere SR, and such that  1 < ˆεSR  0 (Rn) ⩽ 2n−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Also note that ˆεSR  0  is, in fact, the Kelvin transform of the capacitary measure γSr  ( = γBr), where r := 1/R, with respect to the unit sphere S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields  15  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' The inner Gauss variational problem  As before, consider a set A ⊂ Rn such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1) and (P1) are fulfilled, a (signed)  measure ω ∈ M satisfying either (P2) or (P3), and the external field f given by  f := −Uω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  According to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5, the problem of minimizing the Gauss functional If(µ),  If(µ) := ∥µ∥2 + 2  ˆ  f dµ = ∥µ∥2 − 2  ˆ  Uω dµ,  over the class E+  f (A) of all µ ∈ E+(A) with finite If(µ) is uniquely solvable, and its  solution ˆωA, called the inner pseudo-balayage of ω onto A, is uniquely characterized  within E+  f (A) by both (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8) — or, equivalently, by both (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  The rest of the paper is to show that the concept of inner pseudo-balayage serves  as a powerful tool in the inner Gauss variational problem, which reads as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Does there exist λA,f minimizing If(µ) within ˘E+(A)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Here,  ˘E+(A) :=  �  µ ∈ E+(A) : µ(Rn) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Recent results on Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1, which was originated by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Gauss [14], are  reviewed in the monographs [2, 20] (see also numerous references therein);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' for some  of the latest researches on this topic, see [8, 32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' If A = K is compact while f is l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on K, then the existence of  the minimizer λK,f can easily be verified, by use of the fact that the class ˘E+(K) is  vaguely compact, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' [3, Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9, Corollary 3], while the Gauss functional If(·)  is vaguely l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on M+(K), the latter being obvious from the principle of descent and  the vague lower semicontinuity of the mapping µ �→  ´  f dµ on M+(K) (footnote 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  However, such a proof, based on the vague topology only, is no longer applicable if  either A is noncompact, or f is not l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  To investigate Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1 in the general case where A is noncompact and/or  f is not l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=', we have recently developed an approach based on the systematic use  of both the strong and vague topologies on the pre-Hilbert space E, which utilized  essentially the perfectness of the Riesz kernels, see [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' However, if c∗(A) = ∞, then  the analysis performed in [33] was only limited to α ⩽ 2 and ω ⩾ 0, being mainly  based on the theory of inner balayage for positive measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Motivated by this observation, we generalize the approach, suggested in [33], to  arbitrary α ∈ (0, n) and signed ω, by use of the theory of inner pseudo-balayage,  developed in Sections 3, 4 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' The results thereby established are formulated in  Section 6, and are proved in Sections 7–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' It is worth emphasizing that those results  improve substantially many recent ones from [8, 33] (see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3 for some details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' To begin with, observe that under either of assumptions  (P2) or (P3), the Gauss functional If(µ) is finite for all µ ∈ ˘E+(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Thus  ˘E+(A) ⊂ E+  f (A),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1)  and therefore  min  µ∈E+  f (A) If(µ) =: ˆwf(A) ⩽ wf(A) :=  inf  µ∈ ˘E+(A) If(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2)  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' −∞ < wf(A) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  16  Natalia Zorii  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' According to [21, Lemma 5], wf(A) < ∞ is equivalent to the inequality  c∗  �  {x ∈ A : |f|(x) < ∞}  �  > 0,  which indeed holds true by virtue of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1) and the fact that f is finite n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (Here the strengthened version of countable subadditivity for inner capacity has been  utilized, see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=') Finally, combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6) gives wf(A) > −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  □  The solution λA,f to Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1 is unique (if it exists), which can be proved by  use of the convexity of the class ˘E+(A) and the parallelogram identity in the pre-Hil-  bert space E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Such λA,f is said to be the inner f-weighted equilibrium measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  The following theorem, providing characteristic properties of λA,f, can be derived  from the author’s earlier paper [21] (see Theorems 1, 2 and Proposition 1 therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' For λ ∈ ˘E+(A) to be the (unique) solution λA,f to Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1,  it is necessary and sufficient that either of the following two inequalities be fulfilled:  Uλ  f ⩾  ˆ  Uλ  f dλ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on A,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3)  Uλ  f ⩽ wf(A) −  ˆ  f dλ λ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on A,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4)  where  Uλ  f := Uλ + f  is said to be the f-weighted potential of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' If (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3) or (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4) holds true, then actually  ˆ  Uλ  f dλ = wf(A) −  ˆ  f dλ =: cA,f ∈ (−∞, ∞),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5)  cA,f being referred to as the inner f-weighted equilibrium constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='18  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' If f is l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on A (which occurs e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' in case (P3)), then, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4),  U  λA,f  f  ⩽ cA,f on S(λA,f),  which combined with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3) gives  U  λA,f  f  = cA,f n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on A ∩ S(λA,f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' On the existence of λA,f and its properties  In all that follows, except for Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8, we assume A and f to satisfy the  permanent requirements, reminded at the beginning of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' On the solvability of Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Sufficient and/or necessary conditions  for the existence of the (unique) solution λA,f to Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1 are established in  Theorems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7 and Corollaries 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' For λA,f to exist, it is sufficient that19  c∗(A) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1)  Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' λA,f does exist whenever A is quasicompact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' This is obvious since for quasicompact A, both (P1) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1) hold true, by  virtue of [33, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9] and [11, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  □  18Similarly to [20, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' 27], cA,f might also be referred to as the inner modified Robin constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  19See Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2 below for an extension of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1 and Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields  17  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' For λA,f to exist, it is sufficient that  ˆωA(Rn) = 1,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2)  ˆωA being the inner pseudo-balayage of ω onto A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Furthermore, then  λA,f = ˆωA,  wf(A) = ˆwf(A),  cA,f = 0,  cA,f being the inner f-weighted equilibrium constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  On account of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1, in the rest of this section we assume that  c∗(A) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3)  Unless (P2) holds, assume additionally that  lim  |x|→∞, x∈A Uω−(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4)  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1 is unsolvable whenever  ˆωA(Rn) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5)  Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' If ω+ = 0, then Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1 is always unsolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Indeed, if ω = −ω−, then (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5) is fulfilled, for ˆωA = 0 (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  □  Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' If Uω is u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on A, then Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1 is unsolvable whenever  ω+(Rn) < 1/Cn,α,  Cn,α being introduced by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' This follows from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4 by use of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  □  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Unless (P2) holds, assume Uω is continuous20 on A and  lim  |x|→∞, x∈A Uω±(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6)  Then  λA,f exists ⇐⇒ ˆωA(Rn) ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  If moreover  ˆωA(Rn) > 1,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7)  then21  λA,f ̸= ˆωA,  wf(A) ̸= ˆwf(A),  cA,f ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8)  Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Dropping assumption (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3) as well as all those imposed on ω,  consider signed ω ∈ M, compactly supported in A  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Then λA,f exists if and only  if either c∗(A) < ∞, or ˆωA(Rn) ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  In particular, λA,f does not exist if both  c∗(A) = ∞ and ω+(Rn) < 1/Cn,α are fulfilled, Cn,α being introduced by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' This follows by combining Theorems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7 and Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  □  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Thus, if ω ∈ M is compactly supported in A  c while c∗(A) = ∞,  then, according to Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8, Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1 is unsolvable whenever ω+(Rn) is small  enough, whereas ω−(Rn), the total amount of the negative charge, has no influence  on this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' (Note that, when appealing to the electrostatic interpretation  of the problem, the fact just observed agrees with our physical intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=')  20When speaking of a continuous function, we understand that the values are finite numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  21Compare with Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3 as well as with Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  18  Natalia Zorii  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' On the description of the support S(λA,f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' The following Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='10  establishes sufficient conditions for the minimizer λA,f to be of compact support,  whereas Examples 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='11 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='12 analyze their sharpness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Under the hypotheses of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7, assume moreover that  A is not inner α-thin at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Then S(λA,f) is compact whenever (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7) is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Examples 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='11 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='12 provide explicit formulae for S(λA,f) for some specific  A and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' The latter equality in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9) as well as both equalities in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='13) show that  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='10 would fail in general if ˆωA(Rn) > 1 were replaced by ˆωA(Rn) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' For A := Bc  R := {|x| ⩾ R}, R ∈ (0, ∞), and for the unit Dirac  measure ε0 at x = 0, define  ω := ε0/q,  where q := ˆε  Bc  R  0 (Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  As shown in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1, q = 1 if α ⩽ 2, and q > 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4),  ˆωBc  R = ˆε  Bc  R  0 /q,  whence  ˆωBc  R(Rn) = ˆε  Bc  R  0 (Rn)/q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Therefore, by virtue of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3, the solution λBc  R,f to Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1 with f := −Uω  does exist, and moreover  λBc  R,f = ˆωBc  R = ˆε  Bc  R  0 /q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  In view of what was pointed out in Examples 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3, this gives  S(λBc  R,f) =  �  SR  if α ⩾ 2,  Bc  R  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9)  Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Let Bz,r be the closed ball of radius r centered at the point  z := (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' , 0, −r), and let A0 denote the inverse of Bz,r \\ {0} with respect to the  unit sphere S1 centered at the origin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Then A0 is a closed subset of Rn, not α-thin  at infinity, such that 0 ̸∈ A0, and whose boundary ∂A0 is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  In a manner similar to that in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1, we see that  ˆεA0  0  = γ∗  Bz,r,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='10)  γ∗  Bz,r being the Kelvin transform of γBz,r, the capacitary measure on Bz,r, with respect  to S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Noting that γ∗  Bz,r(Rn) = UγBz,r(0) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9)), whereas UγBz,r(0) = 1, we obtain  ˆεA0  0 (Rn) = 1 for any α ∈ (0, n)  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='11)  (compare with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Applying Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3 we therefore infer that the solution  λA0,f to Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1 with A := A0 and f := −Uε0 does exist, and moreover  λA0,f = ˆεA0  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='12)  By virtue of the description of S(γBz,r) [17, Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='13], (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='10) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='12) yield  S(λA0,f) =  �  ∂A0  if α ⩾ 2,  A0  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='13)  Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields  19  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' The results thereby obtained improve substantially many recent ones  from [8, 33], by strengthening their formulations and/or by extending the areas of  their applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' For instance, [8, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6] only deals with closed sets A that  are not thin at infinity, and with external fields f of the form −Uω, where ω := cεx0,  c ∈ (0, ∞), εx0 being the unit Dirac measure at x0 ̸∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' However, even for these  very particular A and ω, all the assertions in [8, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6] are in general weaker  than the relevant ones, established above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' This is caused, in particular, by the fact  that those assertions from [8] are given in terms of ω(Rn), whereas ours — in terms  of ˆωA(Rn) (see Section 4 for the relations between these two values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Regarding the advantages of our current approach in comparison with that sug-  gested in [33], see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3  Due to condition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2), the inner pseudo-balayage ˆωA, minimizing If(µ) over the  class E+  f (A), actually belongs to its proper subclass ˘E+(A), see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Therefore,  wf(A) ⩽ If(ˆωA) = ˆwf(A),  which combined with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2) gives  If(ˆωA) = wf(A) = ˆwf(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Thus ˆωA serves as the (unique) solution to Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' ˆωA = λA,f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Substituting  this equality into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8), we get  ˆ  U  λA,f  f  dλA,f =  ˆ �  U ˆωA − Uω�  dˆωA = 0,  which according to Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4 establishes the remaining relation cA,f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Proofs of Theorems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Extremal measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Let Mf(A) stand for the (nonempty) set of all nets  (µs)s∈S ⊂ ˘E+(A) having the property  lim  s∈S If(µs) = wf(A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1)  those nets (µs)s∈S are said to be minimizing (in Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Using the finiteness  of wf(A) (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3), the convexity of ˘E+(A), and the perfectness of the α-Riesz  kernel, one can see with the aid of arguments similar to those in [33, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2]  that there exists the unique ξA,f ∈ E+ such that, for every (µs)s∈S ∈ Mf(A),  µs → ξA,f strongly and vaguely in E+ (as s ranges through S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2)  This ξ := ξA,f will be referred to as the extremal measure (in Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Due to (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2), we have  ξA,f ∈ E+(A),  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3)  E+(A) being strongly closed by (P1), and moreover  ξA,f(Rn) ⩽ 1,  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4)  the mapping µ �→ µ(Rn) being vaguely l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on M+ [3, Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1, Proposition 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  The following simple observation is crucial to the proofs given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  20  Natalia Zorii  Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1 is solvable if and only if  ξA,f(Rn) = 1 and If(ξA,f) = wf(A),  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5)  and in the affirmative case  λA,f = ξA,f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' The "if" part is evident by (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3), whereas the opposite is implied by the fact  that the trivial net (λA,f) is obviously minimizing, and hence converges strongly to  both λA,f and ξA,f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Since the strong topology on E is Hausdorff, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  □  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Fix a minimizing sequence (µj) ∈ Mf(A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' by (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2),  µj → ξ strongly and vaguely in E+ (as j → ∞),  ξ := ξA,f being the (unique) extremal measure in Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1, whence  sup  j∈N  ∥µj∥ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7)  To show that this ξ serves as the solution to Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1, it is enough to verify (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Since µj → ξ vaguely, applying [3, Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1, Proposition 4] gives  ξ(Rn) ⩽ lim inf  j→∞  µj(Rn) = 1,  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8)  whereas [3, Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4, Corollary 3] yields  ˆ  1K dξ ⩾ lim sup  j→∞  ˆ  1K dµj for every compact K ⊂ Rn,  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9)  the indicator function 1K of K being bounded, of compact support, and u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Combining (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8) and (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9) with  ξ(Rn) = lim  K↑Rn  ˆ  1K dξ,  we get  1 ⩾ ξ(Rn) ⩾ lim sup  (j,K)∈N×C  ˆ  1K dµj = 1 − lim inf  (j,K)∈N×C  ˆ  1A\\K dµj,  N × C being the directed product of the directed sets N and C := CRn [15, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  The former relation in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5) will therefore follow once we establish the equality  lim inf  (j,K)∈N×C  ˆ  1A\\K dµj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='10)  By [17, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6] applied to A \\ K, K ∈ C being arbitrarily chosen, there  exists the (unique) inner capacitary measure γA\\K, minimizing the energy ∥µ∥2 over  the (convex) set ΓA\\K consisting of all µ ∈ E+ with  Uµ ⩾ 1 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on A \\ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  For any K′ ∈ C such that K ⊂ K′, we have ΓA\\K ⊂ ΓA\\K′, and [17, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2]  therefore gives  ∥γA\\K − γA\\K′∥2 ⩽ ∥γA\\K∥2 − ∥γA\\K′∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='11)  Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields  21  Since ∥γA\\K∥2 = c∗(A\\K) [17, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6], ∥γA\\K∥2 decreases as K ranges through  C, which together with (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='11) implies that the net (γA\\K)K∈C ⊂ E+ is Cauchy in the  strong topology on E+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Noting that (γA\\K)K∈C converges vaguely to zero,22 we get  γA\\K → 0 strongly in E+ as K ↑ Rn,  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='12)  the α-Riesz kernel being perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  It follows from the above that  UγA\\K ⩾ 1A\\K n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on A \\ K,  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='13)  and, therefore, µj-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' for all j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Integrating (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='13) with respect to µj we obtain,  by the Cauchy–Schwarz (Bunyakovski) inequality,  ˆ  1A\\K dµj ⩽  ˆ  UγA\\K dµj ⩽ ∥γA\\K∥ · ∥µj∥ for all K ∈ C and j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Combined with (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7) and (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='12), this gives (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='10), hence ξ ∈ ˘E+(A), and consequently  wf(A) ⩽ If(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='14)  To complete the proof of the theorem, it remains to verify the latter relation in  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5), which in view of (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1) and (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='14) is reduced to the inequality  If(ξ) ⩽ lim  j→∞ If(µj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='15)  In case (P2), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='15) follows at once from the strong convergence of (µj) to ξ, by  applying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='12) to each of µj and ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Otherwise, case (P3) holds, and hence f = −Uω  is l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' and bounded on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Thus there is c ∈ (0, ∞) such that f ′ := f + c ⩾ 0 on A,  and [3, Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1, Proposition 4] applied to f ′ gives  ˆ  f dξ + c =  ˆ  f ′ dξ ⩽ lim inf  j→∞  ˆ  f ′ dµj = lim inf  j→∞  ˆ  f dµj + c,  the first and last equalities being valid by virtue of ξ(Rn) = µj(Rn) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Therefore,  ˆ  f dξ ⩽ lim  j→∞  ˆ  f dµj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Multiplied by 2, and then added to  lim  j→∞ ∥µj∥2 = ∥ξ∥2,  this results in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='15), thereby completing the whole proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' A slight generalization of the above proof shows that Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1  and, hence, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2 remain valid for an external field f represented as the sum  f := u + Uϑ,  where ϑ ∈ E, while u : A → (−∞, ∞] is l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=', bounded from below, and such that  c∗({x ∈ A : u(x) < ∞}) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  22Indeed, for any given ϕ ∈ C0(Rn), there exists an open, relatively compact set G ⊂ Rn such  that ϕ(x) = 0 for all x ̸∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Hence, γA\\K(ϕ) = 0 for all K ∈ C with K ⊃ G, and the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  22  Natalia Zorii  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' As c∗(A) = ∞ by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3), there are mutually noninter-  secting, compact sets Kj ⊂ A, j ∈ N, such that |x| ⩾ j for all x ∈ Kj and c(Kj) ⩾ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  If λj := γKj/c(Kj) ∈ ˘E+(Kj) denotes the normalized capacitary measure on Kj, then  ∥λj∥ → 0 as j → ∞,  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='16)  λj → 0 vaguely in E+ as j → ∞,  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='17)  the latter being implied by the fact that for any compact subset K of Rn, we have  K ∩ S(λj) = ∅ for all j large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Define  µj := ˆωA + cjλj,  where cj := 1 − ˆωA(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='18)  Noting from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5) that  0 < cj ⩽ 1 for all j,  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='19)  we get µj ∈ ˘E+(A) for all j, whence  wf(A) ⩽ lim inf  j→∞  If(µj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='20)  On the other hand, If(ˆωA) = ˆwf(A) (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' By  means of a straightforward verification, we derive from (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='16), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='18), and (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='19) that  lim sup  j→∞  If(µj) ⩽ lim sup  j→∞  �  If(ˆωA) + 2cj  ˆ  Uω− dλj  �  ⩽ ˆwf(A) + 2L∞,  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='21)  where  L∞ := lim sup  j→∞  ˆ  Uω− dλj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Let first (P2) take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Applying the Cauchy–Schwarz inequality to the mea-  sures ω−, λj ∈ E+, and then letting j → ∞, we infer from (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='16) that  L∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='22)  Otherwise, (P3) and hence (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4) must be fulfilled, which again results in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='22), λj  being the unit measure supported by A ∩ {|x| ⩾ j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Substituting (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='22) into (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='21),  and then combining the inequality thus obtained with (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='20) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2), we get  lim  j→∞ If(µj) = ˆwf(A) = wf(A),  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='23)  which shows that the sequence (µj) is, in fact, minimizing in Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1, and hence  converges both strongly and vaguely to the extremal measure ξA,f:  µj → ξA,f strongly and vaguely in E+ as j → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  On account of (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='17)–(8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='19), this gives  ˆωA = ξA,f,  the vague topology on M being Hausdorff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Therefore, by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5),  ξA,f(Rn) = ˆωA(Rn) < 1,  and an application of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1 shows that Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1 is indeed unsolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' As shown in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='23), under the assumptions of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4, equal-  ity prevails in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  ˆwf(A) = wf(A)  (compare with Theorems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields  23  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Proofs of Theorems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='10  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Auxiliary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' According to Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2, for every K ∈ CA such that  K ⩾ K0, where c(K0) > 0, there is the (unique) solution λK,f to Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1 with  A := K, whereas by virtue of Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1 below, those λK,f form a minimizing net:  (λK,f)K⩾K0 ∈ Mf(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1)  Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' wf(K) ↓ wf(A) as K ↑ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' For any µ ∈ ˘E+(A), µ(K) ↑ 1 as K ↑ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Applying [10, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2] to each  of the (positive, l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=', µ-integrable) functions κα, Uω+, and Uω−, we therefore get  If(µ) = lim  K↑A If(µ|K) = lim  K↑A If(νK) ⩾ lim  K↑A wf(K),  where  νK := µ|K/µ(K) ∈ ˘E+(K),  K ∈ CA being large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Letting now µ range over ˘E+(A) gives  wf(A) ⩾ lim  K↑A wf(K),  whence the lemma, the opposite being obvious by the monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  □  Unless case (P2) takes place, assume in what follows that the external field  f = −Uω is continuous on A, and that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6) is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' For the extremal measure ξA,f, we have  If(ξA,f) = wf(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' By virtue of (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2) and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1),  λK,f → ξA,f strongly and vaguely in E+ as K ↑ A,  hence  lim  K↑A ∥λK,f∥2 = ∥ξA,f∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3)  If case (P2) takes place, then the strong convergence of (λK,f)K⩾K0 to ξA,f yields, by  applying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='12) to each of λK,f and ξA,f,  lim  K↑A If(λK,f) = If(ξA,f),  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4)  whence, by Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1,  If(ξA,f) = lim  K↑A wf(K) = wf(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  In the remaining case (P3), for any t > 0 choose r so that  |f| < t/2 on A ∩ B  c  r,  which is possible in view of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' On account of (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4),  ����  ˆ  B  c  r  f d(λK,f − ξA,f)  ���� < t for all K ⩾ K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5)  The above r can certainly be chosen so that  ξA,f(Sr) = 0,  the measure ξA,f being bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Then, according to [17, Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5′],  λK,f|Br → ξA,f|Br vaguely in M+ as K ↑ A,  24  Natalia Zorii  whence, by the continuity of f on A,  lim  K↑A  ˆ  f dλK,f|Br =  ˆ  f dξA,f|Br,  which combined with (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5), taken for t > 0 arbitrarily small, results in23  lim  K↑A  ˆ  f dλK,f =  ˆ  f dξA,f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6)  Multiplied by 2, and then added to (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3), this yields (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4), whence (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2), again by  making use of Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  □  Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' λA,f exists if and only if ξA,f(Rn) = 1, and in the affirmative  case λA,f = ξA,f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' This follows by combining Lemmas 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  □  Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' For the extremal measure ξ = ξA,f, we have  Uξ  f ⩾ Cξ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on A,  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7)  Uξ  f ⩽ Cξ on S(ξ),  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8)  where  Cξ :=  ˆ  Uξ  f dξ ∈ (−∞, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' In case (P3), the finiteness of  ´  Uξ  f dξ follows from the boundedness of Uω± on  A, the extremal measure ξ being bounded by (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' while in case (P2), it is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4 applied to each K ∈ CA large enough,  U  λK,f  f  ⩾ cK,f n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on K,  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='10)  U  λK,f  f  ⩽ cK,f on S(λK,f),  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='11)  where  cK,f =  ˆ  U  λK,f  f  dλK,f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Furthermore, by combining (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3) with (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6),  lim  K↑A cK,f = Cξ,  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='12)  Cξ being the (finite) constant appearing in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Fix K∗ ∈ CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' The strong topology on E+ being first-countable, one can choose  a subsequence (λKj,f)j∈N of the net (λK,f)K∈CA such that  λKj,f → ξ strongly (hence vaguely) in E+ as j → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='13)  There is certainly no loss of generality in assuming that  K∗ ⊂ Kj for all j,  for if not, we replace Kj by K′  j := Kj ∪ K∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' then, by the monotonicity of  �  wf(K)  �  ,  the sequence (λK′  j,f)j∈N remains minimizing, and hence also converges strongly to ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Due to the arbitrary choice of K∗ ∈ CA, (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7) will follow once we show that  Uξ  f ⩾ Cξ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on K∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='14)  23In case (P2), (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6) holds true as well, which is obtained by subtracting (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3) from (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Alter-  natively, it can be derived from the strong convergence of the net (λK,f)K⩾K0 to ξA,f, by applying  the Cauchy–Schwarz inequality to the measures ω, λK,f − ξA,f ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields  25  Passing if necessary to a subsequence and changing the notations, we conclude from  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='13), by virtue of [10, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' 166, Remark], that  Uξ = lim  j→∞ UλKj ,f n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='15)  Applying now (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='10) to each Kj, and then letting j → ∞, on account of (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='12) and  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='15) we arrive at (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' (Here the countable subadditivity of inner capacity on  Borel sets has been utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=')  Since (λKj,f) converges to ξ vaguely, see (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='13), for every x ∈ S(ξ) there exist a  subsequence (Kjk) of the sequence (Kj) and points xjk ∈ S(λKjk,f), k ∈ N, such that  xjk approach x as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Thus, according to (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='11),  U  λKjk ,f  f  (xjk) ⩽  ˆ  U  λKjk ,f  f  dλKjk,f for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Letting here k → ∞, and applying (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='12), the continuity of f on A, and the lower  semicontinuity of the mapping (x, µ) �→ Uµ(x) on Rn×M+, M+ being equipped with  the vague topology [10, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1(b)], we get the remaining inequality (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  □  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' We first remark from Theorems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4 that it is  actually enough to consider the case when (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7) is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  For the extremal measure ξ = ξA,f, we infer from (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7) and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8) that  Uξ  f = Cξ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on S(ξ) ∩ A,  whence  Uξ  f = Cξ ξ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=',  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='16)  the measure ξ being of the class E+(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' We claim that then necessarily  Cξ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='17)  Indeed, if this were not true, then (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7) and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='16) would imply, by virtue of Theo-  rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5, that ξ = ˆωA, which however contradicts (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7), for ξ(Rn) ⩽ 1 by (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Integrating (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='16) with respect to ξ we obtain  ˆ  Uξ  f dξ = Cξ · ξ(Rn),  whence, by (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9) and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='17),  ξ(Rn) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Applying Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3 we see that under assumption (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7), λA,f does indeed exist,  and moreover λA,f = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  The equality λA,f = ξ implies, by use of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5), (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='9), and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='17), that  cA,f =  ˆ  U  λA,f  f  dλA,f =  ˆ  Uξ  f dξ = Cξ ̸= 0,  which proves the third relation in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' The first is obvious since, by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7),  λA,f(Rn) = 1 < ˆωA(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Finally, the first relation implies the second, for if not, then wf(A) = ˆwf(A), whence  λA,f = ˆωA, by the uniqueness of ˆωA and the inclusion ˘E+(A) ⊂ E+  f (A), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  26  Natalia Zorii  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' According to Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='7, under the stated assump-  tions there exists the solution λA,f to Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1, and moreover, by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='8),  cA,f ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='18)  Assume to the contrary that S(λA,f) is noncompact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' As seen from Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='5, then  there exists a sequence (xj) ⊂ A such that |xj| → ∞ as j → ∞, and  U  λA,f  f  (xj) = cA,f for all j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  On account of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6), this yields  lim inf  j→∞  UλA,f(xj) = cA,f,  whence cA,f ⩾ 0, which in view of (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='18) shows that, actually,  cA,f > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='19)  But, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='3), U  λA,f  f  ⩾ cA,f n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' on A, which together with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='6) and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='19) gives  lim inf  |x|→∞, x∈A UλA,f (x) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  However, this is impossible in consequence of [16, Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='12(i)], the set A not  being inner α-thin at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  References  [1] Benko, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=', Dragnev, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=', Orive, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=': On point-mass Riesz external fields on the real axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' 491, 124299 (2020)  [2] Borodachov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=', Hardin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=', Saff, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' : Discrete Energy on Rectifiable Sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Springer,  Berlin (2019)  [3] Bourbaki, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=': Integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Chapters 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Springer, Berlin (2004)  [4] Cartan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=': Théorie du potentiel newtonien: énergie, capacité, suites de potentiels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' France 73, 74–106 (1945)  [5] Cartan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=': Théorie générale du balayage en potentiel newtonien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Fourier Grenoble  22, 221–280 (1946)  [6] Deny, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=': Les potentiels d’énergie finie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' 82, 107–183 (1950)  [7] Doob, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' : Classical Potential Theory and Its Probabilistic Counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Springer, Berlin  (1984)  [8] Dragnev, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=', Orive, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=', Saff, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=', Wielonsky F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=': Riesz energy problems with external fields  and related theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Constr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1007/s00365-022-09588-z  [9] Edwards, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' : Functional Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Theory and Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Holt, Rinehart and Winston,  New York (1965)  [10] Fuglede, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=': On the theory of potentials in locally compact spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' 103, 139–215  (1960)  [11] Fuglede, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=': The quasi topology associated with a countably subadditive set function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Fourier Grenoble 21, 123–169 (1971)  [12] Fuglede, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=': Symmetric function kernels and sweeping of measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' 42, 225–259  (2016)  [13] Fuglede, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=', Zorii, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=': Green kernels associated with Riesz kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Fenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  43, 121–145 (2018)  [14] Gauss, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' :  Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnisse des  Quadrats der Entfernung wirkenden Anziehungs– und Abstoßungs–Kräfte (1839).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Werke 5,  197–244 (1867)  Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields  27  [15] Kelley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' : General Topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Princeton, New York (1957)  [16] Kurokawa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=', Mizuta, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=': On the order at infinity of Riesz potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Hiroshima Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' 9,  533–545 (1979)  [17] Landkof, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' : Foundations of Modern Potential Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Springer, Berlin (1972)  [18] Ohtsuka, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=': On potentials in locally compact spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Hiroshima Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' A-I 25,  135–352 (1961)  [19] Riesz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=': Intégrales de Riemann–Liouville et potentiels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Acta Szeged 9, 1–42 (1938)  [20] Saff, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=', Totik, V: Logarithmic Potentials with External Fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Springer, Berlin (1997)  [21] Zorii, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=': Equilibrium potentials with external fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Ukrainian Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' 55, 1423–1444  (2003)  [22] Zorii, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=': Equilibrium problems for potentials with external fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Ukrainian Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' 55,  1588–1618 (2003)  [23] Zorii, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=': Constrained energy problems with external fields for vector measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Nachr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  285, 1144–1165 (2012)  [24] Zorii, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=': Equilibrium problems for infinite dimensional vector potentials with external fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  Potential Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' 38, 397–432 (2013)  [25] Zorii, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=': Necessary and sufficient conditions for the solvability of the Gauss variational problem  for infinite dimensional vector measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Potential Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' 41, 81–115 (2014)  [26] Zorii, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=': A theory of inner Riesz balayage and its applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  68, 41–67 (2020)  [27] Zorii, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=': Harmonic measure, equilibrium measure, and thinness at infinity in the theory of  Riesz potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Potential Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' 57, 447–472 (2022)  [28] Zorii, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=': Balayage of measures on a locally compact space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' 48, 249–277 (2022)  [29] Zorii, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=': On the theory of capacities on locally compact spaces and its interaction with the  theory of balayage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Potential Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1007/s11118-022-10010-3  [30] Zorii, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=':  On the theory of balayage on locally compact spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' Potential Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='1007/s11118-022-10024-x  [31] Zorii, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=': On the role of the point at infinity in Deny’s principle of positivity of mass for Riesz  potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='12418 (2022)  [32] Zorii, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=':  Minimum energy problems with external fields on locally compact spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='  arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='14342 (2022)  [33] Zorii, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=': Minimum Riesz energy problems with external fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content=' arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='05891 (2022)  Institute of Mathematics, Academy of Sciences of Ukraine, Tereshchenkivska 3,  01601, Kyiv-4, Ukraine, natalia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='zorii@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
+page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfh_ga/content/2301.00385v1.pdf'}
diff --git a/X9AzT4oBgHgl3EQfYvxp/content/tmp_files/2301.01340v1.pdf.txt b/X9AzT4oBgHgl3EQfYvxp/content/tmp_files/2301.01340v1.pdf.txt
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+INSCRIBED SQUARES AND RELATION AVOIDING PATHS.
+COLE HUGELMEYER
+Abstract. We develop a connection between the inscribed square problem and the question of
+understanding relation avoiding paths in a complex vector space.
+Our main theorem is that a
+Jordan curve with no inscribed squares would have a seemingly impossible structure which we call
+a square envelope. We will make some conjectures about the nature of relation avoiding paths in
+vector spaces, and show that these conjectures would imply the existence of inscribed squares in
+Jordan curves with finitely many arbitrarily complicated singularities.
+1. Introduction
+A Jordan curve is a continuous injective function γ : S1 Ñ R2 which wraps counterclockwise
+around the region it encloses. An inscribed square of γ is a quadruple of distinct points on impγq
+which form a square in R2.
+Figure 1. An inscribed square of a Jordan curve.
+Conjecture 1 (Toeplitz 1911). Every Jordan curve has an inscribed square.
+Towards this end, we define a bad Jordan curve to be a Jordan curve with no inscribed squares.
+Our main result is that a bad Jordan curve must have a structure which we call a square envelope.
+A square envelope is, loosely speaking, a square moving with time, so that its first two corners are
+always outside the Jordan curve, its second two corners are always inside the Jordan curve, and
+the outside corners wrap completely around the Jordan curve as time progresses.
+To rigorously define a square envelope, we let T : R2 Ñ R2 be the linear map corresponding
+to 90 degree counterclockwise rotation.
+For a pair of points a and b in the plane, we define
+S1pa, bq “ a`Tpb´aq and S2pa, bq “ b`Tpb´aq. If a ‰ b, then the four points a, b, S2pa, bq, S1pa, bq
+form the vertices of a square, labeled counterclockwise.
+1
+arXiv:2301.01340v1  [math.MG]  3 Jan 2023
+
+2
+COLE HUGELMEYER
+Definition 1. Let γ : S1 Ñ R2 be a Jordan curve. A square envelope of γ is a pair of continuous
+functions mapping from R to the plane, e1, e2 : R Ñ R2, such that all of the following are true.
+1) For all x P R, e1pxq and e2pxq are in the open exterior of γ.
+2) For all x P R, S1pe1pxq, e2pxqq and S2pe1pxq, e2pxqq are in the open interior of γ.
+3) limxÑ8 }e2pxq ´ e1pxq} “ limxÑ´8 }e2pxq ´ e1pxq} “ 0.
+4) Let ℓλ be the closed curve obtained by first following e1 from e1p´λq to e1pλq, then a
+straight line from e1pλq to e2pλq, then e2 from e2pλq to e2p´λq, and finally a straight line
+from e2p´λq back to e1p´λq. If p is any point inside γ, then there exists a real number
+λ0 ą 0 such for all λ ą λ0, we have that the winding number of ℓλ around p is equal to one.
+Figure 2. As square envelopes conjecturally do not exist, they are somewhat dif-
+ficult to draw. This is a rough approximation of what one might look like. The
+square must somehow have its first two corners move around the outside of the
+Jordan curve while the other two corners remain inside.
+Theorem 1. Every bad Jordan curve has a square envelope.
+This theorem allows us to make a connection between the inscribed square problem and the
+problem of understanding relation avoiding paths in a vector space. The set of squares is the vector
+space, and the linear relation we avoid is the first two corners of one square touching the second
+two corners of another.
+Let V be a vector space over C, and suppose R1, R2, ..., Rn are linear relations, subspaces of the
+vector space V ˆ V . We then let R be the relation given by the union of sets Yn
+i“1Ri. We will
+assume that R is symmetric, namely aRb ðñ bRa.
+A relation-avoiding path is a continuous function p : r0, 1s Ñ V such that there does not exist
+any pair of times t1, t2 in r0, 1s with ppt1qRppt2q. We define a relation-avoiding origin path to be a
+continuous function p : r0, 8q Ñ V with limtÑ8 pptq “ 0, such that there does not exist any pair
+of times t1, t2 in r0, 8q, for which ppt1qRppt2q. The origin is not included as the endpoint of this
+path because we always have 0R0.
+Definition 2. We say two relation avoiding origin paths p and q are weakly homotopic if there
+exists a continuous function h : r0, 1s ˆ r0, 8q Ñ V such that
+1) limtÑ8 hps, tq “ 0 for all s.
+2) hp0, tq “ pptq for all t.
+3) hp1, tq “ qptq for all t.
+4) There does not exist any ps, tq P r0, 1s ˆ r0, 8q with hps, tq R hps, 0q.
+
+INSCRIBED SQUARES AND RELATION AVOIDING PATHS.
+3
+We say p and q are strongly homotopic if h can be chosen so that hps, ´q is a relation avoiding
+origin path for all s.
+We propose the following conjecture about relation avoiding origin paths.
+Conjecture 2 (SC, Spiral Conjecture). Every relation avoiding origin path p is weakly homotopic
+to a relation avoiding origin path q for which there exist linearly independent vectors x1, ..., xn in
+V and complex numbers a1, ..., an such that
+qptq “
+nÿ
+i“1
+xi ¨ eait
+for all t.
+The following fact provides some evidence for this conjecture.
+Theorem 2. The spiral conjecture is true when V is one dimensional.
+Finally, we will prove the following implication of Conjecture 2.
+Theorem 3. Assume the spiral conjecture.
+Then any Jordan curve γ which is smooth except
+at finitely many points has an inscribed square. The Jordan curve is permitted to be arbitrarily
+complicated near these finitely many singularities.
+We propose that delving into the study of relation avoiding origin paths could be a potential
+approach towards solving the inscribed square problem. At the end of the paper, we will make
+more conjectures about relation avoiding origin paths, which we hope might guide future research.
+2. Square Envelopes
+If γ is a bad Jordan curve, we will define an integer bγ which we call the bad wrapping number
+of γ. This integer is only well-defined for bad Jordan curves. By counting inscribed squares in a
+generic approximation of γ, we will prove that bγ is odd. Then, we will use this fact to construct
+a square envelope of the Jordan curve.
+Let γ be a Jordan curve. We may choose a continuous function f : R2 Ñ R with the property
+that if a is in the interior of γ, then fpaq ă 0, if a is on the image of γ, then fpaq “ 0, and if a is
+outside of γ, then fpaq ą 0. Using this function, we can define a function gf,γ : S1 ˆ S1 Ñ R2, by
+the formula
+gf,γpx, yq “ pfpS1pγpxq, γpyqqq, fpS2pγpxq, γpyqqqq.
+If γ is a bad Jordan curve, then gf,γpx, yq “ p0, 0q if and only if x “ y, because if we have
+x ‰ y and gf,γpx, yq “ p0, 0q, then γpxq, γpyq, S2pγpxq, γpyqq, S1pγpxq, γpyqqq form the vertices of an
+inscribed square.
+Let C “ tpx, yq P S1 ˆ S1 : x ‰ yu.
+Topologically, C is an open cylinder.
+By the above
+proposition, we see that if γ is a bad Jordan curve, then we can restrict the domain of gf,γ to C to
+get a map
+gf,γ|C : C Ñ R2ztp0, 0qu.
+We now fix homotopy equivalences between C, R2ztp0, 0qu, and S1. Our homotopy equivalence
+C Ñ S1 is to simply take the first coordinate of the ordered pair. Our homotopy equivalence
+R2ztp0, 0qu Ñ S1 is radial projection onto the unit circle, which we orient counterclockwise.
+Using these homotopy equivalences, we see that, gf,γ|C induces a map S1 Ñ S1 up to homotopy,
+and therefore gives us an element of π1pS1q » Z. This integer is defined to be bγ.
+bγ does not depend on the choice of f, because given two choices of f, one can be continuously
+transformed into the other while always remaining a valid choice of f. This induces a homotopy
+between the resulting maps S1 Ñ S1, which implies that the value of bγ is constant.
+Lemma 1. For any bad Jordan curve, bγ is odd.
+
+4
+COLE HUGELMEYER
+The proof of this lemma will be given later in this section. The main idea behind the proof is
+that each inscribed square of a generic smooth approximation of γ contributes parity to bγ in a way
+that depends on the cyclic order of the vertices of the square on the Jordan curve. The parity of
+inscribed squares with a given cyclic ordering is independent of the Jordan curve, so we can simply
+calculate the parity of bγ by summing the contributions from each square type. The result is odd
+parity.
+Lemma 1 can be used to prove Theorem 1, which we will do at the end of this section. The main
+idea is that because bγ ‰ 0, the map gf,γ : C Ñ R2ztp0, 0qu wraps nontrivially around the origin.
+This implies that there must be a path between the two ends of the cylinder C that maps into the
+lower left quadrant of the plane under gf,γ. This path is almost a square envelope, except that e1
+and e2 are on the Jordan curve rather than being inside its open exterior. This can be remedied
+by simply pushing the paths off of the curve slightly.
+For a manifold M, let ˜C4pMq denote the manifold of cyclically ordered quadruples of distinct
+points in M. In other words, if C4 distpMq is the subset of M4 consisting of ordered quadruples of
+distinct points, then ˜C4pMq “ C4 distpMq{pZ{4Zq, where Z{4Z acts by cyclic permutation of the
+entries of the 4-tuple. Let Sq denote the the submanifold of ˜C4pR2q consisting of quadruples of
+points which form squares labeled counterclockwise.
+We call a Jordan curve γ : S1 Ñ R2 generic if, within ˜C4pR2q, the subspace ˜C4pimpγqq intersects
+transversely with Sq.
+Let h : S1 ˆ r0, 1s Ñ R2 be a smooth homotopy between two generic Jordan curves for which
+hp¨, tq is a smooth embedding for all t P r0, 1s. We say that h is a generic homotopy if, within
+˜C4pR2q ˆ r0, 1s, the subspace tpq, tq : q P ˜C4pimphp¨, tqqu intersects transversely with the subspace
+Sq ˆ r0, 1s.
+By the transversality theory found in [3], we have the following facts about generic Jordan curves.
+1) Any Jordan curve can be approximated arbitrarily well in the C0 topology by generic Jordan
+curves.
+2) Any two generic Jordan curves have a generic homotopy between them.
+3) For a generic homotopy h, the intersection tpq, tq : q P ˜C4pimphp¨, tqqu X Sq ˆ r0, 1s is a
+compact 1-manifold with boundary. The boundary of this manifold consists of the inscribed
+squares of the two generic Jordan curves at the beginning and end of the homotopy.
+Note that the manifold ˜C4pS1q has three connected components. This corresponds to three types
+of inscribed square.
+Definition 3. Let γ be a Jordan curve. An inscribed square of γ is called type I if the counter-
+clockwise order of points on the square is the same as that of the Jordan curve. Such squares are
+also called gracing squares. A square is called type II if when we label the vertices of the square as
+1, 2, 3, 4 in counterclockwise order, these vertices appear in the order 1, 3, 2, 4 when we go counter-
+clockwise along the Jordan curve. Finally, a square is called type III if the counterclockwise order
+of the vertices is opposite to the counterclockwise order of the Jordan curve.
+Figure 3. Inscribed squares of type I, II, and III respectively.
+
+INSCRIBED SQUARES AND RELATION AVOIDING PATHS.
+5
+Proposition 1. If γ is a generic Jordan curve, then of the inscribed squares of γ, an odd number
+of them are type I, an even number are type II, and an even number are type III.
+Proof. There are three connected components of ˜C4pimpγqq corresponding to the three types of
+inscribed squares when we intersect with Sq. This means that if h is a generic homotopy between
+generic Jordan curves, then the 1-manifold tpq, tq : q P ˜C4pimphp¨, tqqu X Sqˆr0, 1s can be separated
+into three distinct clopen pieces, one for each type of inscribed square. This proves that the parity
+of each type of inscribed square is invariant of which generic Jordan curve we choose. Finally
+we can compute the parities by finding the inscribed squares of any generic Jordan curve. The
+quintessential generic Jordan curve is a non-circular ellipse. This curve has one gracing square, but
+no inscribed square of type II or III.
+□
+Let γ be a generic Jordan curve, and let f be a smooth function R2 Ñ R which has γ as its zero
+set, is negative inside of γ, positive outside of γ, and has non-vanishing gradient on γ. Since γ is
+a smooth embedding, such an f always exists. Now, we will consider gf,γ : pS1q2 Ñ R2, as defined
+in the previous section.
+We see that the zeros of gf,γ consist of the diagonal ∆ “ tpa, aq : a P S1u, as well as four points
+for each inscribed square in ˜C4pimpγqq X Sq, one point for each side of the square. Furthermore, at
+these zeros corresponding to inscribed squares, gf,γ has nonsingular derivative because γ is generic
+and f has non-vanishing gradient on impγq. Let X “ tp P pS1q2 : gf,γppq ‰ p0, 0qu. Let ωγ be the
+cohomology class in H1pX; Z{2Zq given by pulling back the nontrivial class of H1pR2ztp0, 0qu; Z{2Zq
+under gf,γ|X. A small loop around any of the zeroes corresponding to sides of inscribed squares will
+evaluate nontrivially under ω because gf,γ has nonsingular derivative at these zeroes. Furthermore,
+a loop obtained by pushing the diagonal off to one side or the other will evaluate trivially under
+ωγ because it will correspond to a small square sliding around the Jordan curve with two corners
+on γ and the other two corners off to one side of γ, so the corresponding loop in R2ztp0, 0qu will
+remain in just one quadrant of the plane and therefore cannot wrap around the origin.
+Putting this all together, we see that if ℓ is a simple closed curve in X which has the homotopy
+class of the diagonal in S1 ˆ S1, then ωγpℓq is equal to the parity of the number of zeroes of gf,γ in
+either connected component of pS1q2zp∆ Y ℓq.
+Let ∆1 be the anti-diagonal of pS1q2, the loop consisting of pairs pa, bq where a and b are antipodes.
+Then pS1q2zp∆ Y ∆1q has two connected components, which we call A and B. The component A is
+the set of pairs pa, bq for which the angle from a to b is less than π and the component B is the set
+of pairs for which the angle from a to b is greater than π.
+If γ is a bad Jordan curve, and γ1 is a sufficiently C0-close generic approximation, then the parity
+of bγ is equal to ωγ1p∆1q. We can therefore compute bγ by counting the zeroes of gf,γ1 in A.
+Proposition 2. If γ is a bad Jordan curve, and γ1 is a sufficiently C0-close generic approximation,
+then each type I inscribed square of γ1 has exactly three of its corresponding zeroes of gf,γ1 in A,
+each type II inscribed square has exactly two of its corresponding zeroes in A, and each type III
+inscribed square has exactly one of its corresponding zeroes in A.
+Proof. Let γ1, γ2, γ3, ... be a sequence of generic Jordan curves that limit to our bad Jordan curve
+γ in the C0 topology. Let sn be the side length of the largest inscribed square of γn. We know
+that limnÑ8 sn “ 0 because otherwise there would be a convergent subsequence of squares that
+limit to an inscribed square of our bad Jordan curve. We can give S1 a metric by identifying it
+with the unit circle in R2 and using euclidean distance. Using this identification, we let M be the
+set of all pairs pa, bq P pS1q2 which have }a ´ b} ě 1. Let N P Z be sufficiently large that both
+supxPS1 }γNpxq ´ γpxq} and sN are smaller than the quantity ε “ 1
+4 ¨ minpa,bqPM }γpaq ´ γpbq}. We
+claim that γ1 “ γN is an approximation for which the proposition holds.
+Let a, b, c, d be the vertices of an inscribed square of γN, labeled counterclockwise. Let Q be the
+set of four points in S1 which map onto ta, b, c, du under γN. We claim that the diameter of Q is
+
+6
+COLE HUGELMEYER
+less than one. To prove this, suppose there were elements x and y in Q with }x ´ y} ě 1. Then
+px, yq P M, so }γpxq ´ γpyq} ě 4ε, so }γNpxq ´ γNpyq} ě 2ε. However, 2ε ą
+?
+2 ¨ sN and γNpxq and
+γNpyq are vertices of a square with side length at most sN, so this is impossible.
+Since the diameter of Q is less than 1, all four points of Q must lie within some interval I of
+angular length π{3 radians. Orient I counterclockwise, and let w, x, y, z be the four elements of Q
+in the order they appear along I. We know that pw, xq, pw, yq, pw, zq, px, yq, px, zq, and py, zq are in
+A, and all the other ordered pairs are in B.
+We know that γN maps tw, x, y, zu onto ta, b, c, du, but it might not preserve the ordering.
+Without loss of generality, we can assume γNpwq “ a because we can permute the labels a, b, c, d
+cyclically. Then, we have six possibilities to check for the six permutations of the remaining three
+letters. We will now check all of the possibilities.
+1) If pw, x, y, zq ÞÑ pa, b, c, dq, the square is type I and the corresponding zeroes in A are
+pw, xq, px, yq, py, zq.
+2) If pw, x, y, zq ÞÑ pa, c, b, dq, the square is type II and the corresponding zeroes in A are
+pw, yq, px, zq.
+3) If pw, x, y, zq ÞÑ pa, b, d, cq, the square is type II and the corresponding zeroes in A are
+pw, xq, px, zq.
+4) If pw, x, y, zq ÞÑ pa, d, c, bq, the square is type III and the only corresponding zero in A is
+pw, zq.
+5) If pw, x, y, zq ÞÑ pa, d, b, cq, the square is type II and the corresponding zeroes in A are
+pw, yq, py, zq.
+6) If pw, x, y, zq ÞÑ pa, c, d, bq, the square is type II and the corresponding zeroes in A are
+pw, zq, px, yq.
+This confirms the proposition. The type I squares have three corresponding zeroes in A, the type
+II squares have two corresponding zeroes in A, and the type III squares have only one corresponding
+zero in A.
+□
+We can now prove Lemma 1 and Theorem 1.
+Proof of Lemma 1. Let γ be a bad Jordan curve, let γ1 be a sufficiently C0-close generic approxi-
+mation, and let f be a function with γ1 as its zero set as above. We now count zeroes of gf,γ1 in A,
+the parity of which is the parity of bγ. We count the zeroes by those corresponding to the squares
+of each type. We have an odd number of threes, an even number of twos, and an even number of
+ones. This totals to an odd number.
+Proof of Theorem 1. We wish to prove that every bad Jordan curve has a square envelope. We
+have the function gf,γ|C : C Ñ R2ztp0, 0qu, which we know to be homotopically nontrivial. Let
+π : R2ztp0, 0qu Ñ S1 be radial projection onto the unit circle, and let r “ π ˝gf,γ|C. We now take r1
+to be a smooth approximation of r with }rpxq ´ r1pxq} ă 1{4 for all x P C. Let u P S1 be a regular
+value for r1 within distance 1{4 of the point p´1{
+?
+2, ´1{
+?
+2q. Then r´1puq is a 1-dimensional
+manifold, L, which is mapped under r into a circle of radius 1{2 around p´1{
+?
+2, ´1{
+?
+2q, which
+is entirely in the lower left quadrant of the plane. Therefore, gf,γ|C maps L into the lower left
+quadrant of the plane. This implies that the square corresponding to a point of L has its first two
+corners on γ, and the other two in the open interior of γ. Furthermore, since bγ is odd, we see that
+L must have an odd number of connected components which are homeomorphic to R with the two
+ends limiting to the two boundaries of C. Parameterizing such a connected component, we have
+functions e1, e2 : R Ñ R2 with impe1qYimpe2q Ď impγq, and limxÑ˘8 }e1pxq´e2pxq} “ 0, and with
+the property that S1pe1pxq, e2pxqq and S2pe1pxq, e2pxqq are always in the open interior of γ. Since
+the interior of gamma is open, we may choose a continuous function ε : R Ñ Rą0 with the property
+that, for all x, if }a ´ e1pxq} and }b ´ e2pxq} are less than εpxq, then S1pa, bq and S2pa, bq are in the
+
+INSCRIBED SQUARES AND RELATION AVOIDING PATHS.
+7
+open interior of γ as well. Thus, if we choose a continuous deformation of pe1, e2q in which we push
+e1pxq and e2pxq off of the Jordan curve without exceeding a distance of εpxq, we obtain a square
+envelope for γ.
+3. Relation avoiding paths
+In this section, we will prove that if the spiral conjecture is true, then the inscribed square conjec-
+ture holds for Jordan curves which are smooth except at a finite number of arbitrarily complicated
+singularities.
+However, we will first present a proof of Theorem 2, that the spiral conjecture is indeed true in
+the one-dimensional case.
+Proof of Theorem 2. Identifying V with C, our relation avoiding origin paths are paths to the origin
+in the complex plane p : r0, 8q Ñ C. Furthermore, we know that pptq is never zero, because p0, 0q is
+in every linear relation. Thus, without loss of generality, we can ignore any of the Ri corresponding
+to t0u ˆ C or C ˆ t0u. We can therefore rewrite the relation R as
+xRy ðñ Di P t1, ..., nu
+x “ αiy or y “ αix
+where α1, ..., αn are nonzero complex numbers with norm at most one. In particular, our path p
+is relation avoiding if and only if it is disjoint from the paths α1p, α2p, ..., αnp. Without loss of
+generality, we can assume that pp0q “ 1, and that |pptq| ď 1 for all t. The reason we can do this
+is that we can always homotope p to such a path by first contracting it within its image so that
+the starting point is at the point with maximal norm, and then rescaling within C so that the
+starting point becomes 1. We can also assume without loss of generality that p follows the path of
+a logarithmic spiral inside some small neighborhood around 1, because making this happen only
+requires a small perturbation. Working with these assumptions, we fix a branch of the logarithm,
+and let ℓ1, ..., ℓn be paths in upper-half plane such that e2πiℓiptq “ αipptq with ℓip0q “
+1
+2πi lnpαiq,
+and similarly let ℓ be a lift of p with ℓp0q “ 0. The path ℓ divides the upper-half plane, and we
+define U to be the set of all points of Hzimpℓq from which a path to ´8 has even intersection parity
+with ℓ. We then define integers k1, ..., kn, where ki is the largest integer such that ℓip0q ` ki P U.
+We see that ℓ is disjoint from ℓi `k for any index i and integer k, and this gives us paths disjoint
+from ℓ, going up to i8, from every point of the form ℓp0q ` k. Therefore, the homotopy type of ℓ
+can be completely determined by knowing which points of the form ℓip0q ` k are in U. Thus, to
+prove the one-dimensional spiral conjecture, it suffices to prove that there exists a θ P p0, πq so that
+for all i P t1, ..., nu, we have the inequalities argpℓip0q ` kiq ě θ ě argpℓip0q ` ki ` 1q. This θ would
+then denote the angle of a straight line in the upper half plane that exponentiates to the desired
+logarithmic spiral. To prove that such a θ exists, it suffices to prove that for all i and j we have
+argpℓjp0q ` kj ` 1q ď argpℓip0q ` kiq.
+We define a split pair to be a pair of points in the upper half-plane pp, qq, such that p ` ℓ and
+q ` ℓ are both disjoint from ℓ, with p P U and q R U. For instance, pℓip0q ` ki, ℓjp0q ` kj ` 1q is
+always a split pair.
+
+8
+COLE HUGELMEYER
+Figure 4. A depiction of a split pair, disjoint translates of ℓ, one on each side.
+Given a split pair pp, qq, we can construct a new split pair as follows. If imppq ě impqq, then
+pp ´ q, qq is the new split pair. If impqq ą imppq, then pp, q ´ pq is the new split pair. The split
+pair obtained by applying this transformation is called the derived split pair. We claim that the
+derived split pair is always another split pair.
+Figure 5. One way to understand derived split pairs is that one draws a parallel-
+ogram spanning the origin and the two points, then moves the uppermost point to
+the new corner.
+To prove this claim, we consider the case that imppq ě impqq. We have that ℓ ` q is entirely on
+the right side of ℓ, and ℓ`p is entirely on the left side of ℓ. Furthermore, since all of ℓ`p has greater
+imaginary coordinate than impqq, we therefore have that p P ta P U : impaq ě impqqu Ď U ` q.
+This tells us that p ´ q is in U, so pp ´ q, qq is a valid split pair. The argument for the other case
+is similar.
+We say a split pair is good if argppq ą argpqq. We say a split pair is bad otherwise. We see that
+the derived split pair of a bad split pair is bad, and the derived split pair of a good split pair is
+
+INSCRIBED SQUARES AND RELATION AVOIDING PATHS.
+9
+good. To complete the proof of the theorem, it suffices to prove that all split pairs are good, so we
+will consider what occurs under repeated derivation of a bad split pair. First of all, note that if
+imppq{impqq is a rational number, then we will eventually have a split pair with a real number as
+one of the terms, and we can see that any split pair containing a real number is good. Thus, we have
+shown that imppq{impqq is irrational for any bad split pair. Now, we will proceed by contradiction
+to prove that there are no bad split pairs. Suppose that pp, qq is a bad split pair. We will treat the
+cases of argppq ă argpqq and argppq “ argpqq separately.
+Suppose that pp, qq is a bad split pair with argppq ă argpqq. Then, the i-th derived split pair is
+of the form paip ´ biq, ciq ´ dipq, for nonnegative integers ai, bi, ci, di such that
+lim
+iÑ8 ai{bi “ lim
+iÑ8 di{ci “ impqq{imppq
+This tells us that while the imaginary parts of the terms of the derived split pairs approach zero,
+the real parts approach `8 and ´8 respectively. However, this would force the second term to
+eventually be in U, which gives us a contradiction.
+Now, suppose argppq “ argpqq. In this case, the terms of the derived split pairs stay on a fixed
+straight line, and they approach zero. However, earlier in the proof, we assumed without loss of
+generality that p followed the path of a logarithmic spiral in some small neighborhood around 1,
+which means ℓ is a straight line in some small neighborhood around zero. This means that there
+are no bad split pairs in this neighborhood, so we have a contradiction. This completes the proof.
+□
+Before we prove Theorem 3, we need to develop some notation surrounding square envelopes,
+and prove a couple lemmas.
+Definition 4. Let γ : S1 Ñ R2 be a Jordan curve, and let pe1, e2q be a square envelope. We
+write σ˚
+ppe1, e2, γq, where p P t1, 2u and ˚ P t`, ´u, to denote a sign in t`1, ´1u determined as
+follows. For λ P R, let Pλ be the path that starts at e1pλq, then goes along e1 to e1p0q, then goes
+in a straight line from e1p0q to e2p0q, then follows e2 to e2pλq, then goes in a straight line back to
+the starting point e1pλq. We then let npλq be the wrapping number of Pλ around Sppe1p0q, e2p0qq.
+Finally, we set σ˚
+ppe1, e2, γq “ limλÑ˚8p´1qnpλq.
+It is worth observing that these parities are determined by how the Jordan curve winds between
+the vertices of a small square near t “ ˘8. For curves with finitely many singularities, they are
+related to each other by the following lemma.
+Lemma 2. If γ is a Jordan curve which is smooth except at finitely many points, then the following
+facts are true for any square envelope pe1, e2q of γ.
+1) σ`
+1 pe1, e2, γq “ σ`
+2 pe1, e2, γq and σ´
+1 pe1, e2, γq “ σ´
+2 pe1, e2, γq
+2) σ`
+1 pe1, e2, γq “ ´σ´
+1 pe1, e2, γq
+3) If ˚ P t`, ´u is such that σ˚
+1pe1, e2, γq “ ´1, then the limits limtÑ˚8 e1ptq and limtÑ˚8 e2ptq
+exist, and are equal to each other.
+Proof. First of all, by property (4) in the definition of a square envelope, we immediately have that
+σ`
+1 pe1, e2, γq “ ´σ´
+1 pe1, e2, γq and σ`
+2 pe1, e2, γq “ ´σ´
+2 pe1, e2, γq because the defining loops for the
+parities in question compose to give us a loop that wraps around the Jordan curve exactly once.
+Next, we claim that if ˚ P t`, ´u, and the limit limtÑ˚8 e1ptq does not exist, then we have
+σ˚
+1pe1, e2, γq “ σ˚
+2pe1, e2, γq “ 1. To prove this claim, note that for the limit to fail to exist, there
+must be a limit point of e1ptq, t Ñ ˚8 at one of the points where γ is smooth. Taking a sufficiently
+small square of the envelope, in the ˚8 direction, near such a point, we see that the Jordan curve
+must separate the vertices of the square in a way locally equivalent to how a straight line could
+separate the vertices, and there is only one such way that separates the e1, e2 vertices from the
+other two vertices. Since all squares sufficiently near t “ ˘8 will have side length smaller than the
+
+10
+COLE HUGELMEYER
+diameter of some disk inside of the Jordan curve, the paths that define σ˚
+1pe1, e2, γq and σ˚
+2pe1, e2, γq
+cannot wrap around either of the two interior vertices of the square at t “ 0. Therefore, we have
+σ˚
+1pe1, e2, γq “ σ˚
+2pe1, e2, γq “ 1.
+To prove all three parts of the lemma, all that remains is to eliminate the possibility that the limits
+limtÑ˚8 e1ptq and limtÑ˚8 e2ptq exist, but σ˚
+1pe1, e2, γq and σ˚
+2pe1, e2, γq are not equal to one an-
+other. The reason this is impossible is that for this to be the case, the paths e1ptq, e2ptq, S1pe1ptq, e2ptqq,
+and S2pe1ptq, e2ptqq would need to all approach some point in the plane, none of them intersect-
+ing each other, in such a way that as one encircles the point counterclockwise, the paths appear
+in a cyclic order that alternates between the sets te1ptq, e2ptqu and tS1pe1ptq, e2ptqq, S2pe1ptq, e2ptqqu.
+This would then contradict the fact that the simple closed curve γ separates the paths in te1ptq, e2ptqu
+from those in tS1pe1ptq, e2ptqq, S2pe1ptq, e2ptqqu.
+□
+If γ is a Jordan curve which is smooth except at finitely many points, and pe1, e2q is a square
+envelope, we say pe1, e2q is positively oriented if σ`
+1 pe1, e2, γq “ ´1 and negatively oriented if
+σ`
+1 pe1, e2, γq “ `1.
+Note that if pe1ptq, e2ptqq is a negatively oriented square envelope, we can
+obtain a positively oriented one by taking pe1p´tq, e2p´tqq.
+We need one more lemma before we can prove Theorem 3.
+Lemma 3. Let α1, α2, β1, β2 be complex numbers such that the ratio β1{β2 is a positive real
+number. Let t0 and p be arbitrary real numbers with p ‰ 0, and let λ, r1, r2 be real numbers in
+r0, 1q. If there exist real numbers t1 and t2 in r0, 8q such that
+eα1`β1ppt1´t0q ´
+1 ` λr1eippt1´t0q¯
+“ eα2`β2ppt2´t0q ´
+1 ` λr2eippt2´t0q¯
+then there also exist t1
+1 and t1
+2 in r0, 8q such that
+eα1`β1ppt1
+1´t0q ´
+1 ` r1eippt1
+1´t0q¯
+“ eα2`β2ppt1
+2´t0q ´
+1 ` r2eippt1
+2´t0q¯
+Proof. Making the substitutions α1 ´ α2 “ α, ppt1 ´ t0q “ t, and ppt2 ´ t0q “ pβ1{β2qt ` s, the
+equation
+eα1`β1ppt1´t0q ´
+1 ` λr1eippt1´t0q¯
+“ eα2`β2ppt2´t0q ´
+1 ` λr2eippt2´t0q¯
+rearranges to
+eα´β2s “
+1 ` λr1eit
+1 ` λr2eip β1
+β2 t`sq
+Rearranging, and setting a “ r1, bpsq “ ´r2eα`pi´β2qs, and cpsq “ eα`β2s ´ 1, we have
+cpsq “ λ ¨
+ˆ
+a ¨ eit ` bpsq ¨ ei
+´ β1
+β2 t
+¯˙
+Also, let
+Tpsq “ tt : Dt1 ě 0, Dt2 ě 0,
+ppt2 ´ t0q “ pβ1{β2qt ` s,
+ppt1 ´ t0q “ tu
+be the allowable values of t for fixed s, when t1 and t2 are nonnegative. This will always be an
+interval of the form p´8, zs or rz, 8q for some real number z. Now, let Hpsq denote the unique
+minimal simply connected set containing the set
+ta ¨ eit ` bpsq ¨ eipβ1{β2qt : t P Tpsqu
+Then, we see that there exists a pair of nonnegative real numbers t1 and t2 such that
+eα1`β1ppt1´t0q ´
+1 ` λr1eippt1´t0q¯
+“ eα2`β2ppt2´t0q ´
+1 ` λr2eippt2´t0q¯
+if and only if there exists a real number s such that cpsq P λHpsq. Since β1{β2 is positive, when
+β1{β2 is rational, Hpsq is the region enclosed by the outer boundary of an epitrochoid curve. When
+β1{β2 is irrational, Hpsq it is a disk missing some subset of its boundary. In particular, Hpsq is
+
+INSCRIBED SQUARES AND RELATION AVOIDING PATHS.
+11
+always a radial set, meaning that for any x P Hpsq, we have λx P Hpsq for all λ P r0, 1s. Therefore,
+increasing the parameter λ preserves the existence of solutions to our original equation. Setting
+λ “ 1 gives us the desired result.
+Figure 6. The region enclosed by an epitrochoid curve is always a radial set.
+□
+Now, we can prove Theorem 3.
+Proof of Theorem 3. We begin by considering the system of linear relations on C2 given by letting
+px1, y1qRpx2, y2q when
+tx1, y1, x2, y2u X tx1 ` ipy1 ´ x1q, y1 ` ipy1 ´ x1q, x2 ` ipy2 ´ x2q, y2 ` ipy2 ´ x2qu ‰ ∅
+Now, take a positively oriented square envelope pe1, e2q. Due to Lemma 2, restricting the domain
+to r0, 8q immediately gives us a relation avoiding origin path pe1ptq, e2ptqq ´ limtÑ8pe1ptq, e2ptqq
+for R. We are assuming the spiral conjecture, so we may mow take h : r0, 1s ˆ r0, 8q satisfying the
+conditions of the spiral conjecture. This gives us complex numbers a1, a2, and vectors x1, x2 P C2,
+such that pf1ptq, f2ptqq “ fptq “ x1ea1t ` x2ea2t is a relation avoiding origin path for R which has
+the property that the path in C starting at 0, following f2 to f2p0q, going in a straight line from
+f2p0q to f1p0q, then following f1 back to 0 has nontrivial wrapping number parity around the points
+f1p0q `ipf2p0q ´f1p0qq and f2p0q `ipf2p0q´f1p0qq. Without loss of generality, we can assume that
+a1 and a2 have the same real part, because otherwise one term will dominate at large t, reducing
+us to the case of a pure logarithmic spiral, which we will cover later. Similarly, we may assume
+a1 ‰ a2. Now, to prove that no such a1, a2, x1, x2 exist, we see that for the path to be relation
+avoiding, we must be able to find some fixed complex number β, and nonzero real number p, such
+that all four paths f1ptq, f2ptq, f1ptq ` ipf2ptq ´ f1ptqq, and f2ptq ` ipf2ptq ´ f1ptqq are of the form
+eα`qβppt´t0q ´
+1 ` reippt´t0q¯
+with α P C, t0 P R, q P p0, 8q, and r P r0, 1q. Therefore, Lemma 3 allows us to continuously scale
+down all of the r parameters to zero while staying a relation avoiding origin path.
+All that remains is to show that no pure logarithmic spiral fptq “ xeat can be a relation avoiding
+origin path with the stated wrapping number parity condition. To prove this, note that the region
+swept out by a line segment between f1 and f2 would have the same area as the region swept out
+by the line segment between f1 ` ipf2 ´ f1q and f2 ` ipf2 ´ f1q. However, our wrapping number
+
+12
+COLE HUGELMEYER
+condition implies that the former region would need to strictly contain the latter, which would
+contradict them having equal areas.
+□
+To wrap things up, we make a couple more conjectures about relation avoiding origin paths.
+Conjecture 3 (SSC, Strong Spiral Conjecture). Every relation avoiding origin path p is strongly
+homotopic to a relation avoiding origin path q for which there exist linearly independent vectors
+x1, ..., xn in V and complex numbers a1, ..., an such that
+qptq “
+nÿ
+i“1
+xi ¨ eait
+for all t.
+Conjecture 4 (MSC, Monotonic Spiral Conjecture). For any continuous function p : r0, 8q Ñ
+V zt0u with limtÑ8 pptq “ 0, there exists a continuous function h : r0, 1s ˆ r0, 8q Ñ V with the
+following properties.
+1) limtÑ8 hps, tq “ 0 for all s.
+2) hp0, tq “ pptq for all t.
+3) There exist linearly independent vectors x1, ..., xn in V and complex numbers a1, ..., an such
+that
+hp1, tq “
+nÿ
+i“1
+xi ¨ eait
+for all t.
+4) If 0 ď s1 ď s2 ď 1, then Bps2q Ď Bps1q, where
+Bpsq :“
+ď
+pt1,t2qPr0,8q2
+tf P pV ˆ V q1 : fphps, t1q, hps, t2qq “ 0u
+It is not too difficult to prove the implications MSC
+ùñ SSC
+ùñ SC. The strong spiral
+conjecture clearly implies the spiral conjecture, but it might not be so easy to see why the monotonic
+spiral conjecture implies the strong spiral conjecture. The reason is that a small perturbation of
+the homotopy can be made to avoid any relations with codimension greater than one in V ˆ V ,
+and the homotopy for the monotonic spiral conjecture will always increase the set of codimension
+one relations that it avoids.
+Morally speaking, the monotonic spiral conjecture should be true in the one-dimensional case
+because curve shortening flow in a logarithmic metric should have the desired property.
+This
+argument should at least cover the case of smooth paths which are periodic deviations from a
+logarithmic spiral, but the analysis is more complicated for arbitrary continuous paths. It would be
+useful if there was a way to generalize such an argument to higher dimensions, though it is not at
+all obvious how to achieve this. Regardless, we hope that a deep enough understanding of relation
+avoiding paths will lead to new progress on the inscribed square problem.
+
+INSCRIBED SQUARES AND RELATION AVOIDING PATHS.
+13
+References
+[1] Arseniy Akopyan and Sergey Avvakumov. Any cyclic quadrilateral can be inscribed in any closed convex smooth
+curve. Forum of Mathematics, Sigma, 2018.
+[2] H.B. Griffiths. The topology of square pegs in round holes. Proceedings of the London Mathematical Society,
+s3-62(3):647–672, 1991.
+[3] John McCleary Jason Cantarella, Elizabeth Denne. Transversality for configuration spaces and the “square-peg”
+theorem. arXiv:1402.6174 [math.GT], 2014.
+[4] Benjamin Matschke. Equivariant topology methods in discrete geometry. PhD thesis, Freie University at Berlin,
+2011.
+[5] Benjamin Matschke. A survey on the square peg problem. Notices Amer. Math. Soc., 61(4):346–352, 2014.
+[6] Benjamin Matschke. Quadrilaterals inscribed in convex curves. arXiv:1801.01945 [math.MG], 2018.
+[7] M. J. Nielsen and S. E. Wright. Rectangles inscribed in symmetric continua. Geometriae Dedicata, 56(3):285–297,
+1995.
+[8] L. G. Schnirelman. On some geometric properties of closed curves. Usp. Mat. Nauk, 10:34–44, 1944.
+[9] R.E. Schwartz. A trichotomy for rectangles inscribed in jordan loops. arXiv:1804.00740 [math.MG], 2018.
+[10] Terence Tao. An integration approach to the toeplitz square peg problem. Forum of Mathematics, Sigma, (5),
+2017.
+
diff --git a/X9AzT4oBgHgl3EQfYvxp/content/tmp_files/load_file.txt b/X9AzT4oBgHgl3EQfYvxp/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..388186ababa78bb50a9750cb8701bbf507378898
--- /dev/null
+++ b/X9AzT4oBgHgl3EQfYvxp/content/tmp_files/load_file.txt
@@ -0,0 +1,412 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf,len=411
+page_content='INSCRIBED SQUARES AND RELATION AVOIDING PATHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  COLE HUGELMEYER  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We develop a connection between the inscribed square problem and the question of  understanding relation avoiding paths in a complex vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Our main theorem is that a  Jordan curve with no inscribed squares would have a seemingly impossible structure which we call  a square envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We will make some conjectures about the nature of relation avoiding paths in  vector spaces, and show that these conjectures would imply the existence of inscribed squares in  Jordan curves with finitely many arbitrarily complicated singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Introduction  A Jordan curve is a continuous injective function γ : S1 Ñ R2 which wraps counterclockwise  around the region it encloses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' An inscribed square of γ is a quadruple of distinct points on impγq  which form a square in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' An inscribed square of a Jordan curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Conjecture 1 (Toeplitz 1911).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Every Jordan curve has an inscribed square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Towards this end, we define a bad Jordan curve to be a Jordan curve with no inscribed squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Our main result is that a bad Jordan curve must have a structure which we call a square envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  A square envelope is, loosely speaking, a square moving with time, so that its first two corners are  always outside the Jordan curve, its second two corners are always inside the Jordan curve, and  the outside corners wrap completely around the Jordan curve as time progresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  To rigorously define a square envelope, we let T : R2 Ñ R2 be the linear map corresponding  to 90 degree counterclockwise rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  For a pair of points a and b in the plane, we define  S1pa, bq “ a`Tpb´aq and S2pa, bq “ b`Tpb´aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' If a ‰ b, then the four points a, b, S2pa, bq, S1pa, bq  form the vertices of a square, labeled counterclockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='01340v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='MG]  3 Jan 2023  2  COLE HUGELMEYER  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Let γ : S1 Ñ R2 be a Jordan curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' A square envelope of γ is a pair of continuous  functions mapping from R to the plane, e1, e2 : R Ñ R2, such that all of the following are true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  1) For all x P R, e1pxq and e2pxq are in the open exterior of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  2) For all x P R, S1pe1pxq, e2pxqq and S2pe1pxq, e2pxqq are in the open interior of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  3) limxÑ8 }e2pxq ´ e1pxq} “ limxÑ´8 }e2pxq ´ e1pxq} “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  4) Let ℓλ be the closed curve obtained by first following e1 from e1p´λq to e1pλq, then a  straight line from e1pλq to e2pλq, then e2 from e2pλq to e2p´λq, and finally a straight line  from e2p´λq back to e1p´λq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' If p is any point inside γ, then there exists a real number  λ0 ą 0 such for all λ ą λ0, we have that the winding number of ℓλ around p is equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' As square envelopes conjecturally do not exist, they are somewhat dif-  ficult to draw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' This is a rough approximation of what one might look like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' The  square must somehow have its first two corners move around the outside of the  Jordan curve while the other two corners remain inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Every bad Jordan curve has a square envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  This theorem allows us to make a connection between the inscribed square problem and the  problem of understanding relation avoiding paths in a vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' The set of squares is the vector  space, and the linear relation we avoid is the first two corners of one square touching the second  two corners of another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Let V be a vector space over C, and suppose R1, R2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=', Rn are linear relations, subspaces of the  vector space V ˆ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We then let R be the relation given by the union of sets Yn  i“1Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We will  assume that R is symmetric, namely aRb ðñ bRa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  A relation-avoiding path is a continuous function p : r0, 1s Ñ V such that there does not exist  any pair of times t1, t2 in r0, 1s with ppt1qRppt2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We define a relation-avoiding origin path to be a  continuous function p : r0, 8q Ñ V with limtÑ8 pptq “ 0, such that there does not exist any pair  of times t1, t2 in r0, 8q, for which ppt1qRppt2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' The origin is not included as the endpoint of this  path because we always have 0R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We say two relation avoiding origin paths p and q are weakly homotopic if there  exists a continuous function h : r0, 1s ˆ r0, 8q Ñ V such that  1) limtÑ8 hps, tq “ 0 for all s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  2) hp0, tq “ pptq for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  3) hp1, tq “ qptq for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  4) There does not exist any ps, tq P r0, 1s ˆ r0, 8q with hps, tq R hps, 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  INSCRIBED SQUARES AND RELATION AVOIDING PATHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  3  We say p and q are strongly homotopic if h can be chosen so that hps, ´q is a relation avoiding  origin path for all s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  We propose the following conjecture about relation avoiding origin paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Conjecture 2 (SC, Spiral Conjecture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Every relation avoiding origin path p is weakly homotopic  to a relation avoiding origin path q for which there exist linearly independent vectors x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=', xn in  V and complex numbers a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=', an such that  qptq “  nÿ  i“1  xi ¨ eait  for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  The following fact provides some evidence for this conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' The spiral conjecture is true when V is one dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Finally, we will prove the following implication of Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Assume the spiral conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Then any Jordan curve γ which is smooth except  at finitely many points has an inscribed square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' The Jordan curve is permitted to be arbitrarily  complicated near these finitely many singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  We propose that delving into the study of relation avoiding origin paths could be a potential  approach towards solving the inscribed square problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' At the end of the paper, we will make  more conjectures about relation avoiding origin paths, which we hope might guide future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Square Envelopes  If γ is a bad Jordan curve, we will define an integer bγ which we call the bad wrapping number  of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' This integer is only well-defined for bad Jordan curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' By counting inscribed squares in a  generic approximation of γ, we will prove that bγ is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Then, we will use this fact to construct  a square envelope of the Jordan curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Let γ be a Jordan curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We may choose a continuous function f : R2 Ñ R with the property  that if a is in the interior of γ, then fpaq ă 0, if a is on the image of γ, then fpaq “ 0, and if a is  outside of γ, then fpaq ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Using this function, we can define a function gf,γ : S1 ˆ S1 Ñ R2, by  the formula  gf,γpx, yq “ pfpS1pγpxq, γpyqqq, fpS2pγpxq, γpyqqqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  If γ is a bad Jordan curve, then gf,γpx, yq “ p0, 0q if and only if x “ y, because if we have  x ‰ y and gf,γpx, yq “ p0, 0q, then γpxq, γpyq, S2pγpxq, γpyqq, S1pγpxq, γpyqqq form the vertices of an  inscribed square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Let C “ tpx, yq P S1 ˆ S1 : x ‰ yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Topologically, C is an open cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  By the above  proposition, we see that if γ is a bad Jordan curve, then we can restrict the domain of gf,γ to C to  get a map  gf,γ|C : C Ñ R2ztp0, 0qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  We now fix homotopy equivalences between C, R2ztp0, 0qu, and S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Our homotopy equivalence  C Ñ S1 is to simply take the first coordinate of the ordered pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Our homotopy equivalence  R2ztp0, 0qu Ñ S1 is radial projection onto the unit circle, which we orient counterclockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Using these homotopy equivalences, we see that, gf,γ|C induces a map S1 Ñ S1 up to homotopy,  and therefore gives us an element of π1pS1q » Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' This integer is defined to be bγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  bγ does not depend on the choice of f, because given two choices of f, one can be continuously  transformed into the other while always remaining a valid choice of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' This induces a homotopy  between the resulting maps S1 Ñ S1, which implies that the value of bγ is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' For any bad Jordan curve, bγ is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  4  COLE HUGELMEYER  The proof of this lemma will be given later in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' The main idea behind the proof is  that each inscribed square of a generic smooth approximation of γ contributes parity to bγ in a way  that depends on the cyclic order of the vertices of the square on the Jordan curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' The parity of  inscribed squares with a given cyclic ordering is independent of the Jordan curve, so we can simply  calculate the parity of bγ by summing the contributions from each square type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' The result is odd  parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Lemma 1 can be used to prove Theorem 1, which we will do at the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' The main  idea is that because bγ ‰ 0, the map gf,γ : C Ñ R2ztp0, 0qu wraps nontrivially around the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  This implies that there must be a path between the two ends of the cylinder C that maps into the  lower left quadrant of the plane under gf,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' This path is almost a square envelope, except that e1  and e2 are on the Jordan curve rather than being inside its open exterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' This can be remedied  by simply pushing the paths off of the curve slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  For a manifold M, let ˜C4pMq denote the manifold of cyclically ordered quadruples of distinct  points in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' In other words, if C4 distpMq is the subset of M4 consisting of ordered quadruples of  distinct points, then ˜C4pMq “ C4 distpMq{pZ{4Zq, where Z{4Z acts by cyclic permutation of the  entries of the 4-tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Let Sq denote the the submanifold of ˜C4pR2q consisting of quadruples of  points which form squares labeled counterclockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  We call a Jordan curve γ : S1 Ñ R2 generic if, within ˜C4pR2q, the subspace ˜C4pimpγqq intersects  transversely with Sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Let h : S1 ˆ r0, 1s Ñ R2 be a smooth homotopy between two generic Jordan curves for which  hp¨, tq is a smooth embedding for all t P r0, 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We say that h is a generic homotopy if, within  ˜C4pR2q ˆ r0, 1s, the subspace tpq, tq : q P ˜C4pimphp¨, tqqu intersects transversely with the subspace  Sq ˆ r0, 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  By the transversality theory found in [3], we have the following facts about generic Jordan curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  1) Any Jordan curve can be approximated arbitrarily well in the C0 topology by generic Jordan  curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  2) Any two generic Jordan curves have a generic homotopy between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  3) For a generic homotopy h, the intersection tpq, tq : q P ˜C4pimphp¨, tqqu X Sq ˆ r0, 1s is a  compact 1-manifold with boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' The boundary of this manifold consists of the inscribed  squares of the two generic Jordan curves at the beginning and end of the homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Note that the manifold ˜C4pS1q has three connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' This corresponds to three types  of inscribed square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Let γ be a Jordan curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' An inscribed square of γ is called type I if the counter-  clockwise order of points on the square is the same as that of the Jordan curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Such squares are  also called gracing squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' A square is called type II if when we label the vertices of the square as  1, 2, 3, 4 in counterclockwise order, these vertices appear in the order 1, 3, 2, 4 when we go counter-  clockwise along the Jordan curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Finally, a square is called type III if the counterclockwise order  of the vertices is opposite to the counterclockwise order of the Jordan curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Inscribed squares of type I, II, and III respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  INSCRIBED SQUARES AND RELATION AVOIDING PATHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  5  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' If γ is a generic Jordan curve, then of the inscribed squares of γ, an odd number  of them are type I, an even number are type II, and an even number are type III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' There are three connected components of ˜C4pimpγqq corresponding to the three types of  inscribed squares when we intersect with Sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' This means that if h is a generic homotopy between  generic Jordan curves, then the 1-manifold tpq, tq : q P ˜C4pimphp¨, tqqu X Sqˆr0, 1s can be separated  into three distinct clopen pieces, one for each type of inscribed square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' This proves that the parity  of each type of inscribed square is invariant of which generic Jordan curve we choose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Finally  we can compute the parities by finding the inscribed squares of any generic Jordan curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' The  quintessential generic Jordan curve is a non-circular ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' This curve has one gracing square, but  no inscribed square of type II or III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  □  Let γ be a generic Jordan curve, and let f be a smooth function R2 Ñ R which has γ as its zero  set, is negative inside of γ, positive outside of γ, and has non-vanishing gradient on γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Since γ is  a smooth embedding, such an f always exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Now, we will consider gf,γ : pS1q2 Ñ R2, as defined  in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  We see that the zeros of gf,γ consist of the diagonal ∆ “ tpa, aq : a P S1u, as well as four points  for each inscribed square in ˜C4pimpγqq X Sq, one point for each side of the square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Furthermore, at  these zeros corresponding to inscribed squares, gf,γ has nonsingular derivative because γ is generic  and f has non-vanishing gradient on impγq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Let X “ tp P pS1q2 : gf,γppq ‰ p0, 0qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Let ωγ be the  cohomology class in H1pX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Z{2Zq given by pulling back the nontrivial class of H1pR2ztp0, 0qu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Z{2Zq  under gf,γ|X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' A small loop around any of the zeroes corresponding to sides of inscribed squares will  evaluate nontrivially under ω because gf,γ has nonsingular derivative at these zeroes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Furthermore,  a loop obtained by pushing the diagonal off to one side or the other will evaluate trivially under  ωγ because it will correspond to a small square sliding around the Jordan curve with two corners  on γ and the other two corners off to one side of γ, so the corresponding loop in R2ztp0, 0qu will  remain in just one quadrant of the plane and therefore cannot wrap around the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Putting this all together, we see that if ℓ is a simple closed curve in X which has the homotopy  class of the diagonal in S1 ˆ S1, then ωγpℓq is equal to the parity of the number of zeroes of gf,γ in  either connected component of pS1q2zp∆ Y ℓq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Let ∆1 be the anti-diagonal of pS1q2, the loop consisting of pairs pa, bq where a and b are antipodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Then pS1q2zp∆ Y ∆1q has two connected components, which we call A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' The component A is  the set of pairs pa, bq for which the angle from a to b is less than π and the component B is the set  of pairs for which the angle from a to b is greater than π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  If γ is a bad Jordan curve, and γ1 is a sufficiently C0-close generic approximation, then the parity  of bγ is equal to ωγ1p∆1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We can therefore compute bγ by counting the zeroes of gf,γ1 in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' If γ is a bad Jordan curve, and γ1 is a sufficiently C0-close generic approximation,  then each type I inscribed square of γ1 has exactly three of its corresponding zeroes of gf,γ1 in A,  each type II inscribed square has exactly two of its corresponding zeroes in A, and each type III  inscribed square has exactly one of its corresponding zeroes in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Let γ1, γ2, γ3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' be a sequence of generic Jordan curves that limit to our bad Jordan curve  γ in the C0 topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Let sn be the side length of the largest inscribed square of γn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We know  that limnÑ8 sn “ 0 because otherwise there would be a convergent subsequence of squares that  limit to an inscribed square of our bad Jordan curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We can give S1 a metric by identifying it  with the unit circle in R2 and using euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Using this identification, we let M be the  set of all pairs pa, bq P pS1q2 which have }a ´ b} ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Let N P Z be sufficiently large that both  supxPS1 }γNpxq ´ γpxq} and sN are smaller than the quantity ε “ 1  4 ¨ minpa,bqPM }γpaq ´ γpbq}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We  claim that γ1 “ γN is an approximation for which the proposition holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Let a, b, c, d be the vertices of an inscribed square of γN, labeled counterclockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Let Q be the  set of four points in S1 which map onto ta, b, c, du under γN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We claim that the diameter of Q is  6  COLE HUGELMEYER  less than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' To prove this, suppose there were elements x and y in Q with }x ´ y} ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Then  px, yq P M, so }γpxq ´ γpyq} ě 4ε, so }γNpxq ´ γNpyq} ě 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' However, 2ε ą  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  2 ¨ sN and γNpxq and  γNpyq are vertices of a square with side length at most sN, so this is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Since the diameter of Q is less than 1, all four points of Q must lie within some interval I of  angular length π{3 radians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Orient I counterclockwise, and let w, x, y, z be the four elements of Q  in the order they appear along I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We know that pw, xq, pw, yq, pw, zq, px, yq, px, zq, and py, zq are in  A, and all the other ordered pairs are in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  We know that γN maps tw, x, y, zu onto ta, b, c, du, but it might not preserve the ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Without loss of generality, we can assume γNpwq “ a because we can permute the labels a, b, c, d  cyclically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Then, we have six possibilities to check for the six permutations of the remaining three  letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We will now check all of the possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  1) If pw, x, y, zq ÞÑ pa, b, c, dq, the square is type I and the corresponding zeroes in A are  pw, xq, px, yq, py, zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  2) If pw, x, y, zq ÞÑ pa, c, b, dq, the square is type II and the corresponding zeroes in A are  pw, yq, px, zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  3) If pw, x, y, zq ÞÑ pa, b, d, cq, the square is type II and the corresponding zeroes in A are  pw, xq, px, zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  4) If pw, x, y, zq ÞÑ pa, d, c, bq, the square is type III and the only corresponding zero in A is  pw, zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  5) If pw, x, y, zq ÞÑ pa, d, b, cq, the square is type II and the corresponding zeroes in A are  pw, yq, py, zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  6) If pw, x, y, zq ÞÑ pa, c, d, bq, the square is type II and the corresponding zeroes in A are  pw, zq, px, yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  This confirms the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' The type I squares have three corresponding zeroes in A, the type  II squares have two corresponding zeroes in A, and the type III squares have only one corresponding  zero in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  □  We can now prove Lemma 1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Let γ be a bad Jordan curve, let γ1 be a sufficiently C0-close generic approxi-  mation, and let f be a function with γ1 as its zero set as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We now count zeroes of gf,γ1 in A,  the parity of which is the parity of bγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We count the zeroes by those corresponding to the squares  of each type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We have an odd number of threes, an even number of twos, and an even number of  ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' This totals to an odd number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We wish to prove that every bad Jordan curve has a square envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We  have the function gf,γ|C : C Ñ R2ztp0, 0qu, which we know to be homotopically nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Let  π : R2ztp0, 0qu Ñ S1 be radial projection onto the unit circle, and let r “ π ˝gf,γ|C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We now take r1  to be a smooth approximation of r with }rpxq ´ r1pxq} ă 1{4 for all x P C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Let u P S1 be a regular  value for r1 within distance 1{4 of the point p´1{  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  2, ´1{  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Then r´1puq is a 1-dimensional  manifold, L, which is mapped under r into a circle of radius 1{2 around p´1{  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  2, ´1{  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  2q, which  is entirely in the lower left quadrant of the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Therefore, gf,γ|C maps L into the lower left  quadrant of the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' This implies that the square corresponding to a point of L has its first two  corners on γ, and the other two in the open interior of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Furthermore, since bγ is odd, we see that  L must have an odd number of connected components which are homeomorphic to R with the two  ends limiting to the two boundaries of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Parameterizing such a connected component, we have  functions e1, e2 : R Ñ R2 with impe1qYimpe2q Ď impγq, and limxÑ˘8 }e1pxq´e2pxq} “ 0, and with  the property that S1pe1pxq, e2pxqq and S2pe1pxq, e2pxqq are always in the open interior of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Since  the interior of gamma is open, we may choose a continuous function ε : R Ñ Rą0 with the property  that, for all x, if }a ´ e1pxq} and }b ´ e2pxq} are less than εpxq, then S1pa, bq and S2pa, bq are in the  INSCRIBED SQUARES AND RELATION AVOIDING PATHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  7  open interior of γ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Thus, if we choose a continuous deformation of pe1, e2q in which we push  e1pxq and e2pxq off of the Jordan curve without exceeding a distance of εpxq, we obtain a square  envelope for γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Relation avoiding paths  In this section, we will prove that if the spiral conjecture is true, then the inscribed square conjec-  ture holds for Jordan curves which are smooth except at a finite number of arbitrarily complicated  singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  However, we will first present a proof of Theorem 2, that the spiral conjecture is indeed true in  the one-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Identifying V with C, our relation avoiding origin paths are paths to the origin  in the complex plane p : r0, 8q Ñ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Furthermore, we know that pptq is never zero, because p0, 0q is  in every linear relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Thus, without loss of generality, we can ignore any of the Ri corresponding  to t0u ˆ C or C ˆ t0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We can therefore rewrite the relation R as  xRy ðñ Di P t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=', nu  x “ αiy or y “ αix  where α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=', αn are nonzero complex numbers with norm at most one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' In particular, our path p  is relation avoiding if and only if it is disjoint from the paths α1p, α2p, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=', αnp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Without loss of  generality, we can assume that pp0q “ 1, and that |pptq| ď 1 for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' The reason we can do this  is that we can always homotope p to such a path by first contracting it within its image so that  the starting point is at the point with maximal norm, and then rescaling within C so that the  starting point becomes 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We can also assume without loss of generality that p follows the path of  a logarithmic spiral inside some small neighborhood around 1, because making this happen only  requires a small perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Working with these assumptions, we fix a branch of the logarithm,  and let ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=', ℓn be paths in upper-half plane such that e2πiℓiptq “ αipptq with ℓip0q “  1  2πi lnpαiq,  and similarly let ℓ be a lift of p with ℓp0q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' The path ℓ divides the upper-half plane, and we  define U to be the set of all points of Hzimpℓq from which a path to ´8 has even intersection parity  with ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We then define integers k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=', kn, where ki is the largest integer such that ℓip0q ` ki P U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  We see that ℓ is disjoint from ℓi `k for any index i and integer k, and this gives us paths disjoint  from ℓ, going up to i8, from every point of the form ℓp0q ` k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Therefore, the homotopy type of ℓ  can be completely determined by knowing which points of the form ℓip0q ` k are in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Thus, to  prove the one-dimensional spiral conjecture, it suffices to prove that there exists a θ P p0, πq so that  for all i P t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=', nu, we have the inequalities argpℓip0q ` kiq ě θ ě argpℓip0q ` ki ` 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' This θ would  then denote the angle of a straight line in the upper half plane that exponentiates to the desired  logarithmic spiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' To prove that such a θ exists, it suffices to prove that for all i and j we have  argpℓjp0q ` kj ` 1q ď argpℓip0q ` kiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  We define a split pair to be a pair of points in the upper half-plane pp, qq, such that p ` ℓ and  q ` ℓ are both disjoint from ℓ, with p P U and q R U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' For instance, pℓip0q ` ki, ℓjp0q ` kj ` 1q is  always a split pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  8  COLE HUGELMEYER  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' A depiction of a split pair, disjoint translates of ℓ, one on each side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Given a split pair pp, qq, we can construct a new split pair as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' If imppq ě impqq, then  pp ´ q, qq is the new split pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' If impqq ą imppq, then pp, q ´ pq is the new split pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' The split  pair obtained by applying this transformation is called the derived split pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We claim that the  derived split pair is always another split pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' One way to understand derived split pairs is that one draws a parallel-  ogram spanning the origin and the two points, then moves the uppermost point to  the new corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  To prove this claim, we consider the case that imppq ě impqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We have that ℓ ` q is entirely on  the right side of ℓ, and ℓ`p is entirely on the left side of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Furthermore, since all of ℓ`p has greater  imaginary coordinate than impqq, we therefore have that p P ta P U : impaq ě impqqu Ď U ` q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  This tells us that p ´ q is in U, so pp ´ q, qq is a valid split pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' The argument for the other case  is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  We say a split pair is good if argppq ą argpqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We say a split pair is bad otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We see that  the derived split pair of a bad split pair is bad, and the derived split pair of a good split pair is  INSCRIBED SQUARES AND RELATION AVOIDING PATHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  9  good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' To complete the proof of the theorem, it suffices to prove that all split pairs are good, so we  will consider what occurs under repeated derivation of a bad split pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' First of all, note that if  imppq{impqq is a rational number, then we will eventually have a split pair with a real number as  one of the terms, and we can see that any split pair containing a real number is good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Thus, we have  shown that imppq{impqq is irrational for any bad split pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Now, we will proceed by contradiction  to prove that there are no bad split pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Suppose that pp, qq is a bad split pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We will treat the  cases of argppq ă argpqq and argppq “ argpqq separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Suppose that pp, qq is a bad split pair with argppq ă argpqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Then, the i-th derived split pair is  of the form paip ´ biq, ciq ´ dipq, for nonnegative integers ai, bi, ci, di such that  lim  iÑ8 ai{bi “ lim  iÑ8 di{ci “ impqq{imppq  This tells us that while the imaginary parts of the terms of the derived split pairs approach zero,  the real parts approach `8 and ´8 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' However, this would force the second term to  eventually be in U, which gives us a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Now, suppose argppq “ argpqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' In this case, the terms of the derived split pairs stay on a fixed  straight line, and they approach zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' However, earlier in the proof, we assumed without loss of  generality that p followed the path of a logarithmic spiral in some small neighborhood around 1,  which means ℓ is a straight line in some small neighborhood around zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' This means that there  are no bad split pairs in this neighborhood, so we have a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  □  Before we prove Theorem 3, we need to develop some notation surrounding square envelopes,  and prove a couple lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Let γ : S1 Ñ R2 be a Jordan curve, and let pe1, e2q be a square envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We  write σ˚  ppe1, e2, γq, where p P t1, 2u and ˚ P t`, ´u, to denote a sign in t`1, ´1u determined as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' For λ P R, let Pλ be the path that starts at e1pλq, then goes along e1 to e1p0q, then goes  in a straight line from e1p0q to e2p0q, then follows e2 to e2pλq, then goes in a straight line back to  the starting point e1pλq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We then let npλq be the wrapping number of Pλ around Sppe1p0q, e2p0qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Finally, we set σ˚  ppe1, e2, γq “ limλÑ˚8p´1qnpλq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  It is worth observing that these parities are determined by how the Jordan curve winds between  the vertices of a small square near t “ ˘8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' For curves with finitely many singularities, they are  related to each other by the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' If γ is a Jordan curve which is smooth except at finitely many points, then the following  facts are true for any square envelope pe1, e2q of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  1) σ`  1 pe1, e2, γq “ σ`  2 pe1, e2, γq and σ´  1 pe1, e2, γq “ σ´  2 pe1, e2, γq  2) σ`  1 pe1, e2, γq “ ´σ´  1 pe1, e2, γq  3) If ˚ P t`, ´u is such that σ˚  1pe1, e2, γq “ ´1, then the limits limtÑ˚8 e1ptq and limtÑ˚8 e2ptq  exist, and are equal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' First of all, by property (4) in the definition of a square envelope, we immediately have that  σ`  1 pe1, e2, γq “ ´σ´  1 pe1, e2, γq and σ`  2 pe1, e2, γq “ ´σ´  2 pe1, e2, γq because the defining loops for the  parities in question compose to give us a loop that wraps around the Jordan curve exactly once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Next, we claim that if ˚ P t`, ´u, and the limit limtÑ˚8 e1ptq does not exist, then we have  σ˚  1pe1, e2, γq “ σ˚  2pe1, e2, γq “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' To prove this claim, note that for the limit to fail to exist, there  must be a limit point of e1ptq, t Ñ ˚8 at one of the points where γ is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Taking a sufficiently  small square of the envelope, in the ˚8 direction, near such a point, we see that the Jordan curve  must separate the vertices of the square in a way locally equivalent to how a straight line could  separate the vertices, and there is only one such way that separates the e1, e2 vertices from the  other two vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Since all squares sufficiently near t “ ˘8 will have side length smaller than the  10  COLE HUGELMEYER  diameter of some disk inside of the Jordan curve, the paths that define σ˚  1pe1, e2, γq and σ˚  2pe1, e2, γq  cannot wrap around either of the two interior vertices of the square at t “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Therefore, we have  σ˚  1pe1, e2, γq “ σ˚  2pe1, e2, γq “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  To prove all three parts of the lemma, all that remains is to eliminate the possibility that the limits  limtÑ˚8 e1ptq and limtÑ˚8 e2ptq exist, but σ˚  1pe1, e2, γq and σ˚  2pe1, e2, γq are not equal to one an-  other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' The reason this is impossible is that for this to be the case, the paths e1ptq, e2ptq, S1pe1ptq, e2ptqq,  and S2pe1ptq, e2ptqq would need to all approach some point in the plane, none of them intersect-  ing each other, in such a way that as one encircles the point counterclockwise, the paths appear  in a cyclic order that alternates between the sets te1ptq, e2ptqu and tS1pe1ptq, e2ptqq, S2pe1ptq, e2ptqqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  This would then contradict the fact that the simple closed curve γ separates the paths in te1ptq, e2ptqu  from those in tS1pe1ptq, e2ptqq, S2pe1ptq, e2ptqqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  □  If γ is a Jordan curve which is smooth except at finitely many points, and pe1, e2q is a square  envelope, we say pe1, e2q is positively oriented if σ`  1 pe1, e2, γq “ ´1 and negatively oriented if  σ`  1 pe1, e2, γq “ `1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Note that if pe1ptq, e2ptqq is a negatively oriented square envelope, we can  obtain a positively oriented one by taking pe1p´tq, e2p´tqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  We need one more lemma before we can prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Let α1, α2, β1, β2 be complex numbers such that the ratio β1{β2 is a positive real  number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Let t0 and p be arbitrary real numbers with p ‰ 0, and let λ, r1, r2 be real numbers in  r0, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' If there exist real numbers t1 and t2 in r0, 8q such that  eα1`β1ppt1´t0q ´  1 ` λr1eippt1´t0q¯  “ eα2`β2ppt2´t0q ´  1 ` λr2eippt2´t0q¯  then there also exist t1  1 and t1  2 in r0, 8q such that  eα1`β1ppt1  1´t0q ´  1 ` r1eippt1  1´t0q¯  “ eα2`β2ppt1  2´t0q ´  1 ` r2eippt1  2´t0q¯  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Making the substitutions α1 ´ α2 “ α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' ppt1 ´ t0q “ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' and ppt2 ´ t0q “ pβ1{β2qt ` s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' the  equation  eα1`β1ppt1´t0q ´  1 ` λr1eippt1´t0q¯  “ eα2`β2ppt2´t0q ´  1 ` λr2eippt2´t0q¯  rearranges to  eα´β2s “  1 ` λr1eit  1 ` λr2eip β1  β2 t`sq  Rearranging,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' and setting a “ r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' bpsq “ ´r2eα`pi´β2qs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' and cpsq “ eα`β2s ´ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' we have  cpsq “ λ ¨  ˆ  a ¨ eit ` bpsq ¨ ei  ´ β1  β2 t  ¯˙  Also,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' let  Tpsq “ tt : Dt1 ě 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Dt2 ě 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  ppt2 ´ t0q “ pβ1{β2qt ` s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  ppt1 ´ t0q “ tu  be the allowable values of t for fixed s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' when t1 and t2 are nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' This will always be an  interval of the form p´8, zs or rz, 8q for some real number z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Now, let Hpsq denote the unique  minimal simply connected set containing the set  ta ¨ eit ` bpsq ¨ eipβ1{β2qt : t P Tpsqu  Then, we see that there exists a pair of nonnegative real numbers t1 and t2 such that  eα1`β1ppt1´t0q ´  1 ` λr1eippt1´t0q¯  “ eα2`β2ppt2´t0q ´  1 ` λr2eippt2´t0q¯  if and only if there exists a real number s such that cpsq P λHpsq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Since β1{β2 is positive, when  β1{β2 is rational, Hpsq is the region enclosed by the outer boundary of an epitrochoid curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' When  β1{β2 is irrational, Hpsq it is a disk missing some subset of its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' In particular, Hpsq is  INSCRIBED SQUARES AND RELATION AVOIDING PATHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  11  always a radial set, meaning that for any x P Hpsq, we have λx P Hpsq for all λ P r0, 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Therefore,  increasing the parameter λ preserves the existence of solutions to our original equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Setting  λ “ 1 gives us the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' The region enclosed by an epitrochoid curve is always a radial set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  □  Now, we can prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We begin by considering the system of linear relations on C2 given by letting  px1, y1qRpx2, y2q when  tx1, y1, x2, y2u X tx1 ` ipy1 ´ x1q, y1 ` ipy1 ´ x1q, x2 ` ipy2 ´ x2q, y2 ` ipy2 ´ x2qu ‰ ∅  Now, take a positively oriented square envelope pe1, e2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Due to Lemma 2, restricting the domain  to r0, 8q immediately gives us a relation avoiding origin path pe1ptq, e2ptqq ´ limtÑ8pe1ptq, e2ptqq  for R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' We are assuming the spiral conjecture, so we may mow take h : r0, 1s ˆ r0, 8q satisfying the  conditions of the spiral conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' This gives us complex numbers a1, a2, and vectors x1, x2 P C2,  such that pf1ptq, f2ptqq “ fptq “ x1ea1t ` x2ea2t is a relation avoiding origin path for R which has  the property that the path in C starting at 0, following f2 to f2p0q, going in a straight line from  f2p0q to f1p0q, then following f1 back to 0 has nontrivial wrapping number parity around the points  f1p0q `ipf2p0q ´f1p0qq and f2p0q `ipf2p0q´f1p0qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Without loss of generality, we can assume that  a1 and a2 have the same real part, because otherwise one term will dominate at large t, reducing  us to the case of a pure logarithmic spiral, which we will cover later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Similarly, we may assume  a1 ‰ a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Now, to prove that no such a1, a2, x1, x2 exist, we see that for the path to be relation  avoiding, we must be able to find some fixed complex number β, and nonzero real number p, such  that all four paths f1ptq, f2ptq, f1ptq ` ipf2ptq ´ f1ptqq, and f2ptq ` ipf2ptq ´ f1ptqq are of the form  eα`qβppt´t0q ´  1 ` reippt´t0q¯  with α P C, t0 P R, q P p0, 8q, and r P r0, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Therefore, Lemma 3 allows us to continuously scale  down all of the r parameters to zero while staying a relation avoiding origin path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  All that remains is to show that no pure logarithmic spiral fptq “ xeat can be a relation avoiding  origin path with the stated wrapping number parity condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' To prove this, note that the region  swept out by a line segment between f1 and f2 would have the same area as the region swept out  by the line segment between f1 ` ipf2 ´ f1q and f2 ` ipf2 ´ f1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' However, our wrapping number  12  COLE HUGELMEYER  condition implies that the former region would need to strictly contain the latter, which would  contradict them having equal areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  □  To wrap things up, we make a couple more conjectures about relation avoiding origin paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Conjecture 3 (SSC, Strong Spiral Conjecture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Every relation avoiding origin path p is strongly  homotopic to a relation avoiding origin path q for which there exist linearly independent vectors  x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=', xn in V and complex numbers a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=', an such that  qptq “  nÿ  i“1  xi ¨ eait  for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Conjecture 4 (MSC, Monotonic Spiral Conjecture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' For any continuous function p : r0, 8q Ñ  V zt0u with limtÑ8 pptq “ 0, there exists a continuous function h : r0, 1s ˆ r0, 8q Ñ V with the  following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  1) limtÑ8 hps, tq “ 0 for all s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  2) hp0, tq “ pptq for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  3) There exist linearly independent vectors x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=', xn in V and complex numbers a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=', an such  that  hp1, tq “  nÿ  i“1  xi ¨ eait  for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  4) If 0 ď s1 ď s2 ď 1, then Bps2q Ď Bps1q, where  Bpsq :“  ď  pt1,t2qPr0,8q2  tf P pV ˆ V q1 : fphps, t1q, hps, t2qq “ 0u  It is not too difficult to prove the implications MSC  ùñ SSC  ùñ SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' The strong spiral  conjecture clearly implies the spiral conjecture, but it might not be so easy to see why the monotonic  spiral conjecture implies the strong spiral conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' The reason is that a small perturbation of  the homotopy can be made to avoid any relations with codimension greater than one in V ˆ V ,  and the homotopy for the monotonic spiral conjecture will always increase the set of codimension  one relations that it avoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  Morally speaking, the monotonic spiral conjecture should be true in the one-dimensional case  because curve shortening flow in a logarithmic metric should have the desired property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  This  argument should at least cover the case of smooth paths which are periodic deviations from a  logarithmic spiral, but the analysis is more complicated for arbitrary continuous paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' It would be  useful if there was a way to generalize such an argument to higher dimensions, though it is not at  all obvious how to achieve this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Regardless, we hope that a deep enough understanding of relation  avoiding paths will lead to new progress on the inscribed square problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  INSCRIBED SQUARES AND RELATION AVOIDING PATHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  13  References  [1] Arseniy Akopyan and Sergey Avvakumov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Any cyclic quadrilateral can be inscribed in any closed convex smooth  curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Forum of Mathematics, Sigma, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  [2] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Griffiths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' The topology of square pegs in round holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Proceedings of the London Mathematical Society,  s3-62(3):647–672, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  [3] John McCleary Jason Cantarella, Elizabeth Denne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Transversality for configuration spaces and the “square-peg”  theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' arXiv:1402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='6174 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='GT], 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  [4] Benjamin Matschke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Equivariant topology methods in discrete geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' PhD thesis, Freie University at Berlin,  2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  [5] Benjamin Matschke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' A survey on the square peg problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Notices Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=', 61(4):346–352, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  [6] Benjamin Matschke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Quadrilaterals inscribed in convex curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' arXiv:1801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='01945 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='MG], 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Nielsen and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Wright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Rectangles inscribed in symmetric continua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Geometriae Dedicata, 56(3):285–297,  1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  [8] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Schnirelman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' On some geometric properties of closed curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Usp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Nauk, 10:34–44, 1944.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  [9] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Schwartz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' A trichotomy for rectangles inscribed in jordan loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' arXiv:1804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='00740 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='MG], 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content='  [10] Terence Tao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' An integration approach to the toeplitz square peg problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
+page_content=' Forum of Mathematics, Sigma, (5),  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfYvxp/content/2301.01340v1.pdf'}
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+1 
+ 
+Non-Equilibrium Spark Plasma Reactive Doping Enables Highly Adjustable 
+Metal to Insulator Transitions and Improved Mechanical Stability for VO2 
+ 
+Xuanchi Zhou a†, Yuchen Cui a†, Yanlong Shang a, Haifan Li a, Jiaou Wang b, Ye Meng c, 
+Xiaoguang Xu a, Yong Jiang a*, Nuofu Chen d* and Jikun Chen a* 
+ 
+a School of Materials Science and Engineering, University of Science and Technology Beijing, 
+Beijing, 100083, China 
+b Beijing Synchrotron Radiation Facility, Institute of High Energy Physics, Chinese Academy of 
+Sciences, Beijing 100049, China 
+c Institute for Advanced Materials and Technology, University of Science and Technology Beijing, 
+Beijing 100083, China 
+d School of Renewable Energy, North China Electric Power University, Beijing 102206, China 
+ 
+*Authors to whom correspondence should be addressed: jikunchen@ustb.edu.cn (J. Chen), 
+yjiang@ustb.edu.cn (Y. Jiang), nfchen@ncepu.edu.cn (N. Chen). 
+ 
+† X. Zhou and Y. Cui contributed equally to this work. 
+ 
+ 
+
+2 
+ 
+Abstract 
+Although vanadium dioxide (VO2) exhibits the most abrupt metal to insulator 
+transition (MIT) properties near room-temperature, the present regulation of their MIT 
+functionalities is insufficient owing to the high complexity and susception associated 
+with V4+. Herein, we demonstrate a spark plasma assisted reactive sintering (SPARS) 
+approach to simultaneously achieve in situ doping and sintering of VO2 within largely 
+short period (~10 minutes). This enables high convenience and flexibility in 
+regulating the electronic structure of VO2 via dopant elements covering Ti, W, Nb, Mo, 
+Cr and Fe, leading to a wide adjustment in their metal to insulator transition 
+temperature (TMIT) and basic resistivity (ρ). Furthermore, the mechanical strengths of 
+the doped-VO2 were meanwhile largely improved via the compositing effect of high 
+melting-point dopant oxide. The high adjustability in MIT properties and improved 
+mechanical properties further paves the way towards practical applications of VO2 in 
+power electronics, thermochromism and infrared camouflage. 
+ 
+Key words: Quantum materials, Electronic phase transition, Correlated oxides, 
+Correlated electronics; 
+ 
+TOC Graphic 
+ 
+ 
+ 
+ 
+ 
+
+Ti
+大
+W
+Mo
+Fe
+Nb
+Pressure
+Ti Ref.
+★
+W Ref.
+Mo Ref.
+Fe Ref.
+Nb Ref.
+400
+V
+HoleDoping
+350
+Pristine VO
+V
+vO,Particle
+Electric
+TMmO,Particle
+Current
+300
+ElectronDoping
+Sparked
+Plasma
+250
+10
+100
+Nominal Doping Concentration (%)3 
+ 
+1. Introduction 
+The metal to insulator transition (MIT) within d-band correlated oxides brings in 
+abruptions in regulating their electronic and optical properties beyond conventional 
+semiconductors 1-3. Among the existing family of MIT materials, the vanadium 
+dioxide (VO2) exhibits the most abrupt MIT properties near room-temperature that 
+plays irreplaceable role in applications, such as thermochromism 4-6, infrared 
+camouflage 7, 8, and correlated electronic devices 9-12. Nevertheless, these applications 
+of VO2 are still challenged by its material synthesis as the electronic phase diagram of 
+vanadium oxides is extraordinary complex 13. For example, the metal to insulator 
+transition temperature (TMIT) that triggers the MIT is known to be rather sensitive to 
+the valence state of vanadium that is at the intermediate valence state (e.g., V4+) and 
+susceptible to both oxidizing and reducing environment 14-16. Conventionally, the MIT 
+properties and electronic structure of VO2 is adjusted by the elementary substitution, 
+e.g., the TMIT is elevated or descended via substituting V4+ by dopants with lower or 
+higher valence states, respectively 17, 18.  
+ 
+To achieve the abrupt variation in the material resistivity during the MIT of VO2, 
+it is vital to precisely control the concentration and homogeneity in the distribution of 
+the dopant elements, without extrinsically altering the oxygen composition 19. As 
+compared to the vacuum deposition of doped-VO2 thin films via plasmanized or 
+vaporized precursors 20-23, achieving a precise elementary substitution for the bulk 
+materials of VO2 while maintaining their MIT abruption is more difficult. Previously, 
+doped VO2 bulk pellets were made using the following two routes: 1) Firstly 
+synthesizing the powder materials for doped VO2, and afterwards sintering such 
+powder into bulk pellets at precisely controlled atmosphere 24, 25. 2) Co-sintering VO2 
+and dopant oxides at high temperatures over a long period (e.g., several days) at 
+controlled atmosphere 21. Usually, the homogeneity and achievable concentration of 
+the dopant element within VO2 are restricted by its equilibrium phase diagram, while 
+the kinetic process that triggers the diffusion of the dopant elements may easily 
+disturb the desired valence state of vanadium 26, 27. Up to date, an effective strategy to 
+accurately adjust the MIT properties of the VO2 bulk material via conveniently 
+achievable elementary substitution is yet lacking, and this impedes applying VO2 for 
+power electronics as discrete elementary devices. 
+ 
+In this article, we demonstrate a spark plasma assisted reactive sintering (SPARS) 
+strategy that largely elevates the flexibility and effectiveness in the elementary 
+substitution of VO2 as bulk material, covering a large abundance of the dopant 
+elements such as Ti, W, Nb, Mo, Cr, and Fe. The non-equilibrium process related to 
+SPARS largely shorten the diffusion period of the dopants (e.g., 10 minutes), and it 
+enables a broad adjustment of the electronic structure and the MIT functionality of 
+VO2. Compliant variations in the electronic structure of VO2 were probed via the 
+near-edge X-ray absorption fine structure (NEXAFS) analysis that differs to the 
+conventional doping of semiconductors. Furthermore, the unreacted dopant oxides 
+
+4 
+ 
+with high melting point can be meanwhile used to improve the mechanical properties 
+of VO2, paving the way towards their practical applications in power electronics. 
+ 
+2. Experimental Section 
+A spark plasma assisted reactive sintering (SPARS) strategy was used for 
+synthesizing the doped VO2 bulk materials. The as-proposed SPARS approach 
+includes the following two steps. In the first step, the powder of VO2 is homogenously 
+mixed with the transition metal oxides (TMmOn) dopants at the desired stoichiometry, 
+and afterwards pressed inside a graphite die. In the second step, a large electric current 
+(e.g., 300 A/cm2) and high pressure (e.g., 20 MPa) are imparted into the graphite die 
+and trigger the solid-state reaction between VO2 and TMmOn powders at the 
+temperature of 900 C for 10 minutes under a vacuum atmosphere. The as-made VO2 
+bulk material was a pellet with the radius of 10 mm and the thickness of around 4 
+mm.  
+ 
+The crystal structure of samples was characterized by the X-ray diffraction (XRD) 
+(RIGAKU, Smartlab). The valence state of as-made samples was characterized by 
+X-ray photoelectron spectroscopy (XPS) (Kratos, AXIS ULTRADLD) that was 
+equipped with monochromatic Al-Kα source (150 W) and photon energy of 1486.6 eV. 
+All the binding energies as obtained from the XPS were referenced to the C 1s peak 
+from the adventitious carbon at 284.8 eV, while a Shirley function was used to 
+subtract the background. In addition, the valence state of vanadium for VO2 is 
+indicated by the binding energy of V 2p3/2 and V 2p1/2 peaks. To fit the XPS curves, 
+the V 2p3/2 peak can be resolved by the two binding energies at 516.3 and 515.7 eV 
+that represents the valence state of V4+ and V3+, respectively. The V 2p1/2 region can 
+be analogously divided into the two peaks that are respectively located at 523.6 and 
+523.0 eV, according to the previous reports 28, 29. And their morphologies were 
+characterized by scanning electron microscopy (SEM) and energy dispersion 
+spectrum (EDS). The electronic structures of as-made samples were also investigated 
+by the near edge X-ray absorption fine structure (NEXAFS). To characterize their 
+electrical transportation properties, their temperature dependences of resistivity within 
+300-400 K were measured by using a commercialized CTA-system in vacuum, while 
+the one within 1.8-400 K was measured by using the physical property measurement 
+system (PPMS) (Quantum Design). The Vickers hardness of VO2 bulk material were 
+measured by the digital Vickers hardness tester (VTD 512) at room-temperature under 
+the load of 500 kgf. 
+ 
+3. Results and Discussion 
+3.1. Strategy for elementary substitution of VO2 via SPARS 
+To address the center challenge in the precise element doping for bulk VO2, 
+herein a spark plasma assisted in situ doping strategy is used, as illustrated in Fig. 1a. 
+Unlike the previous literatures that uses the reactive spark plasma process for 
+sintering the powder material into bulk pellet, or directly synthesizing compounds 
+from their respective elementary substances 30, 31, herein the reactive spark plasma is 
+
+5 
+ 
+used to achieve the element doping of oxides. Firstly, the powder of VO2 is 
+homogenously mixed with the doped transitional metal oxides (TMmOn, TM = Ti, W, 
+Nb, Mo, Fe, and Cr) and pressed inside a graphite die. Afterwards, a large DC electric 
+current (e.g., around 300 A/cm2) and high pressure (e.g., 20 MPa) are imparted to 
+spark over the powder precursor that triggers their mutual element diffusion and 
+rapidly elevate the temperature to 900 C (e.g., 60% of the melting point of VO2) for 
+their reactive sintering into bulk pellet. Such spark plasma assisted non-equilibrium 
+process brings in the electrical break down process within the sintering that promotes 
+the elementary diffusion and shortens the period associated with the reactive sintering 
+(e.g., 10 minutes). Meanwhile, the residual of the dopant oxides with high melting 
+point within the grain boundary can be used to improve the mechanical properties of 
+the VO2 pellet. Furthermore, as-proposed SPARS approach makes it possible to 
+flexibly adjust the dopant composition and concentration within VO2 with high 
+convenience, by just using the powder of pure VO2 and the dopant oxides as 
+precursors. 
+ 
+3.2. Metal to insulator transition properties for Ti substituted VO2 
+As-proposed SPARS approach was first applied to achieve Ti-doping of VO2, 
+and Fig. 1b shows the X-ray diffraction (XRD) patterns of as-made V1-xTixO2 pellets. 
+In general, the V1-xTixO2 exhibit the same crystal structure to VO2 (M), while their 
+lattice expands with an increasing x as demonstrated in Fig. 1c (see more details of 
+their structure information from the Rietveld refinement in Table S1). It is also worth 
+noticing that the c0/a0 in the lattice constant slightly increases indicated by the 
+Rietveld refinements (see Fig. S1), and this observation is in agreement to the 
+understanding by C. N. R. Rao et al that a local structural distortion upon Ti 
+substitution may also relevant to the variation in MIT properties32. This clearly 
+demonstrates the effective substitution of the lattice vanadium within VO2 by titanium 
+that exhibits larger ionic radius, as is in agreement to the previous report 33. The 
+cross-section morphology of as-sintered V1-xTixO2 pellet as shown in Fig. 1d 
+demonstrates its polycrystallinity with a grain size of ~20 µm, and the same 
+distribution is observed for V and Ti as further confirmed by the energy dispersion 
+spectroscopy (EDS) shown in Figs. 1e and 1f. Furthermore, X-ray photoemission 
+spectroscopy (XPS) analysis shown in Fig. 1g indicates the generation of V3+ valence 
+state via Ti doping of VO2, while the valence state of the Ti remains as 4+ as shown in 
+Fig. 1h. The above understanding about the presence of V3+ valence state via Ti 
+doping is further confirmed by the XPS results for other Ti-substituted samples, as 
+shown in Figs. S2 and S3. Fig. 1i demonstrates the temperature dependence of 
+resistivity (-T) as measured for as-made V1-xTixO2 pellets, where abrupt reductions in 
+their resistivity are observed in -T tendencies by elevating temperature across their 
+TMIT. The elevation in the resistivity for the Ti-substituted VO2 is against the 
+conventional expectation that the presence of V3+ should strengthen the metallic phase 
+that reduces the room-temperature resistivity. It is more likely to be associated with 
+the compositing effect of the residual TiO2 remaining at the grain boundary. 
+
+6 
+ 
+Nevertheless, this is hard to be distinguished in the XRD or EDS mapping, since the 
+particle size of TiO2 (～10-20 nm) is much smaller than VO2 (～20-30 µm) while the 
+crystal structure is also similar for TiO2 and VO2. Similar elevation in the 
+room-temperature resistivity was also previously observed in ref 32, and was 
+attributed to local structure distortion. Therefore, the mechanism related to the Ti 
+doping in VO2 is expected to be more complicated that is worthy to be further 
+explored in the future. In Fig. 1j, the TMIT of as-made V1-xTixO2 pellets were further 
+plotted as a function of doping concentration. As shown TMIT was the averaged ones 
+from the R-T tendency measured by heating up and cooling down, while each critical 
+temperatures were taken when the temperature coefficient of resistance (TCR) reached 
+the maximum, as illustrated in Fig. S4 and Table S2. It can be seen that as-achieved 
+TMIT exhibits a generally reducing tendency with an increasing substitution 
+concentration of Ti. This observation is in general agreement to the previous report 32, 
+and this is attributed to the distorted local structure32 and the presence of V3+ valence 
+state induced by the substitution of titanium. We emphasize that the Ti substitution of 
+VO2 as achieved presently via our SPARS approach reduces the TMIT that is relevant 
+to the Ti doping composition, while the underneath mechanisms could be complicated 
+as is indeed under debate yet according to the previous reports. 
+ 
+3.3. Extending the SPARS strategy to achieved broad regulation in TMIT of VO2  
+To further extend the regulation of TMIT, more dopant oxides (e.g., WO3, MoO3, 
+Nb2O5, Cr2O3 and Fe2O3) were co-sintered with VO2 via the as-proposed SPARS 
+approach, and their respective XRD pattern and representative morphology are shown 
+in Fig. S5 and S6, respectively. It can be seen that the as-synthesized polycrystalline 
+ceramic pellets via the spark plasma assisted sintering process exhibit relatively 
+densified morphology without obvious porosity, and this is further confirmed by its 
+pronounced transition abruption across the TMIT. It can be seen that the as-synthesized 
+polycrystalline doped VO2 ceramic pellets via the spark plasma assisted sintering 
+process exhibit relatively densified morphology, despite the presence of little amount 
+of pores, as indicated in Fig. S7. The as-observed morphology is in consistency with 
+the previous reports on polycrystalline VO2 bulk pellets without doping as synthesized 
+by solid-state reaction or spark plasma sintering 34, 35. This indicates that the 
+as-proposed SPARS strategy can effectively achieve the reactive sintering process that 
+sinters the precursor powders into densified bulk pellet within largely shortened 
+reaction period of 10 minutes. As their -T tendencies shown in Fig. 2a, the TMIT is 
+more broadly adjusted within 240-350 K. In Fig. 2b, as-achieved TMIT were plotted as 
+a function of the substitution concentration, and further compared with the previous 
+reports 21, 32, 36-38. Similar to the previous reports for doped VO2 samples 19, herein 
+substituting the vanadium with elements at higher valence state (e.g., W6+, Mo6+ and 
+Nb5+) reduces the TMIT, while the respective substitution via lower valence elements 
+(e.g., Cr3+ and Fe3+) slightly elevates the TMIT. It is worth noticing that apart from the 
+variation in the valence state of vanadium, the variation in the lattice constant owing 
+
+7 
+ 
+to the different ionic radius of the dopants compared to vanadium may also influence 
+the MIT properties. Nevertheless, in the present work, all the dopant elements (e.g., 
+Ti4+, Nb5+, W6+, Mo6+, Cr3+ and Fe3+) enlarge the lattice constants of VO2 as further 
+summarized in the Table S3, but Cr3+ and Fe3+ increase the TMIT of VO2, while Nb5+, 
+W6+ and Mo6+ reduce its TMIT. This indicates that the regulation in the valence state of 
+vanadium should be a dominant cause to TMIT. In Fig. 2c, the abrupt variations in the 
+resistivity across the MIT (Insul. / Metal.) is further plotted as a function of TMIT, where 
+the magnitude generally decays with their TMIT. It is in particular worth noticing that 
+high transition abruption is achieved for Ti substituted VO2 (e.g., Insul. / Metal. 
+exceeding 102) that is comparable to the presently commercial critical temperature 
+resister (CTR) as reported previously 39. 
+ 
+To evaluate the ability in regulating the TMIT for various dopant elements, the 
+variation in TMIT is linear fitted with the doping concentration (see Fig. S8), and in Fig. 
+2d as-obtained slope (dTMIT / dndopant) is further plotted as a function of the dopant 
+valence state. It can be seen that the variation in TMIT is more effectively reduced by 
+substituting vanadium with the dopant elements at higher valence states (e.g., -17.9 
+K %-1 for W6+ and -9.4 K %-1 for Nb5+). In contrast, the regulation in TMIT is less 
+effective when the substituting elements exhibits the same or lower valence state 
+compared to V4+ (e.g., +2.6 K %-1 for Fe3+ and -0.43 K %-1 for Ti4+).  
+ 
+3.4. Compliant variation in electronic structures as probed by NEXAFS 
+To further qualitatively probe the variations in the electronic structures of VO2 
+associated with the elementary substitution, the near-edge X-ray absorption fine 
+structure (NEXAFS) analysis was performed, as the results shown in Fig. 3a-3b for 
+V-L edge and O-K edge, respectively. In the V-L spectrum of VO2, the V-LⅢ peak 
+(~518 eV) is recognized to be associated with the V 2p3/2 → 3d transition, while the 
+V-LⅡ peak (~524 eV) reflects the V 2p1/2 → 3d transition 40-42. From the perspective 
+of the hybridization between V 3d and O 2p band, the peaks in the O-K edge (e.g., O 
+1s → 2p transition) were respectively corresponding to the t2g and eg orbitals, and the 
+ratio in intensities of t2g / eg was reported to indicate the orbital occupancy 43, 44. 
+Furthermore, the d// orbital associated with the neighboring V ions along the rutile c 
+axis reflects the V-V interaction, while the * orbit pointed in the ligands is 
+corresponding to the electron density that is shifted away from the V-O bond direction. 
+In contrast,  / * orbits with rotational symmetry around V-O bond is associated to 
+the V-O interaction 45. 
+ 
+Substituting the vanadium with elements at higher valence state (e.g., W6+, Mo6+ 
+and Nb5+) results in the leftwards shift of the V-L edge (see Fig. 3a), and this 
+observation is in consistency with the XPS results shown in Fig. S9 (e.g., Fig. S9 a 
+and S9 c) that demonstrate the reduction in the valence state of vanadium towards V3+. 
+
+8 
+ 
+In addition, the as-observed similar peak position of O 1s peak (e.g., ~530.1 eV) for 
+doped VO2 with respect to the pristine one indicates the main presence of lattice 
+oxygen within the material. Meanwhile, the ratio in intensity of t2g / eg is observed to 
+be smaller compared to the pristine VO2, and this demonstrates an enhanced orbital 
+occupancy in t2g band, owing to the electron doping from the higher valence 
+substitution. It is known for VO2 that electron carrier doping strengthens the metallic 
+phase and reduces TMIT, as illustrated in Fig. 3c 20. Therefore, the as-observed 
+reduction in the TMIT via high-valence substitution is associated to the electron 
+occupancy in the t2g band of VO2. In addition, the as-observed reduced intensity of the 
+t2g peak with respect to the eg peak also indicates the weakening of direct V-V 
+interactions upon the cationic substitution 45. It is also interesting to note that similar 
+variations in the electronic structure are observed for Ti substituted VO2, in which 
+case the V-L edge also shifts leftwards while the t2g / eg ratio decreases. The above 
+observation is in agreement to the reduction in TMIT for Ti substituted VO2, although 
+the dopant (Ti4+) exhibits the same valence state compared to the matrix (V4+). It is 
+worth noticing that the reduction as observed in the TMIT of Ti doped VO2 is 
+consistency with the previous reports 32, 46. Nevertheless, the underneath mechanism 
+associated to the regulation in the TMIT via equivalent substitution is still in debate. 
+For example, C. N. R. Rao et al ascribed it to the distorted crystal structure induced 
+by the Ti substitution 32, while I. K. Kristensen attributed such result to the variation 
+in carrier concentrations 46.  
+ 
+In contrast, substituting vanadium with dopant at lower valence state (e.g., Fe3+) 
+results in opposite variations in the vanadium valence state, as the V-L edge shifts 
+rightwards in the V-L edge spectrum. This is in agreement to the XPS result that the 
+valence state of vanadium was slightly enhanced towards higher valence (e.g., +5) 
+valence (see Fig. S9 e), and this is also in general consistency to the observed 
+elevation in TMIT, as illustrated in Fig. 3c. Nevertheless, it is worth noticing that the 
+relative intensity of t2g peak, compared to eg, in the O-K edge is not increased as 
+expected for lower valence element substitution of vanadium, but also reduced. This 
+reveals the presence of a second mechanism that may also influence the electronic 
+structure of the presently made doped VO2 samples, in addition to the substitution of 
+vanadium. This is most likely to be associated to the generation of oxygen vacancy 
+during the SPARS that will also contribute electron carrier from the perspective of 
+anion regulation 47, and as a result induce the elevation in TMIT via Fe substitution, as 
+compared to the previous reports 38. In addition, the oxygen vacancy within VO2 is 
+reported to be formed upon a high-temperature vacuum atmosphere, and this is also 
+the case during the SPARS process in coarse vacuum atmosphere 48. It is also worthy 
+to mention that such potentially generated oxygen vacancy is not the dominate reason 
+for the Ti substitution or higher valence element substitution of vanadium, since a 
+monotonic variation in TMIT with the substitution concentration is indeed observed for 
+all these samples sintered at the same condition. Furthermore, the expected TMIT 
+(around 340 K) and variation in resistivity across MIT (over 102) similar to the 
+previous reports were achieved in as-sintered VO2 pellet without any doping, and this 
+
+9 
+ 
+profoundly demonstrate that the potential generation in oxygen vacancy is not 
+problematic in the present work. From the NEXAFS analysis, it is noteworthy that in 
+contrast to conventional semiconductive doping process that mainly alters the carrier 
+concentration, the elementary substitution of VO2 also results in compliant variation 
+in their electronic structures. 
+ 
+3.5. Meanwhile improving the mechanical strength of the substituted VO2  
+Apart from regulating the electronic structures and MIT properties, it is also 
+worth noticing that the mechanical properties of VO2 pellet can be meanwhile 
+improved along with the elementary substitution process using the SPARS approach 
+via compositing with the residual dopants. Notably, the abrupt variation in the lattice 
+constant of VO2 across the TMIT is the main problem that interrupts their electronic 
+functionality in potential applications as critical temperature resister (CTR). For 
+example, the structural variation promotes the generation and propagation of the 
+cracks within VO2 that reduce the abruption in their MIT and deteriorate the 
+mechanical strength. Therefore, both the MIT properties and the mechanical strength 
+were expected to be improved for the practical electronic applications of VO2. 
+ 
+Fig. 4a compares the Vickers hardness (HV) measured for doped VO2 pellets 
+herein made by SPARS method as a function of the doping concentration. It can be 
+seen that the HV for doped VO2 (e.g., over 500 kgf / mm2) largely exceeds the one 
+observed for the pristine VO2 (e.g., 267.2 kgf / mm2), while HV shows a generally 
+increasing tendency with the doping concentration. This is attributed to the presence 
+of the residual dopant oxide was not fully diffuse into the lattice of VO2 during the 
+SPARS process that strengthens the grain boundary. In Fig. 4b, we further summarize 
+the achievable mechanical strength for the doped VO2 as made in this work plotted as 
+a function of their transition abruption in resistivity across the TMIT. It clearly 
+demonstrates the feasible type and composition in the dopant oxides to achieve abrupt 
+MIT functionality while maintaining good mechanical strength (marked by the square 
+in Fig. 4b). As-observed mechanical strength in the VO2 pellet composited with the 
+high melting-point oxides were comparable to the one observed for VO2 microcrystal 
+as measured by the nanoindentation testing 49. It is worth noticing that the grain 
+boundary within polycrystalline VO2 bulk material inevitably leads to the suppression 
+in its macro mechanical property, as compared with the nanoindentation associated to 
+the nanoscale hardness. 
+ 
+In particular, the co-sintering of VO2 with dopant oxides exhibiting high melting 
+point (e.g., Al2O3, HfO2 and CoO) not only improve the mechanical strength of the 
+pellets, but also enhance the transition abruption in their -T tendencies across TMIT. 
+This is demonstrated in Fig. 4c, where more abrupt MIT functionality with similar 
+TMIT is observed for Hf, Al and Co doped VO2 pellet compared to the pristine VO2 
+pellet. In particular, the magnitude of their basic resistivity at room-temperature (ρ300 K) 
+can be regulated within a broad range across two orders of magnitudes as 
+
+10 
+ 
+demonstrated in Fig. 4d, and this is expected to extend the thermistor functionality. As 
+a representative, Fig. 4e demonstrates the distribution of the elementary distribution of 
+V, Co and O as measured by energy dispersion spectroscopy (EDS) for co-sintered 
+VO2 and CoO with a nominal composition of V0.7Co0.3O2. It can be seen that the 
+cobalt element is not observed in the region associated with the vanadium dioxides, 
+and this differs to the situation for the aforementioned dopant oxides with low melting 
+points (e.g., WO3, Nb2O5, TiO2, etc.).  
+ 
+The above results indicate the less effective diffusion of Co within VO2 during 
+the sintering process owing to the relatively high melting point of CoO (see more 
+details in Fig. S10), and this is in agreement to the small variation in TMIT as observed 
+in Fig. S11. This understanding is further confirmed by the Rietveld refinement of 
+their XRD patterns as shown in Fig. S12, while their corresponding percentage of 
+residual dopants (e.g., Al2O3, CoO, Nb2O5, WO3 and TiO2) were further summarized 
+in Table S4. It demonstrates that the reaction ration (Preact) between the dopant oxides 
+with low melting point (Tmelt) and VO2 (e.g., Preact is of almost ~ 99.6 % for Nb2O5 
+with Tmelt of 1485 C) largely exceeds the ones observed for the dopant oxides with 
+high Tmelt (e.g., Preact is of ~ 80 % for CoO with Tmelt for 1935 C). Nevertheless, the 
+residual dopant oxides, such as Al2O3, HfO2 and CoO, significantly improve the 
+mechanical properties of VO2 without sacrificing the MIT properties. From the 
+perspective of mechanical properties, the presence of compositing oxides at high 
+insulating state (e.g., Al2O3, HfO2 and Co3O4) can effectively prohibit the generation 
+and propagation of the cracks along the grain boundary. Therefore, a more stable 
+electronic functionality for the as-made doped VO2 bulk material is expected, which is 
+a critical issue for the practical applications in electronic devices upon thermoshock. 
+From the perspective of MIT properties, the compositing with the insulating oxides 
+located at the grain boundary can reduce the charge leakage along the cracks within 
+grain boundary for the insulating phase of VO2, and this leads to abrupt transitions in 
+resistivity across the TMIT. Similar effects were also previously reported for the 
+vanadium phosphate glass (VPG) 50 and vanadium-based critical temperature resistor 
+39. Therefore, a more stable electronic functionality for the as-made doped VO2 bulk 
+material is expected, which is a critical issue for the practical applications in 
+electronic devices upon thermoshock 51. 
+ 
+4. Conclusions 
+In summary, we largely improved the effectiveness in regulating the MIT 
+functionality via elementary doping as well as the mechanical properties of VO2 bulk 
+material by developing a spark plasma assisted reactive sintering (SPARS) approach. 
+Compared to the conventional solid-state reactions, the spark plasma induced 
+non-equilibrium process promotes the diffusion of the dopant element from its oxide 
+precursor into the lattice of the powder of VO2 within a largely shortened co-sintering 
+period of ~10 minutes. This brings in high flexibility and convenience to control the 
+dopant type and concentration within bulk VO2, enabling a broad adjustment in the 
+
+11 
+ 
+TMIT (e.g., within 240-350 K). In contrast to conventional semiconductive doping 
+process, compliant variations in the electronic structure are observed within the doped 
+VO2 as probed by NEXAFS, and therefore variation in the MIT properties is more 
+likely to be associated with the doping-induced electronic orbital regulation rather 
+than simply varying the carrier concentration. Furthermore, the mechanical strength of 
+the doped VO2 bulk material can be meanwhile largely improved (e.g., from 267.2 to 
+950 kgf / mm2), owing to compositing effect of the unreacted high melting-point 
+oxides. The high tunability in the MIT functionality and improved mechanical 
+strength as enabled by the SPARS approach pave the way towards practical 
+applications of VO2 as a bulk material in power electronics. 
+
+12 
+ 
+Notes 
+The authors declare no competing financial interest. 
+ 
+Acknowledgements 
+This work was supported by the National Key Research and Development 
+Program of China (No. 2021YFA0718900), the National Natural Science Foundation 
+of China (No. 62074014 and 52073090), and the Beijing New-star Plan of Science 
+and Technology (No. Z191100001119071).  
+ 
+Supporting Information 
+Supporting information is available online. See the supporting information for 
+more details about the synthesis method, the derivation of TMIT, X-ray diffraction 
+(XRD) patterns, the energy dispersion spectrum (EDS), X-ray photoelectron 
+spectroscopy (XPS), the Rietveld refinements and additional analysis results. 
+ 
+Correspondences: Correspondences should be addressed: Prof. Jikun Chen 
+(jikunchen@ustb.edu.cn), Prof. Yong Jiang (yjiang@ustb.edu.cn) and Prof. Nuofu 
+Chen (nfchen@ncepu.edu.cn). 
+ 
+ 
+ 
+
+13 
+ 
+References 
+(1) Ke, Y.; Zhang, B.; Wang, T.; Zhong, Y.; Vu, T. D.; Wang, S.; Liu, Y.; Magdassi, S.; Ye, X.; Zhao, D.; 
+et al. Manipulating atomic defects in plasmonic vanadium dioxide for superior solar and thermal 
+management. Mater. Horiz. 2021, 8 (6), 1700-1710. 
+(2) Morrison, V. R.; Chatelain, R. P.; Tiwari, K. L.; Hendaoui, A.; Bruhacs, A.; Chaker, M.; Siwick, B. J. 
+A photoinduced metal-like phase of monoclinic VO2 revealed by ultrafast electron diffraction. Science 
+2014, 346 (6208), 445-448. 
+(3) Zhou, X.; Li, H.; Meng, F.; Mao, W.; Wang, J.; Jiang, Y.; Fukutani, K.; Wilde, M.; Fugetsu, B.; 
+Sakata, I.; et al. Revealing the Role of Hydrogen in Electron-Doping Mottronics for Strongly 
+Correlated Vanadium Dioxide. J. Phys. Chem. Lett. 2022, 13 (34), 8078-8085. 
+(4) Tang, K.; Dong, K.; Li, J.; Gordon, M. P.; Reichertz, F. G.; Kim, H.; Rho, Y.; Wang, Q.; Lin, C.-Y.; 
+Grigoropoulos, C. P.; et al. Temperature-adaptive radiative coating for all-season household thermal 
+regulation. Science 2021, 374 (6574), 1504-+. 
+(5) Wang, S.; Jiang, T.; Meng, Y.; Yang, R.; Tan, G.; Long, Y. Scalable thermochromic smart windows 
+with passive radiative cooling regulation. Science 2021, 374 (6574), 1501-+. 
+(6) Cui, Y.; Ke, Y.; Liu, C.; Chen, Z.; Wang, N.; Zhang, L.; Zhou, Y.; Wang, S.; Gao, Y.; Long, Y. 
+Thermochromic VO2 for Energy-Efficient Smart Windows. Joule 2018, 2 (9), 1707-1746. 
+(7) Kats, M. A.; Blanchard, R.; Zhang, S.; Genevet, P.; Ko, C.; Ramanathan, S.; Capasso, F. Vanadium 
+Dioxide as a Natural Disordered Metamaterial: Perfect Thermal Emission and Large Broadband 
+Negative Differential Thermal Emittance. Phys. Rev. X 2013, 3 (4). 
+(8) Tang, K.; Wang, X.; Dong, K.; Li, Y.; Li, J.; Sun, B.; Zhang, X.; Dames, C.; Qiu, C.; Yao, J.; et al. A 
+Thermal Radiation Modulation Platform by Emissivity Engineering with Graded Metal-Insulator 
+Transition. Adv. Mater. 2020, 32 (36). 
+(9) Zhang, H.-T.; Park, T. J.; Islam, A. N. M. N.; Tran, D. S. J.; Manna, S.; Wang, Q.; Mondal, S.; Yu, 
+H.; Banik, S.; Cheng, S.; et al. Reconfigurable perovskite nickelate electronics for artificial intelligence. 
+Science (New York, N.Y.) 2022, 375 (6580), 533-539. 
+(10) Zhang, H.-T.; Park, T. J.; Zaluzhnyy, I. A.; Wang, Q.; Wadekar, S. N.; Manna, S.; Andrawis, R.; 
+Sprau, P. O.; Sun, Y.; Zhang, Z.; et al. Perovskite neural trees. Nat. Commun. 2020, 11 (1). 
+(11) Zhang, Z.; Mondal, S.; Mandal, S.; Allred, J. M.; Aghamiri, N. A.; Fali, A.; Zhang, Z.; Zhou, H.; 
+Cao, H.; Rodolakis, F.; et al. Neuromorphic learning with Mott insulator NiO. PNAS 2021, 118 (39). 
+(12) Chen, J.; Hu, H.; Wang, J.; Yajima, T.; Ge, B.; Ke, X.; Dong, H.; Jiang, Y.; Chen, N. Overcoming 
+synthetic metastabilities and revealing metal-to-insulator transition & thermistor bi-functionalities for 
+d-band correlation perovskite nickelates. Mater. Horiz. 2019, 6 (4), 788-795. 
+(13) Park, J. H.; Coy, J. M.; Kasirga, T. S.; Huang, C.; Fei, Z.; Hunter, S.; Cobden, D. H. Measurement 
+of a solid-state triple point at the metal-insulator transition in VO2. Nature 2013, 500 (7463), 431-434. 
+(14) Jeong, J.; Aetukuri, N.; Graf, T.; Schladt, T. D.; Samant, M. G.; Parkin, S. S. P. Suppression of 
+Metal-Insulator Transition in VO2 by Electric Field-Induced Oxygen Vacancy Formation. Science 2013, 
+339 (6126), 1402-1405. 
+(15) Lee, D.; Chung, B.; Shi, Y.; Kim, G. Y.; Campbell, N.; Xue, F.; Song, K.; Choi, S. Y.; Podkaminer, 
+J. P.; Kim, T. H.; et al. Isostructural metal-insulator transition in VO2. Science 2018, 362 (6418), 
+1037-+. 
+(16) Zhou, X.; Wu, Y.; Yan, F.; Zhang, T.; Ke, X.; Meng, K.; Xu, X.; Li, Z.; Miao, J.; Chen, J.; et al. 
+Revealing the high sensitivity in the metal toinsulator transition properties of the pulsed laser deposited 
+VO2 thin films. Ceram. Int. 2021, 47 (18), 25574-25579. 
+
+14 
+ 
+(17) Janod, E.; Tranchant, J.; Corraze, B.; Querre, M.; Stoliar, P.; Rozenberg, M.; Cren, T.; Roditchev, 
+D.; Vinh Ta, P.; Besland, M.-P.; et al. Resistive Switching in Mott Insulators and Correlated Systems. 
+Adv. Funct. Mater. 2015, 25 (40), 6287-6305. 
+(18) Liu, K.; Lee, S.; Yang, S.; Delaire, O.; Wu, J. Recent progresses on physics and applications of 
+vanadium dioxide. Mater. Today 2018, 21 (8), 875-896. 
+(19) Shibuya, K.; Kawasaki, M.; Tokura, Y. Metal-insulator transition in epitaxial V1−xWxO2 (0≤x≤0.33) 
+thin films. Appl. Phys. Lett. 2010, 96 (2). 
+(20) Sakai, E.; Yoshimatsu, K.; Shibuya, K.; Kumigashira, H.; Ikenaga, E.; Kawasaki, M.; Tokura, Y.; 
+Oshima, M. Competition between instabilities of Peierls transition and Mott transition in W-doped VO2 
+thin films. Phys. Rev. B 2011, 84 (19). 
+(21) Takami, H.; Kanki, T.; Ueda, S.; Kobayashi, K.; Tanaka, H. Filling-controlled Mott transition in 
+W-doped VO2. Phys. Rev. B 2012, 85 (20). 
+(22) Holman, K. L.; McQueen, T. M.; Williams, A. J.; Klimczuk, T.; Stephens, P. W.; Zandbergen, H. 
+W.; Xu, Q.; Ronning, F.; Cava, R. J. Insulator to correlated metal transition in V1-xMoxO2. Phys. Rev. B 
+2009, 79 (24). 
+(23) Soltani, M.; Chaker, M.; Haddad, E.; Kruzelecky, R. V.; Margot, J. Effects of Ti-W codoping on 
+the optical and electrical switching of vanadium dioxide thin films grown by a reactive pulsed laser 
+deposition. Appl. Phys. Lett. 2004, 85 (11), 1958-1960. 
+(24) Goncalves, A.; Resende, J.; Marques, A. C.; Pinto, J. V.; Nunes, D.; Marie, A.; Goncalves, R.; 
+Pereira, L.; Martins, R.; Fortunato, E. Smart optically active VO2 nanostructured layers applied in 
+roof-type ceramic tiles for energy efficiency. Sol. Energy Mater. Sol. Cells 2016, 150, 1-9. 
+(25) Wu, Y.; Fan, L.; Liu, Q.; Chen, S.; Huang, W.; Chen, F.; Liao, G.; Zou, C.; Wu, Z. Decoupling the 
+Lattice Distortion and Charge Doping Effects on the Phase Transition Behavior of VO2 by Titanium 
+(Ti4+) Doping. Sci. Rep. 2015, 5. 
+(26) Foltyn, S. R.; Civale, L.; Macmanus-Driscoll, J. L.; Jia, Q. X.; Maiorov, B.; Wang, H.; Maley, M. 
+Materials science challenges for high-temperature superconducting wire. Nat. Mater. 2007, 6 (9), 
+631-642. 
+(27) Florea, M.; Avram, D.; Maraloiu, V. A.; Cojocaru, B.; Tiseanu, C. Heavy doping of ceria by wet 
+impregnation: a viable alternative to bulk doping approaches. Nanoscale 2018, 10 (37), 18043-18054. 
+(28) Silversmit, G.; Depla, D.; Poelman, H.; Marin, G. B.; De Gryse, R. Determination of the V 2p 
+XPS binding energies for different vanadium oxidation states (V5+ to V0+). J. Electron Spectrosc. and 
+Related Phenomena 2004, 135 (2-3), 167-175. 
+(29) Lee, D.; Kim, H.; Kim, J. W.; Lee, I. J.; Kim, Y.; Yun, H.-J.; Lee, J.; Park, S. Hydrogen 
+incorporation induced the octahedral symmetry variation in VO2 films. Appl. Surf. Sci. 2017, 396, 
+36-40. 
+(30) Su, X.; Fu, F.; Yan, Y.; Zheng, G.; Liang, T.; Zhang, Q.; Cheng, X.; Yang, D.; Chi, H.; Tang, X.; et 
+al. Self-propagating high-temperature synthesis for compound thermoelectrics and new criterion for 
+combustion processing. Nat. Commun. 2014, 5. 
+(31) Yang, D.; Su, X.; Li, J.; Bai, H.; Wang, S.; Li, Z.; Tang, H.; Tang, K.; Luo, T.; Yan, Y.; et al. 
+Blocking Ion Migration Stabilizes the High Thermoelectric Performance in Cu2Se Composites. Adv. 
+Mater. 2020, 32 (40). 
+(32) Rao, C. N. R.; Natarajan, M.; Subba Rao, G. V.; Loehman, R. E. Phase transitions and 
+conductivity anomalies in solid solutions of VO2 with TiO2, NbO2, and MoO2. J. Phys. Chem. Solids 
+
+15 
+ 
+1971, 32 (6), 1147-1150. 
+(33) Chen, S.; Liu, J.; Wang, L.; Luo, H.; Gao, Y. Unraveling Mechanism on Reducing Thermal 
+Hysteresis Width of VO2 by Ti Doping: A Joint Experimental and Theoretical Study. J. Phys. Chem. C 
+2014, 118 (33), 18938-18944. 
+(34) Kato, K.; Lee, J.; Fujita, A.; Shirai, T.; Kinemuchi, Y. Influence of strain on latent heat of VO2 
+ceramics. J. Alloy. Compd. 2018, 751, 241-246. 
+(35) Victor, J.-L.; Gaudon, M.; Penin, N.; Chiron, A.; Chung, U. C.; Viraphong, O.; Rougier, A. 
+Innovative sintering process for fabrication of thermochromic smooth VO2 ceramics. J. Alloy. Compd. 
+2022, 890, 161890. 
+(36) Wang, Z.-L.; Zhang, R.; Chen, X.-H.; Fu, Q.-S.; Li, C.-L.; Yuan, S.-L.; Zhao, X.-J.; Tao, H.-Z. Nb 
+doping effect in VO2 studied by investigations of magnetic behavior. Ceram. Int. 2018, 44 (7), 
+8623-8627. 
+(37) Liu, S.-J.; Fang, H.-W.; Su, Y.-T.; Hsieh, J.-H. Metal-insulator transition characteristics of Mo- and 
+Mn-doped VO2 films fabricated by magnetron cosputtering technique. J. J. Appl. Phys. 2014, 53 (6). 
+(38) Victor, J.-L.; Gaudon, M.; Salvatori, G.; Toulemonde, O.; Penin, N.; Rougier, A. Doubling of the 
+Phase Transition Temperature of VO2 by Fe Doping. J. Phys. Chem. Lett. 2021, 12 (32), 7792-7796. 
+(39) Futaki, H. A New Type Semiconductor (Critical Temperature Resistor). J. J. Appl. Phys. 1965, 4 
+(1), 28-41. 
+(40) Aetukuri, N. B.; Gray, A. X.; Drouard, M.; Cossale, M.; Gao, L.; Reid, A. H.; Kukreja, R.; Ohldag, 
+H.; Jenkins, C. A.; Arenholz, E.; et al. Control of the metal-insulator transition in vanadium dioxide by 
+modifying orbital occupancy. Nat. Phys. 2013, 9 (10), 661-666. 
+(41) Wei, J.; Ji, H.; Guo, W.; Nevidomskyy, A. H.; Natelson, D. Hydrogen stabilization of metallic 
+vanadium dioxide in single-crystal nanobeams. Nat. Nanotechnol. 2012, 7 (6), 357-362. 
+(42) Phu Tran Phong, L.; Hofhuis, K.; Rana, A.; Huijben, M.; Hilgenkamp, H.; Rijnders, G. A. J. H. M.; 
+ten Elshof, J. E.; Koster, G.; Gauquelin, N.; Lumbeeck, G.; et al. Tailoring Vanadium Dioxide Film 
+Orientation Using Nanosheets: a Combined Microscopy, Diffraction, Transport, and Soft X-Ray in 
+Transmission Study. Adv. Funct. Mater. 2020, 30 (1). 
+(43) Yoon, H.; Choi, M.; Lim, T.-W.; Kwon, H.; Ihm, K.; Kim, J. K.; Choi, S.-Y.; Son, J. Reversible 
+phase modulation and hydrogen storage in multivalent VO2 epitaxial thin films. Nat. Mater. 2016, 15 
+(10), 1113-+. 
+(44) Chen, Y.; Wang, Z.; Chen, S.; Ren, H.; Wang, L.; Zhang, G.; Lu, Y.; Jiang, J.; Zou, C.; Luo, Y. 
+Non-catalytic hydrogenation of VO2 in acid solution. Nat. Commun. 2018, 9. 
+(45) Ruzmetov, D.; Senanayake, S. D.; Narayanamurti, V.; Ramanathan, S. Correlation between 
+metal-insulator transition characteristics and electronic structure changes in vanadium oxide thin films. 
+Phys. Rev. B 2008, 77 (19). 
+(46) Kristensen, I. K. Semiconductor‐to‐Semiconductor Transition in the Pseudobinary System TiO2–
+VO2. J. Appl. Phys. 1968, 39 (11), 5341-5342. 
+(47) Zhang, W.; Wang, K.; Fan, L.; Liu, L.; Guo, P.; Zou, C.; Wang, J.; Qian, H.; Ibrahim, K.; Yan, W.; 
+et al. Hole Carriers Doping Effect on the Metal-Insulator Transition of N-Incorporated Vanadium 
+Dioxide Thin Films. J. Phys. Chem. C 2014, 118 (24), 12837-12844. 
+(48) Zhang, Z.; Zuo, F.; Wan, C.; Dutta, A.; Kim, J.; Rensberg, J.; Nawrodt, R.; Park, H. H.; Larrabee, 
+T. J.; Guan, X.; et al. Evolution of Metallicity in Vanadium Dioxide by Creation of Oxygen Vacancies. 
+Phys. Rev. Appl. 2017, 7 (3), 034008. 
+
+16 
+ 
+(49) Verma, D.; Singh, D.; Balakrishnan, V. Fracture toughness of VO2 microcrystals across 
+metal-insulator transition. Mater. Lett. 2022, 315. 
+(50) Ivon, A. I.; Kolbunov, V. R.; Chernenko, I. M. Conductivity stabilization by metal and oxide 
+additives in ceramics on the basis of VO2 and glass V2O5–P2O5. J. Non Cryst. Solids 2005, 351 (46), 
+3649-3654. 
+(51) Ivon, A. I.; Kolbunov, V. R.; Chernenko, I. M. Stability of electrical properties of vanadium 
+dioxide based ceramics. J. Eur. Ceram. Soc. 1999, 19 (10), 1883-1888. 
+
+17 
+ 
+Figures and captions 
+ 
+ 
+Fig. 1. (a) Schematic illustration of as-proposed spark plasma assisted reactive sintering (SPARS) 
+doping strategy that achieves the elementary doping and grain boundary modifications of the bulk 
+VO2. (b) X-ray diffraction (XRD) patterns of as-made V1-xTixO2 bulk material (x from 0.1 to 0.4), 
+as indexed to the structure of the VO2 (M) with the P21/c space group. (c) The c-axis lattice 
+constants of as-made V1-xTixO2 plotted as a function of x. (d) The cross-section surface 
+morphology of as-synthesized V0.8Ti0.2O2 bulk material from scanning electron microscopy (SEM). 
+(e), (f) The energy dispersion spectrum (EDS) of (e) V and (f) Ti elements for the as-made 
+V0.8Ti0.2O2 bulk. (g), (h) The X-ray photoemission spectroscopy (XPS) analysis of (g) V 2p peak 
+and (h) O 1s peak for the V0.8Ti0.2O2 bulk. (i) Temperature dependence of the resistivity for 
+as-made V1-xTixO2. (j) The metal to insulator transition temperature (TMIT) of as-made V1-xTixO2 
+plotted as a function of the nominal Ti substitution concentration, compared with the previous 
+report 32. 
+
+(a)
+Pressure
+Microcell
+Sparked
+Plasma
+>>>
+VO,Particle
+Electric
+Bulk
+TMmO,Particle
+Current
+(b)
+(c)
+(d)
+(011)
+Vo.gTio.102
+5.46
+5.44
+Intensity (a.u.)
+Vo.aTio.202
+Vo.7Tio.302
+c
+5.40
+Vo.Tio.02
+VO, (M)
+5.38
+PDF#09-0142
+0
+10
+20
+30
+40
+20
+30
+40
+50
+09
+70
+80
+20(°)
+Nominal DopingConcentration(%)
+(e)
+(f)
+(g)
+VKa1
+Ti Ka1
+VoaTlo.202
+V2p3/2
+Counts
+V2P1/2
+25um
+25um
+512
+516
+520
+524
+528
+(h)
+(i)
+Binding Energy (eV)
+10
+Vo.Tio.202
+345
+-VO2
+oThiswork
+Ti2P3r2
+10°
+Vo.gTio.,02
+340
+ORef.
+Vo.8Tio.202
+Counts
+Vo.7Tio.302
+335
+Vo.6Tio.402
+Vo.5Tio.502
+330
+Ti2P1/2
+TA
+p
+325
+10-3
+320
+.
+10-4
+315
+453
+456
+459
+462
+465
+468
+300
+320
+340
+360
+0
+10
+20
+30
+40
+50
+BindingEnergy (eV)
+T(K)
+Nominal Doping Concentration(%)18 
+ 
+  
+ 
+Fig. 2. (a) The temperature dependence of the resistivity for as-made transitional metal (TM) 
+doped VO2 pellet (TM = Fe, W, Nb, Cr and Mo). (b) The TMIT of as-made doped VO2 plotted as a 
+function of the nominal doping concentration and also compared to the previous reports 19, 22, 32, 36, 
+38. (c) The variation in the resistivity across the metal to insulator transition (Insul. / Metal.) of the 
+doped VO2 made in this work, plotted as a function of their metal to insulator transition 
+temperature (TMIT). (d) The modulation capability in TMIT, as evaluated by dTMIT / dndopant, for 
+various TM dopant elements plotted as a function of their valence state, as also compared with the 
+previous reports 19, 22, 32, 36, 38. The inset shows the zoom in results for Ti, Cr and Fe.  
+
+(a)
+(b)
+-VO2
+Vo.995Wo.0502Vo.9gWo.0102 Vo.98Mo0.0202 Vo.96Mo0.0402
+W
+Mo
+Fe
+Nb
+Vo.95Nbo.0502-VogNbo.1O2
+—Vo.gFeo.102-Vo.sFeo.202
+-Vo.7Cro.30.
+Cr O Ti Ref.
+★
+W Ref.
+Mo Ref.
+Nb Ref.  Fe Ref.
+400
+V
+10-1
+380
+Hole Doping
+360
+Pristine
+V
+10-2
+340
+★★
+MIT
+320
+C
+Heating
+10-3
+300
+Q
+280
+Cooling
+Electron Doping
+10-4
+260
+240
+200
+250
+300
+350
+400
+10
+T (K)
+Nominal Doping Concentration (%)
+(c)
+W
+Mo
+Fe
+Nb 
+Cr
+(d)
+(K %-1)
+5
+This work
+★
+O Ref.
+102
+5
+Mo
+10
+0
+15
+Nb
+-0.4
+Ti
+0
+20
+★
+101.
+W
+240
+320
+330
+340
+350
+3
+4
+5
+6
+TMIT (K)
+Dopant Valence19 
+ 
+ 
+Fig. 3. (a), (b) The near-edge X-ray absorption fine structure (NEXAFS) analysis of (a) V-L edge 
+and (b) O-K edge for the transitional metal (TM) substituted VO2 (TM = Ti, Fe, W, Nb and Mo), 
+compared to the pristine VO2. (c) Illustrating the variation in electronic structures of VO2 via 
+high-valence (e.g., W, Nb and Mo) and low-valence (e.g., Fe and Cr) TM substitutions.  
+ 
+ 
+ 
+
+(a)
+(b)
+V-L Edge
+O-K Edge
+V-LIl
+V-L II
+t2g
+Intensity (a.u.)
+Intensity (a.u.)
+Vo.8 Tio.202 
+_ Vo.gNbo.102
+Vo.8 Tio.202
+- Vo.gNbb.10.
+Vo.8Feo.202 Vo.9gWo.0102
+Vo.8Fe0.202
+V0.99Wo.0102
+VO2
+Vo.96M00.0402
+VO2
+Vo.96M00.04O2
+512
+516
+520
+524
+528
+530
+532
+Photon Energy (eV)
+Photon Energy (eV)
+(c)
+Metallic State
+InsulatingState
+a*
+6
+a*
+元*
+元*
+TMIT
+di
+di*
+元*
+元
+Eft
+MIT
+Ef -
+du
+Electron
+di
+d i
+00000
++Hole
+元
+元
+元
+元
+a
+a
+a
+a
+High-Valence Substitution
+Low-Valence Substitution20 
+ 
+ 
+ 
+Fig. 4. (a) The Vickers hardness (HV) for the transitional metal (TM) substituted VO2 (TM = Ti, Fe, 
+W, Nb, Mo, Hf, Al and Co) made herein, plotted as a function of the nominal doping concentration. 
+(b) Plotting HV as a function of the variation in resistivity across the metal to insulator transition 
+temperature (e.g., Insul. / Metal.) for as-made V1-xTMxO2 (the number of x is marked near the data 
+points) pellet, as compared with the ref. 45. (c) The normalized temperature dependence of the 
+resistivity (-T) of the co-sintered VO2 with transitional metal oxides with high melting points. (d) 
+The basic resistivity at room-temperature of as-made V1-xTMxO2 plotted as a function of the 
+nominal doping concentration. (e) The energy dispersion spectrum (EDS) of V, Co and O elements 
+for the as-made V0.7Co0.3O2 pellet. 
+ 
+
+Ti
+★
+W
+Nb
+Mo
+Fe
+VO2
+OV1-xTi,O2
+V1-xHf,O2
+V1-xCo,02
+V1-xAl,O2
+(a)
+Hf
+Al
+Co
+(b)
+V1-xW,02
+V1-xNb,O2
+V1-xMo,02
+V1-xFe,02
+☆ Ref.
+1000
+1000
+0.3
+0.3
+0.2
+Load
+900
+0.2
+Hy (kgf / mm²)
+/ mm²)
+0.1
+0.4
+800
+800
+0.3
+0.3
+700
+0.1
+0.4
+(kgf /
+0.04
+600
+600
+0.2
+0.01
+500
+0.005
+0.4
+0.05
+400
+400
+VO,
+300
+Pristine VO,
+★
+200
+200
+10
+101
+102
+103
+Nominal Doping Concentration (%)
+Pinsul./ PMetal.
+(c)
+(d)
+100
+10°
+★ Pristine
+Hf
+VO2
+Co
+Hf
+Vo.gHfo.202
+Vo.7Hfo.302
+10-1
+(Pd
+10-2
+-Vo.6Hfo.402
+10-3
+Vo.8Alo.202
+★
+$Vo.7Alo.302
+Vo.7C00.302
+10-4
+10-2
+280
+300
+320
+340
+360
+380
+0
+10
+20
+30
+40
+T (K)
+Nominal Doping Concentration (%)
+(e
+10.7C00.302
+25μm
\ No newline at end of file
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+page_content=' Yuchen Cui a†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Yanlong Shang a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Haifan Li a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Jiaou Wang b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Ye Meng c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Xiaoguang Xu a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Yong Jiang a*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Nuofu Chen d* and Jikun Chen a*  a School of Materials Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' University of Science and Technology Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 100083,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' China  b Beijing Synchrotron Radiation Facility,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Institute of High Energy Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Chinese Academy of  Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' China  c Institute for Advanced Materials and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' University of Science and Technology Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Beijing 100083,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' China  d School of Renewable Energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' North China Electric Power University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Beijing 102206,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' China  Authors to whom correspondence should be addressed: jikunchen@ustb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='cn (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Chen),  yjiang@ustb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='cn (Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Jiang), nfchen@ncepu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='cn (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Chen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  † X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Zhou and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Cui contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  2  Abstract  Although vanadium dioxide (VO2) exhibits the most abrupt metal to insulator  transition (MIT) properties near room-temperature, the present regulation of their MIT  functionalities is insufficient owing to the high complexity and susception associated  with V4+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Herein, we demonstrate a spark plasma assisted reactive sintering (SPARS)  approach to simultaneously achieve in situ doping and sintering of VO2 within largely  short period (~10 minutes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' This enables high convenience and flexibility in  regulating the electronic structure of VO2 via dopant elements covering Ti, W, Nb, Mo,  Cr and Fe, leading to a wide adjustment in their metal to insulator transition  temperature (TMIT) and basic resistivity (ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Furthermore, the mechanical strengths of  the doped-VO2 were meanwhile largely improved via the compositing effect of high  melting-point dopant oxide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' The high adjustability in MIT properties and improved  mechanical properties further paves the way towards practical applications of VO2 in  power electronics, thermochromism and infrared camouflage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Key words: Quantum materials, Electronic phase transition, Correlated oxides,  Correlated electronics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  TOC Graphic  Ti  大  W  Mo  Fe  Nb  Pressure  Ti Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  ★  W Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Mo Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Fe Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Nb Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  400  V  HoleDoping  350  Pristine VO  V  vO,Particle  Electric  TMmO,Particle  Current  300  ElectronDoping  Sparked  Plasma  250  10  100  Nominal Doping Concentration (%)3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Introduction  The metal to insulator transition (MIT) within d-band correlated oxides brings in  abruptions in regulating their electronic and optical properties beyond conventional  semiconductors 1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Among the existing family of MIT materials, the vanadium  dioxide (VO2) exhibits the most abrupt MIT properties near room-temperature that  plays irreplaceable role in applications, such as thermochromism 4-6, infrared  camouflage 7, 8, and correlated electronic devices 9-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Nevertheless, these applications  of VO2 are still challenged by its material synthesis as the electronic phase diagram of  vanadium oxides is extraordinary complex 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' For example, the metal to insulator  transition temperature (TMIT) that triggers the MIT is known to be rather sensitive to  the valence state of vanadium that is at the intermediate valence state (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', V4+) and  susceptible to both oxidizing and reducing environment 14-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Conventionally, the MIT  properties and electronic structure of VO2 is adjusted by the elementary substitution,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', the TMIT is elevated or descended via substituting V4+ by dopants with lower or  higher valence states, respectively 17, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  To achieve the abrupt variation in the material resistivity during the MIT of VO2,  it is vital to precisely control the concentration and homogeneity in the distribution of  the dopant elements, without extrinsically altering the oxygen composition 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' As  compared to the vacuum deposition of doped-VO2 thin films via plasmanized or  vaporized precursors 20-23, achieving a precise elementary substitution for the bulk  materials of VO2 while maintaining their MIT abruption is more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Previously,  doped VO2 bulk pellets were made using the following two routes: 1) Firstly  synthesizing the powder materials for doped VO2, and afterwards sintering such  powder into bulk pellets at precisely controlled atmosphere 24, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2) Co-sintering VO2  and dopant oxides at high temperatures over a long period (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', several days) at  controlled atmosphere 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Usually, the homogeneity and achievable concentration of  the dopant element within VO2 are restricted by its equilibrium phase diagram, while  the kinetic process that triggers the diffusion of the dopant elements may easily  disturb the desired valence state of vanadium 26, 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Up to date, an effective strategy to  accurately adjust the MIT properties of the VO2 bulk material via conveniently  achievable elementary substitution is yet lacking, and this impedes applying VO2 for  power electronics as discrete elementary devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  In this article, we demonstrate a spark plasma assisted reactive sintering (SPARS)  strategy that largely elevates the flexibility and effectiveness in the elementary  substitution of VO2 as bulk material, covering a large abundance of the dopant  elements such as Ti, W, Nb, Mo, Cr, and Fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' The non-equilibrium process related to  SPARS largely shorten the diffusion period of the dopants (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', 10 minutes), and it  enables a broad adjustment of the electronic structure and the MIT functionality of  VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Compliant variations in the electronic structure of VO2 were probed via the  near-edge X-ray absorption fine structure (NEXAFS) analysis that differs to the  conventional doping of semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Furthermore, the unreacted dopant oxides  4  with high melting point can be meanwhile used to improve the mechanical properties  of VO2, paving the way towards their practical applications in power electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Experimental Section  A spark plasma assisted reactive sintering (SPARS) strategy was used for  synthesizing the doped VO2 bulk materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' The as-proposed SPARS approach  includes the following two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' In the first step, the powder of VO2 is homogenously  mixed with the transition metal oxides (TMmOn) dopants at the desired stoichiometry,  and afterwards pressed inside a graphite die.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' In the second step, a large electric current  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', 300 A/cm2) and high pressure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', 20 MPa) are imparted into the graphite die  and trigger the solid-state reaction between VO2 and TMmOn powders at the  temperature of 900 \uf0b0C for 10 minutes under a vacuum atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' The as-made VO2  bulk material was a pellet with the radius of 10 mm and the thickness of around 4  mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  The crystal structure of samples was characterized by the X-ray diffraction (XRD)  (RIGAKU, Smartlab).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' The valence state of as-made samples was characterized by  X-ray photoelectron spectroscopy (XPS) (Kratos, AXIS ULTRADLD) that was  equipped with monochromatic Al-Kα source (150 W) and photon energy of 1486.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='6 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  All the binding energies as obtained from the XPS were referenced to the C 1s peak  from the adventitious carbon at 284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='8 eV, while a Shirley function was used to  subtract the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' In addition, the valence state of vanadium for VO2 is  indicated by the binding energy of V 2p3/2 and V 2p1/2 peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' To fit the XPS curves,  the V 2p3/2 peak can be resolved by the two binding energies at 516.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='3 and 515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='7 eV  that represents the valence state of V4+ and V3+, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' The V 2p1/2 region can  be analogously divided into the two peaks that are respectively located at 523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='6 and  523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='0 eV, according to the previous reports 28, 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' And their morphologies were  characterized by scanning electron microscopy (SEM) and energy dispersion  spectrum (EDS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' The electronic structures of as-made samples were also investigated  by the near edge X-ray absorption fine structure (NEXAFS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' To characterize their  electrical transportation properties, their temperature dependences of resistivity within  300-400 K were measured by using a commercialized CTA-system in vacuum, while  the one within 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='8-400 K was measured by using the physical property measurement  system (PPMS) (Quantum Design).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' The Vickers hardness of VO2 bulk material were  measured by the digital Vickers hardness tester (VTD 512) at room-temperature under  the load of 500 kgf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Results and Discussion  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Strategy for elementary substitution of VO2 via SPARS  To address the center challenge in the precise element doping for bulk VO2,  herein a spark plasma assisted in situ doping strategy is used, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Unlike the previous literatures that uses the reactive spark plasma process for  sintering the powder material into bulk pellet, or directly synthesizing compounds  from their respective elementary substances 30, 31, herein the reactive spark plasma is  5  used to achieve the element doping of oxides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Firstly, the powder of VO2 is  homogenously mixed with the doped transitional metal oxides (TMmOn, TM = Ti, W,  Nb, Mo, Fe, and Cr) and pressed inside a graphite die.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Afterwards, a large DC electric  current (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', around 300 A/cm2) and high pressure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', 20 MPa) are imparted to  spark over the powder precursor that triggers their mutual element diffusion and  rapidly elevate the temperature to 900 \uf0b0C (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', 60% of the melting point of VO2) for  their reactive sintering into bulk pellet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Such spark plasma assisted non-equilibrium  process brings in the electrical break down process within the sintering that promotes  the elementary diffusion and shortens the period associated with the reactive sintering  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', 10 minutes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Meanwhile, the residual of the dopant oxides with high melting  point within the grain boundary can be used to improve the mechanical properties of  the VO2 pellet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Furthermore, as-proposed SPARS approach makes it possible to  flexibly adjust the dopant composition and concentration within VO2 with high  convenience, by just using the powder of pure VO2 and the dopant oxides as  precursors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Metal to insulator transition properties for Ti substituted VO2  As-proposed SPARS approach was first applied to achieve Ti-doping of VO2,  and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 1b shows the X-ray diffraction (XRD) patterns of as-made V1-xTixO2 pellets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  In general, the V1-xTixO2 exhibit the same crystal structure to VO2 (M), while their  lattice expands with an increasing x as demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 1c (see more details of  their structure information from the Rietveld refinement in Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' It is also worth  noticing that the c0/a0 in the lattice constant slightly increases indicated by the  Rietveld refinements (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' S1), and this observation is in agreement to the  understanding by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Rao et al that a local structural distortion upon Ti  substitution may also relevant to the variation in MIT properties32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' This clearly  demonstrates the effective substitution of the lattice vanadium within VO2 by titanium  that exhibits larger ionic radius, as is in agreement to the previous report 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' The  cross-section morphology of as-sintered V1-xTixO2 pellet as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 1d  demonstrates its polycrystallinity with a grain size of ~20 µm, and the same  distribution is observed for V and Ti as further confirmed by the energy dispersion  spectroscopy (EDS) shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 1e and 1f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Furthermore, X-ray photoemission  spectroscopy (XPS) analysis shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 1g indicates the generation of V3+ valence  state via Ti doping of VO2, while the valence state of the Ti remains as 4+ as shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 1h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' The above understanding about the presence of V3+ valence state via Ti  doping is further confirmed by the XPS results for other Ti-substituted samples, as  shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' S2 and S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 1i demonstrates the temperature dependence of  resistivity (\uf072-T) as measured for as-made V1-xTixO2 pellets, where abrupt reductions in  their resistivity are observed in \uf072-T tendencies by elevating temperature across their  TMIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' The elevation in the resistivity for the Ti-substituted VO2 is against the  conventional expectation that the presence of V3+ should strengthen the metallic phase  that reduces the room-temperature resistivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' It is more likely to be associated with  the compositing effect of the residual TiO2 remaining at the grain boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  6  Nevertheless, this is hard to be distinguished in the XRD or EDS mapping, since the  particle size of TiO2 (～10-20 nm) is much smaller than VO2 (～20-30 µm) while the  crystal structure is also similar for TiO2 and VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Similar elevation in the  room-temperature resistivity was also previously observed in ref 32, and was  attributed to local structure distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Therefore, the mechanism related to the Ti  doping in VO2 is expected to be more complicated that is worthy to be further  explored in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 1j, the TMIT of as-made V1-xTixO2 pellets were further  plotted as a function of doping concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' As shown TMIT was the averaged ones  from the R-T tendency measured by heating up and cooling down, while each critical  temperatures were taken when the temperature coefficient of resistance (TCR) reached  the maximum, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' S4 and Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' It can be seen that as-achieved  TMIT exhibits a generally reducing tendency with an increasing substitution  concentration of Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' This observation is in general agreement to the previous report 32,  and this is attributed to the distorted local structure32 and the presence of V3+ valence  state induced by the substitution of titanium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' We emphasize that the Ti substitution of  VO2 as achieved presently via our SPARS approach reduces the TMIT that is relevant  to the Ti doping composition, while the underneath mechanisms could be complicated  as is indeed under debate yet according to the previous reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Extending the SPARS strategy to achieved broad regulation in TMIT of VO2  To further extend the regulation of TMIT, more dopant oxides (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', WO3, MoO3,  Nb2O5, Cr2O3 and Fe2O3) were co-sintered with VO2 via the as-proposed SPARS  approach, and their respective XRD pattern and representative morphology are shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' S5 and S6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' It can be seen that the as-synthesized polycrystalline  ceramic pellets via the spark plasma assisted sintering process exhibit relatively  densified morphology without obvious porosity, and this is further confirmed by its  pronounced transition abruption across the TMIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' It can be seen that the as-synthesized  polycrystalline doped VO2 ceramic pellets via the spark plasma assisted sintering  process exhibit relatively densified morphology, despite the presence of little amount  of pores, as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' The as-observed morphology is in consistency with  the previous reports on polycrystalline VO2 bulk pellets without doping as synthesized  by solid-state reaction or spark plasma sintering 34, 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' This indicates that the  as-proposed SPARS strategy can effectively achieve the reactive sintering process that  sinters the precursor powders into densified bulk pellet within largely shortened  reaction period of 10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' As their \uf072-T tendencies shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2a, the TMIT is  more broadly adjusted within 240-350 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2b, as-achieved TMIT were plotted as  a function of the substitution concentration, and further compared with the previous  reports 21, 32, 36-38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Similar to the previous reports for doped VO2 samples 19, herein  substituting the vanadium with elements at higher valence state (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', W6+, Mo6+ and  Nb5+) reduces the TMIT, while the respective substitution via lower valence elements  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', Cr3+ and Fe3+) slightly elevates the TMIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' It is worth noticing that apart from the  variation in the valence state of vanadium, the variation in the lattice constant owing  7  to the different ionic radius of the dopants compared to vanadium may also influence  the MIT properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Nevertheless, in the present work, all the dopant elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=',  Ti4+, Nb5+, W6+, Mo6+, Cr3+ and Fe3+) enlarge the lattice constants of VO2 as further  summarized in the Table S3, but Cr3+ and Fe3+ increase the TMIT of VO2, while Nb5+,  W6+ and Mo6+ reduce its TMIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' This indicates that the regulation in the valence state of  vanadium should be a dominant cause to TMIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2c, the abrupt variations in the  resistivity across the MIT (\uf072Insul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' / \uf072Metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=') is further plotted as a function of TMIT, where  the magnitude generally decays with their TMIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' It is in particular worth noticing that  high transition abruption is achieved for Ti substituted VO2 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', \uf072Insul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' / \uf072Metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  exceeding 102) that is comparable to the presently commercial critical temperature  resister (CTR) as reported previously 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  To evaluate the ability in regulating the TMIT for various dopant elements, the  variation in TMIT is linear fitted with the doping concentration (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' S8), and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  2d as-obtained slope (dTMIT / dndopant) is further plotted as a function of the dopant  valence state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' It can be seen that the variation in TMIT is more effectively reduced by  substituting vanadium with the dopant elements at higher valence states (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', -17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='9  K %-1 for W6+ and -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='4 K %-1 for Nb5+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' In contrast, the regulation in TMIT is less  effective when the substituting elements exhibits the same or lower valence state  compared to V4+ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='6 K %-1 for Fe3+ and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='43 K %-1 for Ti4+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Compliant variation in electronic structures as probed by NEXAFS  To further qualitatively probe the variations in the electronic structures of VO2  associated with the elementary substitution, the near-edge X-ray absorption fine  structure (NEXAFS) analysis was performed, as the results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 3a-3b for  V-L edge and O-K edge, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' In the V-L spectrum of VO2, the V-LⅢ peak  (~518 eV) is recognized to be associated with the V 2p3/2 → 3d transition, while the  V-LⅡ peak (~524 eV) reflects the V 2p1/2 → 3d transition 40-42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' From the perspective  of the hybridization between V 3d and O 2p band, the peaks in the O-K edge (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', O  1s → 2p transition) were respectively corresponding to the t2g and eg orbitals, and the  ratio in intensities of t2g / eg was reported to indicate the orbital occupancy 43, 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Furthermore, the d// orbital associated with the neighboring V ions along the rutile c  axis reflects the V-V interaction, while the \uf070* orbit pointed in the ligands is  corresponding to the electron density that is shifted away from the V-O bond direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  In contrast, \uf073 / \uf073* orbits with rotational symmetry around V-O bond is associated to  the V-O interaction 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Substituting the vanadium with elements at higher valence state (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', W6+, Mo6+  and Nb5+) results in the leftwards shift of the V-L edge (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 3a), and this  observation is in consistency with the XPS results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' S9 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' S9 a  and S9 c) that demonstrate the reduction in the valence state of vanadium towards V3+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  8  In addition, the as-observed similar peak position of O 1s peak (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', ~530.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='1 eV) for  doped VO2 with respect to the pristine one indicates the main presence of lattice  oxygen within the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Meanwhile, the ratio in intensity of t2g / eg is observed to  be smaller compared to the pristine VO2, and this demonstrates an enhanced orbital  occupancy in t2g band, owing to the electron doping from the higher valence  substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' It is known for VO2 that electron carrier doping strengthens the metallic  phase and reduces TMIT, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 3c 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Therefore, the as-observed  reduction in the TMIT via high-valence substitution is associated to the electron  occupancy in the t2g band of VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' In addition, the as-observed reduced intensity of the  t2g peak with respect to the eg peak also indicates the weakening of direct V-V  interactions upon the cationic substitution 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' It is also interesting to note that similar  variations in the electronic structure are observed for Ti substituted VO2, in which  case the V-L edge also shifts leftwards while the t2g / eg ratio decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' The above  observation is in agreement to the reduction in TMIT for Ti substituted VO2, although  the dopant (Ti4+) exhibits the same valence state compared to the matrix (V4+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' It is  worth noticing that the reduction as observed in the TMIT of Ti doped VO2 is  consistency with the previous reports 32, 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Nevertheless, the underneath mechanism  associated to the regulation in the TMIT via equivalent substitution is still in debate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  For example, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Rao et al ascribed it to the distorted crystal structure induced  by the Ti substitution 32, while I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Kristensen attributed such result to the variation  in carrier concentrations 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  In contrast, substituting vanadium with dopant at lower valence state (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', Fe3+)  results in opposite variations in the vanadium valence state, as the V-L edge shifts  rightwards in the V-L edge spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' This is in agreement to the XPS result that the  valence state of vanadium was slightly enhanced towards higher valence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', +5)  valence (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' S9 e), and this is also in general consistency to the observed  elevation in TMIT, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Nevertheless, it is worth noticing that the  relative intensity of t2g peak, compared to eg, in the O-K edge is not increased as  expected for lower valence element substitution of vanadium, but also reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' This  reveals the presence of a second mechanism that may also influence the electronic  structure of the presently made doped VO2 samples, in addition to the substitution of  vanadium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' This is most likely to be associated to the generation of oxygen vacancy  during the SPARS that will also contribute electron carrier from the perspective of  anion regulation 47, and as a result induce the elevation in TMIT via Fe substitution, as  compared to the previous reports 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' In addition, the oxygen vacancy within VO2 is  reported to be formed upon a high-temperature vacuum atmosphere, and this is also  the case during the SPARS process in coarse vacuum atmosphere 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' It is also worthy  to mention that such potentially generated oxygen vacancy is not the dominate reason  for the Ti substitution or higher valence element substitution of vanadium, since a  monotonic variation in TMIT with the substitution concentration is indeed observed for  all these samples sintered at the same condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Furthermore, the expected TMIT  (around 340 K) and variation in resistivity across MIT (over 102) similar to the  previous reports were achieved in as-sintered VO2 pellet without any doping, and this  9  profoundly demonstrate that the potential generation in oxygen vacancy is not  problematic in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' From the NEXAFS analysis, it is noteworthy that in  contrast to conventional semiconductive doping process that mainly alters the carrier  concentration, the elementary substitution of VO2 also results in compliant variation  in their electronic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Meanwhile improving the mechanical strength of the substituted VO2  Apart from regulating the electronic structures and MIT properties, it is also  worth noticing that the mechanical properties of VO2 pellet can be meanwhile  improved along with the elementary substitution process using the SPARS approach  via compositing with the residual dopants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Notably, the abrupt variation in the lattice  constant of VO2 across the TMIT is the main problem that interrupts their electronic  functionality in potential applications as critical temperature resister (CTR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' For  example, the structural variation promotes the generation and propagation of the  cracks within VO2 that reduce the abruption in their MIT and deteriorate the  mechanical strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Therefore, both the MIT properties and the mechanical strength  were expected to be improved for the practical electronic applications of VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 4a compares the Vickers hardness (HV) measured for doped VO2 pellets  herein made by SPARS method as a function of the doping concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' It can be  seen that the HV for doped VO2 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', over 500 kgf / mm2) largely exceeds the one  observed for the pristine VO2 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', 267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='2 kgf / mm2), while HV shows a generally  increasing tendency with the doping concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' This is attributed to the presence  of the residual dopant oxide was not fully diffuse into the lattice of VO2 during the  SPARS process that strengthens the grain boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 4b, we further summarize  the achievable mechanical strength for the doped VO2 as made in this work plotted as  a function of their transition abruption in resistivity across the TMIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' It clearly  demonstrates the feasible type and composition in the dopant oxides to achieve abrupt  MIT functionality while maintaining good mechanical strength (marked by the square  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' As-observed mechanical strength in the VO2 pellet composited with the  high melting-point oxides were comparable to the one observed for VO2 microcrystal  as measured by the nanoindentation testing 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' It is worth noticing that the grain  boundary within polycrystalline VO2 bulk material inevitably leads to the suppression  in its macro mechanical property, as compared with the nanoindentation associated to  the nanoscale hardness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  In particular, the co-sintering of VO2 with dopant oxides exhibiting high melting  point (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', Al2O3, HfO2 and CoO) not only improve the mechanical strength of the  pellets, but also enhance the transition abruption in their \uf072-T tendencies across TMIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  This is demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 4c, where more abrupt MIT functionality with similar  TMIT is observed for Hf, Al and Co doped VO2 pellet compared to the pristine VO2  pellet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' In particular, the magnitude of their basic resistivity at room-temperature (ρ300 K)  can be regulated within a broad range across two orders of magnitudes as  10  demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 4d, and this is expected to extend the thermistor functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' As  a representative, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 4e demonstrates the distribution of the elementary distribution of  V, Co and O as measured by energy dispersion spectroscopy (EDS) for co-sintered  VO2 and CoO with a nominal composition of V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='7Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='3O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' It can be seen that the  cobalt element is not observed in the region associated with the vanadium dioxides,  and this differs to the situation for the aforementioned dopant oxides with low melting  points (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', WO3, Nb2O5, TiO2, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  The above results indicate the less effective diffusion of Co within VO2 during  the sintering process owing to the relatively high melting point of CoO (see more  details in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' S10), and this is in agreement to the small variation in TMIT as observed  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' This understanding is further confirmed by the Rietveld refinement of  their XRD patterns as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' S12, while their corresponding percentage of  residual dopants (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', Al2O3, CoO, Nb2O5, WO3 and TiO2) were further summarized  in Table S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' It demonstrates that the reaction ration (Preact) between the dopant oxides  with low melting point (Tmelt) and VO2 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', Preact is of almost ~ 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='6 % for Nb2O5  with Tmelt of 1485 \uf0b0C) largely exceeds the ones observed for the dopant oxides with  high Tmelt (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', Preact is of ~ 80 % for CoO with Tmelt for 1935 \uf0b0C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Nevertheless, the  residual dopant oxides, such as Al2O3, HfO2 and CoO, significantly improve the  mechanical properties of VO2 without sacrificing the MIT properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' From the  perspective of mechanical properties, the presence of compositing oxides at high  insulating state (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', Al2O3, HfO2 and Co3O4) can effectively prohibit the generation  and propagation of the cracks along the grain boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Therefore, a more stable  electronic functionality for the as-made doped VO2 bulk material is expected, which is  a critical issue for the practical applications in electronic devices upon thermoshock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  From the perspective of MIT properties, the compositing with the insulating oxides  located at the grain boundary can reduce the charge leakage along the cracks within  grain boundary for the insulating phase of VO2, and this leads to abrupt transitions in  resistivity across the TMIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Similar effects were also previously reported for the  vanadium phosphate glass (VPG) 50 and vanadium-based critical temperature resistor  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Therefore, a more stable electronic functionality for the as-made doped VO2 bulk  material is expected, which is a critical issue for the practical applications in  electronic devices upon thermoshock 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Conclusions  In summary, we largely improved the effectiveness in regulating the MIT  functionality via elementary doping as well as the mechanical properties of VO2 bulk  material by developing a spark plasma assisted reactive sintering (SPARS) approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Compared to the conventional solid-state reactions, the spark plasma induced  non-equilibrium process promotes the diffusion of the dopant element from its oxide  precursor into the lattice of the powder of VO2 within a largely shortened co-sintering  period of ~10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' This brings in high flexibility and convenience to control the  dopant type and concentration within bulk VO2, enabling a broad adjustment in the  11  TMIT (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', within 240-350 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' In contrast to conventional semiconductive doping  process, compliant variations in the electronic structure are observed within the doped  VO2 as probed by NEXAFS, and therefore variation in the MIT properties is more  likely to be associated with the doping-induced electronic orbital regulation rather  than simply varying the carrier concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Furthermore, the mechanical strength of  the doped VO2 bulk material can be meanwhile largely improved (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', from 267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='2 to  950 kgf / mm2), owing to compositing effect of the unreacted high melting-point  oxides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' The high tunability in the MIT functionality and improved mechanical  strength as enabled by the SPARS approach pave the way towards practical  applications of VO2 as a bulk material in power electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  12  Notes  The authors declare no competing financial interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Acknowledgements  This work was supported by the National Key Research and Development  Program of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2021YFA0718900), the National Natural Science Foundation  of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 62074014 and 52073090), and the Beijing New-star Plan of Science  and Technology (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Z191100001119071).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Supporting Information  Supporting information is available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' See the supporting information for  more details about the synthesis method, the derivation of TMIT, X-ray diffraction  (XRD) patterns, the energy dispersion spectrum (EDS), X-ray photoelectron  spectroscopy (XPS), the Rietveld refinements and additional analysis results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Correspondences: Correspondences should be addressed: Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Jikun Chen  (jikunchen@ustb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='cn), Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Yong Jiang (yjiang@ustb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='cn) and Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Nuofu  Chen (nfchen@ncepu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  13  References  (1) Ke, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Wang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Zhong, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Vu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Magdassi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Ye, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Zhao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Manipulating atomic defects in plasmonic vanadium dioxide for superior solar and thermal  management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Horiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2021, 8 (6), 1700-1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (2) Morrison, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Chatelain, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Tiwari, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Hendaoui, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Bruhacs, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Chaker, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Siwick, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  A photoinduced metal-like phase of monoclinic VO2 revealed by ultrafast electron diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Science  2014, 346 (6208), 445-448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (3) Zhou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Meng, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Mao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Jiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Fukutani, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Wilde, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Fugetsu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Sakata, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Revealing the Role of Hydrogen in Electron-Doping Mottronics for Strongly  Correlated Vanadium Dioxide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2022, 13 (34), 8078-8085.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (4) Tang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Dong, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Gordon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Reichertz, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Kim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Rho, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Wang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Lin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Grigoropoulos, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Temperature-adaptive radiative coating for all-season household thermal  regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Science 2021, 374 (6574), 1504-+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (5) Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Jiang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Meng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Yang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Tan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Long, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Scalable thermochromic smart windows  with passive radiative cooling regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Science 2021, 374 (6574), 1501-+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (6) Cui, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Ke, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Wang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Zhou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Gao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Long, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Thermochromic VO2 for Energy-Efficient Smart Windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Joule 2018, 2 (9), 1707-1746.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (7) Kats, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Blanchard, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Genevet, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Ko, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Ramanathan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Capasso, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Vanadium  Dioxide as a Natural Disordered Metamaterial: Perfect Thermal Emission and Large Broadband  Negative Differential Thermal Emittance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' X 2013, 3 (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (8) Tang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Dong, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Sun, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Dames, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Qiu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Yao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' A  Thermal Radiation Modulation Platform by Emissivity Engineering with Graded Metal-Insulator  Transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2020, 32 (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (9) Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Park, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Islam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Tran, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Manna, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Wang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Mondal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Yu,  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Banik, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Cheng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Reconfigurable perovskite nickelate electronics for artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Science (New York, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=') 2022, 375 (6580), 533-539.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (10) Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content=' Park, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Zaluzhnyy, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Wang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Wadekar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Manna, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Andrawis, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Sprau, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Sun, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Perovskite neural trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2020, 11 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (11) Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Mondal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Mandal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Allred, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Aghamiri, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Fali, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Zhou, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Cao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Rodolakis, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Neuromorphic learning with Mott insulator NiO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' PNAS 2021, 118 (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (12) Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Hu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Yajima, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Ge, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Ke, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Dong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Jiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Chen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Overcoming  synthetic metastabilities and revealing metal-to-insulator transition & thermistor bi-functionalities for  d-band correlation perovskite nickelates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Horiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2019, 6 (4), 788-795.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (13) Park, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Coy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Kasirga, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Huang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Fei, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Hunter, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Cobden, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Measurement  of a solid-state triple point at the metal-insulator transition in VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Nature 2013, 500 (7463), 431-434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (14) Jeong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Aetukuri, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Graf, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Schladt, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Samant, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Parkin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Suppression of  Metal-Insulator Transition in VO2 by Electric Field-Induced Oxygen Vacancy Formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Science 2013,  339 (6126), 1402-1405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (15) Lee, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Chung, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Shi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Kim, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Campbell, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Xue, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Song, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Choi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Podkaminer,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Kim, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Isostructural metal-insulator transition in VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Science 2018, 362 (6418),  1037-+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (16) Zhou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Yan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Zhang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Ke, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Meng, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Xu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Miao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Revealing the high sensitivity in the metal toinsulator transition properties of the pulsed laser deposited  VO2 thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Ceram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2021, 47 (18), 25574-25579.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  14  (17) Janod, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Tranchant, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Corraze, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Querre, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Stoliar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Rozenberg, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Cren, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Roditchev,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Vinh Ta, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Besland, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Resistive Switching in Mott Insulators and Correlated Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2015, 25 (40), 6287-6305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (18) Liu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Delaire, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Recent progresses on physics and applications of  vanadium dioxide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Today 2018, 21 (8), 875-896.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (19) Shibuya, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Kawasaki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Tokura, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Metal-insulator transition in epitaxial V1−xWxO2 (0≤x≤0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='33)  thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2010, 96 (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (20) Sakai, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Yoshimatsu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Shibuya, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Kumigashira, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Ikenaga, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Kawasaki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Tokura, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Oshima, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Competition between instabilities of Peierls transition and Mott transition in W-doped VO2  thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' B 2011, 84 (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (21) Takami, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Kanki, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Ueda, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Kobayashi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Tanaka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Filling-controlled Mott transition in  W-doped VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' B 2012, 85 (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (22) Holman, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content=' B  2009, 79 (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content=' 2004, 85 (11), 1958-1960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content=' 2015, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Macmanus-Driscoll, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content='  Materials science challenges for high-temperature superconducting wire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2007, 6 (9),  631-642.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' De Gryse, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Determination of the V 2p  XPS binding energies for different vanadium oxidation states (V5+ to V0+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Electron Spectrosc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' and  Related Phenomena 2004, 135 (2-3), 167-175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (29) Lee, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Kim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Lee, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Kim, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Yun, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Park, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Hydrogen  incorporation induced the octahedral symmetry variation in VO2 films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2017, 396,  36-40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (30) Su, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Fu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Yan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Zheng, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Liang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Zhang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Cheng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Yang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Chi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Tang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Self-propagating high-temperature synthesis for compound thermoelectrics and new criterion for  combustion processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2014, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (31) Yang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Su, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Bai, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Tang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Tang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Luo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Yan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Blocking Ion Migration Stabilizes the High Thermoelectric Performance in Cu2Se Composites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2020, 32 (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (32) Rao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Natarajan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Subba Rao, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Loehman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Phase transitions and  conductivity anomalies in solid solutions of VO2 with TiO2, NbO2, and MoO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Solids  15  1971, 32 (6), 1147-1150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (33) Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Luo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Gao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Unraveling Mechanism on Reducing Thermal  Hysteresis Width of VO2 by Ti Doping: A Joint Experimental and Theoretical Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' C  2014, 118 (33), 18938-18944.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (34) Kato, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Fujita, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Shirai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Kinemuchi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Influence of strain on latent heat of VO2  ceramics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Alloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Compd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2018, 751, 241-246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (35) Victor, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Gaudon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Penin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Chiron, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Chung, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Viraphong, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Rougier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Innovative sintering process for fabrication of thermochromic smooth VO2 ceramics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Alloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Compd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  2022, 890, 161890.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (36) Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Zhang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Fu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Yuan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Zhao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Tao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Nb  doping effect in VO2 studied by investigations of magnetic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Ceram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2018, 44 (7),  8623-8627.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (37) Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Fang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Su, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Hsieh, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Metal-insulator transition characteristics of Mo- and  Mn-doped VO2 films fabricated by magnetron cosputtering technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2014, 53 (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (38) Victor, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Gaudon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Salvatori, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Toulemonde, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Penin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Rougier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Doubling of the  Phase Transition Temperature of VO2 by Fe Doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2021, 12 (32), 7792-7796.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (39) Futaki, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' A New Type Semiconductor (Critical Temperature Resistor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 1965, 4  (1), 28-41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (40) Aetukuri, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Gray, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Drouard, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Cossale, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Gao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Reid, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Kukreja, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Ohldag,  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Jenkins, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Arenholz, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Control of the metal-insulator transition in vanadium dioxide by  modifying orbital occupancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2013, 9 (10), 661-666.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (41) Wei, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Ji, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Guo, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Nevidomskyy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Natelson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Hydrogen stabilization of metallic  vanadium dioxide in single-crystal nanobeams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Nanotechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2012, 7 (6), 357-362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (42) Phu Tran Phong, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Hofhuis, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Rana, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Huijben, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Hilgenkamp, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Rijnders, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  ten Elshof, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Koster, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Gauquelin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Lumbeeck, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Tailoring Vanadium Dioxide Film  Orientation Using Nanosheets: a Combined Microscopy, Diffraction, Transport, and Soft X-Ray in  Transmission Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2020, 30 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (43) Yoon, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Choi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Lim, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Kwon, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Ihm, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Choi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Son, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Reversible  phase modulation and hydrogen storage in multivalent VO2 epitaxial thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2016, 15  (10), 1113-+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (44) Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Ren, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Lu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Zou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Luo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Non-catalytic hydrogenation of VO2 in acid solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2018, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (45) Ruzmetov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Senanayake, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Narayanamurti, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Ramanathan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Correlation between  metal-insulator transition characteristics and electronic structure changes in vanadium oxide thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' B 2008, 77 (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (46) Kristensen, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Semiconductor‐to‐Semiconductor Transition in the Pseudobinary System TiO2–  VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 1968, 39 (11), 5341-5342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (47) Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Wang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Fan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Guo, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Zou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Qian, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Ibrahim, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Yan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Hole Carriers Doping Effect on the Metal-Insulator Transition of N-Incorporated Vanadium  Dioxide Thin Films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' C 2014, 118 (24), 12837-12844.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (48) Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Zuo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Wan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Dutta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Rensberg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Nawrodt, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Park, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Larrabee,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Guan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Evolution of Metallicity in Vanadium Dioxide by Creation of Oxygen Vacancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2017, 7 (3), 034008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  16  (49) Verma, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Singh, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Balakrishnan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Fracture toughness of VO2 microcrystals across  metal-insulator transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2022, 315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (50) Ivon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Kolbunov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Chernenko, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Conductivity stabilization by metal and oxide  additives in ceramics on the basis of VO2 and glass V2O5–P2O5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Non Cryst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Solids 2005, 351 (46),  3649-3654.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (51) Ivon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Kolbunov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Chernenko, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Stability of electrical properties of vanadium  dioxide based ceramics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Ceram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 1999, 19 (10), 1883-1888.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  17  Figures and captions  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' (a) Schematic illustration of as-proposed spark plasma assisted reactive sintering (SPARS)  doping strategy that achieves the elementary doping and grain boundary modifications of the bulk  VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' (b) X-ray diffraction (XRD) patterns of as-made V1-xTixO2 bulk material (x from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='4),  as indexed to the structure of the VO2 (M) with the P21/c space group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' (c) The c-axis lattice  constants of as-made V1-xTixO2 plotted as a function of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' (d) The cross-section surface  morphology of as-synthesized V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='8Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='2O2 bulk material from scanning electron microscopy (SEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (e), (f) The energy dispersion spectrum (EDS) of (e) V and (f) Ti elements for the as-made  V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='8Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='2O2 bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' (g), (h) The X-ray photoemission spectroscopy (XPS) analysis of (g) V 2p peak  and (h) O 1s peak for the V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='8Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='2O2 bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' (i) Temperature dependence of the resistivity for  as-made V1-xTixO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' (j) The metal to insulator transition temperature (TMIT) of as-made V1-xTixO2  plotted as a function of the nominal Ti substitution concentration, compared with the previous  report 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (a)  Pressure  Microcell  Sparked  Plasma  >>>  VO,Particle  Electric  Bulk  TMmO,Particle  Current  (b)  (c)  (d)  (011)  Vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='gTio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='102  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='46  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='44  Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=')  Vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='aTio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='202  Vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='7Tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='302  c  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='40  Vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='Tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='02  VO, (M)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='38  PDF#09-0142  0  10  20  30  40  20  30  40  50  09  70  80  20(°)  Nominal DopingConcentration(%)  (e)  (f)  (g)  VKa1  Ti Ka1  VoaTlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='202  V2p3/2  Counts  V2P1/2  25um  25um  512  516  520  524  528  (h)  (i)  Binding Energy (eV)  10  Vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='Tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='202  345  VO2  oThiswork  Ti2P3r2  10°  Vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='gTio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=',02  340  ORef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='8Tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='202  Counts  Vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='7Tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='302  335  Vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='6Tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='402  Vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='5Tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='502  330  Ti2P1/2  TA  p  325  10-3  320  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  10-4  315  453  456  459  462  465  468  300  320  340  360  0  10  20  30  40  50  BindingEnergy (eV)  T(K)  Nominal Doping Concentration(%)18  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' (a) The temperature dependence of the resistivity for as-made transitional metal (TM)  doped VO2 pellet (TM = Fe, W, Nb, Cr and Mo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' (b) The TMIT of as-made doped VO2 plotted as a  function of the nominal doping concentration and also compared to the previous reports 19, 22, 32, 36,  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' (c) The variation in the resistivity across the metal to insulator transition (\uf072Insul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' / \uf072Metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=') of the  doped VO2 made in this work, plotted as a function of their metal to insulator transition  temperature (TMIT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' (d) The modulation capability in TMIT, as evaluated by dTMIT / dndopant, for  various TM dopant elements plotted as a function of their valence state, as also compared with the  previous reports 19, 22, 32, 36, 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' The inset shows the zoom in results for Ti, Cr and Fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (a)  (b)  VO2  Vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content='0202 Vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content='0402  W  Mo  Fe  Nb  Vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content='0502-VogNbo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='1O2  —Vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='gFeo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='102-Vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content='202  Vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='7Cro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Cr O Ti Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  ★  W Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Mo Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  Nb Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' Fe Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  400  V  10-1  380  Hole Doping  360  Pristine  V  10-2  340  ★★  MIT  320  C  Heating  10-3  300  Q  280  Cooling  Electron Doping  10-4  260  240  200  250  300  350  400  10  T (K)  Nominal Doping Concentration (%)  (c)  W  Mo  Fe  Nb  Cr  (d)  (K %-1)  5  This work  ★  O Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  102  5  Mo  10  0  15  Nb  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='4  Ti  0  20  ★  101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  W  240  320  330  340  350  3  4  5  6  TMIT (K)  Dopant Valence19  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' (a), (b) The near-edge X-ray absorption fine structure (NEXAFS) analysis of (a) V-L edge  and (b) O-K edge for the transitional metal (TM) substituted VO2 (TM = Ti, Fe, W, Nb and Mo),  compared to the pristine VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' (c) Illustrating the variation in electronic structures of VO2 via  high-valence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content=', W, Nb and Mo) and low-valence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content=', Fe and Cr) TM substitutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content=')  Vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='8 Tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='202  _ Vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content='  Vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content='0102  VO2  Vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content='0402  VO2  Vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content='04O2  512  516  520  524  528  530  532  Photon Energy (eV)  Photon Energy (eV)  (c)  Metallic State  InsulatingState  a*  6  a*  元*  元*  TMIT  di  di*  元*  元  Eft  MIT  Ef -  du  Electron  di  d i  00000  +Hole  元  元  元  元  a  a  a  a  High-Valence Substitution  Low-Valence Substitution20  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content=' (a) The Vickers hardness (HV) for the transitional metal (TM) substituted VO2 (TM = Ti, Fe,  W, Nb, Mo, Hf, Al and Co) made herein, plotted as a function of the nominal doping concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='  (b) Plotting HV as a function of the variation in resistivity across the metal to insulator transition  temperature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=', \uf072Insul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' / \uf072Metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=') for as-made V1-xTMxO2 (the number of x is marked near the data  points) pellet, as compared with the ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' (c) The normalized temperature dependence of the  resistivity (\uf072-T) of the co-sintered VO2 with transitional metal oxides with high melting points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' (d)  The basic resistivity at room-temperature of as-made V1-xTMxO2 plotted as a function of the  nominal doping concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content=' (e) The energy dispersion spectrum (EDS) of V, Co and O elements  for the as-made V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content='  Ti  ★  W  Nb  Mo  Fe  VO2  OV1-xTi,O2  V1-xHf,O2  V1-xCo,02  V1-xAl,O2  (a)  Hf  Al  Co  (b)  V1-xW,02  V1-xNb,O2  V1-xMo,02  V1-xFe,02  ☆ Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+page_content='2  Load  900  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
+page_content='2  Hy (kgf / mm²)  / mm²)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfvfnv/content/2301.00634v1.pdf'}
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+A Survey of Explainable AI in Deep Visual Modeling: Methods and Metrics
+Naveed Akhtar
+The University of Western Australia
+naveed.akhtar@uwa.edu.au
+Abstract
+Deep visual models have widespread applications
+in high-stake domains. Hence, their black-box na-
+ture is currently attracting a large interest of the
+research community. We present the first survey
+in Explainable AI that focuses on the methods and
+metrics for interpreting deep visual models. Cover-
+ing the landmark contributions along the state-of-
+the-art, we not only provide a taxonomic organi-
+sation of the existing techniques, but also excavate
+a range of evaluation metrics and collate them as
+measures of different properties of model explana-
+tions. Along the insightful discussion on the cur-
+rent trends, we also discuss the challenges and fu-
+ture avenues for this research direction.
+1
+Introduction
+Visual computational models have widespread applications,
+ranging from casual use in handheld devices to high-stake
+tasks in forensics, surveillance, autonomous driving and med-
+ical diagnosis etc. Contemporary visual models rely heavily
+on deep learning, which is a black-box technology. Hence,
+the opacity of deep visual models is becoming a major hur-
+dle in enabling fairness, transparency and accountability of
+the AI systems that rely on these models. Given the emerg-
+ing development and deployment policies for AI around the
+globe, it is apparent that the future of deep visual models in
+practice hinges on addressing their black-box nature.
+Historically, visual modeling has inspired numerous break-
+throughs in deep learning research owing to the convenient
+interpretability of visual data. The same is currently fueling
+a high research activity in explainable deep visual modeling.
+Since vision research is already a proven source of many no-
+table advances in deep learning, systematically framing this
+research activity can be pivotal for the future of Explainable
+AI (XAI). Whereas the existing literature provides reviews of
+the broader domain of XAI, it is still void of a taxonomic
+treatment of the techniques devised to render deep visual
+models more transparent. This paper fills this critical research
+gap by making the following important contributions.
+• It provides the first comprehensive review of the literature
+in XAI that focuses on deep visual models1. The review
+1We only focus on the natural image domain, which is also the
+Figure 1: The literature in visual model explanation is currently
+dominated by post-hoc techniques (Top) that treat pre-trained mod-
+els under black- and white-box setups. Backpropagated gradients
+and model activations are common tools to generate visual expla-
+nation in white-box setup, whereas model-agnostic techniques nor-
+mally query the models to compute a mask for salient input regions
+in black-box settings. (Bottom) Techniques inducing inherently ex-
+plainable models rely on non-traditional building blocks for neural
+modeling or augment the standard blocks with interpretable mod-
+ules. Their outputs may still require post-hoc tools for explanation.
+covers the influential techniques developed at the advent
+of the modern deep learning era, as well as the most re-
+cent state-of-the-art methods appearing in the flagship re-
+search sources of the domain. It pays a special attention to
+the landmark contributions. The literature is taxonomized
+following the technical methodologies adopted by the ap-
+proaches. Thereby, providing a clear picture of the broad
+strategies leveraged by the existing techniques.
+• We find that quantification of visual model interpretation
+performance still lacks the desired clarity in the literature
+due to the intrinsic difficulties in measuring the abstract
+subjective notion of explainablity. To help the research
+community addressing that, we excavate a range of eval-
+uation metrics from the literature, and organize them as
+quantitative measures of different properties of explana-
+tion. This forms the second major part of our literature
+source of methods for other image domains, e.g., medical imaging.
+arXiv:2301.13445v1  [cs.CV]  31 Jan 2023
+
+Post-hoc model interpretation tools
+White-box
+propagation
+Attribution
+Back
+dpu
+indul
+Activation
+Saliency
+dpu
+Standard
+Model
+agnostic
+Saliency
+mask
+Pre-trained
+Black-box
+Non-traditional
+Augmented
+(from scratch)
+indul
+Post
+-hoc
+Inherentlyexplainablemodelsreview.
+It is envisaged that this first-of-its-kind broad
+methodical treatment of the evaluation metrics will sig-
+nificantly contribute towards the much needed systematic
+evaluation of the future techniques.
+• Due to the absence of a focused review, the current lit-
+erature in XAI for visual modeling suffers from obvious
+inconsistencies in technical terminologies and providing
+the correct context to the techniques. We address this by
+providing accessible understanding of the terminologies
+along with insightful discussions throughout the paper.
+• The paper finally highlights major open challenges for this
+direction and provides the future outlook by building on
+the insights from the reviewed literature.
+2
+Overview and Terminologies
+This section introduces the common terminologies used in
+the reviewed literature and formalizes the problem from the
+perspective of systematically categorizing the existing con-
+tributions. The existing literature is often found referring to
+the same concepts using different terminologies. This section
+also collates these terminologies for the community.
+We find that the visual modelling domain is not too par-
+ticular about discriminating between the terms explanation
+and interpretation. Without any obvious reasons, the even-
+tual outcomes of the methods are sometimes referred to as
+explanations, whereas the same outcomes are called inter-
+pretations in other cases. This survey favours the term ex-
+planation for the output of a technique. Notably, the works
+contributing towards the intrinsic transparency of the models
+generally prefer the term explainable for the models. Meth-
+ods aiming for intrinsically explainable models pursue their
+goals as discussed below.
+Let M : M(I) → y be a visual model that maps an image
+I ∈ Rh×w×c having ‘c’ channels to the output y. In vision
+literature, the explanation task mainly considers classifiers as
+the target models, where y ∈ RK is a K-class prediction vec-
+tor. Methods seeking to improve the inherent explainability
+of the models see them as a hierarchy of functions, i.e.,
+M(I)=fL(ΘL, fL−1(ΘL−1, fL−2...(Θ2, f1(Θ1, I))...)→y
+where Θℓ denotes the model parameters (including biases)
+for the ℓth function fℓ. These methods take one of two ap-
+proaches. (i) Directly use easy to interpret functions for all
+fℓ - see § 3.1, or modify some fℓ for more transparency
+- see § 3.2 (Internal component augmentation).
+Models
+implementing this strategy are sometimes also called self-
+explainable models. (ii) Plug in an external explainer module
+E(M, I) in the function hierarchy to generate the explanation
+- § 3.2 (External component augmentation). This explainer is
+induced along the model. Collectively, the approaches under
+both (i)&(ii) are referred to as ante-hoc methods, as opposed
+to the post-hoc techniques, which are discussed next.
+The post-hoc methods consider M to be fixed, i.e., they
+do not interfere with the design or induction process of the
+model. To understand the common post-hoc visual explana-
+tion objective, consider a set PI = {p1
+I, p2
+I, ..., pw×h
+I
+} ⊂ Rc
+that contains the pixels of the input I. The explanation objec-
+tive here is to seek an ordered array WI ⊂ R, s.t. |WI| = |PI|
+and the ith element of this array, i.e. wi
+I ∈ WI, encodes a
+weight for the corresponding element in PI. The weights
+in WI are intended to be human-understandable. The post-
+hoc explanation methods generally implement the function
+S : S(M, I) → WI. Here, WI can be a binary array that
+represents a mask for I that suppresses the irrelevant pi
+I ∈ PI
+in the transform M(I) → y. This objective is more per-
+tinent to the model-agnostic post-hoc methods - see § 4.1.
+These methods avoid using any internal signal of the model,
+e.g., neuron activations, thereby assuming a black-box setup
+which does not allow access to the model details.
+When access to the model is available, the weights wi
+I ∈
+WI take real values to quantify the importance of the input
+pixels. To that end, a wide range of methods rely mainly
+on the activations of the model’s internal neurons to estimate
+WI. We call such methods neuron activation-based tech-
+niques - see § 4.2. The output of such methods is commonly
+known as a saliency map. The activation-based techniques
+often also use model gradients. However, those gradients are
+generally not backpropagated right until the input. The meth-
+ods that compute the model gradients w.r.t. the input are cat-
+egorised as the backpropagation-based methods in this sur-
+vey - see § 4.3. The output of these methods is more com-
+monly termed attribution map. Note that, attribution maps
+and saliency maps are technically ‘synonyms’. We find multi-
+ple papers in this domain that use these terms interchangeably
+to describe the generated visual explanations. Other common
+terms that refer to the same concept in the literature are rele-
+vance scores, feature contributions and importance maps.
+Whereas the ante-hoc methods render the model intrinsi-
+cally more transparent, the interpretation process may still use
+post-hoc tools or need human domain experts to explain the
+predictions. Within the vision domain, the preferred mode
+of explanations are visualizations. However, rarely, we also
+encounter text as the explanations [Hendricks et al., 2016],
+[Park et al., 2018]. We also find that the common objective
+of the post-hoc techniques is to generate input-specific ex-
+planations. A rare exception to that is input-agnostic saliency
+mapping [Akhtar and Jalwana, 2023] that generates visual ex-
+planations that encode generic understanding of the model of
+human-defined concepts.
+3
+Inherently explainable models
+We find that there is a prevailing impression in the community
+that explainability and performance of deep visual models are
+inversely correlated. However, Rudin [2019] rightly noted a
+lack of concrete evidence in this regard, and argued in favor of
+developing inherently explainable models. To that end, recent
+papers devise approaches along two broad directions: (a) de-
+signing explainable models from scratch, and (b) augmenting
+black-box models with interpretable components.
+3.1
+Explainable model design from scratch
+Along this direction, a category of methods explores the lore
+of generalized additive models (GAMs) [Hastie, 2017]. The
+central idea is to implement the intended transformation (i.e.,
+modeling) of the input as a set of additive sub-models that
+deal with the input features separately. This is used to pre-
+serve the interpretability of the overall transformation [Agar-
+wal et al., 2021], [Chang et al., 2021], [Dubey et al., 2022],
+[Alvarez Melis and Jaakkola, 2018]. However, the lack of
+
+Figure 2:
+Flowchart
+for
+general guidance to select
+explanation tools for visual
+model application domains.
+The chart follows termi-
+nologies used to categorise
+the literature in this survey.
+Best viewed enlarged.
+expressive power of such composite modelling, and its lim-
+ited scalability are major hurdles in the practicability of this
+strategy. Taking a different approach, Blazek et al. [2021]
+proposed essence neural networks that employ non-standard
+building blocks of Support Vector Machines based neurons.
+With a systematic architectural design of the underlying neu-
+ral network, they showed remarkable generalizable abilities
+of the approach. However, the design process of their net-
+work remains cumbersome and relies heavily on an active in-
+volvement of the domain experts. Li et al. [2021] proposed
+to build the classification stage of a visual classifier using a
+slot attention mechanism that learns a semantically meaning-
+ful internal representation. However, their method must still
+rely on a backbone module in the early stage of the network,
+which compromises the explainability of the overall model.
+It is a common knowledge that a key strength of deep learn-
+ing, especially in the vision domain, is its ability to represent
+the data with discernible low- and high-level features through
+a hierarchical structure.
+Methods aiming at fully explain-
+able models must account for such a sensible hierarchical
+treatment of the domain data in the network structure manu-
+ally. This generally requires expert knowledge of the domain,
+which makes fully explainable modeling a less attractive op-
+tion for many problems.
+3.2
+Augmenting neural models for explainability
+In general, explainability of visual models is seen as a con-
+tinuum rather than a discrete measure. This perspective has
+encouraged researchers to devise methods for improving the
+‘extent’ of interpretability of the models. Generally, this is
+pursued by augmenting the conventional neural models with
+(i) interpretable internal components, or (ii) external compo-
+nents providing the desired explanations.
+Internal component augmentation:
+To improve the in-
+trinsic explainability of visual models, Chen et al. [2020]
+introduced a concept whitening module that aims at align-
+ing the axes of the latent space of the model with apriori
+known concepts. This alignment also involves a process of
+de-correlating the concepts space for intelligibility. Wang et
+al. [2021] also explored a closely related mechanism with a
+plug-in embedding space for the model that is spanned by
+basis concepts. Here, the basis concepts are kept category-
+aware, and the within-category concepts are kept orthogonal
+to each other for their human-understandability. These are in-
+deed useful methods, however they come at the cost of further
+complicating the training process of neural modeling.
+Concept bottleneck modelling [Koh et al., 2020] is an-
+other interesting framework that lets a neural model first learn
+human-interpretable (sub-)concepts, and makes the final pre-
+diction by further processing those concepts. Architecturally,
+it introduces a concept layer in the network that also enables
+human intervention during the prediction stage in [Chauhan
+et al., 2022]. Concept bottleneck models (CBMs) are cur-
+rently attracting notable attention of the research community.
+Bahadori et al. [2020] performed causal reasoning to debias
+such models, whereas Antognini et al. [2021] used this frame-
+work for the application of textual rationalization. Working
+on the principles similar to CBMs, ProtoPNetwork [Chen et
+al., 2019] also introduces a prototype layer in the network
+that associates high-level sub-concepts to its neurons.
+Notice that some approaches discussed in § 3.1 have a close
+conceptual resemblance to CMBs. For instance, like CBMs,
+only the later layers of the visual models in [Blazek and Lin,
+2021] are interpretable. This is because, their technique must
+rely on deep features (instead of image pixels) to construct
+the explainable (sub-)network in the case of visual modeling.
+However, unlike [Blazek and Lin, 2021], the techniques using
+internal component augmentation employ the standard neu-
+rons along some basic arithmetic operations to improve the
+interpretability of the network layer(s). It is notable though,
+both paradigms remain interested in the deeper layers of the
+neural network for model explainability. This is true even for
+[Kim et al., 2022], which specifically attempts to make Vi-
+sion Transformers more explainable by using neural trees as
+a decoder of the representation learned by the model, where
+the nodes of the trees are contextual Transformer modules.
+External component augmentation:
+Internal component
+augmentation techniques embed modules in the neural net-
+work such that they do not form a new branch of the graph. In
+contrast, some recent techniques introduce explanation mod-
+ules as external branches of the network. For instance, Sarkar
+et al. [2022] proposed a framework to learn an external expla-
+nation module in the context of concept-based models [Koh
+et al., 2020]. Their explanation module uses an external con-
+cept decoder to improve model interpretability.
+Similarly,
+[Lang et al., 2021] trains an external StyleGAN as an ex-
+plainer for a visual classifier. Human-understandable expla-
+nations are produced in this approach by relying on semanti-
+cally meaningful dimensions learned by the style-space of the
+external module. Another example of external explainer can
+be found in [Stalder et al., 2022], which is claimed to pro-
+duce sharp masks of relevant image regions using a trainable
+model. Though useful, an apparent drawback of the external
+module-based methods is that the module can inadvertently
+affect both the model performance and its explanations. For
+the later case, the explanation can unintentionally be encod-
+ing the module’s behavior instead of explaining the model.
+
+Yes
+Use backpropagation
+Accessto
+Visual
+Use post-hoc
+Use white-box
+Yes
+model
+resolution
+orneuron-concept
+Highcomputational cost
+tools
+methods
+details
+critical
+associationmethods
+No
+Use saliency
+No
+Low computational cost
+No
+mapping methods
+Use model-agnostic
+Train standard
+methods (black-box)
+Develop
+network
+Start
+own
+model
+Explainability
+Train explainable
+No
+No
+Useexternal component
+model from scratch
+augmentation
+Yes
+Expert
+Yes
+Preference
+Performance
+Is model
+Yes
+knowledge
+Augment existing
+Exp. vs. Perf.
+fidelity
+Useinternalcomponent
+Assuming sufficient
+available
+network architectures
+critical?
+augmentation
+computational resources & data4
+Post-hoc explanations
+It is hard to identify any study that rigorously establishes an
+inverse correlation between the model explainability and its
+performance. Nevertheless, it is widely presumed that this
+inverse relation exists. Consequently, the research commu-
+nity is relatively more active in devising post-hoc explana-
+tion tools as compared to developing intrinsically explainable
+models. Post-hoc tools include a category of methods that
+leverage the model information when it is available - see § 4.2
+and 4.3, whereas another class of methods can handle the sce-
+nario when such information is unavailable - see §4.1. This
+flexibility, along the fact that these methods do not interfere
+with model performance, makes post-hoc tools very practical.
+4.1
+Model-agnostic methods
+For the cases where the model information is unavailable,
+i.e., black-box scenarios, post-hoc explanation methods query
+the model in some form to identify the important pixels in
+the input in a model-agnostic manner. For instance, Fong
+et al. [2019] used queries to compute extremal perturbations
+to identify the most salient input pixels. Local Interpretable
+Model-agnostic Explanations (LIME) is an early attempt of
+model-agnostic methods that approximates the model behav-
+ior within a local neighborhood around the input by fitting
+a simple interpretable model to the perturbed versions of the
+input [Ribeiro et al., 2016]. However, LIME is known to be
+unstable. Dhurandhar et al. [2022] recently proposed a fix to
+this instability. Randomized Input Sampling for Explanation
+(RISE) [Petsiuk et al., 2018] is yet another popular model-
+agnostic explanation method. Lundberg & Lee [2017] pro-
+posed SHapley Additive exPlanations (SHAP) that leverage
+Shapely values [Shapley, 1997] to estimate the contribution
+of each feature to the prediction. Recently, it is claimed that
+the marginal contribution computed by the Shapely values is
+sub-optimal for model explanation. Hence, a weightedSHAP
+is introduced in [Kwon and Zou, 2022].
+Model-agnostic strategy is the only scheme available for
+explaining a model in a black-box setup.
+Since the ap-
+proaches do not leverage model information, they solve a
+harder problem to generate explanations. Thus, we often find
+these methods relying on heuristics to control their solution
+space. These heuristics can have unintended effects on the
+generated explanations. In general, despite using heuristics,
+these methods remain computationally expensive.
+When the internal information of the model is available,
+there are two important signals that can be exploited to ex-
+plain the model behavior; namely, (i) internal activations and
+(ii) backpropagated gradients. We discuss the methods ex-
+ploiting (i) and (ii) in § 4.2 and §4.3, respectively.
+4.2
+Neuron activation-based techniques
+There is a range of methods that strongly rely on the activa-
+tion values of the internal neurons of the model to explain
+their predictions. It is emphasized though that the activations
+may not be the only internal signals used by these methods.
+Saliency methods:
+A popular stream of techniques com-
+putes saliency maps for the inputs as visual explanations us-
+ing internal neuron activations of the models. Class Activa-
+tion Mapping (CAM) [Zhou et al., 2016] is one of the earliest
+methods in this direction, which employs the global average
+pooling neuron activations to weakly localize the objects in
+the inputs for saliency mapping. Oquob et al. [2015] used the
+max pooling neurons for a similar purpose. Whereas multiple
+other techniques exist for activation-based saliency mapping,
+GradCAM [Selvaraju et al., 2017] is a landmark work, which
+combines the model gradients with the activations of its in-
+ternal neurons to compute the saliency maps. Multiple vari-
+ants of GradCAM have appeared in the literature, e.g., Grad-
+CAM++ [Chattopadhay et al., 2018], Score-CAM [Wang et
+al., 2020], Smooth-GradCAM++ [Omeiza et al., 2019], all
+of which use neuron activation signals. Generally, there is a
+large disparity between the sizes of the activation maps of the
+internal layers of a model and the input. This requires inter-
+polation of the activation signals for computing the eventual
+saliency map, which results in poor resolution of the maps.
+Recently, Jalwana et al. [2021] addressed this weakness with
+their CAMERAS method that employs a multi-scale mapping
+for high-resolution saliency mapping.
+Despite a considerable number of existing methods for
+activation-based saliency mapping, we still regularly witness
+new techniques emerging along this line of research. On the
+other hand, there are still many other popular classic tech-
+niques, e.g., Layerwise Relevance Propagation (LRP) [Bach
+et al., 2015], DeepLift [Shrikumar et al., 2017] that rely on
+activations of multiple layers to compute the saliency maps.
+With the paradigm shift in visual modeling in favor of Trans-
+formers, these classical methods are also being extended to
+explain the Transformer based models [Ali et al., 2022].
+Neuron-concept
+association
+methods:
+Whereas
+tech-
+niques like CBMs [Koh et al., 2020] encourage neuron-
+concept association by design, we also find methods that aim
+at identifying such an association in a post-hoc manner. For
+instance, Bau et al. [2017] aimed at quantifying the alignment
+between the model neurons and known semantic concepts.
+Similarly, Mu et al. [2020] searched for logical forms of ex-
+planation defined by a composition of the concepts learned
+by polysemantic neurons. A neuron-concept explainer is also
+developed in [Wang et al., 2022] that utilizes both saliency
+maps and Shapley values. Previously, Shapley values of neu-
+rons are also leveraged by [Ghorbani and Zou, 2020] to an-
+alyze the importance of neurons for a given output. On the
+other hand, activations are also used by the popular TCAV
+method [Kim et al., 2018] to identify the sensitivity of the
+neurons to a given set of concept.
+4.3
+Backpropagation-based methods
+Gradients of a model w.r.t. input are widely considered a
+natural analogue of its coefficients in the modern deep neu-
+ral networks. We find a range of methods leveraging back-
+propagated model gradients for explanation.
+Simonyan et
+al. [2014] were among the first to estimate input feature rel-
+evance for the prediction with model gradients.
+Guided-
+backpropagation [Springenberg et al., 2015], smoothGrad
+[Smilkov et al., 2017], fullGrad [Srinivas and Fleuret, 2019]
+and multiple other methods in the literature later that used
+the backpropagated gradients to estimate the feature attribu-
+tion scores. It is noteworthy that backpropagated gradients
+are sometimes also employed by other categories of methods.
+
+For instance, the saliency methods, e.g., Grad-CAM, CAM-
+ERAS, in § 4.2 also use model gradients. However, those
+gradients are not fully propagated back to the input.
+Within the methods mainly relying on the backpropagated
+gradients, there is an important sub-category of path attribu-
+tion methods. Inspired by game-theory, these methods accu-
+mulate the backpropagated gradients of the images defined
+by a path between the input and a reference image, where
+the latter signifies the absence of the features in the input.
+The pioneering work by Sundrarajan et al. [2017] claims that
+the path attribution strategy can preserve certain desirable ax-
+iomatic properties for the explanations. Their claims were
+further qualified recently in [Lundstrom et al., 2022]. The
+original work has inspired a range of methods employing the
+path attribution framework [Yang et al., 2023], [Erion et al.,
+2021]. However, Stumfels et al. [2020] have also noted that
+the path attribution strategy can suffer from certain artifacts
+resulting from the reference image itself. Our close inspec-
+tion revealed that despite their axiomatic nature, these meth-
+ods sometimes compute counter-intuitive results. Moreover,
+the literature often fails to establish that the eventual methods
+actually retain the claimed properties.
+5
+Evaluation metrics
+Evaluating the performance of a model explanation tool is
+not straightforward. Due to the abstract subjective nature of
+the notion of explanation and the lack of ground-truth, ob-
+jectively evaluating a method’s performance requires quanti-
+fying multiple aspects of the computed explanation. Hence,
+development of metrics to evaluate explanation methods is
+currently actively pursued. We excavate numerous metrics
+from different contributions and organise them based on the
+explanation properties quantified by them.
+5.1
+Quantifying the model fidelity
+Fidelity of the explanation to the model is a measure of the
+extent to which the explanation captures the true behavior of
+the model. It is the most essential property of the computed
+explanation. Broadly, the metrics quantifying this property
+rely on estimating some form of correlation between the rel-
+evance of the input features (as computed by the explanation
+method) and the output variation when those features are per-
+turbed/removed. The core intuition is that for a model-faithful
+explanation, the output variations should be stronger for the
+perturbations to the more relevant features.
+As an early example of evaluating the model fidelity, Bach
+et al. [2015] developed a metric of pixel flipping where the
+fidelity is evaluated by observing the change in the predic-
+tion scores by flipping the input pixels with high (absolute)
+scores. The metric was later generalized to region perturba-
+tion in [Samek et al., 2016] where a random sample from a
+uniform distribution is used to perturb a set of input pixels
+to analyze the effects on predictions. Similarly, Alvarez et
+al. [2018] removed the pixels from the input and measured
+the correlation between the prediction scores and the attri-
+bution scores. In [Ancona et al., 2017], SensitivityN metric
+is proposed to measure the correlation between the sum of a
+subset of computed pixel attributions and the change in the
+prediction when the corresponding pixels are perturbed. This
+notion is further generalized in [Yeh et al., 2019] with an In-
+fidelity measure as the Expected difference between the dot
+product of the input perturbation to the attribution, and the
+prediction change resulting from the perturbation.
+The key strategy of the above metrics is to perturb (or re-
+move) input features and analyze the effects on the output.
+However, Hooker et al. [2019] showed that this results in a
+distribution shift problem for the models because their train-
+ing process does not account for any pixel removal or per-
+turbation.
+This compromises the reliability of the evalua-
+tion metrics. Hence, they proposed to iteratively retrain the
+model after the removal of each pixel on the modified training
+data. Their Remove and Retrain (ROAR) is one of the widely
+adopted metric due to its global (in terms of training data)
+nature. Nevertheless, its evaluation can be computationally
+prohibitive. To address that, Rong et al. [2022] proposed a
+Remove and Debias (ROAD) metric that avoids the retraining
+with a de-biasing step. The authors claim a desirable prop-
+erty of consistency between the ROAD scores when the most
+relevant pixels are removed first, or the least relevant pixels
+are removed first - a property not exhibited by ROAR.
+Whereas multiple metrics exist for model fidelity, there is
+still no single agreed-upon metric that is considered to ac-
+curately and comprehensively quantify this property.
+The
+emerging relevant trends include combining the existing met-
+rics [Yang et al., 2023] and quantifying the model fidelity
+with other properties such as consistency and sufficiency of
+the computed attribution [Dasgupta et al., 2022] or proposing
+novel settings to evaluate this property [Rao et al., 2022]. A
+comprehensive and reliable evaluation of this property is still
+an open challenge for the vision domain.
+5.2
+Quantifying the localisation ability
+An accurate explanation must assign high relevance to the
+foreground object features. This intuition has led to a range of
+metrics that aim at leveraging ground truth in quantifying the
+explanation performance. Their key idea is to measure how
+well the computed relevance scores correlate with the bound-
+ing boxes, segmentation masks, or the grid cells that contain
+the object of interest. In that sense, we can collectively see
+them as localisation quantifying metrics.
+Pointing game [Zhang et al., 2018] is one of the popular
+examples in this direction, which computes the percentage
+of the most relevant pixel for every sample being located on
+the foreground object. Similarly, Kohlbrenner et al. [2020]
+proposed an attribution localization metric that computes the
+fraction of the cumulative relevance score attributed to the
+foreground object in an input. The same measure is termed
+relevance mass in [Arras et al., 2022], where another metric
+of relevance rank is also introduced, which estimates the frac-
+tion of the object mask pixels common to the top K relevant
+pixels, where K is the mask size. In [Arras et al., 2022], mo-
+saics of different data samples are use to quantify the fraction
+of the relevance assigned to the foreground object.
+Whereas the localisation based metrics can leverage the
+ground truth, their objective is agnostic to model fidelity. An
+explanation method enforcing stronger attribution to the fore-
+ground, irrespective of the actual behavior of the model, will
+achieve higher score for these metrics. Hence, relying solely
+
+Table 1: Representative evaluation metrics and their ability to measure the extent of different properties of explanations. More *s indicate
+better ability. Refer to § 5 for the description of the properties. GT denotes Ground Truth usage, and Comput. is computational efficiency.
+Metric
+Work
+Model Fidelity
+Localisation
+Stability
+Conciseness
+Sanity preservation
+Axiomatic properties
+GT
+Comput.
+Pixel Flipping
+Bach et al. [2015]
+****
+*
+
+*
+**
+
+
+***
+SensitivityN
+Ancona et al. [2017]
+****
+*
+
+*
+**
+***
+
+*
+Insertion/deletion
+Petsiuk et al. [2018]
+****
+*
+
+*
+**
+
+
+***
+ROAR
+Hooker et al. [2019]
+****
+*
+
+*
+**
+
+
+*
+ROAD
+Rong et al. [2022]
+****
+*
+
+*
+**
+
+
+**
+Pointing game
+Zhang et al. [2018]
+
+****
+
+*
+
+
+
+***
+Stability
+Alvarez et al. [2018]
+**
+
+****
+
+*
+
+
+**
+Sensitivity
+Yeh et al. [2019]
+**
+
+****
+
+*
+
+
+**
+Sparseness
+Chalasani et al. [2020]
+*
+*
+
+****
+*
+
+
+**
+Effective complexity
+Nguyen et al. [2020]
+*
+*
+
+****
+*
+
+
+**
+Cascading randomization
+Adeboya et al. [2018]
+***
+*
+
+
+****
+
+
+**
+Completeness
+Sundararajan et al. [2017]
+**
+
+
+*
+**
+****
+
+*
+on the localisation based metrics can be misleading.
+5.3
+Quantifying the stability
+A correct explanation should not change drastically when the
+input is changed only slightly. Following this intuition mul-
+tiple metrics have emerged that quantify some form of sta-
+bility of the computed explanations. For instance, Avarez
+et al. [2018] measured the local Lipschitz continuity of the
+explanation to observe its stability. Similarly, Motavon et
+al. [2018] computed the maximum ℓ1-distance between the
+explanations of two inputs normalized by the Euclidean dis-
+tance between them, to observe the explanation stability. In
+[Yeh et al., 2019], an estimate of the maximum distance be-
+tween the explanations of the neighborhood inputs is consid-
+ered to quantify the stability. Dasgupta et al. [2022] quanti-
+fied the probability of the similarity between the explanations
+of similar samples. In [Agarwal et al., 2022], relative changes
+in the explanations are estimated with respect to the input,
+model embeddings and the model output logits to propose
+three measures of stability, namely; relative input stability,
+relative representation stability and relative output stability
+of the explanations. The authors argue to discard the assump-
+tion that the model is a black-box (for black-box methods)
+and propose to leverage full model information to ensure a
+more reliable evaluation of the explanation method.
+Whereas the notion of stability is understood in the same
+sense across different papers, we do not find consensus in its
+naming convention. For instance, stability is referred to as
+continuity in [Montavon et al., 2018], sensitivity in [Yeh et
+al., 2019] and consistency in [Dasgupta et al., 2022]. It is
+also worth highlighting that the prevailing existing notion of
+stability is underpinned by the assumption that the model in
+question is smooth. Stability measures for the explanation of
+a model not following this assumption can be problematic.
+5.4
+Evaluating other desirable properties
+Besides the above, we also find a number of other metrics
+in the literature that address other desirable properties of the
+explanations. We summarise them below.
+Conciseness:
+An effective explanation should be concise.
+Following this intuition, Chalasani et al. [2020] quantified the
+explanation conciseness by estimating its sparseness with the
+Gini Index, whereas Bhatt et al. [2020] evaluated the explana-
+tion complexity to measure the conciseness using a fractional
+contribution distribution of the input features for the expla-
+nation. Similarly, Nguyen et al. [2020] also measured an ef-
+fective complexity of the explanation, where low complexity
+implies less reliance on more input features. Again, whereas
+different naming conventions are used by the original works,
+they are all concerned with conciseness.
+Sanity preservation:
+Adeboya et al. [2018] first showed
+that multiple saliency mapping methods fail basic sanity
+checks. Thereafter, sanity preservation is often used as an
+evaluation criterion for the saliency methods [Jalwana et al.,
+2021]. Cascading randomization [Adebayo et al., 2018] is
+the common test in this regard that alters model weights in a
+cascading manner to visually observe the effects of this ran-
+domization on the computed explanation. Sixt et al. [2020]
+also proposed a test for the backpropagation based methods
+which observes the effects of randomizing the later layers on
+the explanation. Such sanity preservation tests can often be
+viewed as convenient qualitative (in some cases, quantitative)
+measures of other properties, e.g., model fidelity.
+Axiomatic properties:
+In the above, we are able to or-
+ganize the evaluation metrics systematically. This owes to
+the fact that these metrics are essentially developed to eval-
+uating specific properties of the explanation. Nevertheless,
+those properties themselves remain abstract notions. In con-
+trast, there are a few properties that have concrete mathe-
+matical definitions, e.g., completeness, linearity, input invari-
+ance [Sundararajan et al., 2017], [Kindermans et al., 2019].
+Measuring the extent to which these axiomatic properties are
+satisfied by the explanations, is also employed as an evalua-
+tion criterion. However, it is more relevant to the game-theory
+inspired path attribution methods [Sundararajan et al., 2017].
+We refer to [Lundstrom et al., 2022] for the details on the ax-
+iomatic properties. More recently, Khakzar et al. [2022] also
+proposed an empirical framework that evaluates explanations
+using multiple non-game theoretic axioms.
+6
+Challenges and future directions
+The notion of explanation is abstract and subjective.
+Its
+ill-defined nature poses multiple challenges for the research
+community, especially in the vision domain where the quali-
+tative aspects of perceptual data confound even the objective
+of explanation. With this first literature review of XAI for vi-
+sual models, we are able to identify the key challenges for this
+direction. They are listed below with pointers to the potential
+avenues to explore for their solution.
+Post-hoc methods not addressing model opacity:
+The
+very aim of XAI for deep learning is to address the black-
+box nature of the models.
+However, despite explaining a
+model prediction with the post-hoc techniques, the model re-
+tains nearly the same level of opacity because the explanation
+
+relates only to a single sample. Current literature is domi-
+nated by post-hoc techniques, which places a question mark
+on the current objective of computing input-specific explana-
+tions pursued by these methods. Ideally, this dominant stream
+of methods should also help in reducing the model opacity
+by explaining their general behavior. Very recently, interest-
+ing results in this regard are reported in [Akhtar and Jalwana,
+2023], where the saliency maps are computed to capture the
+model’s sensitivity to generic geometric patterns in an input-
+agnostic manner. We can expect future post-hoc techniques
+to explore this concept further.
+Correlation vs causation:
+It is often the case that “corre-
+lation is not causation”. The current literature in vision do-
+main predominantly sees the explanation problem as com-
+puting correlation between the input features and the model
+prediction. For instance, the saliency maps - see § 4.2 - sim-
+ply highlight the image regions having high relevance to the
+model prediction. Better correlation is also the criterion to
+improve the model fidelity scores - see § 5.1. Though useful,
+correlation alone is insufficient to truly explain the model.
+On one hand, it is imperative to develop techniques that ac-
+count for causation; on the other, we need evaluation metrics
+to measure the causal capacity of the existing methods. We
+anticipate these methods to be more sophisticated and evalua-
+tion metrics to be more comprehensive than the existing ones.
+Inspirational domain mismatch:
+Visual data has its pecu-
+liar nature. Incidentally, many techniques for explaining the
+visual models are inspired by other domains. For instance,
+path-based methods (in § 4.3) and axiomatic properties evalu-
+ation (in § 5.4) have game-theoretic roots. Similarly, Shapely
+values [Shapley, 1997] are Economy related concepts. The
+inherent mismatch between the vision domain and the inspir-
+ing domains can lead to problems. However, this aspect is
+often overlooked in the literature. For example, it is very hard
+to define ‘absence’ of a feature in an image to correctly iden-
+tify the ‘reference’ for that image in path based methods - see
+§ 4.3. This is not an issue in other game-theory applications.
+Interestingly, we also witnessed that the eventual attribution
+maps computed by the existing path based methods seldom
+exhibit the acclaimed axiomatic properties in a strict sense.
+The future explorations must address such domain mismatch
+problems diligently to make the explanations reliable.
+Performance vs transparency dilemma:
+Indeed, we find
+a trend that more human-understandable models have lower
+performance. Models not accounting for explainability are
+currently the top performers in the vision domain. This ob-
+servation is likely a major contributor to the common belief
+that explainability and performance have an inverse relation.
+However, we do not find rigorous studies in the current lit-
+erature that analytically (or more systematically) strengthen
+or refute this idea. A few leading vision research labs even
+believe explainability to be an unnecessary trait in practice.
+However, regulatory authorities concerning the development
+and deployment of AI have a consensus that the systems need
+to be transparent/explainable. Resolving this conflicting situ-
+ation demands a comprehensive investigation of the ‘perfor-
+mance vs transparency’ dilemma. We can anticipate some
+research activity addressing this in the future.
+The (in)consistency of methods and metrics:
+Due to the
+absence of concrete ground truth for the explainability task in
+general, the methods are hard to evaluate. The issue is further
+riled by the inconsistency of the available evaluation metrics.
+For instance, statistical unreliability and inconsistency of a
+few such metrics is already proven in [Tomsett et al., 2020].
+Implying, comparative rankings of the saliency methods is
+untrustworthy under these metrics. At the same time, expla-
+nation methods themselves are known to sometimes fail basic
+sanity checks [Adebayo et al., 2018]. This dicephalous nature
+of the problem demands further research that fully focuses on
+each sub-problem individually. Instead of making minor con-
+tributions of evaluation metrics along proposing new expla-
+nation methods (a common trend in the literature), thorough
+works that fully focus on devising the evaluation metrics and
+schemes are needed. There is also an obvious need to invent
+techniques to provide ground truth for evaluations.
+References
+[Adebayo et al., 2018] Julius
+Adebayo,
+Justin
+Gilmer,
+Michael Muelly, Ian Goodfellow, Moritz Hardt, and Been
+Kim. Sanity checks for saliency maps. NeurIPS, 2018.
+[Agarwal et al., 2021] Rishabh
+Agarwal,
+Levi
+Melnick,
+Nicholas Frosst, Ben Lengerich, Rich Caruana, and Ge-
+offrey Hinton. Neural additive models: Interpretable ma-
+chine learning with neural nets. NeurIPS, 2021.
+[Agarwal et al., 2022] Chirag Agarwal, Nari Johnson, Mar-
+tin Pawelczyk, Eshika Saxena, Marinka Zitnik, and
+Himabindu Lakkaraju. Rethinking stability for attribution-
+based explanations. arXiv:2203.06877, 2022.
+[Akhtar and Jalwana, 2023] Naveed Akhtar and Moham-
+mad Jalwana.
+Rethinking interpretation: Input-agnostic
+saliency mapping of deep visual classifiers. In AAAI, 2023.
+[Ali et al., 2022] Ameen Ali,
+Thomas Schnake,
+Oliver
+Eberle, Gr´egoire Montavon, Klaus-Robert M¨uller, and
+Lior Wolf.
+Xai for transformers:
+better explanations
+through conservative propagation. ICML, 2022.
+[Alvarez Melis and Jaakkola, 2018] David
+Alvarez
+Melis
+and Tommi Jaakkola. Towards robust interpretability with
+self-explaining neural networks. NeurIPS, 2018.
+[Ancona et al., 2017] Marco Ancona, Enea Ceolini, Cengiz
+¨Oztireli, and Markus Gross. Towards better understand-
+ing of gradient-based attribution methods for deep neural
+networks. arXiv preprint arXiv:1711.06104, 2017.
+[Antognini and Faltings, 2021] Diego
+Antognini
+and
+Boi
+Faltings.
+Rationalization
+through
+concepts.
+arXiv:2105.04837, 2021.
+[Arras et al., 2022] Leila Arras, Ahmed Osman, and Woj-
+ciech Samek.
+Clevr-xai: a benchmark dataset for the
+ground truth evaluation of neural network explanations.
+Information Fusion, 81:14–40, 2022.
+[Bach et al., 2015] Sebastian
+Bach,
+Alexander
+Binder,
+Gr´egoire Montavon, Frederick Klauschen, Klaus-Robert
+M¨uller, and Wojciech Samek. On pixel-wise explanations
+for non-linear classifier decisions by layer-wise relevance
+propagation. PloS one, 10(7):e0130140, 2015.
+
+[Bahadori and Heckerman, 2020] Mohammad Bahadori and
+David Heckerman. Debiasing concept-based explanations
+with causal analysis. arXiv:2007.11500, 2020.
+[Bau et al., 2017] David Bau, Bolei Zhou, Aditya Khosla,
+Aude Oliva, and Antonio Torralba. Network dissection:
+Quantifying interpretability of deep visual representations.
+In IEEE CVPR, 2017.
+[Bhatt et al., 2020] Umang
+Bhatt,
+Adrian
+Weller,
+and
+Jos´e MF Moura. Evaluating and aggregating feature-based
+model explanations. arXiv:2005.00631, 2020.
+[Blazek and Lin, 2021] Paul Blazek and Milo Lin. Explain-
+able neural networks that simulate reasoning. Nature Com-
+putational Science, 1(9):607–618, 2021.
+[Chalasani et al., 2020] Prasad Chalasani,
+Jiefeng Chen,
+Xi Wu, and Somesh Jha. Concise explanations of neural
+networks using adversarial training. In ICML, 2020.
+[Chang et al., 2021] Chun-Hao Chang, Rich Caruana, and
+Anna Goldenberg. Neural generalized additive model for
+interpretable deep learning. arXiv:2106.01613, 2021.
+[Chattopadhay et al., 2018] Aditya Chattopadhay, Anirban
+Sarkar, Prantik Howlader, and Vineeth Balasubramanian.
+Grad-cam++: Generalized gradient-based visual explana-
+tions for deep convolutional networks. In WACV, 2018.
+[Chauhan et al., 2022] Kushal Chauhan, Rishabh Tiwari, Jan
+Freyberg, and Pradeep Shenoy. Interactive concept bottle-
+neck models. arXiv:2212.07430, 2022.
+[Chen et al., 2019] Chaofan Chen, Oscar Li, Daniel Tao,
+Alina Barnett, Cynthia Rudin, and Jonathan K Su. This
+looks like that:
+deep learning for interpretable image
+recognition. NeurIPS, 2019.
+[Chen et al., 2020] Zhi Chen, Yijie Bei, and Cynthia Rudin.
+Concept whitening for interpretable image recognition.
+Nature Machine Intelligence, 2(12):772–782, 2020.
+[Dasgupta et al., 2022] Sanjoy Dasgupta, Nave Frost, and
+Michal Moshkovitz. Framework for evaluating faithful-
+ness of local explanations. arXiv:2202.00734, 2022.
+[Dhurandhar et al., 2022] Amit
+Dhurandhar,
+Karthikeyan
+Ramamurthy, and Karthikeyan Shanmugam.
+Is this the
+right neighborhood? accurate and query efficient model
+agnostic explanations. In NeurIPS, 2022.
+[Dubey et al., 2022] Abhimanyu Dubey, Filip Radenovic,
+and Dhruv Mahajan. Scalable interpretability via polyno-
+mials. NeurIPS, 2022.
+[Erion et al., 2021] Gabriel Erion, Joseph D Janizek, Pascal
+Sturmfels, Scott M Lundberg, and Su-In Lee. Improving
+performance of deep learning models with axiomatic attri-
+bution priors and expected gradients. Nature MI, 2021.
+[Fong et al., 2019] Ruth Fong, Mandela Patrick, and Andrea
+Vedaldi. Understanding deep networks via extremal per-
+turbations and smooth masks. In ICCV, 2019.
+[Ghorbani and Zou, 2020] Amirata Ghorbani and James Y
+Zou. Neuron shapley: Discovering the responsible neu-
+rons. NeurIPS, 2020.
+[Hastie, 2017] Trevor Hastie. Generalized additive models.
+In Statistical models, pages 249–307. Routledge, 2017.
+[Hendricks et al., 2016] Lisa
+Hendricks,
+Zeynep
+Akata,
+Marcus Rohrbach, Jeff Donahue, and Trevor Darrell. Gen-
+erating visual explanations. In ECCV, 2016.
+[Hooker et al., 2019] Sara Hooker, Dumitru Erhan, Pieter
+Kindermans, and Been Kim.
+A benchmark for inter-
+pretability methods in deep neural nets. NeurIPS, 2019.
+[Jalwana et al., 2021] Mohammad Jalwana, Naveed Akhtar,
+Mohammed Bennamoun, and Ajmal Mian. Cameras: En-
+hanced resolution and sanity preserving class activation
+mapping for image saliency. In IEEE CVPR, 2021.
+[Khakzar et al., 2022] Ashkan Khakzar, Pedram Khorsandi,
+Rozhin Nobahari, and Nassir Navab. Do explanations ex-
+plain? model knows best. In IEEE CVPR, 2022.
+[Kim et al., 2018] Been Kim, Martin Wattenberg, Justin
+Gilmer, Carrie Cai, James Wexler, Fernanda Viegas, et al.
+Interpretability beyond feature attribution: Quantitative
+testing with concept activation vectors. In ICML, 2018.
+[Kim et al., 2022] Sangwon Kim, Jaeyeal Nam, and By-
+oung Chul Ko. Vit-net: Interpretable vision transformers
+with neural tree decoder. In ICML, 2022.
+[Kindermans et al., 2019] Pieter-Jan
+Kindermans,
+Sara
+Hooker, Julius Adebayo, Maximilian Alber, Kristof T
+Sch¨utt, Sven D¨ahne, Dumitru Erhan, and Been Kim. The
+(un) reliability of saliency methods. In Explainable AI:
+IEVDL, pages 267–280. Springer, 2019.
+[Koh et al., 2020] Pang Wei Koh, Thao Nguyen, Yew Siang
+Tang, Stephen Mussmann, Emma Pierson, Been Kim, and
+Percy Liang. Concept bottleneck models. In ICML, 2020.
+[Kohlbrenner et al., 2020] Maximilian
+Kohlbrenner,
+Shinichi Bauer, Alexander Binder, and Sebastian La-
+puschkin.
+Towards best practice in explaining neural
+network decisions with lrp. In IJCNN, 2020.
+[Kwon and Zou, 2022] Yongchan Kwon and James Zou.
+WeightedSHAP: analyzing and improving shapley based
+feature attributions. arXiv:2209.13429, 2022.
+[Lang et al., 2021] Oran Lang, Yossi Gandelsman, Michal
+Yarom, Yoav Wald, Gal Elidan, Avinatan Hassidim,
+William T Freeman, Phillip Isola, Amir Globerson, Michal
+Irani, et al. Explaining in style: Training a gan to explain
+a classifier in stylespace. In ICCV, 2021.
+[Li et al., 2021] Liangzhi Li, Bowen Wang, Manisha Verma,
+Yuta Nakashima, Ryo Kawasaki, and Hajime Nagahara.
+Scouter: Slot attention-based classifier for explainable im-
+age recognition. In ICCV, 2021.
+[Lundberg and Lee, 2017] Scott Lundberg and Su-Im Lee.
+A unified approach to interpreting model predictions.
+NeurIPS, 2017.
+[Lundstrom et al., 2022] Daniel
+D
+Lundstrom,
+Tianjian
+Huang, and Meisam Razaviyayn. A rigorous study of inte-
+grated gradients method and extensions to internal neuron
+attributions. In ICML, 2022.
+
+[Montavon et al., 2018] Gr´egoire
+Montavon,
+Wojciech
+Samek, and Klaus-Robert M¨uller.
+Methods for inter-
+preting and understanding deep neural networks. Digital
+signal processing, 73:1–15, 2018.
+[Mu and Andreas, 2020] Jesse Mu and Jacob Andreas. Com-
+positional explanations of neurons. NeurIPS, 2020.
+[Nguyen and Mart´ınez, 2020] An-phi
+Nguyen
+and
+Mar´ıa Rodr´ıguez Mart´ınez.
+On quantitative aspects
+of model interpretability. arXiv:2007.07584, 2020.
+[Omeiza et al., 2019] Daniel Omeiza,
+Skyler Speakman,
+Celia Cintas, and Komminist Weldermariam. An enhanced
+inference level visualization technique for deep convolu-
+tional neural network models. arXiv:1908.01224, 2019.
+[Oquab et al., 2015] Maxime Oquab, L´eon Bottou, Ivan
+Laptev, and Josef Sivic. Is object localization for free?-
+weakly-supervised learning with convolutional neural net-
+works. In IEEE CVPR, 2015.
+[Park et al., 2018] Dong Park, Lisa Anne Hendricks, Zeynep
+Akata, Anna Rohrbach, Bernt Schiele, Trevor Darrell, and
+Marcus Rohrbach. Multimodal explanations: Justifying
+decisions and pointing to the evidence. In CVPR, 2018.
+[Petsiuk et al., 2018] Vitali Petsiuk, Abir Das, and Kate
+Saenko. Rise: Randomized input sampling for explana-
+tion of black-box models. arXiv:1806.07421, 2018.
+[Rao et al., 2022] Sukrut Rao, Moritz B¨ohle, and Bernt
+Schiele. Towards better understanding attribution meth-
+ods. In IEEE CVPR, 2022.
+[Ribeiro et al., 2016] Marco Tulio Ribeiro, Sameer Singh,
+and Carlos Guestrin. ” why should i trust you?” explaining
+the predictions of any classifier. In ACM SIGKDD, 2016.
+[Rong et al., 2022] Yao Rong,
+Tobias Leemann,
+Vadim
+Borisov, Gjergji Kasneci, and Enkelejda Kasneci.
+A
+consistent and efficient evaluation strategy for attribution
+methods. In ICML, 2022.
+[Rudin, 2019] Cynthia Rudin.
+Stop explaining black box
+machine learning models for high stakes decisions and use
+interpretable models instead. NMI, 1(5):206–215, 2019.
+[Samek et al., 2016] Wojciech Samek, Alexander Binder,
+Gr´egoire Montavon, and Klaus-Robert M¨uller. Evaluat-
+ing the visualization of what a deep neural network has
+learned. IEEE TNNLS, 28(11):2660–2673, 2016.
+[Sarkar et al., 2022] Anirban Sarkar, Deepak Vijaykeerthy,
+Anindya Sarkar, and Vineeth N Balasubramanian.
+A
+framework for learning ante-hoc explainable models via
+concepts. In IEEE CVPR, 2022.
+[Selvaraju et al., 2017] Ramprasaath
+Selvaraju,
+Michael
+Cogswell, Abhishek Das, Devi Parikh, and Dhruv Batra.
+Grad-cam: Visual explanations from deep networks via
+gradient-based localization. In ICCV, 2017.
+[Shapley, 1997] Lloyd S Shapley.
+A value for n-person
+games. Classics in game theory, 69, 1997.
+[Shrikumar et al., 2017] Avanti Shrikumar, Peyton Green-
+side, and Anshul Kundaje. Learning important features
+through propagating activations. In ICML, 2017.
+[Simonyan et al., 2014] Karen Simonyan, Andrea Vedaldi,
+and Andrew Zisserman.
+Deep inside convolutional
+networks:
+Visualising image classification models and
+saliency maps. ICLR, 2014.
+[Sixt et al., 2020] Leon Sixt, Maximilian Granz, and Tim
+Landgraf. When explanations lie: Why many modified
+bp attributions fail. In ICML, 2020.
+[Smilkov et al., 2017] Daniel Smilkov, Nikhil Thorat, Been
+Kim, and Martin Wattenberg.
+Smoothgrad: removing
+noise by adding noise. arXiv:1706.03825, 2017.
+[Springenberg et al., 2015] Jost Springenberg, Alexey Doso-
+vitskiy, Thomas Brox, and Martin Riedmiller. Striving for
+simplicity: The all convolutional net. ICLR, 2015.
+[Srinivas and Fleuret, 2019] Suraj
+Srinivas
+and
+Franc¸ois
+Fleuret. Full-gradient representation for neural network
+visualization. NeurIPS, 2019.
+[Stalder et al., 2022] Steven Stalder, Nathana¨el Perraudin,
+Radhakrishna
+Achanta,
+Fernando
+Perez-Cruz,
+and
+Michele Volpi. What you see is what you classify: Black
+box attributions. NeurIPS, 2022.
+[Sturmfels et al., 2020] Pascal Sturmfels, Scott Lundberg,
+and Su-In Lee. Visualizing the impact of feature attribu-
+tion baselines. Distill, 2020.
+[Sundararajan et al., 2017] Mukund Sundararajan,
+Ankur
+Taly, and Qiqi Yan. Axiomatic attribution for deep net-
+works. In ICML, 2017.
+[Tomsett et al., 2020] Richard
+Tomsett,
+Dan
+Harborne,
+Supriyo Chakraborty, Prudhvi Gurram, and Alun Preece.
+Sanity checks for saliency metrics. In AAAI, 2020.
+[Wang et al., 2020] Haofan Wang, Zifan Wang, Mengnan
+Du, Fan Yang, Zijian Zhang, Sirui Ding, Piotr Mardziel,
+and Xia Hu. Score-weighted visual explanations for con-
+volutional neural networks. In IEEE CVPRw, 2020.
+[Wang et al., 2021] Jiaqi Wang, Huafeng Liu, Xinyue Wang,
+and Liping Jing. Interpretable image recognition by con-
+structing transparent embedding space. In ICCV, 2021.
+[Wang et al., 2022] Andong Wang, Wei-Ning Lee, and Xi-
+aojuan Qi. Hint: Hierarchical neuron concept explainer.
+In IEEE CVPR, 2022.
+[Yang et al., 2023] Peiyu Yang, Naveed Akhtar, Zeyi Wen,
+and Ajmal Mian. Local path integration for attribution. In
+AAAI Conference on Artificial Intelligence. AAAI, 2023.
+[Yeh et al., 2019] Chih-Kuan Yeh, Cheng-Yu Hsieh, Arun
+Suggala, David Inouye, and Pradeep Ravikumar. On the
+(in)fidelity and sensitivity of explanations. NeurIPS, 2019.
+[Zhang et al., 2018] Jianming Zhang, Sarah Adel Bargal,
+Zhe Lin, Jonathan Brandt, Xiaohui Shen, and Stan
+Sclaroff. Top-down neural attention by excitation back-
+prop. IJCV, 126(10):1084–1102, 2018.
+[Zhou et al., 2016] Bolei Zhou,
+Agata Lapedriza,
+Aude
+Oliva, and Antonio Torralba. Learning deep features for
+discriminative localization. In IEEE CVPR, 2016.
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf,len=804
+page_content='A Survey of Explainable AI in Deep Visual Modeling: Methods and Metrics  Naveed Akhtar  The University of Western Australia  naveed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='akhtar@uwa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='au  Abstract  Deep visual models have widespread applications  in high-stake domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Hence, their black-box na-  ture is currently attracting a large interest of the  research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' We present the first survey  in Explainable AI that focuses on the methods and  metrics for interpreting deep visual models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Cover-  ing the landmark contributions along the state-of-  the-art, we not only provide a taxonomic organi-  sation of the existing techniques, but also excavate  a range of evaluation metrics and collate them as  measures of different properties of model explana-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Along the insightful discussion on the cur-  rent trends, we also discuss the challenges and fu-  ture avenues for this research direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  1  Introduction  Visual computational models have widespread applications,  ranging from casual use in handheld devices to high-stake  tasks in forensics, surveillance, autonomous driving and med-  ical diagnosis etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Contemporary visual models rely heavily  on deep learning, which is a black-box technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Hence,  the opacity of deep visual models is becoming a major hur-  dle in enabling fairness, transparency and accountability of  the AI systems that rely on these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Given the emerg-  ing development and deployment policies for AI around the  globe, it is apparent that the future of deep visual models in  practice hinges on addressing their black-box nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Historically, visual modeling has inspired numerous break-  throughs in deep learning research owing to the convenient  interpretability of visual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' The same is currently fueling  a high research activity in explainable deep visual modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Since vision research is already a proven source of many no-  table advances in deep learning, systematically framing this  research activity can be pivotal for the future of Explainable  AI (XAI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Whereas the existing literature provides reviews of  the broader domain of XAI, it is still void of a taxonomic  treatment of the techniques devised to render deep visual  models more transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' This paper fills this critical research  gap by making the following important contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  It provides the first comprehensive review of the literature  in XAI that focuses on deep visual models1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' The review  1We only focus on the natural image domain, which is also the  Figure 1: The literature in visual model explanation is currently  dominated by post-hoc techniques (Top) that treat pre-trained mod-  els under black- and white-box setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Backpropagated gradients  and model activations are common tools to generate visual expla-  nation in white-box setup, whereas model-agnostic techniques nor-  mally query the models to compute a mask for salient input regions  in black-box settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' (Bottom) Techniques inducing inherently ex-  plainable models rely on non-traditional building blocks for neural  modeling or augment the standard blocks with interpretable mod-  ules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Their outputs may still require post-hoc tools for explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  covers the influential techniques developed at the advent  of the modern deep learning era, as well as the most re-  cent state-of-the-art methods appearing in the flagship re-  search sources of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' It pays a special attention to  the landmark contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' The literature is taxonomized  following the technical methodologies adopted by the ap-  proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Thereby, providing a clear picture of the broad  strategies leveraged by the existing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  We find that quantification of visual model interpretation  performance still lacks the desired clarity in the literature  due to the intrinsic difficulties in measuring the abstract  subjective notion of explainablity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' To help the research  community addressing that, we excavate a range of eval-  uation metrics from the literature, and organize them as  quantitative measures of different properties of explana-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' This forms the second major part of our literature  source of methods for other image domains, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', medical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='13445v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='CV]  31 Jan 2023  Post-hoc model interpretation tools  White-box  propagation  Attribution  Back  dpu  indul  Activation  Saliency  dpu  Standard  Model  agnostic  Saliency  mask  Pre-trained  Black-box  Non-traditional  Augmented  (from scratch)  indul  Post  hoc  Inherentlyexplainablemodelsreview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  It is envisaged that this first-of-its-kind broad  methodical treatment of the evaluation metrics will sig-  nificantly contribute towards the much needed systematic  evaluation of the future techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Due to the absence of a focused review, the current lit-  erature in XAI for visual modeling suffers from obvious  inconsistencies in technical terminologies and providing  the correct context to the techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' We address this by  providing accessible understanding of the terminologies  along with insightful discussions throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  The paper finally highlights major open challenges for this  direction and provides the future outlook by building on  the insights from the reviewed literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  2  Overview and Terminologies  This section introduces the common terminologies used in  the reviewed literature and formalizes the problem from the  perspective of systematically categorizing the existing con-  tributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' The existing literature is often found referring to  the same concepts using different terminologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' This section  also collates these terminologies for the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  We find that the visual modelling domain is not too par-  ticular about discriminating between the terms explanation  and interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Without any obvious reasons, the even-  tual outcomes of the methods are sometimes referred to as  explanations, whereas the same outcomes are called inter-  pretations in other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' This survey favours the term ex-  planation for the output of a technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Notably, the works  contributing towards the intrinsic transparency of the models  generally prefer the term explainable for the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Meth-  ods aiming for intrinsically explainable models pursue their  goals as discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Let M : M(I) → y be a visual model that maps an image  I ∈ Rh×w×c having ‘c’ channels to the output y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In vision  literature, the explanation task mainly considers classifiers as  the target models, where y ∈ RK is a K-class prediction vec-  tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Methods seeking to improve the inherent explainability  of the models see them as a hierarchy of functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=',  M(I)=fL(ΘL, fL−1(ΘL−1, fL−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='(Θ2, f1(Θ1, I)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=')→y  where Θℓ denotes the model parameters (including biases)  for the ℓth function fℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' These methods take one of two ap-  proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' (i) Directly use easy to interpret functions for all  fℓ - see § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='1, or modify some fℓ for more transparency  see § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='2 (Internal component augmentation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Models  implementing this strategy are sometimes also called self-  explainable models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' (ii) Plug in an external explainer module  E(M, I) in the function hierarchy to generate the explanation  § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='2 (External component augmentation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' This explainer is  induced along the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Collectively, the approaches under  both (i)&(ii) are referred to as ante-hoc methods, as opposed  to the post-hoc techniques, which are discussed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  The post-hoc methods consider M to be fixed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', they  do not interfere with the design or induction process of the  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' To understand the common post-hoc visual explana-  tion objective, consider a set PI = {p1  I, p2  I, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', pw×h  I  } ⊂ Rc  that contains the pixels of the input I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' The explanation objec-  tive here is to seek an ordered array WI ⊂ R, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' |WI| = |PI|  and the ith element of this array, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' wi  I ∈ WI, encodes a  weight for the corresponding element in PI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' The weights  in WI are intended to be human-understandable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' The post-  hoc explanation methods generally implement the function  S : S(M, I) → WI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Here, WI can be a binary array that  represents a mask for I that suppresses the irrelevant pi  I ∈ PI  in the transform M(I) → y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' This objective is more per-  tinent to the model-agnostic post-hoc methods - see § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  These methods avoid using any internal signal of the model,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', neuron activations, thereby assuming a black-box setup  which does not allow access to the model details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  When access to the model is available, the weights wi  I ∈  WI take real values to quantify the importance of the input  pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' To that end, a wide range of methods rely mainly  on the activations of the model’s internal neurons to estimate  WI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' We call such methods neuron activation-based tech-  niques - see § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' The output of such methods is commonly  known as a saliency map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' The activation-based techniques  often also use model gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' However, those gradients are  generally not backpropagated right until the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' The meth-  ods that compute the model gradients w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' the input are cat-  egorised as the backpropagation-based methods in this sur-  vey - see § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' The output of these methods is more com-  monly termed attribution map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Note that, attribution maps  and saliency maps are technically ‘synonyms’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' We find multi-  ple papers in this domain that use these terms interchangeably  to describe the generated visual explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Other common  terms that refer to the same concept in the literature are rele-  vance scores, feature contributions and importance maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Whereas the ante-hoc methods render the model intrinsi-  cally more transparent, the interpretation process may still use  post-hoc tools or need human domain experts to explain the  predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Within the vision domain, the preferred mode  of explanations are visualizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' However, rarely, we also  encounter text as the explanations [Hendricks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2016],  [Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' We also find that the common objective  of the post-hoc techniques is to generate input-specific ex-  planations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' A rare exception to that is input-agnostic saliency  mapping [Akhtar and Jalwana, 2023] that generates visual ex-  planations that encode generic understanding of the model of  human-defined concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  3  Inherently explainable models  We find that there is a prevailing impression in the community  that explainability and performance of deep visual models are  inversely correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' However, Rudin [2019] rightly noted a  lack of concrete evidence in this regard, and argued in favor of  developing inherently explainable models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' To that end, recent  papers devise approaches along two broad directions: (a) de-  signing explainable models from scratch, and (b) augmenting  black-box models with interpretable components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='1  Explainable model design from scratch  Along this direction, a category of methods explores the lore  of generalized additive models (GAMs) [Hastie, 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' The  central idea is to implement the intended transformation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=',  modeling) of the input as a set of additive sub-models that  deal with the input features separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' This is used to pre-  serve the interpretability of the overall transformation [Agar-  wal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2021], [Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2021], [Dubey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022],  [Alvarez Melis and Jaakkola, 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' However, the lack of  Figure 2:  Flowchart  for  general guidance to select  explanation tools for visual  model application domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  The chart follows termi-  nologies used to categorise  the literature in this survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Best viewed enlarged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  expressive power of such composite modelling, and its lim-  ited scalability are major hurdles in the practicability of this  strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Taking a different approach, Blazek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2021]  proposed essence neural networks that employ non-standard  building blocks of Support Vector Machines based neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  With a systematic architectural design of the underlying neu-  ral network, they showed remarkable generalizable abilities  of the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' However, the design process of their net-  work remains cumbersome and relies heavily on an active in-  volvement of the domain experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2021] proposed  to build the classification stage of a visual classifier using a  slot attention mechanism that learns a semantically meaning-  ful internal representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' However, their method must still  rely on a backbone module in the early stage of the network,  which compromises the explainability of the overall model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  It is a common knowledge that a key strength of deep learn-  ing, especially in the vision domain, is its ability to represent  the data with discernible low- and high-level features through  a hierarchical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Methods aiming at fully explain-  able models must account for such a sensible hierarchical  treatment of the domain data in the network structure manu-  ally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' This generally requires expert knowledge of the domain,  which makes fully explainable modeling a less attractive op-  tion for many problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='2  Augmenting neural models for explainability  In general, explainability of visual models is seen as a con-  tinuum rather than a discrete measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' This perspective has  encouraged researchers to devise methods for improving the  ‘extent’ of interpretability of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Generally, this is  pursued by augmenting the conventional neural models with  (i) interpretable internal components, or (ii) external compo-  nents providing the desired explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Internal component augmentation:  To improve the in-  trinsic explainability of visual models, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2020]  introduced a concept whitening module that aims at align-  ing the axes of the latent space of the model with apriori  known concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' This alignment also involves a process of  de-correlating the concepts space for intelligibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Wang et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2021] also explored a closely related mechanism with a  plug-in embedding space for the model that is spanned by  basis concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Here, the basis concepts are kept category-  aware, and the within-category concepts are kept orthogonal  to each other for their human-understandability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' These are in-  deed useful methods, however they come at the cost of further  complicating the training process of neural modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Concept bottleneck modelling [Koh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2020] is an-  other interesting framework that lets a neural model first learn  human-interpretable (sub-)concepts, and makes the final pre-  diction by further processing those concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Architecturally,  it introduces a concept layer in the network that also enables  human intervention during the prediction stage in [Chauhan  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Concept bottleneck models (CBMs) are cur-  rently attracting notable attention of the research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Bahadori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2020] performed causal reasoning to debias  such models, whereas Antognini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2021] used this frame-  work for the application of textual rationalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Working  on the principles similar to CBMs, ProtoPNetwork [Chen et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2019] also introduces a prototype layer in the network  that associates high-level sub-concepts to its neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Notice that some approaches discussed in § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='1 have a close  conceptual resemblance to CMBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' For instance, like CBMs,  only the later layers of the visual models in [Blazek and Lin,  2021] are interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' This is because, their technique must  rely on deep features (instead of image pixels) to construct  the explainable (sub-)network in the case of visual modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  However, unlike [Blazek and Lin, 2021], the techniques using  internal component augmentation employ the standard neu-  rons along some basic arithmetic operations to improve the  interpretability of the network layer(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' It is notable though,  both paradigms remain interested in the deeper layers of the  neural network for model explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' This is true even for  [Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022], which specifically attempts to make Vi-  sion Transformers more explainable by using neural trees as  a decoder of the representation learned by the model, where  the nodes of the trees are contextual Transformer modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  External component augmentation:  Internal component  augmentation techniques embed modules in the neural net-  work such that they do not form a new branch of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In  contrast, some recent techniques introduce explanation mod-  ules as external branches of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' For instance, Sarkar  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2022] proposed a framework to learn an external expla-  nation module in the context of concept-based models [Koh  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Their explanation module uses an external con-  cept decoder to improve model interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Similarly,  [Lang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2021] trains an external StyleGAN as an ex-  plainer for a visual classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Human-understandable expla-  nations are produced in this approach by relying on semanti-  cally meaningful dimensions learned by the style-space of the  external module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Another example of external explainer can  be found in [Stalder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022], which is claimed to pro-  duce sharp masks of relevant image regions using a trainable  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Though useful, an apparent drawback of the external  module-based methods is that the module can inadvertently  affect both the model performance and its explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' For  the later case, the explanation can unintentionally be encod-  ing the module’s behavior instead of explaining the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='Use backpropagation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='Accessto  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='Visual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='Use post-hoc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='Use white-box  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='resolution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='orneuron-concept  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='Highcomputational cost  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='tools  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='methods  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='details  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='critical  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='associationmethods  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='No  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='Use saliency  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='No  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='Low computational cost  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='No  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='mapping methods  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='Use model-agnostic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='Train standard  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='methods (black-box)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='Develop  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='network  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='Start  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='own  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='Explainability  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='Train explainable  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='No  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='No  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='Useexternal component  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='model from scratch  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='augmentation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='Expert  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='Preference  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='Performance  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='Is model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='knowledge  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='Augment existing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Perf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  fidelity  Useinternalcomponent  Assuming sufficient  available  network architectures  critical?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  augmentation  computational resources & data4  Post-hoc explanations  It is hard to identify any study that rigorously establishes an  inverse correlation between the model explainability and its  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Nevertheless, it is widely presumed that this  inverse relation exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Consequently, the research commu-  nity is relatively more active in devising post-hoc explana-  tion tools as compared to developing intrinsically explainable  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Post-hoc tools include a category of methods that  leverage the model information when it is available - see § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='2  and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='3, whereas another class of methods can handle the sce-  nario when such information is unavailable - see §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' This  flexibility, along the fact that these methods do not interfere  with model performance, makes post-hoc tools very practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='1  Model-agnostic methods  For the cases where the model information is unavailable,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', black-box scenarios, post-hoc explanation methods query  the model in some form to identify the important pixels in  the input in a model-agnostic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' For instance, Fong  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2019] used queries to compute extremal perturbations  to identify the most salient input pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Local Interpretable  Model-agnostic Explanations (LIME) is an early attempt of  model-agnostic methods that approximates the model behav-  ior within a local neighborhood around the input by fitting  a simple interpretable model to the perturbed versions of the  input [Ribeiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' However, LIME is known to be  unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Dhurandhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2022] recently proposed a fix to  this instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Randomized Input Sampling for Explanation  (RISE) [Petsiuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2018] is yet another popular model-  agnostic explanation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Lundberg & Lee [2017] pro-  posed SHapley Additive exPlanations (SHAP) that leverage  Shapely values [Shapley, 1997] to estimate the contribution  of each feature to the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Recently, it is claimed that  the marginal contribution computed by the Shapely values is  sub-optimal for model explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Hence, a weightedSHAP  is introduced in [Kwon and Zou, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Model-agnostic strategy is the only scheme available for  explaining a model in a black-box setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Since the ap-  proaches do not leverage model information, they solve a  harder problem to generate explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Thus, we often find  these methods relying on heuristics to control their solution  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' These heuristics can have unintended effects on the  generated explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In general, despite using heuristics,  these methods remain computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  When the internal information of the model is available,  there are two important signals that can be exploited to ex-  plain the model behavior;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' namely, (i) internal activations and  (ii) backpropagated gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' We discuss the methods ex-  ploiting (i) and (ii) in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='2 and §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='2  Neuron activation-based techniques  There is a range of methods that strongly rely on the activa-  tion values of the internal neurons of the model to explain  their predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' It is emphasized though that the activations  may not be the only internal signals used by these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Saliency methods:  A popular stream of techniques com-  putes saliency maps for the inputs as visual explanations us-  ing internal neuron activations of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Class Activa-  tion Mapping (CAM) [Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2016] is one of the earliest  methods in this direction, which employs the global average  pooling neuron activations to weakly localize the objects in  the inputs for saliency mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Oquob et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2015] used the  max pooling neurons for a similar purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Whereas multiple  other techniques exist for activation-based saliency mapping,  GradCAM [Selvaraju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2017] is a landmark work, which  combines the model gradients with the activations of its in-  ternal neurons to compute the saliency maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Multiple vari-  ants of GradCAM have appeared in the literature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', Grad-  CAM++ [Chattopadhay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2018], Score-CAM [Wang et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2020], Smooth-GradCAM++ [Omeiza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2019], all  of which use neuron activation signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Generally, there is a  large disparity between the sizes of the activation maps of the  internal layers of a model and the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' This requires inter-  polation of the activation signals for computing the eventual  saliency map, which results in poor resolution of the maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Recently, Jalwana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2021] addressed this weakness with  their CAMERAS method that employs a multi-scale mapping  for high-resolution saliency mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Despite a considerable number of existing methods for  activation-based saliency mapping, we still regularly witness  new techniques emerging along this line of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' On the  other hand, there are still many other popular classic tech-  niques, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', Layerwise Relevance Propagation (LRP) [Bach  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2015], DeepLift [Shrikumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2017] that rely on  activations of multiple layers to compute the saliency maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  With the paradigm shift in visual modeling in favor of Trans-  formers, these classical methods are also being extended to  explain the Transformer based models [Ali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Neuron-concept  association  methods:  Whereas  tech-  niques like CBMs [Koh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2020] encourage neuron-  concept association by design, we also find methods that aim  at identifying such an association in a post-hoc manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' For  instance, Bau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2017] aimed at quantifying the alignment  between the model neurons and known semantic concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Similarly, Mu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2020] searched for logical forms of ex-  planation defined by a composition of the concepts learned  by polysemantic neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' A neuron-concept explainer is also  developed in [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022] that utilizes both saliency  maps and Shapley values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Previously, Shapley values of neu-  rons are also leveraged by [Ghorbani and Zou, 2020] to an-  alyze the importance of neurons for a given output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' On the  other hand, activations are also used by the popular TCAV  method [Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2018] to identify the sensitivity of the  neurons to a given set of concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='3  Backpropagation-based methods  Gradients of a model w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' input are widely considered a  natural analogue of its coefficients in the modern deep neu-  ral networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' We find a range of methods leveraging back-  propagated model gradients for explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Simonyan et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2014] were among the first to estimate input feature rel-  evance for the prediction with model gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Guided-  backpropagation [Springenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2015], smoothGrad  [Smilkov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2017], fullGrad [Srinivas and Fleuret, 2019]  and multiple other methods in the literature later that used  the backpropagated gradients to estimate the feature attribu-  tion scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' It is noteworthy that backpropagated gradients  are sometimes also employed by other categories of methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  For instance, the saliency methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', Grad-CAM, CAM-  ERAS, in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='2 also use model gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' However, those  gradients are not fully propagated back to the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Within the methods mainly relying on the backpropagated  gradients, there is an important sub-category of path attribu-  tion methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Inspired by game-theory, these methods accu-  mulate the backpropagated gradients of the images defined  by a path between the input and a reference image, where  the latter signifies the absence of the features in the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  The pioneering work by Sundrarajan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2017] claims that  the path attribution strategy can preserve certain desirable ax-  iomatic properties for the explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Their claims were  further qualified recently in [Lundstrom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' The  original work has inspired a range of methods employing the  path attribution framework [Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2023], [Erion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=',  2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' However, Stumfels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2020] have also noted that  the path attribution strategy can suffer from certain artifacts  resulting from the reference image itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Our close inspec-  tion revealed that despite their axiomatic nature, these meth-  ods sometimes compute counter-intuitive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Moreover,  the literature often fails to establish that the eventual methods  actually retain the claimed properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  5  Evaluation metrics  Evaluating the performance of a model explanation tool is  not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Due to the abstract subjective nature of  the notion of explanation and the lack of ground-truth, ob-  jectively evaluating a method’s performance requires quanti-  fying multiple aspects of the computed explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Hence,  development of metrics to evaluate explanation methods is  currently actively pursued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' We excavate numerous metrics  from different contributions and organise them based on the  explanation properties quantified by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='1  Quantifying the model fidelity  Fidelity of the explanation to the model is a measure of the  extent to which the explanation captures the true behavior of  the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' It is the most essential property of the computed  explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Broadly, the metrics quantifying this property  rely on estimating some form of correlation between the rel-  evance of the input features (as computed by the explanation  method) and the output variation when those features are per-  turbed/removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' The core intuition is that for a model-faithful  explanation, the output variations should be stronger for the  perturbations to the more relevant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  As an early example of evaluating the model fidelity, Bach  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2015] developed a metric of pixel flipping where the  fidelity is evaluated by observing the change in the predic-  tion scores by flipping the input pixels with high (absolute)  scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' The metric was later generalized to region perturba-  tion in [Samek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2016] where a random sample from a  uniform distribution is used to perturb a set of input pixels  to analyze the effects on predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Similarly, Alvarez et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2018] removed the pixels from the input and measured  the correlation between the prediction scores and the attri-  bution scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In [Ancona et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2017], SensitivityN metric  is proposed to measure the correlation between the sum of a  subset of computed pixel attributions and the change in the  prediction when the corresponding pixels are perturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' This  notion is further generalized in [Yeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2019] with an In-  fidelity measure as the Expected difference between the dot  product of the input perturbation to the attribution, and the  prediction change resulting from the perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  The key strategy of the above metrics is to perturb (or re-  move) input features and analyze the effects on the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  However, Hooker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2019] showed that this results in a  distribution shift problem for the models because their train-  ing process does not account for any pixel removal or per-  turbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  This compromises the reliability of the evalua-  tion metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Hence, they proposed to iteratively retrain the  model after the removal of each pixel on the modified training  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Their Remove and Retrain (ROAR) is one of the widely  adopted metric due to its global (in terms of training data)  nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Nevertheless, its evaluation can be computationally  prohibitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' To address that, Rong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2022] proposed a  Remove and Debias (ROAD) metric that avoids the retraining  with a de-biasing step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' The authors claim a desirable prop-  erty of consistency between the ROAD scores when the most  relevant pixels are removed first, or the least relevant pixels  are removed first - a property not exhibited by ROAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Whereas multiple metrics exist for model fidelity, there is  still no single agreed-upon metric that is considered to ac-  curately and comprehensively quantify this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  The  emerging relevant trends include combining the existing met-  rics [Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2023] and quantifying the model fidelity  with other properties such as consistency and sufficiency of  the computed attribution [Dasgupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022] or proposing  novel settings to evaluate this property [Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' A  comprehensive and reliable evaluation of this property is still  an open challenge for the vision domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='2  Quantifying the localisation ability  An accurate explanation must assign high relevance to the  foreground object features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' This intuition has led to a range of  metrics that aim at leveraging ground truth in quantifying the  explanation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Their key idea is to measure how  well the computed relevance scores correlate with the bound-  ing boxes, segmentation masks, or the grid cells that contain  the object of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In that sense, we can collectively see  them as localisation quantifying metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Pointing game [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2018] is one of the popular  examples in this direction, which computes the percentage  of the most relevant pixel for every sample being located on  the foreground object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Similarly, Kohlbrenner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2020]  proposed an attribution localization metric that computes the  fraction of the cumulative relevance score attributed to the  foreground object in an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' The same measure is termed  relevance mass in [Arras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022], where another metric  of relevance rank is also introduced, which estimates the frac-  tion of the object mask pixels common to the top K relevant  pixels, where K is the mask size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In [Arras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022], mo-  saics of different data samples are use to quantify the fraction  of the relevance assigned to the foreground object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Whereas the localisation based metrics can leverage the  ground truth, their objective is agnostic to model fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' An  explanation method enforcing stronger attribution to the fore-  ground, irrespective of the actual behavior of the model, will  achieve higher score for these metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Hence, relying solely  Table 1: Representative evaluation metrics and their ability to measure the extent of different properties of explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' More *s indicate  better ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Refer to § 5 for the description of the properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' GT denotes Ground Truth usage, and Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' is computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Metric  Work  Model Fidelity  Localisation  Stability  Conciseness  Sanity preservation  Axiomatic properties  GT  Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Pixel Flipping  Bach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2015]  **** \x17 **  \x17  \x17  ***  SensitivityN  Ancona et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2017]  **** \x17 **  ***  \x17 Insertion/deletion  Petsiuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2018]  **** \x17 **  \x17  \x17  ***  ROAR  Hooker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2019]  **** \x17 **  \x17  \x17 ROAD  Rong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2022]  **** \x17 **  \x17  \x17  **  Pointing game  Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2018]  \x17  ****  \x17 \x17  \x17  \x13  ***  Stability  Alvarez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2018]  **  \x17  ****  \x17 \x17  \x17  **  Sensitivity  Yeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2019]  **  \x17  ****  \x17 \x17  \x17  **  Sparseness  Chalasani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2020] \x17  **** \x17  \x17  **  Effective complexity  Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2020] \x17  **** \x17  \x17  **  Cascading randomization  Adeboya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2018]  *** \x17  \x17  ****  \x17  \x17  **  Completeness  Sundararajan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2017]  **  \x17  \x17 **  ****  \x17 on the localisation based metrics can be misleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='3  Quantifying the stability  A correct explanation should not change drastically when the  input is changed only slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Following this intuition mul-  tiple metrics have emerged that quantify some form of sta-  bility of the computed explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' For instance, Avarez  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2018] measured the local Lipschitz continuity of the  explanation to observe its stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Similarly, Motavon et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2018] computed the maximum ℓ1-distance between the  explanations of two inputs normalized by the Euclidean dis-  tance between them, to observe the explanation stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In  [Yeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2019], an estimate of the maximum distance be-  tween the explanations of the neighborhood inputs is consid-  ered to quantify the stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Dasgupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2022] quanti-  fied the probability of the similarity between the explanations  of similar samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In [Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022], relative changes  in the explanations are estimated with respect to the input,  model embeddings and the model output logits to propose  three measures of stability, namely;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' relative input stability,  relative representation stability and relative output stability  of the explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' The authors argue to discard the assump-  tion that the model is a black-box (for black-box methods)  and propose to leverage full model information to ensure a  more reliable evaluation of the explanation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Whereas the notion of stability is understood in the same  sense across different papers, we do not find consensus in its  naming convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' For instance, stability is referred to as  continuity in [Montavon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2018], sensitivity in [Yeh et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2019] and consistency in [Dasgupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' It is  also worth highlighting that the prevailing existing notion of  stability is underpinned by the assumption that the model in  question is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Stability measures for the explanation of  a model not following this assumption can be problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='4  Evaluating other desirable properties  Besides the above, we also find a number of other metrics  in the literature that address other desirable properties of the  explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' We summarise them below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Conciseness:  An effective explanation should be concise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Following this intuition, Chalasani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2020] quantified the  explanation conciseness by estimating its sparseness with the  Gini Index, whereas Bhatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2020] evaluated the explana-  tion complexity to measure the conciseness using a fractional  contribution distribution of the input features for the expla-  nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Similarly, Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2020] also measured an ef-  fective complexity of the explanation, where low complexity  implies less reliance on more input features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Again, whereas  different naming conventions are used by the original works,  they are all concerned with conciseness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Sanity preservation:  Adeboya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2018] first showed  that multiple saliency mapping methods fail basic sanity  checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Thereafter, sanity preservation is often used as an  evaluation criterion for the saliency methods [Jalwana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=',  2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Cascading randomization [Adebayo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2018] is  the common test in this regard that alters model weights in a  cascading manner to visually observe the effects of this ran-  domization on the computed explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Sixt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2020]  also proposed a test for the backpropagation based methods  which observes the effects of randomizing the later layers on  the explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Such sanity preservation tests can often be  viewed as convenient qualitative (in some cases, quantitative)  measures of other properties, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', model fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Axiomatic properties:  In the above, we are able to or-  ganize the evaluation metrics systematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' This owes to  the fact that these metrics are essentially developed to eval-  uating specific properties of the explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Nevertheless,  those properties themselves remain abstract notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In con-  trast, there are a few properties that have concrete mathe-  matical definitions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', completeness, linearity, input invari-  ance [Sundararajan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2017], [Kindermans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Measuring the extent to which these axiomatic properties are  satisfied by the explanations, is also employed as an evalua-  tion criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' However, it is more relevant to the game-theory  inspired path attribution methods [Sundararajan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  We refer to [Lundstrom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022] for the details on the ax-  iomatic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' More recently, Khakzar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' [2022] also  proposed an empirical framework that evaluates explanations  using multiple non-game theoretic axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  6  Challenges and future directions  The notion of explanation is abstract and subjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Its  ill-defined nature poses multiple challenges for the research  community, especially in the vision domain where the quali-  tative aspects of perceptual data confound even the objective  of explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' With this first literature review of XAI for vi-  sual models, we are able to identify the key challenges for this  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' They are listed below with pointers to the potential  avenues to explore for their solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Post-hoc methods not addressing model opacity:  The  very aim of XAI for deep learning is to address the black-  box nature of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  However, despite explaining a  model prediction with the post-hoc techniques, the model re-  tains nearly the same level of opacity because the explanation  relates only to a single sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Current literature is domi-  nated by post-hoc techniques, which places a question mark  on the current objective of computing input-specific explana-  tions pursued by these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Ideally, this dominant stream  of methods should also help in reducing the model opacity  by explaining their general behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Very recently, interest-  ing results in this regard are reported in [Akhtar and Jalwana,  2023], where the saliency maps are computed to capture the  model’s sensitivity to generic geometric patterns in an input-  agnostic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' We can expect future post-hoc techniques  to explore this concept further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Correlation vs causation:  It is often the case that “corre-  lation is not causation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' The current literature in vision do-  main predominantly sees the explanation problem as com-  puting correlation between the input features and the model  prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' For instance, the saliency maps - see § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='2 - sim-  ply highlight the image regions having high relevance to the  model prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Better correlation is also the criterion to  improve the model fidelity scores - see § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Though useful,  correlation alone is insufficient to truly explain the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  On one hand, it is imperative to develop techniques that ac-  count for causation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' on the other, we need evaluation metrics  to measure the causal capacity of the existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' We  anticipate these methods to be more sophisticated and evalua-  tion metrics to be more comprehensive than the existing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Inspirational domain mismatch:  Visual data has its pecu-  liar nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Incidentally, many techniques for explaining the  visual models are inspired by other domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' For instance,  path-based methods (in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='3) and axiomatic properties evalu-  ation (in § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='4) have game-theoretic roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Similarly, Shapely  values [Shapley, 1997] are Economy related concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' The  inherent mismatch between the vision domain and the inspir-  ing domains can lead to problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' However, this aspect is  often overlooked in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' For example, it is very hard  to define ‘absence’ of a feature in an image to correctly iden-  tify the ‘reference’ for that image in path based methods - see  § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' This is not an issue in other game-theory applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Interestingly, we also witnessed that the eventual attribution  maps computed by the existing path based methods seldom  exhibit the acclaimed axiomatic properties in a strict sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  The future explorations must address such domain mismatch  problems diligently to make the explanations reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Performance vs transparency dilemma:  Indeed, we find  a trend that more human-understandable models have lower  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Models not accounting for explainability are  currently the top performers in the vision domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' This ob-  servation is likely a major contributor to the common belief  that explainability and performance have an inverse relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  However, we do not find rigorous studies in the current lit-  erature that analytically (or more systematically) strengthen  or refute this idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' A few leading vision research labs even  believe explainability to be an unnecessary trait in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  However, regulatory authorities concerning the development  and deployment of AI have a consensus that the systems need  to be transparent/explainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Resolving this conflicting situ-  ation demands a comprehensive investigation of the ‘perfor-  mance vs transparency’ dilemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' We can anticipate some  research activity addressing this in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  The (in)consistency of methods and metrics:  Due to the  absence of concrete ground truth for the explainability task in  general, the methods are hard to evaluate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' The issue is further  riled by the inconsistency of the available evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  For instance, statistical unreliability and inconsistency of a  few such metrics is already proven in [Tomsett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Implying, comparative rankings of the saliency methods is  untrustworthy under these metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' At the same time, expla-  nation methods themselves are known to sometimes fail basic  sanity checks [Adebayo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' This dicephalous nature  of the problem demands further research that fully focuses on  each sub-problem individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Instead of making minor con-  tributions of evaluation metrics along proposing new expla-  nation methods (a common trend in the literature), thorough  works that fully focus on devising the evaluation metrics and  schemes are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' There is also an obvious need to invent  techniques to provide ground truth for evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  References  [Adebayo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2018] Julius  Adebayo,  Justin  Gilmer,  Michael Muelly, Ian Goodfellow, Moritz Hardt, and Been  Kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Sanity checks for saliency maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' NeurIPS, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2021] Rishabh  Agarwal,  Levi  Melnick,  Nicholas Frosst, Ben Lengerich, Rich Caruana, and Ge-  offrey Hinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Neural additive models: Interpretable ma-  chine learning with neural nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' NeurIPS, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022] Chirag Agarwal, Nari Johnson, Mar-  tin Pawelczyk, Eshika Saxena, Marinka Zitnik, and  Himabindu Lakkaraju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Rethinking stability for attribution-  based explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='06877, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Akhtar and Jalwana, 2023] Naveed Akhtar and Moham-  mad Jalwana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Rethinking interpretation: Input-agnostic  saliency mapping of deep visual classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In AAAI, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Ali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022] Ameen Ali,  Thomas Schnake,  Oliver  Eberle, Gr´egoire Montavon, Klaus-Robert M¨uller, and  Lior Wolf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Xai for transformers:  better explanations  through conservative propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' ICML, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Alvarez Melis and Jaakkola, 2018] David  Alvarez  Melis  and Tommi Jaakkola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Towards robust interpretability with  self-explaining neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' NeurIPS, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Ancona et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2017] Marco Ancona, Enea Ceolini, Cengiz  ¨Oztireli, and Markus Gross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Towards better understand-  ing of gradient-based attribution methods for deep neural  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' arXiv preprint arXiv:1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='06104, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Antognini and Faltings, 2021] Diego  Antognini  and  Boi  Faltings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Rationalization  through  concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='04837, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Arras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022] Leila Arras, Ahmed Osman, and Woj-  ciech Samek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Clevr-xai: a benchmark dataset for the  ground truth evaluation of neural network explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Information Fusion, 81:14–40, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Bach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2015] Sebastian  Bach,  Alexander  Binder,  Gr´egoire Montavon, Frederick Klauschen, Klaus-Robert  M¨uller, and Wojciech Samek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' On pixel-wise explanations  for non-linear classifier decisions by layer-wise relevance  propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' PloS one, 10(7):e0130140, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Bahadori and Heckerman, 2020] Mohammad Bahadori and  David Heckerman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Debiasing concept-based explanations  with causal analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='11500, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Bau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2017] David Bau, Bolei Zhou, Aditya Khosla,  Aude Oliva, and Antonio Torralba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Network dissection:  Quantifying interpretability of deep visual representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  In IEEE CVPR, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Bhatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2020] Umang  Bhatt,  Adrian  Weller,  and  Jos´e MF Moura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Evaluating and aggregating feature-based  model explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='00631, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Blazek and Lin, 2021] Paul Blazek and Milo Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Explain-  able neural networks that simulate reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Nature Com-  putational Science, 1(9):607–618, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Chalasani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2020] Prasad Chalasani,  Jiefeng Chen,  Xi Wu, and Somesh Jha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Concise explanations of neural  networks using adversarial training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In ICML, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2021] Chun-Hao Chang, Rich Caruana, and  Anna Goldenberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Neural generalized additive model for  interpretable deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='01613, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Chattopadhay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2018] Aditya Chattopadhay, Anirban  Sarkar, Prantik Howlader, and Vineeth Balasubramanian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Grad-cam++: Generalized gradient-based visual explana-  tions for deep convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In WACV, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Chauhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022] Kushal Chauhan, Rishabh Tiwari, Jan  Freyberg, and Pradeep Shenoy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Interactive concept bottle-  neck models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='07430, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2019] Chaofan Chen, Oscar Li, Daniel Tao,  Alina Barnett, Cynthia Rudin, and Jonathan K Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' This  looks like that:  deep learning for interpretable image  recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' NeurIPS, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2020] Zhi Chen, Yijie Bei, and Cynthia Rudin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Concept whitening for interpretable image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Nature Machine Intelligence, 2(12):772–782, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Dasgupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022] Sanjoy Dasgupta, Nave Frost, and  Michal Moshkovitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Framework for evaluating faithful-  ness of local explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='00734, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Dhurandhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022] Amit  Dhurandhar,  Karthikeyan  Ramamurthy, and Karthikeyan Shanmugam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Is this the  right neighborhood?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' accurate and query efficient model  agnostic explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In NeurIPS, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Dubey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022] Abhimanyu Dubey, Filip Radenovic,  and Dhruv Mahajan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Scalable interpretability via polyno-  mials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' NeurIPS, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Erion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2021] Gabriel Erion, Joseph D Janizek, Pascal  Sturmfels, Scott M Lundberg, and Su-In Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Improving  performance of deep learning models with axiomatic attri-  bution priors and expected gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Nature MI, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Fong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2019] Ruth Fong, Mandela Patrick, and Andrea  Vedaldi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Understanding deep networks via extremal per-  turbations and smooth masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In ICCV, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Ghorbani and Zou, 2020] Amirata Ghorbani and James Y  Zou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Neuron shapley: Discovering the responsible neu-  rons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' NeurIPS, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Hastie, 2017] Trevor Hastie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Generalized additive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  In Statistical models, pages 249–307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Routledge, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Hendricks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2016] Lisa  Hendricks,  Zeynep  Akata,  Marcus Rohrbach, Jeff Donahue, and Trevor Darrell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Gen-  erating visual explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In ECCV, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Hooker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2019] Sara Hooker, Dumitru Erhan, Pieter  Kindermans, and Been Kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  A benchmark for inter-  pretability methods in deep neural nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' NeurIPS, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Jalwana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2021] Mohammad Jalwana, Naveed Akhtar,  Mohammed Bennamoun, and Ajmal Mian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Cameras: En-  hanced resolution and sanity preserving class activation  mapping for image saliency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In IEEE CVPR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Khakzar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022] Ashkan Khakzar, Pedram Khorsandi,  Rozhin Nobahari, and Nassir Navab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Do explanations ex-  plain?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' model knows best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In IEEE CVPR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2018] Been Kim, Martin Wattenberg, Justin  Gilmer, Carrie Cai, James Wexler, Fernanda Viegas, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Interpretability beyond feature attribution: Quantitative  testing with concept activation vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In ICML, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022] Sangwon Kim, Jaeyeal Nam, and By-  oung Chul Ko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Vit-net: Interpretable vision transformers  with neural tree decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In ICML, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Kindermans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2019] Pieter-Jan  Kindermans,  Sara  Hooker, Julius Adebayo, Maximilian Alber, Kristof T  Sch¨utt, Sven D¨ahne, Dumitru Erhan, and Been Kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' The  (un) reliability of saliency methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In Explainable AI:  IEVDL, pages 267–280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Springer, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Koh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2020] Pang Wei Koh, Thao Nguyen, Yew Siang  Tang, Stephen Mussmann, Emma Pierson, Been Kim, and  Percy Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Concept bottleneck models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In ICML, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Kohlbrenner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2020] Maximilian  Kohlbrenner,  Shinichi Bauer, Alexander Binder, and Sebastian La-  puschkin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Towards best practice in explaining neural  network decisions with lrp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In IJCNN, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Kwon and Zou, 2022] Yongchan Kwon and James Zou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  WeightedSHAP: analyzing and improving shapley based  feature attributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='13429, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Lang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2021] Oran Lang, Yossi Gandelsman, Michal  Yarom, Yoav Wald, Gal Elidan, Avinatan Hassidim,  William T Freeman, Phillip Isola, Amir Globerson, Michal  Irani, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Explaining in style: Training a gan to explain  a classifier in stylespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In ICCV, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2021] Liangzhi Li, Bowen Wang, Manisha Verma,  Yuta Nakashima, Ryo Kawasaki, and Hajime Nagahara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Scouter: Slot attention-based classifier for explainable im-  age recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In ICCV, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Lundberg and Lee, 2017] Scott Lundberg and Su-Im Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  A unified approach to interpreting model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  NeurIPS, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Lundstrom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022] Daniel  D  Lundstrom,  Tianjian  Huang, and Meisam Razaviyayn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' A rigorous study of inte-  grated gradients method and extensions to internal neuron  attributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In ICML, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Montavon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2018] Gr´egoire  Montavon,  Wojciech  Samek, and Klaus-Robert M¨uller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Methods for inter-  preting and understanding deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Digital  signal processing, 73:1–15, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Mu and Andreas, 2020] Jesse Mu and Jacob Andreas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Com-  positional explanations of neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' NeurIPS, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Nguyen and Mart´ınez, 2020] An-phi  Nguyen  and  Mar´ıa Rodr´ıguez Mart´ınez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  On quantitative aspects  of model interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='07584, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Omeiza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2019] Daniel Omeiza,  Skyler Speakman,  Celia Cintas, and Komminist Weldermariam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' An enhanced  inference level visualization technique for deep convolu-  tional neural network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' arXiv:1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='01224, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Oquab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2015] Maxime Oquab, L´eon Bottou, Ivan  Laptev, and Josef Sivic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Is object localization for free?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='-  weakly-supervised learning with convolutional neural net-  works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In IEEE CVPR, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2018] Dong Park, Lisa Anne Hendricks, Zeynep  Akata, Anna Rohrbach, Bernt Schiele, Trevor Darrell, and  Marcus Rohrbach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Multimodal explanations: Justifying  decisions and pointing to the evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In CVPR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Petsiuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2018] Vitali Petsiuk, Abir Das, and Kate  Saenko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Rise: Randomized input sampling for explana-  tion of black-box models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' arXiv:1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='07421, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022] Sukrut Rao, Moritz B¨ohle, and Bernt  Schiele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Towards better understanding attribution meth-  ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In IEEE CVPR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Ribeiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2016] Marco Tulio Ribeiro, Sameer Singh,  and Carlos Guestrin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' ” why should i trust you?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' explaining  the predictions of any classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In ACM SIGKDD, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Rong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022] Yao Rong,  Tobias Leemann,  Vadim  Borisov, Gjergji Kasneci, and Enkelejda Kasneci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  A  consistent and efficient evaluation strategy for attribution  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In ICML, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Rudin, 2019] Cynthia Rudin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Stop explaining black box  machine learning models for high stakes decisions and use  interpretable models instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' NMI, 1(5):206–215, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Samek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2016] Wojciech Samek, Alexander Binder,  Gr´egoire Montavon, and Klaus-Robert M¨uller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Evaluat-  ing the visualization of what a deep neural network has  learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' IEEE TNNLS, 28(11):2660–2673, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Sarkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022] Anirban Sarkar, Deepak Vijaykeerthy,  Anindya Sarkar, and Vineeth N Balasubramanian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  A  framework for learning ante-hoc explainable models via  concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In IEEE CVPR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Selvaraju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2017] Ramprasaath  Selvaraju,  Michael  Cogswell, Abhishek Das, Devi Parikh, and Dhruv Batra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Grad-cam: Visual explanations from deep networks via  gradient-based localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In ICCV, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Shapley, 1997] Lloyd S Shapley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  A value for n-person  games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Classics in game theory, 69, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Shrikumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2017] Avanti Shrikumar, Peyton Green-  side, and Anshul Kundaje.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Learning important features  through propagating activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In ICML, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Simonyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2014] Karen Simonyan, Andrea Vedaldi,  and Andrew Zisserman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Deep inside convolutional  networks:  Visualising image classification models and  saliency maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' ICLR, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Sixt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2020] Leon Sixt, Maximilian Granz, and Tim  Landgraf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' When explanations lie: Why many modified  bp attributions fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In ICML, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Smilkov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2017] Daniel Smilkov, Nikhil Thorat, Been  Kim, and Martin Wattenberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Smoothgrad: removing  noise by adding noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' arXiv:1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='03825, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Springenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2015] Jost Springenberg, Alexey Doso-  vitskiy, Thomas Brox, and Martin Riedmiller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Striving for  simplicity: The all convolutional net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' ICLR, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Srinivas and Fleuret, 2019] Suraj  Srinivas  and  Franc¸ois  Fleuret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Full-gradient representation for neural network  visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' NeurIPS, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Stalder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022] Steven Stalder, Nathana¨el Perraudin,  Radhakrishna  Achanta,  Fernando  Perez-Cruz,  and  Michele Volpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' What you see is what you classify: Black  box attributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' NeurIPS, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Sturmfels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2020] Pascal Sturmfels, Scott Lundberg,  and Su-In Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Visualizing the impact of feature attribu-  tion baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Distill, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Sundararajan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2017] Mukund Sundararajan,  Ankur  Taly, and Qiqi Yan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Axiomatic attribution for deep net-  works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In ICML, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Tomsett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2020] Richard  Tomsett,  Dan  Harborne,  Supriyo Chakraborty, Prudhvi Gurram, and Alun Preece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  Sanity checks for saliency metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In AAAI, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2020] Haofan Wang, Zifan Wang, Mengnan  Du, Fan Yang, Zijian Zhang, Sirui Ding, Piotr Mardziel,  and Xia Hu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Score-weighted visual explanations for con-  volutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In IEEE CVPRw, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2021] Jiaqi Wang, Huafeng Liu, Xinyue Wang,  and Liping Jing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Interpretable image recognition by con-  structing transparent embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In ICCV, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2022] Andong Wang, Wei-Ning Lee, and Xi-  aojuan Qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Hint: Hierarchical neuron concept explainer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  In IEEE CVPR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2023] Peiyu Yang, Naveed Akhtar, Zeyi Wen,  and Ajmal Mian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Local path integration for attribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In  AAAI Conference on Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' AAAI, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Yeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2019] Chih-Kuan Yeh, Cheng-Yu Hsieh, Arun  Suggala, David Inouye, and Pradeep Ravikumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' On the  (in)fidelity and sensitivity of explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' NeurIPS, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2018] Jianming Zhang, Sarah Adel Bargal,  Zhe Lin, Jonathan Brandt, Xiaohui Shen, and Stan  Sclaroff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Top-down neural attention by excitation back-  prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' IJCV, 126(10):1084–1102, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content='  [Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=', 2016] Bolei Zhou,  Agata Lapedriza,  Aude  Oliva, and Antonio Torralba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' Learning deep features for  discriminative localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
+page_content=' In IEEE CVPR, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFQT4oBgHgl3EQf8Dba/content/2301.13445v1.pdf'}
diff --git a/aNE2T4oBgHgl3EQfEwbs/content/tmp_files/2301.03640v1.pdf.txt b/aNE2T4oBgHgl3EQfEwbs/content/tmp_files/2301.03640v1.pdf.txt
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+Precision predictions for exotic lepton production at the Large Hadron Collider
+A. H. Ajjath,1, ∗ Benjamin Fuks,1, † Hua-Sheng Shao,1, ‡ and Yehudi Simon1, §
+1Sorbonne Universit´e, CNRS, Laboratoire de Physique Th´eorique et Hautes ´Energies, LPTHE, F-75005 Paris, France
+We calculate total and differential cross sections for the pair production, at the Large Hadron
+Collider, of exotic leptons that could emerge from models with vector-like leptons and in Type-III see-
+saw scenarios. Our predictions include next-to-leading-order QCD corrections, and we subsequently
+match them with either parton showers, or threshold resummation at the next-to-next-to-leading
+logarithmic accuracy. Our results show an important increase of the cross sections relative to the
+leading-order predictions, exhibit a distortion of the shapes for various differential distributions, and
+feature a significant reduction of the scale uncertainties. Our predictions have been obtained from
+new FeynRules model implementations and associated UFO model libraries. This completes the
+set of next-to-leading-order implementations of new physics models featuring extra leptons that are
+publicly available on the FeynRules model database.
+I.
+INTRODUCTION
+Several extensions of the Standard Model (SM) predict
+the existence of new exotic heavy leptons. These arise in
+particular in composite models [1–3], grand unified the-
+ories [4, 5], supersymmetric models [6–8], left-right sym-
+metric models [9–12], dark matter scenarios [13] or in
+Type I [14–21] and Type-III [21, 22] seesaw models. In
+composite scenarios, the new physics particle spectrum
+features vector-like leptons (VLLs) transforming as elec-
+troweak SU(2)L singlets or doublets. In contrast, in see-
+saw and left-right models, neutrino masses are generated
+from Yukawa interactions of new electroweak singlets or
+triplets of fermions with the SM Higgs field and lepton
+weak doublet. Consequently, searches for new heavy lep-
+tons consist of an important component of the experi-
+mental beyond the SM (BSM) search programme at the
+Large Hadron Collider (LHC).
+The ATLAS and CMS collaborations have explored
+the associated parameter spaces, both for promptly-
+decaying [23–38] and long-lived [25, 39] extra leptons,
+and for a variety of mass ranges. Limits have been set on
+the mixing properties of long-lived heavy charge-neutral
+leptons for masses of 3–15 GeV, while short-lived neutral
+and charged leptons must have masses of at least about
+950 GeV in Type-III seesaw models (for branching ratios
+of 1 in the final state considered). Moreover, in left-right
+models neutral lepton masses must be larger than 3 TeV
+for WR boson masses smaller than 4–5 TeV, and indirect
+probes for heavy neutrinos via vector-boson-fusion pro-
+cesses additionally constrain masses ranging up to 20 TeV
+for large mixings with the SM leptons and neutrinos. Fi-
+nally, VLLs are restricted to be heavier than about 800
+GeV or lighter than about 100 GeV.
+From the theoretical side, total and differential cross
+section calculations for collider processes involving ex-
+tra neutrinos are known at next-to-leading order (NLO)
+∗ aabdulhameed@lpthe.jussieu.fr
+† fuks@lpthe.jussieu.fr
+‡ huasheng.shao@lpthe.jussieu.fr
+§ ysimon@lpthe.jussieu.fr
+in QCD in a generic simplified model describing the dy-
+namics of the heavy neutrinos [40, 41], and for an ef-
+fective left-right-symmetric scenario [42].
+In addition,
+NLO-QCD predictions matched with threshold resum-
+mation at the next-to-leading-logarithmic (NLL) accu-
+racy [43–45] and approximate next-to-next-to-leading-
+order cross section matched with next-to-next-to-leading-
+logarithmic (NNLL) threshold resummation [46] can also
+be obtained from electroweakino pair-production pro-
+cesses in the Minimal Supersymmetric Standard Model,
+after decoupling all supersymmetric states but the pro-
+duced electroweakinos.
+Beside differential and total
+rates, precision collider simulations in which NLO calcu-
+lations are matched with parton showers (PS) are avail-
+able for heavy neutrino simplified models [40–42]. How-
+ever, simulations for Type-III seesaw scenarios and com-
+posite VLL setups are only available at leading order
+(LO) so far.
+The goal of this paper is to fill this gap, and to re-
+port about the development of two new publicly available
+UFO [47] model libraries allowing for event generation
+at the NLO+PS accuracy. Our implementation are suit-
+able in particular to describe VLL and Type-III seesaw
+lepton production processes from computations achieved
+by means of the precision Monte Carlo event generator
+MadGraph5 aMC@NLO [48, 49] (MG5aMC). More-
+over, we take the opportunity to update predictions for
+the total rates and the invariant mass distributions of
+the BSM Drell-Yan-like processes inherent to the models
+considered, and present results at NLO in QCD matched
+with threshold resummation at NNLL, following the for-
+malism of [50–52].
+The rest of this paper is organised as follows. In sec-
+tion II, we introduce two effective theoretical frameworks
+suitable for our calculations, a first one dedicated to
+a simplified VLL model and a second one to Type-III
+seesaw scenarios, and we report details about their im-
+plementation.In section III, we briefly describe the for-
+malism that we use to resum the large threshold loga-
+rithms. In section IV, we make use of the MG5aMC
+platform to study the phenomenology of the two mod-
+els at the NLO+PS accuracy, as well as of an in-house
+arXiv:2301.03640v1  [hep-ph]  9 Jan 2023
+
+2
+programme to handle predictions at NLO+NNLL in the
+strong coupling. We compute total rates for the produc-
+tion of extra leptons (section IV.1), invariant-mass dis-
+tributions (section IV.2), and we additionally present for
+the first time NLO+PS-accurate differential distributions
+relevant for experimental searches for VLLs and Type-III
+seesaw fermions (section IV.3). We summarise our work
+and conclude in section V.
+II.
+THEORETICAL MODELS AND THE
+IMPLEMENTATIONS
+In order to study the phenomenology of the considered
+models, we construct two effective frameworks, one for
+each of the models.
+We minimally extend the SM in
+terms of fields and interactions, so that the resulting new
+physics parameter spaces are of small dimensionality.
+II.1.
+A simplified model for VLL phenomenology
+We begin by considering an extension of the SM in
+which the theory field content includes a set of VLL
+fields. They are all colour singlet, but each of them lies
+in a different SU(2)L representation.
+They are corre-
+spondingly assigned different hypercharge U(1)Y quan-
+tum numbers. In order to be as model independent as
+possible, we adopt a simplified model approach and fo-
+cus on extra leptons that are either electrically neutral
+or with an electric charge Q = ±1. These leptons are
+organised in (vector-like) SU(2)L doublets and singlets,
+L0 =
+�
+�N 0
+E0
+�
+� ,
+˜E0 ,
+˜N 0 ,
+(1)
+where in this notation the superscript ‘0’ indicates that
+the fields are gauge eigenstates, and the tilde above a field
+indicates that it is an SU(2)L singlet. We next implement
+the mixing between the SM and the new lepton fields.
+To this aim, we introduce an effective parametrisation
+apt to capture the main phenomenological features of the
+vector-like fields in a model-independent way, following
+guidelines introduced for vector-like quark setups [53, 54].
+In practice, we assume that the mixing between the
+SM fields and the new leptons is small, so that the gauge
+interactions of the different fields are unaffected at the
+first order. Moreover, we implement the off-diagonal in-
+teractions of the exotic leptons with the SM ones through
+generic free parameters, which further open the VLL de-
+cay channels into a SM lepton and an electroweak boson.
+The corresponding Lagrangian, given in terms of mass
+eigenstates (the superscript ‘0’ being therefore dropped),
+Field
+Spin
+Representation
+Name
+L0
+(1/2, 1/2)
+(1, 2)−1/2
+VLL0
+˜
+N 0
+(1/2, 1/2)
+(1, 1)0
+VLN0
+˜E0
+(1/2, 1/2)
+(1, 1)−1
+VLE0
+TABLE I. Gauge eigenstates complementing the SM field
+content, their spin given as their representation under the
+SO(1, 3) group (second column), their SU(3)c × SU(2)L ×
+U(1)Y (third column) representation and their name in the
+FeynRules implementation (last column).
+reads
+LVLL = LSM + i¯L /DL − mN ¯NN − mE ¯EE
++ i ¯˜N /∂ ˜N − m ˜
+N ¯˜N ˜N + i ¯˜E /D ˜E − m ˜
+E ¯˜E ˜E
++
+�
+Ψ=E, ˜
+E
+�
+h¯Ψ
+�
+ˆκ
+Ψ
+LPL + ˆκ
+Ψ
+RPR
+�
+ℓ + g
+√
+2
+¯Ψ /W
+−κ
+Ψ
+LPLνℓ
++
+g
+2cW
+¯Ψ/Z
+�
+˜κ
+Ψ
+LPL + ˜κ
+Ψ
+RPR
+�
+ℓ + H.c.
+�
++
+�
+Ψ=N, ˜
+N
+�
+h¯Ψˆκ
+Ψ
+LPLνℓ +
+g
+2cW
+¯Ψ/Z˜κ
+Ψ
+LPLνℓ
++ g
+√
+2
+¯Ψ /W
++�
+κ
+Ψ
+LPL + κ
+Ψ
+RPR
+�
+ℓ + H.c.
+�
+,
+(2)
+where LSM is the SM Lagrangian, and the parameters
+mN, mE, m ˜
+N and m ˜
+E stand for the masses of the four
+new fields in the physical basis, assuming that the masses
+of the doublet component fields can be different after
+electroweak symmetry breaking and particle mixing. The
+first two lines in this Lagrangian include, additionally to
+the SM Lagrangian, all gauge-invariant kinetic and mass
+terms for the new states. The gauge-covariant derivative
+operator Dµ is defined, for a generic field ψ, by
+Dµψ = ∂µψ − ig′BµY ψ − igW k
+µT kψ .
+(3)
+Here, the coupling constants g and g′ respectively stand
+for the weak SU(2)L and hypercharge U(1)Y coupling
+constants, and Bµ and W k
+µ are the associated gauge
+fields. The action of the hypercharge operator Y on the
+field ψ can be deduced from table I, as the representa-
+tion to adopt for the SU(2) generators T k. In particular,
+T k = 0 for an SU(2)L singlet, and T k = σk/2 for an
+SU(2)L doublet, with σk being the Pauli matrices. In
+other words, we approximate mass eigenstates by gauge
+eigenstates in all kinetic terms, i.e. L0 ≈ L, ˜E0 ≈ ˜E and
+˜N 0 ≈ ˜N.
+The last four lines of the Lagrangian (2) collect the
+effective interactions of each of the four VLLs considered
+with a SM lepton (ℓ standing for the charged lepton field
+and νℓ for the neutrino one), and either the Higgs boson
+h, the W boson or the Z boson. In (2), all flavour indices
+are understood so that each of the κ, ˆκ and ˜κ couplings
+has to be seen as a vector in the flavour space. Moreover,
+cW refers to the cosine of the electroweak mixing angle.
+
+3
+Field
+Spin
+Name
+PDG
+Mass
+Width
+N
+(1/2, 1/2)
+VLLN
+9000001
+MVLLN
+WVLLN
+E
+(1/2, 1/2)
+VLLE
+9000002
+MVLLE
+WVLLE
+˜
+N
+(1/2, 1/2)
+VLN
+9000003
+MVLN
+WVLN
+˜E
+(1/2, 1/2)
+VLE
+9000004
+MVLE
+WVLE
+TABLE II. Mass eigenstates supplementing the SM, with
+their spin quantum number (second column), name used in
+the FeynRules convention (third column) and adopted PDG
+identifier (fourth column). In the last two columns, we pro-
+vide the FeynRules symbols associated with the particle
+masses and widths.
+In the following, the exact values of the κ, ˆκ and ˜κ cou-
+pling vectors are irrelevant, provided that they are not
+too large to guarantee that the VLL states have a narrow
+width, and not too small so that they can promptly de-
+cay into a lepton+electroweak boson system within LHC
+detector scales. In a hadron collision process in which
+the new leptons are pair produced, such couplings indeed
+only appear in the heavy particle decays.
+In order to allow for phenomenological studies of
+the model, we implement it in the FeynRules pack-
+age [55, 56], starting from the SM implementation that
+is shipped with the programme.
+We include the def-
+initions of the gauge eigenstates of (1), together with
+the corresponding mass eigenstates appearing in the La-
+grangian (2). Information on these fields, their names
+in the FeynRules conventions, and the Particle Data
+Group (PDG) identifiers that we have adopted for the
+physical fields, are provided in tables I and II. These ta-
+bles also include the symbols associated with the mass
+and width of the physical fields. All BSM couplings ap-
+pearing in (2) have been implemented as three-vectors in
+the flavour space, following the convention of table III.
+This table also includes information on the Les Houches
+block structure used to organise all model external pa-
+rameters [57], as required by all high-energy physics pro-
+grammes relying on FeynRules for model implementa-
+tion. Moreover, their specific contributions to any pro-
+cess can be turned off through a dedicated interaction
+order named VLL (see the FeynRules manual [56]).
+II.2.
+An effective Type-III seesaw Lagrangian
+In Type-III seesaw models, neutrino masses are gen-
+erated through the interactions of the SM Higgs field Φ
+with the SM leptons and at least two generations of ex-
+tra fermions lying in the adjoint representation of SU(2)L
+and with zero hypercharge. In the following, we make use
+of two-component Weyl fermion notation for all fields,
+and omit all SM and BSM generation indices for clarity.
+In such a formalism, the Lagrangian of the model is ex-
+pressed in terms of the SM weak doublet of left-handed
+leptons LL, the SM weak singlet of right-handed charged
+leptons Ec
+R (ER being thus the corresponding left-handed
+Couplings
+Names
+Les Houches blocks
+(ˆκE
+L)i, (ˆκE
+R)i
+KLLEH[i], KRLEH[i]
+KLLEH, KRLEH
+(ˆκ
+˜
+E
+L)i, (ˆκ
+˜
+E
+R)i
+KLEH[i], KREH[i]
+KLEH, KREH
+(ˆκN
+L )i
+KLLNH[i]
+KLLNH
+(ˆκ
+˜
+N
+L )i
+KLNH[i]
+KLNH
+(˜κE
+L)i, (˜κE
+R)i
+KLLEZ[i], KRLEZ[i]
+KLLEZ, KRLEZ
+(˜κ
+˜
+E
+L)i, (˜κ
+˜
+E
+R)i
+KLEZ[i], KREZ[i]
+KLEZ, KREZ
+(˜κN
+L )i
+KLLNZ[i]
+KLLNZ
+(˜κ
+˜
+N
+L )i
+KLNZ[i]
+KLNZ
+(κE
+L)i
+KLLEW[i]
+KLLEW
+(κ
+˜
+E
+L)i
+KLEW[i]
+KLEW
+(κN
+L )i, (κN
+R)i
+KLLNW[i], KRLNW[i]
+KLLNW, KRLNW
+(κ
+˜
+N
+L )i, (κ
+˜
+N
+R)i
+KLNW[i], KRNW[i]
+KLNW, KRNW
+TABLE III. Three-point VLL coupling strengths to a SM lep-
+ton and an electroweak boson, given together with the associ-
+ated FeynRules symbol and the corresponding Les Houches
+block. The indice i denotes a generation index ranging from
+1 to 3.
+Weyl spinor), and the weak triplet of extra lepton Σk
+(with k = 1, 2, 3 being an SU(2)L adjoint index). The
+three gauge eigenstates Σk can be conveniently related
+to states of definite electric charge E± ≡ Σ± (of charge
+Q = ±1) and N ≡ Σ0 (of charge Q = 0) by introducing
+the matrix representation for the SU(2)L triplets Σij,
+with i and j referring to fundamental indices of SU(2)L.
+We obtain,
+Σi
+j =
+1
+√
+2(σk)i
+jΣk =
+� 1
+√
+2N
+E+
+E−
+− 1
+√
+2N
+�
+.
+(4)
+The Type-III Lagrangian is given by
+LTypeIII = LSM + Lkin
++
+�
+yℓ Φ†LL.ER + 2yΣ Φ·
+�
+ΣkT kLL
+�
++ H.c.
+�
+.
+(5)
+In our notation, LSM stands for the reduced SM La-
+grangian in which all terms involving a leptonic field have
+been removed, and Lkin collects all gauge-invariant ki-
+netic terms for the (two-component) leptonic fields LL,
+ER and Σ, and a mass term for the Σ field (of mass mΣ).
+Moreover, the matrices T k = σk/2 stand for the gener-
+ators of SU(2)L in the fundamental representation, and
+the explicit scalar product appearing on the second line
+refers to the SU(2)-invariant product of two fields lying
+in its fundamental representation. In Type-III models,
+charged lepton and neutrino masses are driven by the SM
+leptonic 3 × 3 Yukawa matrix yℓ, and the heavy neutrino
+Yukawa matrix yΣ whose size depends on the number of
+generations of new fermions.
+From the LHC physics point of view, we can simplify
+the model presented above by emphasising the focus on
+the lightest of all Σ states, that are assumed to be the
+
+4
+only ones to which the LHC would be sensitive.
+This
+strategy follows that introduced in ref. [58]. Under such
+an assumption, the relevant part of the Yukawa matrix yΣ
+(in the Σ-flavour × SM-flavour space) becomes a vector
+in the SM-flavour space, that we take real for simplicity.
+The mass eigenstates of the model hence include three
+generations of physical up-type and down-type quarks,
+and four generations of physical charged leptons (ℓ′) and
+Majorana neutrinos (ν′).
+After electroweak symmetry
+breaking, the Lagrangian (5) induces a mixing between
+the three SM leptons and the new states Σ, rendering at
+least two neutrinos massive. Introducing the three 4 × 4
+mixing matrices in the lepton flavour space U ℓ
+L, U ℓ
+R and
+U 0, lepton gauge and mass eigenstates are related by [59],
+�
+EL
+E−
+�
+= U ℓ
+L ℓ′
+L ,
+�
+ER
+E+c
+�
+= U ℓ
+R ℓ′
+R ,
+�
+νL
+N
+�
+= U ν ν′ ,
+(6)
+where the three-component vector in the flavour space
+EL (νL) stands for the down-type (up-type) component
+of the weak doublet of left-handed SM leptons. As the
+new fermion masses of O(mΣ) are expected to be heavy
+compared with the neutrino masses of O(yΣv), with v
+being the SM Higgs vacuum expectation value, the three
+mixing matrices can be expanded at the first order in
+yv/mΣ [60, 61],
+U ℓ
+L =
+�
+� 1 − ε
+v
+mΣ yΣ
+− v
+mΣ y†
+Σ 1 − ε′
+�
+� ,
+U ℓ
+R =
+�
+�
+1
+Mℓv
+m2
+Σ yΣ
+− Mℓv
+m2
+Σ y†
+Σ
+1
+�
+� ,
+U ν =
+�
+�
+�
+1 − 1
+2ε
+�
+UPMNS
+v
+√
+2mΣ yΣ
+−
+v
+√
+2mΣ y†
+Σ
+1 − 1
+2ε′
+�
+� .
+(7)
+These expressions depend on the quantities ε and ε′, that
+are 3 × 3 and scalar objects in the SM flavour space re-
+spectively,
+ε =
+v2
+2m2
+Σ
+yΣy†
+Σ
+and
+ε′ =
+v2
+2m2
+Σ
+y†
+ΣyΣ ,
+(8)
+as well as on the SM lepton mass matrix Mℓ (that is diag-
+onal in the flavour space) and on the unitary Pontecorvo-
+Maki-Nakagawa-Sakata (PMNS) matrix UPMNS.
+The
+latter can be defined from the neutrino oscillation pa-
+rameters, namely the neutrino mixing angles θij (with
+i, j = 1, 2, 3), the Dirac CP-violating phase ϕCP and the
+two Majorana CP-violating phases ϕ1 and ϕ2.
+We implement the Type-III model described above
+in the FeynRules package [55, 56] following the same
+method that has been used for the Type II seesaw im-
+plementation [62]. We begin with the implementation of
+the SM shipped with FeynRules, from which all lepton
+definitions and related Lagrangian terms have been mod-
+ified. In practice, we have modified all lepton and neu-
+trino definitions so that two-component left-handed Weyl
+fermions [63] could be instead used for the gauge eigen-
+states of the model (Ec
+R, LL). Next, we add definitions
+Field
+Spin
+Representation
+# generations
+Name
+LL
+(1/2, 0)
+(1, 2)−1/2
+3
+LLw
+ER
+(1/2, 0)
+(1, 1)1
+3
+ERw
+Σk
+(1/2, 0)
+(1, 3)0
+1
+Sigw
+TABLE IV. Gauge eigenstates associated with the leptonic
+sector of the Type-III seesaw model, their spin given as their
+representation under the SO(1, 3) group (second column),
+their SU(3)c ×SU(2)L ×U(1)Y representation (third column),
+and their name in the FeynRules implementation (last col-
+umn).
+Field
+Spin
+Name
+PDG
+Mass
+Width
+e−
+(1/2, 1/2)
+e
+11
+Me
+–
+µ−
+(1/2, 1/2)
+mu
+13
+MMu
+–
+τ −
+(1/2, 1/2)
+ta
+15
+MTA
+–
+E−
+(1/2, 1/2)
+SigM
+9000017
+MSigma
+WSigM
+ν1
+(1/2, 1/2)
+v1
+12
+Mv1
+–
+ν2
+(1/2, 1/2)
+v2
+14
+Mv2
+–
+ν3
+(1/2, 1/2)
+v3
+16
+Mv3
+–
+N
+(1/2, 1/2)
+Sig0
+9000018
+MSigma
+WSig0
+TABLE V. Mass eigenstates that either supplement the SM
+or whose definition is altered relatively to the SM, with their
+spin representation (second column), name used in the Feyn-
+Rules convention (third column) and adopted PDG identi-
+fier (fourth column). In the last two columns, we provide the
+FeynRules symbols associated with the particle masses and
+widths.
+for the fermionic triplets Σk, still using left-handed Weyl
+fermions, and incorporate the mixing relations (6) for
+the definition of the four physical charged lepton states
+and the four physical neutrino states in terms of all gauge
+eigenstates. Finally, we map the physical two-component
+fermions of the model into the corresponding Dirac fields
+(charged leptons) and Majorana fields (neutrinos). More
+information on all fields included in the model imple-
+mentation is given in tables IV and V (representation,
+names in the FeynRules conventions, PDG identifiers,
+symbols for masses and widths).
+The new physics parameters yΣ and mΣ appearing in
+the Lagrangian (5) are implemented in a standard way,
+together with the neutrino oscillation parameters dictat-
+ing the values of the PMNS matrix. Moreover, we assume
+a normal neutrino mass hierarchy and set the masses of
+the three lightest neutrinos mν1, mν2 and mν3 from the
+value of the smallest neutrino mass (mν1 in our case), and
+the neutrino squared mass differences ∆m2
+21 and ∆m2
+31,
+mν2 =
+�
+m2ν1 + ∆m2
+21 and mν3 =
+�
+m2ν1 + ∆m2
+31 . (9)
+More information on the free parameters of the lep-
+tonic/neutrino sector of the model is provided in table VI
+(FeynRules names and Les Houches block structure).
+
+5
+Parameter
+Name
+LH block
+LH counter
+(yΣ)e
+ySigma[1]
+YSIGMA
+1
+(yΣ)µ
+ySigma[2]
+YSIGMA
+2
+(yΣ)τ
+ySigma[3]
+YSIGMA
+3
+mΣ
+MSigma
+MASS
+9000017
+mν1
+Mv1
+MASS
+12
+∆m2
+21
+dmsq21
+MNU
+2
+∆m2
+31
+dmsq31
+MNU
+3
+θ12
+th12
+PMNS
+1
+θ23
+th23
+PMNS
+2
+θ13
+th13
+PMNS
+3
+ϕCP
+delCP
+PMNS
+4
+ϕ1
+PhiM1
+PMNS
+5
+ϕ2
+PhiM2
+PMNS
+6
+TABLE VI. External parameters defining the leptonic sec-
+tor of the Type-III seesaw model, including the neutrino pa-
+rameters in the context of a normal mass hierarchy (so that
+mν1 < mν2 < mν3). Each parameter is given together with
+the symbol used in the FeynRules implementation, and the
+corresponding Les Houches (LH) block and counter informa-
+tion.
+II.3.
+From Lagrangian to events at the LHC
+In order to handle LHC simulations at NLO-QCD
+matched with PS, we make use of the two FyenRules
+model implementations detailed in secitons II.1 and II.2,
+and jointly use them with the MoGRe package (ver-
+sion 1.1) [64], NloCt (version 1.0.1) [65] and Fey-
+nArts (version 3.9) [66]. This allows us to renormalise
+the bare Lagrangians (2) and (5) relatively to O(αs) QCD
+interactions, and generate UFO model files [47] includ-
+ing both tree-level interactions, UV counterterms, and
+the so-called R2 Feynman rules required for the numer-
+ical evaluation of the numerators of one-loop integrands
+in a four-dimensional spacetime. Such UFO models can
+subsequently be used with MG5aMC [48, 49] for LO and
+NLO calculations in QCD, as well as by Herwig++ [67]
+and Sherpa [68] at LO.
+Before closing this subsection, we provide informa-
+tion on the MG5aMC framework [48, 49] which employ
+to carry out fixed-order (N)LO and (N)LO+PS calcula-
+tions. MG5aMC handles infrared singularities inherent
+to NLO calculations via the FKS method [69, 70], in an
+automated way through the MadFKS module [71, 72].
+The evaluation of UV-renormalised one-loop amplitudes
+is achieved by switching dynamically between several
+integral-reduction techniques that work either at the in-
+tegrand level (like the OPP method [73] or Laurent-
+series expansion [74]) or through tensor-integral reduc-
+tion [75–77]. This has been automated in the MadLoop
+module [48, 78], that exploits the public codes Cut-
+Tools [79], Ninja [80, 81] and Collier [82]. Moreover,
+one-loop computations have been optimised at the inte-
+grand level through an in-house procedure inspired by
+the idea of OpenLoops [83]. Finally, NLO+PS predic-
+tions are obtained by matching fixed-order calculations
+with PS according to the MC@NLO method [84].
+III.
+IMPROVING THEORY ACCURACY
+BEYOND NLO: THRESHOLD RESUMMATION
+For the Drell-Yan-like processes considered, it is well-
+known that large logarithms spoil the convergence of
+the perturbative series when the invariant mass M of
+the final-state system approaches the hadronic centre-of-
+mass energy √s. This calls for a proper resummation
+of soft-gluon radiation, or at least a matching with PS
+as achieved in the MG5aMC framework.
+In this sec-
+tion, we briefly describe, for the convenience of readers,
+the theoretical formalism that we adopt for fixed-order
+calculations matched with threshold resummation. Ad-
+ditional details and an extensive description can be found
+in section 2 of [85].
+At the partonic level, the corresponding kinematic re-
+gion is defined in terms of the partonic scaling variable
+z = M 2/ˆs when z → 1, with
+√
+ˆs being the partonic
+centre-of-mass energy. In this limit, the perturbative co-
+efficients in the cross sections get contributions in the
+form of αs ln(1 − z) originating from soft gluon emission,
+which could be of O(1) and thus potentially spoil the
+perturbative convergence of the usual series in αs. This
+issue can be resolved by reorganising the perturbative
+expansion in an alternative manner, and resumming the
+large logarithms in αs ln(1 − z) to all orders in αs.
+Resummation calculations are conveniently carried out
+in the Mellin N-space conjugate to z, where the z → 1
+limit thus corresponds to the large N region. The re-
+summed partonic cross section in the Mellin space, de-
+noted by ∆res
+q¯q (N, M 2, µ2
+F ) with µF being the factorisa-
+tion scale, is defined in eq. (2.47) from [85] for a generic
+process with a colourless final state. For a Drell-Yan-like
+process and at the kth logarithmic accuracy (NkLL), it
+reads
+∆res
+q¯q (N, M 2, µ2
+F )
+���
+NkLL = ˜g0,q¯q(M 2, µ2
+F , µ2
+R)
+���
+NkLO
+× exp
+�
+g1,q¯q(ω) ln N +
+k+1
+�
+j=2
+aj−2
+s
+(µ2
+R)gj,q¯q(ω)
+�
+.
+(10)
+In this expression, the ˜g0,q¯q|NkLO factor collects the N-
+independent terms of the first k + 1 coefficients of the
+usual as perturbative expansion (in Mellin space), where
+we have introduced the short-hand notation as(µ2
+R) =
+αs(µ2
+R)/4π with µR denoting the renormalisation scale.
+This factor is process dependent, and it gets contribu-
+tions from virtual corrections and soft real emission. In
+the exponent, the process-independent (i.e. universal) co-
+efficients gj,q¯q(ω) with j > 0 receive contributions from
+the threshold logarithmic terms originating from real
+
+6
+emission.
+Given ω = 2β0as(µ2
+R) ln N ∼ O(1) with β0
+being the first coefficient of the QCD beta function, they
+effectively resum these logarithmic contributions to all
+orders in αs. We refer to appendix A for the analytical
+form of the various coefficients appearing in (10) and that
+are relevant for NLO+NNLL calculations for the Drell-
+Yan-like processes considered.
+As the separation of the N-independent and N-
+dependent pieces in (10) is not unambiguous, different
+resummation schemes have been proposed and described
+in section 2.4 of [85]. In the present paper, we consider
+the so-called N1 resummation scheme for simplicity.
+Resummed calculations must then have to be matched
+with fixed-order predictions at (N)LO. This is achieved
+by adding the resummed and (N)LO results and sub-
+tracting of all double-counted contributions. The latter
+correspond to the (N)LO soft-virtual terms of the par-
+tonic cross section, which can be obtained by expand-
+ing ∆res
+q¯q (N, M 2, µ2
+F ) at O(αb(+1)
+s
+), where b stands for the
+power in αs of the LO contributions (that is 0 here).
+IV.
+CROSS SECTIONS FOR EXTRA LEPTON
+PRODUCTION AT THE LHC
+In this section, we compute total and differential cross
+sections relevant for the production of additional leptons
+such as those appearing in the models introduced in sec-
+tion II. Predictions at (N)LO and (N)LO+PS are ob-
+tained within the MG5aMC framework (version 3.3.0),
+using the UFO models developed in this work, and we
+employ Pythia 8.2 [86] to deal with the simulation of
+the QCD environement (parton showering and hadronisa-
+tion). On the other hand, total rate calculations match-
+ing fixed-order predictions with soft-gluon resummation
+are derived with an in-house code.
+We define the electroweak sector through three inde-
+pendent input parameters that we choose to be the Z-
+boson mass mZ, the electromagnetic coupling constant
+evaluated at the Z-pole α(mZ), and the Fermi constant
+GF ,
+mZ = 91.1876 GeV ,
+α−1(mZ) = 127.9 ,
+GF = 1.1663787 · 10−5 GeV−2 .
+(11)
+In addition, the CKM matrix is taken diagonal, the
+pole mass of the top quark mt = 172.7 GeV, and we
+consider nq = 5 active quark flavours.
+Moreover, the
+widths of all particles appearing in the relevant diagrams
+have been set to zero.
+Our predictions make use of
+the CT18NNLO [87] set of parton distribution functions
+(PDFs), which are provided by LHAPDF [88] that we
+also use to control the renormalisation group running of
+the strong coupling αs. For predictions in the Type-III
+seesaw model, we safely set the elements of the ε matrix
+to zero, as they turn to be negligible once bounds from
+flavour and electroweak precision data are accounted
+for [89].
+The central value of the renormalisation and factorisa-
+tion scales is set to the invariant mass M of the produced
+di-lepton system, µR = µF = M. Scale uncertainties are
+then evaluated through the usual seven-point variation
+method, in which the renormalisation and factorisation
+scales are varied independently by a factor of two up and
+down relative to their central value with the two extreme
+cases µR/µF = 4 or 1/4 being excluded.
+IV.1.
+Total cross sections at the LHC
+We dedicate this section to an overview of the be-
+haviour of the total cross sections for exotic lepton pro-
+duction at the LHC with √s = 14 TeV, as a function
+of the lepton mass. We study the production of a pair
+of electrically-charged VLLs, and we consider both the
+cases of an SU(2)L singlet and doublet of VLLs,
+pp → ˜E+ ˜E− ,
+pp → E+E− .
+(12)
+Moreover, we also explore the production of a pair of
+singly-charged Type-III leptons that are mostly weak
+triplets,
+pp → E+E− ≡ Σ+Σ− .
+(13)
+For the last process, we introduced the abusive nota-
+tion Σ± ≡ E± to make an explicit distinction between
+the VLLs appearing in the model of section II.1 (pro-
+cess (12) and notation of table II), and those inherent to
+the Type-III seesaw model of section II.2 (process (13)
+and notation of table V). We do not consider any other
+pair-production mechanism (i.e. the production of a pair
+of neutral or doubly-charged leptons, or of an associated
+pair of leptons of different charges), as cross section pre-
+dictions are not expected to exhibit a fundamentally dif-
+ferent behaviour due to the purely-electroweak nature of
+the processes involved. However, as our UFO model files
+are public and MG5aMC is a general-purpose event gen-
+erator, interested readers can study by themselves any
+other process, both at (N)LO and (N)LO+PS.
+In the left panel of figure 1, we report LO production
+cross sections for the three processes of eqs. (12) and
+(13). The cross sections are found to span about 7 or-
+ders of magnitude for exotic lepton masses varying from
+200 GeV to 2.5 TeV. Cross sections around 100–1000 fb
+are found for small lepton masses of a few hundreds of
+GeV, whereas the production rates drop to the 0.001–
+1 fb regime for leptons of 1–2 TeV, making the potential
+observation of such BSM particles at the LHC more chal-
+lenging. Moreover, for a given lepton mass, the produc-
+tion of a pair of weak-triplet states (pp → Σ+Σ−; red) is
+favoured over that of weak-doublet states (pp → E+E−;
+blue), while the latter is favoured over the production of
+weak-singlet states (pp → ˜E+ ˜E−; green). Such a hierar-
+chy, as well as the relative differences observed between
+the rates that are factors of a few, can be understood
+from the different SU(2)L representations of the fields
+
+7
+FIG. 1.
+Total cross sections for the production of charged
+leptons typical from VLL and Type-III seesaw models, pre-
+sented as a function of the lepton mass. We consider SU(2)L
+singlet (green), doublet (blue) and triplet (red) leptons, and
+the LHC at 14 TeV. Predictions are shown at LO (left panel),
+as well as in the form of LO and NLO K-factors together with
+the associated scale uncertainties (right panel).
+considered, together with the Drell-Yan-like nature of the
+lepton pair-production mechanism.
+In the right panel of figure 1, we present the corre-
+sponding K-factors, that we define, for a given lepton
+mass, as the ratio of a cross section to the associated LO
+one at central scale. K-factors are shown both at LO
+(shaded area) and NLO (hatched area), together with
+the associated scale uncertainties. We observe mild K-
+factor values at NLO, which vary in the 1.15–1.40 range
+as a function of the exotic lepton mass. Moreover, the
+K-factors are found to be (almost) independent of the
+process. Such a result is not surprising as the underly-
+ing Born contribution factorises in the case of a Drell-
+Yan-like process.
+Uncertainties are significantly larger
+at LO than at NLO, and vary in the last case from a
+few percents at small lepton masses to about 10% for
+larger masses. This behaviour stems from the typically
+larger invariant masses associated with heavier di-lepton
+systems, that naturally enhance the importance of the
+threshold logarithms that ought to be resummed.
+In the upper panel of figure 2, we show the produc-
+tion rates obtained after matching NLO predictions with
+soft gluon resummation at the NNLL accuracy, these re-
+sults being currently the best theoretical predictions of
+the total cross sections for the processes considered. In
+order to estimate the associated impact, we present, in
+the three lower panels of figure 2, the ratio of the NLO,
+NLO+NLL and NLO+NNLL rates to the NLO one for
+the three processes. By virtue of the factorisation prop-
+erties of the Born contributions, the predictions for these
+ratios are mostly independent of the process. We observe
+a mild increase of the total rate once threshold resumma-
+tion is included, although this increase is mostly driven
+FIG. 2. Total NLO+NNLL cross sections (upper panel) for
+the production of vector-like and Type-III seesaw leptons, pre-
+sented as a function of the mass of the exotic lepton and for
+the LHC at a centre-of-mass energy of 14 TeV. We also dis-
+play the ratios of the NLO (green), NLO+NLL (red) and
+NLO+NNLL (blue) rates to the NLO ones (with a central
+scale choice), together with the associated scale uncertainties.
+by the leading and next-to-leading logarithmic contri-
+butions. NNLL contributions indeed barely modify the
+total rates.
+Numerically, these enhancements with re-
+spect to the NLO predictions are found to be about 5%
+for scenarios with light leptons, and range to about 10%
+for scenarios featuring heavier leptons. In addition, it is
+observed that the resummed results (at NLO+NLL and
+NLO+NNLL) lie outside the error bands of the NLO ones
+when a lepton mass of a few hundreds GeV is considered.
+This situation is similar to the case of the SM Drell-Yan
+process.
+As typical from resummation calculations, the most
+notable feature of the NLO+NNLL rate concerns the
+drastic reduction of the scale uncertainties inherent to
+the predictions, that are reduced below the percent level
+in average, regardless of the lepton mass. The necessity
+of incorporating soft gluon resummation in the predic-
+tions is made even more evident for scenarios featuring
+heavy exotic leptons. Here, the scale uncertainty bands
+at NLO+(N)NLL drastically decrease, which could be
+anticipated as we deal with a perturbative calculations in
+which αs is even smaller (the scale at which it is evaluated
+being trivially larger). This essentially cures the counter-
+intuitive large scale uncertainties observed at NLO.
+
+LOtotalcrosssection
+1.50
+P→-+
+103
+Vs=14 TeV
+9 1.25
+LO E
+olOL
+102
+NLOE
+1.00
+LO E
+101
+NLOE
+1.50
+LO E
+Pp→E-E+
+100
+NLO E
+91.25
+6
+1.00
+10-2
+1.50
++-d
+10-3
+91.25
+oloL
+10-4
+1.00
+1000
+2000
+1000
+2000
+mE,mé, mz[GeV]
+mE,mE, mz[GeV]NLO+NNLLtotalcrosssection
+103
+VS = 14 TeV
+102
+101
+100
+P→+
+6
+PP→E-E+
+10-1
+NLO
+10-2
+NLO+ NLL
+Pp→EE+
+10-3
+NLO+NNLL
+Pp-→E-E+
+1.0
+oloNLO
+1.1
+Pp
+-E-E+
+1.0
+G/ONLO
+1.1
+1.0
+150
+400
+650
+900115014001650190021502400
+mE,mE,mz[GeV]8
+FIG. 3. Invariant mass spectrum for the processes (12), together with the associated scale uncertainties (upper inset of each
+panel). We show fixed-order results at LO (red) and NLO (blue), as well as after matching them with threshold resummation
+at LO+LL (cyan), NLO+NLL (yellow) and NLO+NNLL (purple), and we consider new lepton masses of 600 GeV (top row),
+900 GeV (middle row) and 1.5 TeV (bottom row). We additionally present the bin-by-bin ratios of the NLO, NLO+NLL and
+NLO+NNLL spectra to the NLO+NNLL ones, with the associated uncertainties (lower inset in each panel).
+
+PP → E-E+@LHC14
+10-3
+mE=600GeV
+[fb/GeV]
+10-4
+LO
+NLO
+LO+LL
+10-5
+NLO+ NLL
+NLO + NNLL
+1.00
+doNLO+NNLL
+op
+0.95
+0.90
+1500
+2000
+2500
+3000
+3500
+4000
+M [GeV]10-2
+PP →E- E+@LHC14
+mE=600GeV
+ [fb/GeV]
+10-3
+LO
+NLO
+LO+LL
+NLO + NLL
+10-5
+NLO+ NNLL
+1.00
+doNLO+NNLL
+op
+0.95
+0.90
+1500
+2000
+2500
+3000
+3500
+4000
+M [GeV]Pp → E- E+@ LHC14
+mE=900GeV
+10-4
+[fb/GeV]
+LO
+NLO
+10-5
+TT+OT
+NLO+NLL
+NLO+NNLL
+1.00
+doNLO+NNLL
+op
+0.95
+0.90
+M [GeV]Pp → E- E+@ LHC14
+mE=900GeV
+[fb/GeV]
+10
+LO
+NLO
+LO+LL
+10-5
+NLO+NLL
+NLO+NNLL
+1.00
+doNLO+NNLL
+op
+0.95
+0.90
+M [GeV]1e-5
+1.0
+PP → E- E+@ LHC14
+0.8
+mE=1500GeV
+[fb/GeV]
+0.6
+0.4
+LO
+NLO+NLL
+0.2
+NLO
+NLO+NNLL
+LO +LL
+0.0
+1.00
+op
+0.95
+0.90
+3000
+3200
+3400
+3600
+3800
+4000
+M [GeV]1e-5
+PP→E-E+@LHC14
+2.0
+mE=1500GeV
+[fb/GeV]
+1.5
+1.0
+LO
+NLO+NLL
+0.5
+NLO
+NLO+NNLL
+LO +LL
+0.0
+1.00
+op
+0.95
+0.90
+3000
+3200
+3400
+3600
+3800
+4000
+M [GeV]9
+IV.2.
+Invariant mass distributions
+In this section, we consider again the processes (12)
+and (13), and we calculate the associated invariant-mass
+distributions dσ/dM, with M standing for the invari-
+ant mass of the di-lepton system. As an illustration, we
+choose three typical exotic lepton masses of 600 GeV,
+900 GeV and 1.5 TeV, which correspond to three scenar-
+ios that have not been fully excluded yet by the LHC ex-
+periments. The definition of these scenarios is driven by
+current experimental search results. The CMS collabora-
+tion has indeed observed an excess of 2.8σ when searching
+for VLLs with a mass of 600 GeV [38], whereas VLLs of
+900 GeV can already be probed by existing run 2 CMS
+and ATLAS searches.
+In contrast, extra leptons with
+a mass of 1500 GeV lie beyond the current reach, but
+they could potentially be probed at future LHC runs. In
+addition, all chosen extra lepton masses lead to PDF un-
+certainties under fair control [64]. The results are shown
+in figures 3 and 4.
+In the upper inset in each plot, we display fixed-
+order predictions and the associated scale uncertainties
+at LO (red) and NLO (blue), as well as after match-
+ing them with threshold resummation at LO+LL (cyan),
+NLO+NLL (yellow) and NLO+NNLL (purple). We have
+checked that for invariant mass distributions, (N)LO+PS
+results coincide with (N)LO calculations since this ob-
+servable is insensitive to PS effects. (N)LO+PS predic-
+tions are therefore not included in the figures. In general,
+we observe a rapid increase of the differential cross sec-
+tion close to the kinematic threshold M0 equals to twice
+the heavy lepton mass, until it peaks at M ≈ 1.06M0
+before falling off towards higher invariant-mass values.
+For any given M value, fixed-order LO predictions
+(red) are found to be notably smaller. Their matching
+with predictions including the resummation of the LL
+contributions (cyan) yields a significant enhancement of
+the central value. It reaches 15%–25% in the peak re-
+gion, and 30% at larger invariant masses where the rele-
+vant phase space region is closer to the partonic threshold
+(z → 1). Resummation effects are therefore more promi-
+nent in the latter case. Nevertheless, scale uncertainties
+in both cases remain large, and the two sets of predic-
+tions do not overlap within their error bands. This effect
+can be tamed down after including NLO corrections. In
+this case, both NLO+NLL and NLO+NNLL predictions
+are found to agree with each other once uncertainties are
+accounted for, whereas NLO spectra are slightly smaller
+for all considered M values.
+In order to better assess the impact of threshold re-
+summation, we display in the lower insets of figures 3
+and 4 bin-by-bin ratios of the NLO, NLO+NLL and
+NLO+NNLL rates to the most precise NLO+NNLL pre-
+dictions evaluated with central scale choices. The bands
+represent again the associated scale uncertainties.
+We
+observe that the increase of the differential NLO cross
+section induced by NLL or NNLL resummation (or equiv-
+alently, the decrease of the NLO cross sections, shown
+FIG. 4. Same as figure 3 but for Type-III seesaw leptons and
+the process (13).
+in blue, relative to the most precise NLO+NNLL pre-
+dictions) depends on the invariant mass of the di-lepton
+system M, and therefore indirectly on the mass of the
+lepton species produced that fix the kinematic produc-
+tion threshold M0. Preferred configurations hence natu-
+
+Pp → Z- E+@ LHC14
+10-3
+ms=900GeV
+[fb/GeV]
+LO
+10-4
+NLO
+TT+OT
+NLO+NLL
+NLO+NNLL
+1.00
+doNLO+NNLL
+op
+0.95
+0.90
+M [GeV]1e-5
+Pp → -Z+@ LHC14
+6
+mz=-1500GeV
+[fb/GeV]
+4
+LO
+NLO+NLL
+NLO
+NLO + NNLL
+LO +LL
+0
+1.00
+op
+0.95
+0.90
+3000
+3200
+3400
+3600
+3800
+4000
+M [GeV]Pp → -Z+@ LHC14
+10-2
+m==600GeV
+[fb/GeV]
+10-3
+LO
+NLO
+10
+TT+OT
+NLO+NLL
+NLO+NNLL
+1.00
+doNLO+NNLL
+op
+0.95
+0.90
+1500
+2000
+2500
+3000
+3500
+4000
+M [GeV]10
+FIG. 5. Transverse momentum spectra for the processes (12), together with the associated scale uncertainties (upper inset
+in each panel). We show fixed-order results at NLO (blue), as well as LO+PS (green) and NLO+PS (olive) predictions. We
+consider lepton masses of 600 GeV (top row), 900 GeV (middle row) and 1.5 TeV (bottom row), and we additionally present
+the corresponding bin-by-bin ratios to the NLO spectra with the associated scale uncertainties (lower inset in each panel).
+
+PP-→E-E+ @LHC14
+mE=600 GeV
+10-3
+[fb/GeV]
+10-5
+NLO
+10-7
+LO+PS
+NLO +PS
+do/doNLO
+1
+0
+500
+1000
+1500
+2000
+Pr [GeV]10-1
+PP-→E-E+ @LHC14
+mE=600 GeV
+[fb/GeV]
+10-3
+10-5
+NLO
+LO +PS
+10-7
+NLO +PS
+do/doNLO
+0
+500
+1000
+1500
+2000
+Pr [GeV]PP→E- E+ @ LHC14
+10-3
+mE=900GeV
+[fb/GeV]
+10-5
+NLO
+LO+PS
+NLO +PS
+2
+do/doNLO
+0
+0
+500
+1000
+1500
+2000
+Pr [GeV]PP→E-E+ @LHC14
+mE=900-GeV
+[fb/GeV]
+10-5
+NLO
+10-7
+LO +PS
+NLO +PS
+do/doNLO
+0
+0
+500
+1000
+1500
+2000
+Pr [GeV]PP→E- E+ @ LHC14
+m=1500GeV
+10~5
+[fb/GeV]
+10-7
+NLO
+LO +PS
+10-9
+NLO +PS
+2
+do/doNLO
+0
+500
+1000
+1500
+2000
+Pr [GeV]PP→E-E+ @LHC14
+10-
+mE=1500 GeV
+[fb/GeV]
+10-6
+NLO
+10-8
+LO +PS
+NLO +PS
+2
+do/doNLO
+0
+500
+1000
+1500
+2000
+Pr [GeV]11
+rally target larger z values closer to 1 for heavy lepton
+production than for light lepton production. Resumma-
+tion effects are therefore expected to be more important
+for heavy leptons, when considering M values lying at a
+given relative distance from M0. Again, we observe that
+NLO+(N)NLL results are generally outside the NLO er-
+ror bands.
+Nevertheless, the shape of the spectrum is stabilised
+after including threshold resummation at NLL (yellow).
+NNLL resummation (purple) only yields a mild increase
+of the rate by less than 1%, which is largely indepen-
+dent of the di-lepton invariant mass M.
+This shows
+that a good perturbative convergence has been achieved
+at NLO+NNLL. Moreover, NLO+NNLL predictions are
+crucial to reduce the scale uncertainties to be less than
+0.5%, hence motivating using more precise predictions
+for BSM signals when available.
+IV.3.
+Transverse momentum spectra
+We now turn to the study of the distribution in the
+transverse momentum (pT ) of the lepton pair, which is
+a typical observable that shows that the inclusion of PS
+is essential. Although the MG5aMC framework practi-
+cally allows for investigations of any observable, we only
+focus, for the sake of an example, on this distribution
+in the pT of the di-lepton system. In figures 5 and 6,
+we present dσ/dpT distributions at NLO (blue), LO+PS
+(green) and NLO+PS (olive), whereas the LO distribu-
+tions are trivially located at pT = 0. As in section IV.2,
+we choose three benchmark scenarios featuring extra lep-
+tons with masses of 600 GeV, 900 GeV and 1.5 TeV re-
+spectively. The different pT spectra are shown in the up-
+per insets of the figures, whereas the lower insets display
+their bin-by-bin ratios to the (fixed-order) NLO predic-
+tions.
+While NLO predictions (blue) in principle diverge at
+small pT due to uncancelled soft and/or collinear singu-
+larities originating from real emission, the integration of
+the differential cross section within a given bin regularises
+this divergence, the bin-by-bin results shown in the fig-
+ures being normalised by the bin size. We therefore ob-
+serve a finite cross section with a pronounced maximum
+in the low pT region, with pT ≲ 30 GeV (which corre-
+sponds to the first bin shown in the plots). The cross
+sections in the small pT regime are therefore expected to
+be better described by matching fixed-order predictions
+with PS. Showering, as well as the non-perturbative in-
+trinsic transverse momentum kT of the constituents of
+the protons, should additionally distort the shapes of the
+spectra in the low pT regime. NLO+PS predictions are
+hence smaller than NLO ones at small pT ≲ 30 GeV, be-
+fore becoming considerably greater at intermediate trans-
+verse momenta (pT ∈ [60, 500] GeV). At larger pT val-
+ues, QCD radiation encoded in the real matrix elements
+dominates, so that NLO+PS predictions agree with NLO
+ones.
+FIG. 6. Same as figure 5 but for Type-III seesaw leptons.
+This last effect can be even more evidenced by study-
+ing the LO+PS curves (green). Non-zero pT values are
+here purely arising from shower effects, which gives rise
+to a too soft spectrum in the high-pT regime. In conclu-
+sion, NLO+PS should be the best for describing such an
+observable, while NLO (LO+PS) predictions fail at low
+
+Pp-→Z- + @ LHC14
+mz=600 GeV
+10-2
+[fb/GeV]
+10-4
+NLO
+10-6
+LO +PS
+NLO +PS
+1.5
+do/doNLO
+1.0
+0.5
+0.0
+0
+500
+1000
+1500
+2000
+Pr [GeV]PP→Z-+ @ LHC14
+ms=900 GeV
+10-3
+[fb/GeV]
+10-5
+NLO
+LO+PS
+10-7
+NLO +PS
+2
+do/doNLO
+0
+500
+1000
+1500
+2000
+Pr [GeV]Pp→Z-Z+ @ LHC14
+10
+mz=.1500.GeV
+[fb/GeV]
+10-6
+NLO
+10-8
+LO+PS
+NLO +PS
+2
+do/doNLO
+0
+500
+1000
+1500
+2000
+Pr [GeV]12
+(high) pT . Finally, we can note that multiplying LO+PS
+predictions by an overall K-factor, as traditionally done
+in many experimental and phenomenological studies, is
+unjustified in the aim of an accurate signal description,
+and will even yield qualitatively wrong results at high pT .
+V.
+CONCLUSION
+Numerous extensions of the Standard Model feature
+additional leptons that carry a variety of different elec-
+tric charges.
+They are consequently actively searched
+for at collider experiments. In this work, we have stud-
+ied the production of these extra leptons in effective
+frameworks representative of the TeV-scale phenomenol-
+ogy of several models featuring additional leptons. By
+providing FeynRules implementations and the associ-
+ated UFO libraries for Type-III seesaw models and new
+physics scenarios with VLLs, we complete the set of pub-
+licly available models suitable for calculations relevant
+for the production of new leptons at colliders beyond
+the LO or LO+PS accuracy.
+For the fist time, VLL
+and Type-III lepton production can now be simulated
+at NLO+PS accuracy, as was already the case for other
+neutrino mass models or left-right-symmetric scenarios
+which both involve new non-coloured fermions. The cor-
+responding model files can be downloaded from https:
+//feynrules.irmp.ucl.ac.be/wiki/NLOModels.
+We have reported the most precise calculations of to-
+tal rates to date for the production of a pair of Type-III
+leptons or VLLs lying in the trivial or fundamental repre-
+sentations of SU(2)L. Our predictions include both NLO
+QCD corrections and threshold resummation effects at
+NNLL. Higher-order QCD effects increase the production
+rates by 25%–30%, the exact value depending on the sce-
+nario and the extra lepton mass, and scale uncertainties
+are reduced below 1%. We have additionally investigated
+the impact of these corrections on the distributions in the
+invariant mass of the produced heavy lepton system and
+observed a notable increase in the differential rates and
+a significant reduction of the scale uncertainties.
+Finally, we have made use of the designed UFO models
+to highlight the joint impact of NLO corrections and PS
+matching on an example of observable relevant for exist-
+ing searches for extra leptons. We have chosen the distri-
+bution in the transverse momentum of the di-lepton sys-
+tem. We have illustrated how NLO+PS improves fixed-
+order NLO computations at low pT , and provides a better
+modelling of the physical spectra at intermediate pT val-
+ues below 1 TeV. These calculations have been achieved
+fully automatically, in the MG5aMC framework, in a
+setup similar to that used in ATLAS and CMS studies as
+well as in many existing phenomenological explorations.
+Upgrades of existing studies and searches to include NLO
+corrections matched with PS should therefore be straight-
+forward.
+ACKNOWLEDGMENTS
+The authors are grateful to M. Cirelli and G. Soyez for
+motivating discussions during the course of this project.
+We acknowledge support from the European Union’s
+Horizon 2020 research and innovation programme (grant
+agreement No. 824093,
+STRONG-2020,
+EU Virtual
+Access “NLOAccess”), the European Research Coun-
+cil (ERC grant agreement ID 101041109, “BOSON”),
+the French ANR (grants ANR-20-CE31-0015, “PrecisO-
+nium”, and ANR-21-CE31-0013, “DMwithLLPatLHC”),
+and the CNRS IEA (grant No. 205210, “GlueGraph”).
+Appendix A: Resummation coefficients
+In this section, we provide analytical results for the
+resummation coefficients appearing in (10) in the case
+of a Drell-Yan-like process. Those comprise the process-
+dependent term ˜g0,q¯q, as well as the universal coefficients
+of the exponent gk,q¯q with k > 0. The latter are iden-
+tical to those given in appendix A of [90] (the λ param-
+eter of [90] being replaced by ω). We then provide be-
+low the expression for the process-dependent coefficient
+˜g0,q¯q(M 2, µ2
+F , µ2
+R) that can be expanded in as as
+˜g0,q¯q(M 2, µ2
+F , µ2
+R) =
+dˆσ(0)
+q¯q
+d ln M 2
+�
+1 +
+�
+k=1
+ak
+s(µ2
+R)˜g(k)
+0,q¯q
+�
+,
+(A1)
+where
+dˆσ(0)
+q¯q
+d ln M 2 is the Born partonic cross section differen-
+tiating with respect to ln M 2. The ˜g(1)
+0,q¯q and ˜g(2)
+0,q¯q coeffi-
+cients in QCD (for nq = 5) relevant for resummation at
+NNLL explicitly read
+˜g(1)
+0,q¯q = −64
+3
++ 64
+3 ζ2 − 8Lfr + 8Lqr ,
+˜g(2)
+0,q¯q = −1291
+9
++ 64ζ2
+9
++ 368ζ2
+2
+3
++ 4528ζ3
+27
++
+188L2
+fr
+3
++ 4L2
+qr
+3
++ Lfr
+�1324
+9
+− 1888ζ2
+9
++ 32ζ3
+3
+�
++ Lqr
+�148
+9
+− 64Lfr + 416ζ2
+9
+− 32ζ3
+3
+�
+,
+(A2)
+with Lqr = ln M 2
+µ2
+R , Lfr = ln µ2
+F
+µ2
+R and ζn being the Riemann
+zeta function ζ(n).
+In the processes (12) and (13), the di-lepton system
+is produced either through an s-channel virtual photon
+exchange, or through an s-channel Z-boson exchange. In
+other words, the Born partonic cross section can be split
+into three pieces, namely the square of photon-exchange
+diagram, that of the Z-exchange diagram, and the inter-
+ference between the two diagrams. Mathematically, this
+
+13
+can be written as
+ˆσ(0)
+q¯q = ˆσ(0),γ
+q¯q
++ ˆσ(0),Z
+q¯q
++ ˆσ(0),int
+q¯q
+.
+(A3)
+The photon exchange contribution depends on the elec-
+tric charge of the initial-state quarks Qq, and on that of
+the final-state leptons Qℓ,
+ˆσ(0),γ
+q¯q
+= Q2
+qQ2
+ℓ
+4πα2
+9M 2
+�
+1 − 4m2
+ℓ
+M 2
+�
+1 + 2m2
+ℓ
+M 2
+�
+.
+(A4)
+In this expression, mℓ is the mass of the produced lepton,
+and it is thus respectively given by m ˜
+E, mE and mΣ in
+the three processes shown in (12) and (13). As all exotic
+leptons produced in these processes satisfy Qℓ = −1,
+ˆσ(0),γ
+q¯q
+is identical in all three cases.
+The Z-boson exchange contribution and the interfer-
+ence term read
+ˆσ(0),Z
+q¯q
+= f 2
+ℓ
+(V 2
+q + A2
+q)
+4
+M 4
+(M 2 − m2
+Z)2 + Γ2
+Zm2
+Z
+ˆσ(0),γ
+q¯q
+Q2q
+,
+ˆσ(0),int
+q¯q
+= fℓQqVq
+M 2(M 2 − m2
+Z)
+(M 2 − m2
+Z)2 + Γ2
+Zm2
+Z
+ˆσ(0),γ
+q¯q
+Q2q
+,
+(A5)
+where Vq = Iq −2s2
+WQq and Aq = Iq represent the vector
+and axial couplings between the initial quarks and the Z
+boson respectively. Here, the parameters cW and sW are
+the cosine and sine of the electroweak mixing angle, and
+Iq is the weak isospin quantum number of the quark q,
+that is thus equal to 1/2 for up-type quarks and −1/2 for
+down-type quarks. In addition, ΓZ denotes the width of
+the Z boson, and the prefactor fℓ depends on the quan-
+tum numbers of the lepton ℓ produced. It is thus given
+by
+fℓ =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+−c−2
+W
+for
+ℓ = ˜E ,
+c2
+W −s2
+W
+2s2
+W c2
+W
+for
+ℓ = E ,
+s−2
+W
+for
+ℓ = Σ .
+(A6)
+[1] G. Panico and A. Wulzer, The Composite
+Nambu-Goldstone Higgs, vol. 913. Springer, 2016.
+arXiv:1506.01961 [hep-ph].
+[2] G. Cacciapaglia, C. Pica, and F. Sannino,
+“Fundamental Composite Dynamics: A Review,” Phys.
+Rept. 877 (2020) 1–70, arXiv:2002.04914 [hep-ph].
+[3] G. Cacciapaglia, A. Deandrea, and K. Sridhar, “Review
+of fundamental composite dynamics,” Eur. Phys. J. ST
+231 no. 7, (2022) 1221–1222.
+[4] A. P. Morais, R. Pasechnik, and W. Porod, “Grand
+Unified Origin of Gauge Interactions and Families
+Replication in the Standard Model,” Universe 7 no. 12,
+(2021) 461, arXiv:2001.04804 [hep-ph].
+[5] A. P. Morais, R. Pasechnik, and W. Porod, “Prospects
+for new physics from gauge left-right-colour-family
+grand unification hypothesis,” Eur. Phys. J. C 80
+no. 12, (2020) 1162, arXiv:2001.06383 [hep-ph].
+[6] S. P. Martin, “Extra vector-like matter and the lightest
+Higgs scalar boson mass in low-energy supersymmetry,”
+Phys. Rev. D 81 (2010) 035004, arXiv:0910.2732
+[hep-ph].
+[7] M. Endo, K. Hamaguchi, S. Iwamoto, and N. Yokozaki,
+“Higgs Mass and Muon Anomalous Magnetic Moment
+in Supersymmetric Models with Vector-Like Matters,”
+Phys. Rev. D 84 (2011) 075017, arXiv:1108.3071
+[hep-ph].
+[8] J. Y. Araz, S. Banerjee, M. Frank, B. Fuks, and
+A. Goudelis, “Dark matter and collider signals in an
+MSSM extension with vector-like multiplets,” Phys.
+Rev. D 98 no. 11, (2018) 115009, arXiv:1810.07224
+[hep-ph].
+[9] J. C. Pati and A. Salam, “Lepton Number as the
+Fourth Color,” Phys. Rev. D 10 (1974) 275–289.
+[Erratum: Phys.Rev.D 11, 703–703 (1975)].
+[10] R. N. Mohapatra and J. C. Pati, “A Natural Left-Right
+Symmetry,” Phys. Rev. D 11 (1975) 2558.
+[11] G. Senjanovic and R. N. Mohapatra, “Exact Left-Right
+Symmetry and Spontaneous Violation of Parity,” Phys.
+Rev. D 12 (1975) 1502.
+[12] R. N. Mohapatra, F. E. Paige, and D. P. Sidhu,
+“Symmetry Breaking and Naturalness of Parity
+Conservation in Weak Neutral Currents in Left-Right
+Symmetric Gauge Theories,” Phys. Rev. D 17 (1978)
+2462.
+[13] J. Halverson, N. Orlofsky, and A. Pierce, “Vectorlike
+Leptons as the Tip of the Dark Matter Iceberg,” Phys.
+Rev. D 90 no. 1, (2014) 015002, arXiv:1403.1592
+[hep-ph].
+[14] P. Minkowski, “µ → eγ at a Rate of One Out of 109
+Muon Decays?,” Phys. Lett. B 67 (1977) 421–428.
+[15] R. N. Mohapatra and G. Senjanovic, “Neutrino Mass
+and Spontaneous Parity Nonconservation,” Phys. Rev.
+Lett. 44 (1980) 912.
+[16] T. Yanagida, “Horizontal gauge symmetry and masses
+of neutrinos,” Conf. Proc. C 7902131 (1979) 95–99.
+[17] M. Gell-Mann, P. Ramond, and R. Slansky, “Complex
+Spinors and Unified Theories,” Conf. Proc. C 790927
+(1979) 315–321, arXiv:1306.4669 [hep-th].
+[18] S. L. Glashow, “The Future of Elementary Particle
+Physics,” NATO Sci. Ser. B 61 (1980) 687.
+[19] R. E. Shrock, “General Theory of Weak Leptonic and
+Semileptonic Decays. 1. Leptonic Pseudoscalar Meson
+Decays, with Associated Tests For, and Bounds on,
+Neutrino Masses and Lepton Mixing,” Phys. Rev. D 24
+
+14
+(1981) 1232.
+[20] J. Schechter and J. W. F. Valle, “Neutrino Masses in
+SU(2) x U(1) Theories,” Phys. Rev. D 22 (1980) 2227.
+[21] Y. Cai, T. Han, T. Li, and R. Ruiz, “Lepton Number
+Violation: Seesaw Models and Their Collider Tests,”
+Front. in Phys. 6 (2018) 40, arXiv:1711.02180
+[hep-ph].
+[22] R. Foot, H. Lew, X. G. He, and G. C. Joshi, “Seesaw
+Neutrino Masses Induced by a Triplet of Leptons,” Z.
+Phys. C 44 (1989) 441.
+[23] ATLAS Collaboration, M. Aaboud et al., “Search for
+heavy Majorana or Dirac neutrinos and right-handed W
+gauge bosons in final states with two charged leptons
+and two jets at √s = 13 TeV with the ATLAS
+detector,” JHEP 01 (2019) 016, arXiv:1809.11105
+[hep-ex].
+[24] ATLAS Collaboration, M. Aaboud et al., “Search for a
+right-handed gauge boson decaying into a
+high-momentum heavy neutrino and a charged lepton in
+pp collisions with the ATLAS detector at √s = 13
+TeV,” Phys. Lett. B 798 (2019) 134942,
+arXiv:1904.12679 [hep-ex].
+[25] ATLAS Collaboration, G. Aad et al., “Search for
+heavy neutral leptons in decays of W bosons produced
+in 13 TeV pp collisions using prompt and displaced
+signatures with the ATLAS detector,” JHEP 10 (2019)
+265, arXiv:1905.09787 [hep-ex].
+[26] ATLAS Collaboration, G. Aad et al., “Search for
+type-III seesaw heavy leptons in dilepton final states in
+pp collisions at √s = 13 TeV with the ATLAS
+detector,” Eur. Phys. J. C 81 no. 3, (2021) 218,
+arXiv:2008.07949 [hep-ex].
+[27] ATLAS Collaboration, G. Aad et al., “Search for new
+phenomena in three- or four-lepton events in pp
+collisions at √s =13 TeV with the ATLAS detector,”
+Phys. Lett. B 824 (2022) 136832, arXiv:2107.00404
+[hep-ex].
+[28] ATLAS Collaboration, G. Aad et al., “Search for
+type-III seesaw heavy leptons in leptonic final states in
+pp collisions at √s = 13 TeV with the ATLAS
+detector,” arXiv:2202.02039 [hep-ex].
+[29] CMS Collaboration, A. M. Sirunyan et al., “Search for
+Evidence of the Type-III Seesaw Mechanism in
+Multilepton Final States in Proton-Proton Collisions at
+√s = 13 TeV,” Phys. Rev. Lett. 119 no. 22, (2017)
+221802, arXiv:1708.07962 [hep-ex].
+[30] CMS Collaboration, A. M. Sirunyan et al., “Search for
+heavy neutral leptons in events with three charged
+leptons in proton-proton collisions at √s = 13 TeV,”
+Phys. Rev. Lett. 120 no. 22, (2018) 221801,
+arXiv:1802.02965 [hep-ex].
+[31] CMS Collaboration, A. M. Sirunyan et al., “Search for
+a heavy right-handed W boson and a heavy neutrino in
+events with two same-flavor leptons and two jets at
+√s = 13 TeV,” JHEP 05 (2018) 148,
+arXiv:1803.11116 [hep-ex].
+[32] CMS Collaboration, A. M. Sirunyan et al., “Search for
+heavy Majorana neutrinos in same-sign dilepton
+channels in proton-proton collisions at √s = 13 TeV,”
+JHEP 01 (2019) 122, arXiv:1806.10905 [hep-ex].
+[33] CMS Collaboration, A. M. Sirunyan et al., “Search for
+heavy neutrinos and third-generation leptoquarks in
+hadronic states of two τ leptons and two jets in
+proton-proton collisions at √s = 13 TeV,” JHEP 03
+(2019) 170, arXiv:1811.00806 [hep-ex].
+[34] CMS Collaboration, A. M. Sirunyan et al., “Search for
+vector-like leptons in multilepton final states in
+proton-proton collisions at √s = 13 TeV,” Phys. Rev. D
+100 no. 5, (2019) 052003, arXiv:1905.10853 [hep-ex].
+[35] CMS Collaboration, A. M. Sirunyan et al., “Search for
+physics beyond the standard model in multilepton final
+states in proton-proton collisions at √s = 13 TeV,”
+JHEP 03 (2020) 051, arXiv:1911.04968 [hep-ex].
+[36] CMS Collaboration, A. Tumasyan et al., “Search for a
+right-handed W boson and a heavy neutrino in
+proton-proton collisions at √s = 13 TeV,” JHEP 04
+(2022) 047, arXiv:2112.03949 [hep-ex].
+[37] CMS Collaboration, “Search for resonant and
+nonresonant production of pairs of dijet resonances in
+proton-proton collisions at √s = 13 TeV,”
+arXiv:2206.09997 [hep-ex].
+[38] CMS Collaboration, “Search for pair-produced
+vector-like leptons in final states with third-generation
+leptons and at least three b quark jets in proton-proton
+collisions at √s = 13 TeV,” arXiv:2208.09700
+[hep-ex].
+[39] ATLAS Collaboration, “Search for heavy neutral
+leptons in decays of W bosons using a dilepton
+displaced vertex in √s = 13 TeV pp collisions with the
+ATLAS detector,” arXiv:2204.11988 [hep-ex].
+[40] C. Degrande, O. Mattelaer, R. Ruiz, and J. Turner,
+“Fully-Automated Precision Predictions for Heavy
+Neutrino Production Mechanisms at Hadron Colliders,”
+Phys. Rev. D 94 no. 5, (2016) 053002,
+arXiv:1602.06957 [hep-ph].
+[41] B. Fuks, J. Neundorf, K. Peters, R. Ruiz, and
+M. Saimpert, “Majorana neutrinos in same-sign
+W ±W ± scattering at the LHC: Breaking the TeV
+barrier,” Phys. Rev. D 103 no. 5, (2021) 055005,
+arXiv:2011.02547 [hep-ph].
+[42] O. Mattelaer, M. Mitra, and R. Ruiz, “Automated
+Neutrino Jet and Top Jet Predictions at
+Next-to-Leading-Order with Parton Shower Matching in
+Effective Left-Right Symmetric Models,”
+arXiv:1610.08985 [hep-ph].
+[43] J. Debove, B. Fuks, and M. Klasen, “Threshold
+resummation for gaugino pair production at hadron
+colliders,” Nucl. Phys. B 842 (2011) 51–85,
+arXiv:1005.2909 [hep-ph].
+[44] B. Fuks, M. Klasen, D. R. Lamprea, and M. Rothering,
+“Gaugino production in proton-proton collisions at a
+center-of-mass energy of 8 TeV,” JHEP 10 (2012) 081,
+arXiv:1207.2159 [hep-ph].
+[45] B. Fuks, M. Klasen, D. R. Lamprea, and M. Rothering,
+“Precision predictions for electroweak superpartner
+production at hadron colliders with Resummino,” Eur.
+Phys. J. C 73 (2013) 2480, arXiv:1304.0790 [hep-ph].
+[46] J. Fiaschi and M. Klasen, “Higgsino and gaugino pair
+production at the LHC with aNNLO+NNLL precision,”
+Phys. Rev. D 102 no. 9, (2020) 095021,
+arXiv:2006.02294 [hep-ph].
+[47] C. Degrande, C. Duhr, B. Fuks, D. Grellscheid,
+O. Mattelaer, and T. Reiter, “UFO - The Universal
+FeynRules Output,” Comput. Phys. Commun. 183
+(2012) 1201–1214, arXiv:1108.2040 [hep-ph].
+[48] J. Alwall, R. Frederix, S. Frixione, V. Hirschi,
+F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer,
+P. Torrielli, and M. Zaro, “The automated computation
+
+15
+of tree-level and next-to-leading order differential cross
+sections, and their matching to parton shower
+simulations,” JHEP 07 (2014) 079, arXiv:1405.0301
+[hep-ph].
+[49] R. Frederix, S. Frixione, V. Hirschi, D. Pagani, H. S.
+Shao, and M. Zaro, “The automation of next-to-leading
+order electroweak calculations,” JHEP 07 (2018) 185,
+arXiv:1804.10017 [hep-ph]. [Erratum: JHEP 11, 085
+(2021)].
+[50] G. F. Sterman, “Summation of Large Corrections to
+Short Distance Hadronic Cross-Sections,” Nucl. Phys. B
+281 (1987) 310–364.
+[51] S. Catani and L. Trentadue, “Resummation of the QCD
+Perturbative Series for Hard Processes,” Nucl. Phys. B
+327 (1989) 323–352.
+[52] A. Vogt, “Next-to-next-to-leading logarithmic threshold
+resummation for deep inelastic scattering and the
+Drell-Yan process,” Phys. Lett. B 497 (2001) 228–234,
+arXiv:hep-ph/0010146.
+[53] M. Buchkremer, G. Cacciapaglia, A. Deandrea, and
+L. Panizzi, “Model Independent Framework for
+Searches of Top Partners,” Nucl. Phys. B 876 (2013)
+376–417, arXiv:1305.4172 [hep-ph].
+[54] B. Fuks and H.-S. Shao, “QCD next-to-leading-order
+predictions matched to parton showers for vector-like
+quark models,” Eur. Phys. J. C 77 no. 2, (2017) 135,
+arXiv:1610.04622 [hep-ph].
+[55] N. D. Christensen, P. de Aquino, C. Degrande,
+C. Duhr, B. Fuks, M. Herquet, F. Maltoni, and
+S. Schumann, “A Comprehensive approach to new
+physics simulations,” Eur. Phys. J. C 71 (2011) 1541,
+arXiv:0906.2474 [hep-ph].
+[56] A. Alloul, N. D. Christensen, C. Degrande, C. Duhr,
+and B. Fuks, “FeynRules 2.0 - A complete toolbox for
+tree-level phenomenology,” Comput. Phys. Commun.
+185 (2014) 2250–2300, arXiv:1310.1921 [hep-ph].
+[57] P. Z. Skands et al., “SUSY Les Houches accord:
+Interfacing SUSY spectrum calculators, decay packages,
+and event generators,” JHEP 07 (2004) 036,
+arXiv:hep-ph/0311123.
+[58] C. Biggio and F. Bonnet, “Implementation of the Type
+III Seesaw Model in FeynRules/MadGraph and
+Prospects for Discovery with Early LHC Data,” Eur.
+Phys. J. C 72 (2012) 1899, arXiv:1107.3463 [hep-ph].
+[59] T. Li and X.-G. He, “Neutrino Masses and Heavy
+Triplet Leptons at the LHC: Testability of Type III
+Seesaw,” Phys. Rev. D 80 (2009) 093003,
+arXiv:0907.4193 [hep-ph].
+[60] A. Abada, C. Biggio, F. Bonnet, M. B. Gavela, and
+T. Hambye, “Low energy effects of neutrino masses,”
+JHEP 12 (2007) 061, arXiv:0707.4058 [hep-ph].
+[61] A. Abada, C. Biggio, F. Bonnet, M. B. Gavela, and
+T. Hambye, “mu —> e gamma and tau —> l gamma
+decays in the fermion triplet seesaw model,” Phys. Rev.
+D 78 (2008) 033007, arXiv:0803.0481 [hep-ph].
+[62] B. Fuks, M. Nemevˇsek, and R. Ruiz, “Doubly Charged
+Higgs Boson Production at Hadron Colliders,” Phys.
+Rev. D 101 no. 7, (2020) 075022, arXiv:1912.08975
+[hep-ph].
+[63] C. Duhr and B. Fuks, “A superspace module for the
+FeynRules package,” Comput. Phys. Commun. 182
+(2011) 2404–2426, arXiv:1102.4191 [hep-ph].
+[64] S. Frixione, B. Fuks, V. Hirschi, K. Mawatari, H.-S.
+Shao, P. A. Sunder, and M. Zaro, “Automated
+simulations beyond the Standard Model:
+supersymmetry,” JHEP 12 (2019) 008,
+arXiv:1907.04898 [hep-ph].
+[65] C. Degrande, “Automatic evaluation of UV and R2
+terms for beyond the Standard Model Lagrangians: a
+proof-of-principle,” Comput. Phys. Commun. 197
+(2015) 239–262, arXiv:1406.3030 [hep-ph].
+[66] T. Hahn, “Generating Feynman diagrams and
+amplitudes with FeynArts 3,” Comput. Phys. Commun.
+140 (2001) 418–431, arXiv:hep-ph/0012260.
+[67] J. Bellm et al., “Herwig 7.0/Herwig++ 3.0 release
+note,” Eur. Phys. J. C 76 no. 4, (2016) 196,
+arXiv:1512.01178 [hep-ph].
+[68] T. Gleisberg, S. Hoeche, F. Krauss, M. Schonherr,
+S. Schumann, F. Siegert, and J. Winter, “Event
+generation with SHERPA 1.1,” JHEP 02 (2009) 007,
+arXiv:0811.4622 [hep-ph].
+[69] S. Frixione, Z. Kunszt, and A. Signer, “Three jet
+cross-sections to next-to-leading order,” Nucl. Phys. B
+467 (1996) 399–442, arXiv:hep-ph/9512328.
+[70] S. Frixione, “A General approach to jet cross-sections in
+QCD,” Nucl. Phys. B 507 (1997) 295–314,
+arXiv:hep-ph/9706545.
+[71] R. Frederix, S. Frixione, F. Maltoni, and T. Stelzer,
+“Automation of next-to-leading order computations in
+QCD: The FKS subtraction,” JHEP 10 (2009) 003,
+arXiv:0908.4272 [hep-ph].
+[72] R. Frederix, S. Frixione, A. S. Papanastasiou,
+S. Prestel, and P. Torrielli, “Off-shell single-top
+production at NLO matched to parton showers,” JHEP
+06 (2016) 027, arXiv:1603.01178 [hep-ph].
+[73] G. Ossola, C. G. Papadopoulos, and R. Pittau,
+“Reducing full one-loop amplitudes to scalar integrals
+at the integrand level,” Nucl. Phys. B 763 (2007)
+147–169, arXiv:hep-ph/0609007.
+[74] P. Mastrolia, E. Mirabella, and T. Peraro, “Integrand
+reduction of one-loop scattering amplitudes through
+Laurent series expansion,” JHEP 06 (2012) 095,
+arXiv:1203.0291 [hep-ph]. [Erratum: JHEP 11, 128
+(2012)].
+[75] G. Passarino and M. J. G. Veltman, “One Loop
+Corrections for e+ e- Annihilation Into mu+ mu- in the
+Weinberg Model,” Nucl. Phys. B 160 (1979) 151–207.
+[76] A. I. Davydychev, “A Simple formula for reducing
+Feynman diagrams to scalar integrals,” Phys. Lett. B
+263 (1991) 107–111.
+[77] A. Denner and S. Dittmaier, “Reduction schemes for
+one-loop tensor integrals,” Nucl. Phys. B 734 (2006)
+62–115, arXiv:hep-ph/0509141.
+[78] V. Hirschi, R. Frederix, S. Frixione, M. V. Garzelli,
+F. Maltoni, and R. Pittau, “Automation of one-loop
+QCD corrections,” JHEP 05 (2011) 044,
+arXiv:1103.0621 [hep-ph].
+[79] G. Ossola, C. G. Papadopoulos, and R. Pittau,
+“CutTools: A Program implementing the OPP
+reduction method to compute one-loop amplitudes,”
+JHEP 03 (2008) 042, arXiv:0711.3596 [hep-ph].
+[80] T. Peraro, “Ninja: Automated Integrand Reduction via
+Laurent Expansion for One-Loop Amplitudes,”
+Comput. Phys. Commun. 185 (2014) 2771–2797,
+arXiv:1403.1229 [hep-ph].
+[81] V. Hirschi and T. Peraro, “Tensor integrand reduction
+via Laurent expansion,” JHEP 06 (2016) 060,
+arXiv:1604.01363 [hep-ph].
+
+16
+[82] A. Denner, S. Dittmaier, and L. Hofer, “Collier: a
+fortran-based Complex One-Loop LIbrary in Extended
+Regularizations,” Comput. Phys. Commun. 212 (2017)
+220–238, arXiv:1604.06792 [hep-ph].
+[83] F. Cascioli, P. Maierhofer, and S. Pozzorini, “Scattering
+Amplitudes with Open Loops,” Phys. Rev. Lett. 108
+(2012) 111601, arXiv:1111.5206 [hep-ph].
+[84] S. Frixione and B. R. Webber, “Matching NLO QCD
+computations and parton shower simulations,” JHEP
+06 (2002) 029, arXiv:hep-ph/0204244.
+[85] A. H. Ajjath and H.-S. Shao, “N3LO+N3LL QCD
+improved Higgs pair cross sections,” arXiv:2209.03914
+[hep-ph].
+[86] T. Sj¨ostrand, S. Ask, J. R. Christiansen, R. Corke,
+N. Desai, P. Ilten, S. Mrenna, S. Prestel, C. O.
+Rasmussen, and P. Z. Skands, “An introduction to
+PYTHIA 8.2,” Comput. Phys. Commun. 191 (2015)
+159–177, arXiv:1410.3012 [hep-ph].
+[87] T.-J. Hou et al., “New CTEQ global analysis of
+quantum chromodynamics with high-precision data
+from the LHC,” Phys. Rev. D 103 no. 1, (2021) 014013,
+arXiv:1912.10053 [hep-ph].
+[88] A. Buckley, J. Ferrando, S. Lloyd, K. Nordstr¨om,
+B. Page, M. R¨ufenacht, M. Sch¨onherr, and G. Watt,
+“LHAPDF6: parton density access in the LHC
+precision era,” Eur. Phys. J. C 75 (2015) 132,
+arXiv:1412.7420 [hep-ph].
+[89] C. Biggio, E. Fernandez-Martinez, M. Filaci,
+J. Hernandez-Garcia, and J. Lopez-Pavon, “Global
+Bounds on the Type-III Seesaw,” JHEP 05 (2020) 022,
+arXiv:1911.11790 [hep-ph].
+[90] A. H. Ajjath, A. Chakraborty, G. Das, P. Mukherjee,
+and V. Ravindran, “Resummed prediction for Higgs
+boson production through bb annihilation at N3LL,”
+JHEP 11 (2019) 006, arXiv:1905.03771 [hep-ph].
+
diff --git a/aNE2T4oBgHgl3EQfEwbs/content/tmp_files/load_file.txt b/aNE2T4oBgHgl3EQfEwbs/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..a885f32363eb00355b83808f1043721c0de6276f
--- /dev/null
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@@ -0,0 +1,1183 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf,len=1182
+page_content='Precision predictions for exotic lepton production at the Large Hadron Collider  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Ajjath,1, ∗ Benjamin Fuks,1, † Hua-Sheng Shao,1, ‡ and Yehudi Simon1, §  1Sorbonne Universit´e, CNRS, Laboratoire de Physique Th´eorique et Hautes ´Energies, LPTHE, F-75005 Paris, France  We calculate total and differential cross sections for the pair production, at the Large Hadron  Collider, of exotic leptons that could emerge from models with vector-like leptons and in Type-III see-  saw scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Our predictions include next-to-leading-order QCD corrections, and we subsequently  match them with either parton showers, or threshold resummation at the next-to-next-to-leading  logarithmic accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Our results show an important increase of the cross sections relative to the  leading-order predictions, exhibit a distortion of the shapes for various differential distributions, and  feature a significant reduction of the scale uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Our predictions have been obtained from  new FeynRules model implementations and associated UFO model libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' This completes the  set of next-to-leading-order implementations of new physics models featuring extra leptons that are  publicly available on the FeynRules model database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  INTRODUCTION  Several extensions of the Standard Model (SM) predict  the existence of new exotic heavy leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' These arise in  particular in composite models [1–3], grand unified the-  ories [4, 5], supersymmetric models [6–8], left-right sym-  metric models [9–12], dark matter scenarios [13] or in  Type I [14–21] and Type-III [21, 22] seesaw models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In  composite scenarios, the new physics particle spectrum  features vector-like leptons (VLLs) transforming as elec-  troweak SU(2)L singlets or doublets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In contrast, in see-  saw and left-right models, neutrino masses are generated  from Yukawa interactions of new electroweak singlets or  triplets of fermions with the SM Higgs field and lepton  weak doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Consequently, searches for new heavy lep-  tons consist of an important component of the experi-  mental beyond the SM (BSM) search programme at the  Large Hadron Collider (LHC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  The ATLAS and CMS collaborations have explored  the associated parameter spaces, both for promptly-  decaying [23–38] and long-lived [25, 39] extra leptons,  and for a variety of mass ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Limits have been set on  the mixing properties of long-lived heavy charge-neutral  leptons for masses of 3–15 GeV, while short-lived neutral  and charged leptons must have masses of at least about  950 GeV in Type-III seesaw models (for branching ratios  of 1 in the final state considered).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Moreover, in left-right  models neutral lepton masses must be larger than 3 TeV  for WR boson masses smaller than 4–5 TeV, and indirect  probes for heavy neutrinos via vector-boson-fusion pro-  cesses additionally constrain masses ranging up to 20 TeV  for large mixings with the SM leptons and neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Fi-  nally, VLLs are restricted to be heavier than about 800  GeV or lighter than about 100 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  From the theoretical side, total and differential cross  section calculations for collider processes involving ex-  tra neutrinos are known at next-to-leading order (NLO)  ∗ aabdulhameed@lpthe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='jussieu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='fr  † fuks@lpthe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='jussieu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='fr  ‡ huasheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='shao@lpthe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='jussieu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='fr  § ysimon@lpthe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='jussieu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='fr  in QCD in a generic simplified model describing the dy-  namics of the heavy neutrinos [40, 41], and for an ef-  fective left-right-symmetric scenario [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  In addition,  NLO-QCD predictions matched with threshold resum-  mation at the next-to-leading-logarithmic (NLL) accu-  racy [43–45] and approximate next-to-next-to-leading-  order cross section matched with next-to-next-to-leading-  logarithmic (NNLL) threshold resummation [46] can also  be obtained from electroweakino pair-production pro-  cesses in the Minimal Supersymmetric Standard Model,  after decoupling all supersymmetric states but the pro-  duced electroweakinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Beside differential and total  rates, precision collider simulations in which NLO calcu-  lations are matched with parton showers (PS) are avail-  able for heavy neutrino simplified models [40–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' How-  ever, simulations for Type-III seesaw scenarios and com-  posite VLL setups are only available at leading order  (LO) so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  The goal of this paper is to fill this gap, and to re-  port about the development of two new publicly available  UFO [47] model libraries allowing for event generation  at the NLO+PS accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Our implementation are suit-  able in particular to describe VLL and Type-III seesaw  lepton production processes from computations achieved  by means of the precision Monte Carlo event generator  MadGraph5 aMC@NLO [48, 49] (MG5aMC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' More-  over, we take the opportunity to update predictions for  the total rates and the invariant mass distributions of  the BSM Drell-Yan-like processes inherent to the models  considered, and present results at NLO in QCD matched  with threshold resummation at NNLL, following the for-  malism of [50–52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  The rest of this paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In sec-  tion II, we introduce two effective theoretical frameworks  suitable for our calculations, a first one dedicated to  a simplified VLL model and a second one to Type-III  seesaw scenarios, and we report details about their im-  plementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='In section III, we briefly describe the for-  malism that we use to resum the large threshold loga-  rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In section IV, we make use of the MG5aMC  platform to study the phenomenology of the two mod-  els at the NLO+PS accuracy, as well as of an in-house  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='03640v1  [hep-ph]  9 Jan 2023  2  programme to handle predictions at NLO+NNLL in the  strong coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' We compute total rates for the produc-  tion of extra leptons (section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1), invariant-mass dis-  tributions (section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='2), and we additionally present for  the first time NLO+PS-accurate differential distributions  relevant for experimental searches for VLLs and Type-III  seesaw fermions (section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' We summarise our work  and conclude in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  THEORETICAL MODELS AND THE  IMPLEMENTATIONS  In order to study the phenomenology of the considered  models, we construct two effective frameworks, one for  each of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  We minimally extend the SM in  terms of fields and interactions, so that the resulting new  physics parameter spaces are of small dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  A simplified model for VLL phenomenology  We begin by considering an extension of the SM in  which the theory field content includes a set of VLL  fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' They are all colour singlet, but each of them lies  in a different SU(2)L representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  They are corre-  spondingly assigned different hypercharge U(1)Y quan-  tum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In order to be as model independent as  possible, we adopt a simplified model approach and fo-  cus on extra leptons that are either electrically neutral  or with an electric charge Q = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' These leptons are  organised in (vector-like) SU(2)L doublets and singlets,  L0 =  �  �N 0  E0  �  � ,  ˜E0 ,  ˜N 0 ,  (1)  where in this notation the superscript ‘0’ indicates that  the fields are gauge eigenstates, and the tilde above a field  indicates that it is an SU(2)L singlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' We next implement  the mixing between the SM and the new lepton fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  To this aim, we introduce an effective parametrisation  apt to capture the main phenomenological features of the  vector-like fields in a model-independent way, following  guidelines introduced for vector-like quark setups [53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  In practice, we assume that the mixing between the  SM fields and the new leptons is small, so that the gauge  interactions of the different fields are unaffected at the  first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Moreover, we implement the off-diagonal in-  teractions of the exotic leptons with the SM ones through  generic free parameters, which further open the VLL de-  cay channels into a SM lepton and an electroweak boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  The corresponding Lagrangian, given in terms of mass  eigenstates (the superscript ‘0’ being therefore dropped),  Field  Spin  Representation  Name  L0  (1/2, 1/2)  (1, 2)−1/2  VLL0  ˜  N 0  (1/2, 1/2)  (1, 1)0  VLN0  ˜E0  (1/2, 1/2)  (1, 1)−1  VLE0  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Gauge eigenstates complementing the SM field  content, their spin given as their representation under the  SO(1, 3) group (second column), their SU(3)c × SU(2)L ×  U(1)Y (third column) representation and their name in the  FeynRules implementation (last column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  reads  LVLL = LSM + i¯L /DL − mN ¯NN − mE ¯EE  + i ¯˜N /∂ ˜N − m ˜  N ¯˜N ˜N + i ¯˜E /D ˜E − m ˜  E ¯˜E ˜E  +  �  Ψ=E, ˜  E  �  h¯Ψ  �  ˆκ  Ψ  LPL + ˆκ  Ψ  RPR  �  ℓ + g  √  2  ¯Ψ /W  −κ  Ψ  LPLνℓ  +  g  2cW  ¯Ψ/Z  �  ˜κ  Ψ  LPL + ˜κ  Ψ  RPR  �  ℓ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  �  +  �  Ψ=N, ˜  N  �  h¯Ψˆκ  Ψ  LPLνℓ +  g  2cW  ¯Ψ/Z˜κ  Ψ  LPLνℓ  + g  √  2  ¯Ψ /W  +�  κ  Ψ  LPL + κ  Ψ  RPR  �  ℓ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  �  ,  (2)  where LSM is the SM Lagrangian, and the parameters  mN, mE, m ˜  N and m ˜  E stand for the masses of the four  new fields in the physical basis, assuming that the masses  of the doublet component fields can be different after  electroweak symmetry breaking and particle mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' The  first two lines in this Lagrangian include, additionally to  the SM Lagrangian, all gauge-invariant kinetic and mass  terms for the new states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' The gauge-covariant derivative  operator Dµ is defined, for a generic field ψ, by  Dµψ = ∂µψ − ig′BµY ψ − igW k  µT kψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  (3)  Here, the coupling constants g and g′ respectively stand  for the weak SU(2)L and hypercharge U(1)Y coupling  constants, and Bµ and W k  µ are the associated gauge  fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' The action of the hypercharge operator Y on the  field ψ can be deduced from table I, as the representa-  tion to adopt for the SU(2) generators T k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In particular,  T k = 0 for an SU(2)L singlet, and T k = σk/2 for an  SU(2)L doublet, with σk being the Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In  other words, we approximate mass eigenstates by gauge  eigenstates in all kinetic terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' L0 ≈ L, ˜E0 ≈ ˜E and  ˜N 0 ≈ ˜N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  The last four lines of the Lagrangian (2) collect the  effective interactions of each of the four VLLs considered  with a SM lepton (ℓ standing for the charged lepton field  and νℓ for the neutrino one), and either the Higgs boson  h, the W boson or the Z boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In (2), all flavour indices  are understood so that each of the κ, ˆκ and ˜κ couplings  has to be seen as a vector in the flavour space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Moreover,  cW refers to the cosine of the electroweak mixing angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  3  Field  Spin  Name  PDG  Mass  Width  N  (1/2, 1/2)  VLLN  9000001  MVLLN  WVLLN  E  (1/2, 1/2)  VLLE  9000002  MVLLE  WVLLE  ˜  N  (1/2, 1/2)  VLN  9000003  MVLN  WVLN  ˜E  (1/2, 1/2)  VLE  9000004  MVLE  WVLE  TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Mass eigenstates supplementing the SM, with  their spin quantum number (second column), name used in  the FeynRules convention (third column) and adopted PDG  identifier (fourth column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In the last two columns, we pro-  vide the FeynRules symbols associated with the particle  masses and widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  In the following, the exact values of the κ, ˆκ and ˜κ cou-  pling vectors are irrelevant, provided that they are not  too large to guarantee that the VLL states have a narrow  width, and not too small so that they can promptly de-  cay into a lepton+electroweak boson system within LHC  detector scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In a hadron collision process in which  the new leptons are pair produced, such couplings indeed  only appear in the heavy particle decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  In order to allow for phenomenological studies of  the model, we implement it in the FeynRules pack-  age [55, 56], starting from the SM implementation that  is shipped with the programme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  We include the def-  initions of the gauge eigenstates of (1), together with  the corresponding mass eigenstates appearing in the La-  grangian (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Information on these fields, their names  in the FeynRules conventions, and the Particle Data  Group (PDG) identifiers that we have adopted for the  physical fields, are provided in tables I and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' These ta-  bles also include the symbols associated with the mass  and width of the physical fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' All BSM couplings ap-  pearing in (2) have been implemented as three-vectors in  the flavour space, following the convention of table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  This table also includes information on the Les Houches  block structure used to organise all model external pa-  rameters [57], as required by all high-energy physics pro-  grammes relying on FeynRules for model implementa-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Moreover, their specific contributions to any pro-  cess can be turned off through a dedicated interaction  order named VLL (see the FeynRules manual [56]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  An effective Type-III seesaw Lagrangian  In Type-III seesaw models, neutrino masses are gen-  erated through the interactions of the SM Higgs field Φ  with the SM leptons and at least two generations of ex-  tra fermions lying in the adjoint representation of SU(2)L  and with zero hypercharge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In the following, we make use  of two-component Weyl fermion notation for all fields,  and omit all SM and BSM generation indices for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  In such a formalism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' the Lagrangian of the model is ex-  pressed in terms of the SM weak doublet of left-handed  leptons LL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' the SM weak singlet of right-handed charged  leptons Ec  R (ER being thus the corresponding left-handed  Couplings  Names  Les Houches blocks  (ˆκE  L)i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' (ˆκE  R)i  KLLEH[i],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' KRLEH[i]  KLLEH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' KRLEH  (ˆκ  ˜  E  L)i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' (ˆκ  ˜  E  R)i  KLEH[i],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' KREH[i]  KLEH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' KREH  (ˆκN  L )i  KLLNH[i]  KLLNH  (ˆκ  ˜  N  L )i  KLNH[i]  KLNH  (˜κE  L)i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' (˜κE  R)i  KLLEZ[i],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' KRLEZ[i]  KLLEZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' KRLEZ  (˜κ  ˜  E  L)i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' (˜κ  ˜  E  R)i  KLEZ[i],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' KREZ[i]  KLEZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' KREZ  (˜κN  L )i  KLLNZ[i]  KLLNZ  (˜κ  ˜  N  L )i  KLNZ[i]  KLNZ  (κE  L)i  KLLEW[i]  KLLEW  (κ  ˜  E  L)i  KLEW[i]  KLEW  (κN  L )i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' (κN  R)i  KLLNW[i],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' KRLNW[i]  KLLNW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' KRLNW  (κ  ˜  N  L )i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' (κ  ˜  N  R)i  KLNW[i],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' KRNW[i]  KLNW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' KRNW  TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Three-point VLL coupling strengths to a SM lep-  ton and an electroweak boson, given together with the associ-  ated FeynRules symbol and the corresponding Les Houches  block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' The indice i denotes a generation index ranging from  1 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Weyl spinor), and the weak triplet of extra lepton Σk  (with k = 1, 2, 3 being an SU(2)L adjoint index).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' The  three gauge eigenstates Σk can be conveniently related  to states of definite electric charge E± ≡ Σ± (of charge  Q = ±1) and N ≡ Σ0 (of charge Q = 0) by introducing  the matrix representation for the SU(2)L triplets Σij,  with i and j referring to fundamental indices of SU(2)L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  We obtain,  Σi  j =  1  √  2(σk)i  jΣk =  � 1  √  2N  E+  E−  − 1  √  2N  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  (4)  The Type-III Lagrangian is given by  LTypeIII = LSM + Lkin  +  �  yℓ Φ†LL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='ER + 2yΣ Φ·  �  ΣkT kLL  �  + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  (5)  In our notation, LSM stands for the reduced SM La-  grangian in which all terms involving a leptonic field have  been removed, and Lkin collects all gauge-invariant ki-  netic terms for the (two-component) leptonic fields LL,  ER and Σ, and a mass term for the Σ field (of mass mΣ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Moreover, the matrices T k = σk/2 stand for the gener-  ators of SU(2)L in the fundamental representation, and  the explicit scalar product appearing on the second line  refers to the SU(2)-invariant product of two fields lying  in its fundamental representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In Type-III models,  charged lepton and neutrino masses are driven by the SM  leptonic 3 × 3 Yukawa matrix yℓ, and the heavy neutrino  Yukawa matrix yΣ whose size depends on the number of  generations of new fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  From the LHC physics point of view, we can simplify  the model presented above by emphasising the focus on  the lightest of all Σ states, that are assumed to be the  4  only ones to which the LHC would be sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  This  strategy follows that introduced in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Under such  an assumption, the relevant part of the Yukawa matrix yΣ  (in the Σ-flavour × SM-flavour space) becomes a vector  in the SM-flavour space, that we take real for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  The mass eigenstates of the model hence include three  generations of physical up-type and down-type quarks,  and four generations of physical charged leptons (ℓ′) and  Majorana neutrinos (ν′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  After electroweak symmetry  breaking, the Lagrangian (5) induces a mixing between  the three SM leptons and the new states Σ, rendering at  least two neutrinos massive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Introducing the three 4 × 4  mixing matrices in the lepton flavour space U ℓ  L, U ℓ  R and  U 0, lepton gauge and mass eigenstates are related by [59],  �  EL  E−  �  = U ℓ  L ℓ′  L ,  �  ER  E+c  �  = U ℓ  R ℓ′  R ,  �  νL  N  �  = U ν ν′ ,  (6)  where the three-component vector in the flavour space  EL (νL) stands for the down-type (up-type) component  of the weak doublet of left-handed SM leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' As the  new fermion masses of O(mΣ) are expected to be heavy  compared with the neutrino masses of O(yΣv), with v  being the SM Higgs vacuum expectation value, the three  mixing matrices can be expanded at the first order in  yv/mΣ [60, 61],  U ℓ  L =  �  � 1 − ε  v  mΣ yΣ  − v  mΣ y†  Σ 1 − ε′  �  � ,  U ℓ  R =  �  �  1  Mℓv  m2  Σ yΣ  − Mℓv  m2  Σ y†  Σ  1  �  � ,  U ν =  �  �  �  1 − 1  2ε  �  UPMNS  v  √  2mΣ yΣ  −  v  √  2mΣ y†  Σ  1 − 1  2ε′  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  (7)  These expressions depend on the quantities ε and ε′, that  are 3 × 3 and scalar objects in the SM flavour space re-  spectively,  ε =  v2  2m2  Σ  yΣy†  Σ  and  ε′ =  v2  2m2  Σ  y†  ΣyΣ ,  (8)  as well as on the SM lepton mass matrix Mℓ (that is diag-  onal in the flavour space) and on the unitary Pontecorvo-  Maki-Nakagawa-Sakata (PMNS) matrix UPMNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  The  latter can be defined from the neutrino oscillation pa-  rameters, namely the neutrino mixing angles θij (with  i, j = 1, 2, 3), the Dirac CP-violating phase ϕCP and the  two Majorana CP-violating phases ϕ1 and ϕ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  We implement the Type-III model described above  in the FeynRules package [55, 56] following the same  method that has been used for the Type II seesaw im-  plementation [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' We begin with the implementation of  the SM shipped with FeynRules, from which all lepton  definitions and related Lagrangian terms have been mod-  ified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In practice, we have modified all lepton and neu-  trino definitions so that two-component left-handed Weyl  fermions [63] could be instead used for the gauge eigen-  states of the model (Ec  R, LL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Next, we add definitions  Field  Spin  Representation  # generations  Name  LL  (1/2, 0)  (1, 2)−1/2  3  LLw  ER  (1/2, 0)  (1, 1)1  3  ERw  Σk  (1/2, 0)  (1, 3)0  1  Sigw  TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Gauge eigenstates associated with the leptonic  sector of the Type-III seesaw model, their spin given as their  representation under the SO(1, 3) group (second column),  their SU(3)c ×SU(2)L ×U(1)Y representation (third column),  and their name in the FeynRules implementation (last col-  umn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Field  Spin  Name  PDG  Mass  Width  e−  (1/2, 1/2)  e  11  Me  –  µ−  (1/2, 1/2)  mu  13  MMu  –  τ −  (1/2, 1/2)  ta  15  MTA  –  E−  (1/2, 1/2)  SigM  9000017  MSigma  WSigM  ν1  (1/2, 1/2)  v1  12  Mv1  –  ν2  (1/2, 1/2)  v2  14  Mv2  –  ν3  (1/2, 1/2)  v3  16  Mv3  –  N  (1/2, 1/2)  Sig0  9000018  MSigma  WSig0  TABLE V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Mass eigenstates that either supplement the SM  or whose definition is altered relatively to the SM, with their  spin representation (second column), name used in the Feyn-  Rules convention (third column) and adopted PDG identi-  fier (fourth column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In the last two columns, we provide the  FeynRules symbols associated with the particle masses and  widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  for the fermionic triplets Σk, still using left-handed Weyl  fermions, and incorporate the mixing relations (6) for  the definition of the four physical charged lepton states  and the four physical neutrino states in terms of all gauge  eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Finally, we map the physical two-component  fermions of the model into the corresponding Dirac fields  (charged leptons) and Majorana fields (neutrinos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' More  information on all fields included in the model imple-  mentation is given in tables IV and V (representation,  names in the FeynRules conventions, PDG identifiers,  symbols for masses and widths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  The new physics parameters yΣ and mΣ appearing in  the Lagrangian (5) are implemented in a standard way,  together with the neutrino oscillation parameters dictat-  ing the values of the PMNS matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Moreover, we assume  a normal neutrino mass hierarchy and set the masses of  the three lightest neutrinos mν1, mν2 and mν3 from the  value of the smallest neutrino mass (mν1 in our case), and  the neutrino squared mass differences ∆m2  21 and ∆m2  31,  mν2 =  �  m2ν1 + ∆m2  21 and mν3 =  �  m2ν1 + ∆m2  31 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' (9)  More information on the free parameters of the lep-  tonic/neutrino sector of the model is provided in table VI  (FeynRules names and Les Houches block structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  5  Parameter  Name  LH block  LH counter  (yΣ)e  ySigma[1]  YSIGMA  1  (yΣ)µ  ySigma[2]  YSIGMA  2  (yΣ)τ  ySigma[3]  YSIGMA  3  mΣ  MSigma  MASS  9000017  mν1  Mv1  MASS  12  ∆m2  21  dmsq21  MNU  2  ∆m2  31  dmsq31  MNU  3  θ12  th12  PMNS  1  θ23  th23  PMNS  2  θ13  th13  PMNS  3  ϕCP  delCP  PMNS  4  ϕ1  PhiM1  PMNS  5  ϕ2  PhiM2  PMNS  6  TABLE VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' External parameters defining the leptonic sec-  tor of the Type-III seesaw model, including the neutrino pa-  rameters in the context of a normal mass hierarchy (so that  mν1 < mν2 < mν3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Each parameter is given together with  the symbol used in the FeynRules implementation, and the  corresponding Les Houches (LH) block and counter informa-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  From Lagrangian to events at the LHC  In order to handle LHC simulations at NLO-QCD  matched with PS, we make use of the two FyenRules  model implementations detailed in secitons II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1 and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='2,  and jointly use them with the MoGRe package (ver-  sion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1) [64], NloCt (version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1) [65] and Fey-  nArts (version 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='9) [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' This allows us to renormalise  the bare Lagrangians (2) and (5) relatively to O(αs) QCD  interactions, and generate UFO model files [47] includ-  ing both tree-level interactions, UV counterterms, and  the so-called R2 Feynman rules required for the numer-  ical evaluation of the numerators of one-loop integrands  in a four-dimensional spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Such UFO models can  subsequently be used with MG5aMC [48, 49] for LO and  NLO calculations in QCD, as well as by Herwig++ [67]  and Sherpa [68] at LO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Before closing this subsection, we provide informa-  tion on the MG5aMC framework [48, 49] which employ  to carry out fixed-order (N)LO and (N)LO+PS calcula-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' MG5aMC handles infrared singularities inherent  to NLO calculations via the FKS method [69, 70], in an  automated way through the MadFKS module [71, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  The evaluation of UV-renormalised one-loop amplitudes  is achieved by switching dynamically between several  integral-reduction techniques that work either at the in-  tegrand level (like the OPP method [73] or Laurent-  series expansion [74]) or through tensor-integral reduc-  tion [75–77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' This has been automated in the MadLoop  module [48, 78], that exploits the public codes Cut-  Tools [79], Ninja [80, 81] and Collier [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Moreover,  one-loop computations have been optimised at the inte-  grand level through an in-house procedure inspired by  the idea of OpenLoops [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Finally, NLO+PS predic-  tions are obtained by matching fixed-order calculations  with PS according to the MC@NLO method [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  IMPROVING THEORY ACCURACY  BEYOND NLO: THRESHOLD RESUMMATION  For the Drell-Yan-like processes considered, it is well-  known that large logarithms spoil the convergence of  the perturbative series when the invariant mass M of  the final-state system approaches the hadronic centre-of-  mass energy √s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' This calls for a proper resummation  of soft-gluon radiation, or at least a matching with PS  as achieved in the MG5aMC framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  In this sec-  tion, we briefly describe, for the convenience of readers,  the theoretical formalism that we adopt for fixed-order  calculations matched with threshold resummation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Ad-  ditional details and an extensive description can be found  in section 2 of [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  At the partonic level, the corresponding kinematic re-  gion is defined in terms of the partonic scaling variable  z = M 2/ˆs when z → 1, with  √  ˆs being the partonic  centre-of-mass energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In this limit, the perturbative co-  efficients in the cross sections get contributions in the  form of αs ln(1 − z) originating from soft gluon emission,  which could be of O(1) and thus potentially spoil the  perturbative convergence of the usual series in αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' This  issue can be resolved by reorganising the perturbative  expansion in an alternative manner, and resumming the  large logarithms in αs ln(1 − z) to all orders in αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Resummation calculations are conveniently carried out  in the Mellin N-space conjugate to z, where the z → 1  limit thus corresponds to the large N region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' The re-  summed partonic cross section in the Mellin space, de-  noted by ∆res  q¯q (N, M 2, µ2  F ) with µF being the factorisa-  tion scale, is defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='47) from [85] for a generic  process with a colourless final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' For a Drell-Yan-like  process and at the kth logarithmic accuracy (NkLL), it  reads  ∆res  q¯q (N, M 2, µ2  F )  ���  NkLL = ˜g0,q¯q(M 2, µ2  F , µ2  R)  ���  NkLO  × exp  �  g1,q¯q(ω) ln N +  k+1  �  j=2  aj−2  s  (µ2  R)gj,q¯q(ω)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  (10)  In this expression, the ˜g0,q¯q|NkLO factor collects the N-  independent terms of the first k + 1 coefficients of the  usual as perturbative expansion (in Mellin space), where  we have introduced the short-hand notation as(µ2  R) =  αs(µ2  R)/4π with µR denoting the renormalisation scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  This factor is process dependent, and it gets contribu-  tions from virtual corrections and soft real emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In  the exponent, the process-independent (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' universal) co-  efficients gj,q¯q(ω) with j > 0 receive contributions from  the threshold logarithmic terms originating from real  6  emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Given ω = 2β0as(µ2  R) ln N ∼ O(1) with β0  being the first coefficient of the QCD beta function, they  effectively resum these logarithmic contributions to all  orders in αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' We refer to appendix A for the analytical  form of the various coefficients appearing in (10) and that  are relevant for NLO+NNLL calculations for the Drell-  Yan-like processes considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  As the separation of the N-independent and N-  dependent pieces in (10) is not unambiguous, different  resummation schemes have been proposed and described  in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='4 of [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In the present paper, we consider  the so-called N1 resummation scheme for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Resummed calculations must then have to be matched  with fixed-order predictions at (N)LO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' This is achieved  by adding the resummed and (N)LO results and sub-  tracting of all double-counted contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' The latter  correspond to the (N)LO soft-virtual terms of the par-  tonic cross section, which can be obtained by expand-  ing ∆res  q¯q (N, M 2, µ2  F ) at O(αb(+1)  s  ), where b stands for the  power in αs of the LO contributions (that is 0 here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  CROSS SECTIONS FOR EXTRA LEPTON  PRODUCTION AT THE LHC  In this section, we compute total and differential cross  sections relevant for the production of additional leptons  such as those appearing in the models introduced in sec-  tion II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Predictions at (N)LO and (N)LO+PS are ob-  tained within the MG5aMC framework (version 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0),  using the UFO models developed in this work, and we  employ Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='2 [86] to deal with the simulation of  the QCD environement (parton showering and hadronisa-  tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' On the other hand, total rate calculations match-  ing fixed-order predictions with soft-gluon resummation  are derived with an in-house code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  We define the electroweak sector through three inde-  pendent input parameters that we choose to be the Z-  boson mass mZ, the electromagnetic coupling constant  evaluated at the Z-pole α(mZ), and the Fermi constant  GF ,  mZ = 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1876 GeV ,  α−1(mZ) = 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='9 ,  GF = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1663787 · 10−5 GeV−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  (11)  In addition, the CKM matrix is taken diagonal, the  pole mass of the top quark mt = 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='7 GeV, and we  consider nq = 5 active quark flavours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Moreover, the  widths of all particles appearing in the relevant diagrams  have been set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Our predictions make use of  the CT18NNLO [87] set of parton distribution functions  (PDFs), which are provided by LHAPDF [88] that we  also use to control the renormalisation group running of  the strong coupling αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' For predictions in the Type-III  seesaw model, we safely set the elements of the ε matrix  to zero, as they turn to be negligible once bounds from  flavour and electroweak precision data are accounted  for [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  The central value of the renormalisation and factorisa-  tion scales is set to the invariant mass M of the produced  di-lepton system, µR = µF = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Scale uncertainties are  then evaluated through the usual seven-point variation  method, in which the renormalisation and factorisation  scales are varied independently by a factor of two up and  down relative to their central value with the two extreme  cases µR/µF = 4 or 1/4 being excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Total cross sections at the LHC  We dedicate this section to an overview of the be-  haviour of the total cross sections for exotic lepton pro-  duction at the LHC with √s = 14 TeV, as a function  of the lepton mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' We study the production of a pair  of electrically-charged VLLs, and we consider both the  cases of an SU(2)L singlet and doublet of VLLs,  pp → ˜E+ ˜E− ,  pp → E+E− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  (12)  Moreover, we also explore the production of a pair of  singly-charged Type-III leptons that are mostly weak  triplets,  pp → E+E− ≡ Σ+Σ− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  (13)  For the last process, we introduced the abusive nota-  tion Σ± ≡ E± to make an explicit distinction between  the VLLs appearing in the model of section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1 (pro-  cess (12) and notation of table II), and those inherent to  the Type-III seesaw model of section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='2 (process (13)  and notation of table V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' We do not consider any other  pair-production mechanism (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' the production of a pair  of neutral or doubly-charged leptons, or of an associated  pair of leptons of different charges), as cross section pre-  dictions are not expected to exhibit a fundamentally dif-  ferent behaviour due to the purely-electroweak nature of  the processes involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' However, as our UFO model files  are public and MG5aMC is a general-purpose event gen-  erator, interested readers can study by themselves any  other process, both at (N)LO and (N)LO+PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  In the left panel of figure 1, we report LO production  cross sections for the three processes of eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' (12) and  (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' The cross sections are found to span about 7 or-  ders of magnitude for exotic lepton masses varying from  200 GeV to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='5 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Cross sections around 100–1000 fb  are found for small lepton masses of a few hundreds of  GeV, whereas the production rates drop to the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='001–  1 fb regime for leptons of 1–2 TeV, making the potential  observation of such BSM particles at the LHC more chal-  lenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Moreover, for a given lepton mass, the produc-  tion of a pair of weak-triplet states (pp → Σ+Σ−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' red) is  favoured over that of weak-doublet states (pp → E+E−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  blue), while the latter is favoured over the production of  weak-singlet states (pp → ˜E+ ˜E−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Such a hierar-  chy, as well as the relative differences observed between  the rates that are factors of a few, can be understood  from the different SU(2)L representations of the fields  7  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Total cross sections for the production of charged  leptons typical from VLL and Type-III seesaw models, pre-  sented as a function of the lepton mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' We consider SU(2)L  singlet (green), doublet (blue) and triplet (red) leptons, and  the LHC at 14 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Predictions are shown at LO (left panel),  as well as in the form of LO and NLO K-factors together with  the associated scale uncertainties (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  considered, together with the Drell-Yan-like nature of the  lepton pair-production mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  In the right panel of figure 1, we present the corre-  sponding K-factors, that we define, for a given lepton  mass, as the ratio of a cross section to the associated LO  one at central scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' K-factors are shown both at LO  (shaded area) and NLO (hatched area), together with  the associated scale uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' We observe mild K-  factor values at NLO, which vary in the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='15–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='40 range  as a function of the exotic lepton mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Moreover, the  K-factors are found to be (almost) independent of the  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Such a result is not surprising as the underly-  ing Born contribution factorises in the case of a Drell-  Yan-like process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Uncertainties are significantly larger  at LO than at NLO, and vary in the last case from a  few percents at small lepton masses to about 10% for  larger masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' This behaviour stems from the typically  larger invariant masses associated with heavier di-lepton  systems, that naturally enhance the importance of the  threshold logarithms that ought to be resummed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  In the upper panel of figure 2, we show the produc-  tion rates obtained after matching NLO predictions with  soft gluon resummation at the NNLL accuracy, these re-  sults being currently the best theoretical predictions of  the total cross sections for the processes considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In  order to estimate the associated impact, we present, in  the three lower panels of figure 2, the ratio of the NLO,  NLO+NLL and NLO+NNLL rates to the NLO one for  the three processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' By virtue of the factorisation prop-  erties of the Born contributions, the predictions for these  ratios are mostly independent of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' We observe  a mild increase of the total rate once threshold resumma-  tion is included, although this increase is mostly driven  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Total NLO+NNLL cross sections (upper panel) for  the production of vector-like and Type-III seesaw leptons, pre-  sented as a function of the mass of the exotic lepton and for  the LHC at a centre-of-mass energy of 14 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' We also dis-  play the ratios of the NLO (green), NLO+NLL (red) and  NLO+NNLL (blue) rates to the NLO ones (with a central  scale choice), together with the associated scale uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  by the leading and next-to-leading logarithmic contri-  butions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' NNLL contributions indeed barely modify the  total rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Numerically, these enhancements with re-  spect to the NLO predictions are found to be about 5%  for scenarios with light leptons, and range to about 10%  for scenarios featuring heavier leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In addition, it is  observed that the resummed results (at NLO+NLL and  NLO+NNLL) lie outside the error bands of the NLO ones  when a lepton mass of a few hundreds GeV is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  This situation is similar to the case of the SM Drell-Yan  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  As typical from resummation calculations, the most  notable feature of the NLO+NNLL rate concerns the  drastic reduction of the scale uncertainties inherent to  the predictions, that are reduced below the percent level  in average, regardless of the lepton mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' The necessity  of incorporating soft gluon resummation in the predic-  tions is made even more evident for scenarios featuring  heavy exotic leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Here, the scale uncertainty bands  at NLO+(N)NLL drastically decrease, which could be  anticipated as we deal with a perturbative calculations in  which αs is even smaller (the scale at which it is evaluated  being trivially larger).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' This essentially cures the counter-  intuitive large scale uncertainties observed at NLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  LOtotalcrosssection  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='50  P→-+  103  Vs=14 TeV  9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='25  LO E  olOL  102  NLOE  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='00  LO E  101  NLOE  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='50  LO E  Pp→E-E+  100  NLO E  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='25  6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='00  10-2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='50  +-d  10-3  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='25  oloL  10-4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='00  1000  2000  1000  2000  mE,mé, mz[GeV]  mE,mE, mz[GeV]NLO+NNLLtotalcrosssection  103  VS = 14 TeV  102  101  100  P→+  6  PP→E-E+  10-1  NLO  10-2  NLO+ NLL  Pp→EE+  10-3  NLO+NNLL  Pp-→E-E+  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0  oloNLO  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1  Pp  E-E+  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0  G/ONLO  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0  150  400  650  900115014001650190021502400  mE,mE,mz[GeV]8  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Invariant mass spectrum for the processes (12), together with the associated scale uncertainties (upper inset of each  panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' We show fixed-order results at LO (red) and NLO (blue), as well as after matching them with threshold resummation  at LO+LL (cyan), NLO+NLL (yellow) and NLO+NNLL (purple), and we consider new lepton masses of 600 GeV (top row),  900 GeV (middle row) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='5 TeV (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' We additionally present the bin-by-bin ratios of the NLO, NLO+NLL and  NLO+NNLL spectra to the NLO+NNLL ones, with the associated uncertainties (lower inset in each panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  PP → E-E+@LHC14  10-3  mE=600GeV  [fb/GeV]  10-4  LO  NLO  LO+LL  10-5  NLO+ NLL  NLO + NNLL  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='00  doNLO+NNLL  op  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='90  1500  2000  2500  3000  3500  4000  M [GeV]10-2  PP →E- E+@LHC14  mE=600GeV  [fb/GeV]  10-3  LO  NLO  LO+LL  NLO + NLL  10-5  NLO+ NNLL  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='00  doNLO+NNLL  op  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='90  1500  2000  2500  3000  3500  4000  M [GeV]Pp → E- E+@ LHC14  mE=900GeV  10-4  [fb/GeV]  LO  NLO  10-5  TT+OT  NLO+NLL  NLO+NNLL  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='00  doNLO+NNLL  op  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='90  M [GeV]Pp → E- E+@ LHC14  mE=900GeV  [fb/GeV]  10  LO  NLO  LO+LL  10-5  NLO+NLL  NLO+NNLL  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='00  doNLO+NNLL  op  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='90  M [GeV]1e-5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0  PP → E- E+@ LHC14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='8  mE=1500GeV  [fb/GeV]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='4  LO  NLO+NLL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='2  NLO  NLO+NNLL  LO +LL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='00  op  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='90  3000  3200  3400  3600  3800  4000  M [GeV]1e-5  PP→E-E+@LHC14  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0  mE=1500GeV  [fb/GeV]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0  LO  NLO+NLL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='5  NLO  NLO+NNLL  LO +LL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='00  op  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='90  3000  3200  3400  3600  3800  4000  M [GeV]9  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Invariant mass distributions  In this section, we consider again the processes (12)  and (13), and we calculate the associated invariant-mass  distributions dσ/dM, with M standing for the invari-  ant mass of the di-lepton system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' As an illustration, we  choose three typical exotic lepton masses of 600 GeV,  900 GeV and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='5 TeV, which correspond to three scenar-  ios that have not been fully excluded yet by the LHC ex-  periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' The definition of these scenarios is driven by  current experimental search results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' The CMS collabora-  tion has indeed observed an excess of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='8σ when searching  for VLLs with a mass of 600 GeV [38], whereas VLLs of  900 GeV can already be probed by existing run 2 CMS  and ATLAS searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  In contrast, extra leptons with  a mass of 1500 GeV lie beyond the current reach, but  they could potentially be probed at future LHC runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In  addition, all chosen extra lepton masses lead to PDF un-  certainties under fair control [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' The results are shown  in figures 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  In the upper inset in each plot, we display fixed-  order predictions and the associated scale uncertainties  at LO (red) and NLO (blue), as well as after match-  ing them with threshold resummation at LO+LL (cyan),  NLO+NLL (yellow) and NLO+NNLL (purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' We have  checked that for invariant mass distributions, (N)LO+PS  results coincide with (N)LO calculations since this ob-  servable is insensitive to PS effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' (N)LO+PS predic-  tions are therefore not included in the figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In general,  we observe a rapid increase of the differential cross sec-  tion close to the kinematic threshold M0 equals to twice  the heavy lepton mass, until it peaks at M ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='06M0  before falling off towards higher invariant-mass values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  For any given M value, fixed-order LO predictions  (red) are found to be notably smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Their matching  with predictions including the resummation of the LL  contributions (cyan) yields a significant enhancement of  the central value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' It reaches 15%–25% in the peak re-  gion, and 30% at larger invariant masses where the rele-  vant phase space region is closer to the partonic threshold  (z → 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Resummation effects are therefore more promi-  nent in the latter case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Nevertheless, scale uncertainties  in both cases remain large, and the two sets of predic-  tions do not overlap within their error bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' This effect  can be tamed down after including NLO corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In  this case, both NLO+NLL and NLO+NNLL predictions  are found to agree with each other once uncertainties are  accounted for, whereas NLO spectra are slightly smaller  for all considered M values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  In order to better assess the impact of threshold re-  summation, we display in the lower insets of figures 3  and 4 bin-by-bin ratios of the NLO, NLO+NLL and  NLO+NNLL rates to the most precise NLO+NNLL pre-  dictions evaluated with central scale choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' The bands  represent again the associated scale uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  We  observe that the increase of the differential NLO cross  section induced by NLL or NNLL resummation (or equiv-  alently, the decrease of the NLO cross sections, shown  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Same as figure 3 but for Type-III seesaw leptons and  the process (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  in blue, relative to the most precise NLO+NNLL pre-  dictions) depends on the invariant mass of the di-lepton  system M, and therefore indirectly on the mass of the  lepton species produced that fix the kinematic produc-  tion threshold M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Preferred configurations hence natu-  Pp → Z- E+@ LHC14  10-3  ms=900GeV  [fb/GeV]  LO  10-4  NLO  TT+OT  NLO+NLL  NLO+NNLL  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='00  doNLO+NNLL  op  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='90  M [GeV]1e-5  Pp → -Z+@ LHC14  6  mz=-1500GeV  [fb/GeV]  4  LO  NLO+NLL  NLO  NLO + NNLL  LO +LL  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='00  op  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='90  3000  3200  3400  3600  3800  4000  M [GeV]Pp → -Z+@ LHC14  10-2  m==600GeV  [fb/GeV]  10-3  LO  NLO  10  TT+OT  NLO+NLL  NLO+NNLL  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='00  doNLO+NNLL  op  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='90  1500  2000  2500  3000  3500  4000  M [GeV]10  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Transverse momentum spectra for the processes (12), together with the associated scale uncertainties (upper inset  in each panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' We show fixed-order results at NLO (blue), as well as LO+PS (green) and NLO+PS (olive) predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' We  consider lepton masses of 600 GeV (top row), 900 GeV (middle row) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='5 TeV (bottom row), and we additionally present  the corresponding bin-by-bin ratios to the NLO spectra with the associated scale uncertainties (lower inset in each panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='PP-→E-E+ @LHC14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='mE=600 GeV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='10-3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='[fb/GeV]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='10-5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='NLO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='10-7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='LO+PS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='NLO +PS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='do/doNLO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='2000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='Pr [GeV]10-1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='PP-→E-E+ @LHC14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='mE=600 GeV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='[fb/GeV]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='10-3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='10-5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='NLO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='LO +PS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='10-7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='NLO +PS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='do/doNLO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='2000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='Pr [GeV]PP→E- E+ @ LHC14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='10-3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='mE=900GeV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='[fb/GeV]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='10-5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='NLO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='LO+PS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='NLO +PS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='do/doNLO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='2000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='Pr [GeV]PP→E-E+ @LHC14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='mE=900-GeV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='[fb/GeV]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='10-5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='NLO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='10-7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='LO +PS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='NLO +PS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='do/doNLO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
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+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='2000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='Pr [GeV]PP→E- E+ @ LHC14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='m=1500GeV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='10~5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='[fb/GeV]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='10-7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='NLO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='LO +PS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='10-9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='NLO +PS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='do/doNLO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='2000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='Pr [GeV]PP→E-E+ @LHC14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='10-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='mE=1500 GeV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='[fb/GeV]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='10-6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='NLO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='10-8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='LO +PS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='NLO +PS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='do/doNLO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='2000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='Pr [GeV]11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='rally target larger z values closer to 1 for heavy lepton  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='production than for light lepton production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Resumma-  tion effects are therefore expected to be more important  for heavy leptons, when considering M values lying at a  given relative distance from M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Again, we observe that  NLO+(N)NLL results are generally outside the NLO er-  ror bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Nevertheless, the shape of the spectrum is stabilised  after including threshold resummation at NLL (yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  NNLL resummation (purple) only yields a mild increase  of the rate by less than 1%, which is largely indepen-  dent of the di-lepton invariant mass M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  This shows  that a good perturbative convergence has been achieved  at NLO+NNLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Moreover, NLO+NNLL predictions are  crucial to reduce the scale uncertainties to be less than  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='5%, hence motivating using more precise predictions  for BSM signals when available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Transverse momentum spectra  We now turn to the study of the distribution in the  transverse momentum (pT ) of the lepton pair, which is  a typical observable that shows that the inclusion of PS  is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Although the MG5aMC framework practi-  cally allows for investigations of any observable, we only  focus, for the sake of an example, on this distribution  in the pT of the di-lepton system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In figures 5 and 6,  we present dσ/dpT distributions at NLO (blue), LO+PS  (green) and NLO+PS (olive), whereas the LO distribu-  tions are trivially located at pT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' As in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='2,  we choose three benchmark scenarios featuring extra lep-  tons with masses of 600 GeV, 900 GeV and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='5 TeV re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' The different pT spectra are shown in the up-  per insets of the figures, whereas the lower insets display  their bin-by-bin ratios to the (fixed-order) NLO predic-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  While NLO predictions (blue) in principle diverge at  small pT due to uncancelled soft and/or collinear singu-  larities originating from real emission, the integration of  the differential cross section within a given bin regularises  this divergence, the bin-by-bin results shown in the fig-  ures being normalised by the bin size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' We therefore ob-  serve a finite cross section with a pronounced maximum  in the low pT region, with pT ≲ 30 GeV (which corre-  sponds to the first bin shown in the plots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' The cross  sections in the small pT regime are therefore expected to  be better described by matching fixed-order predictions  with PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Showering, as well as the non-perturbative in-  trinsic transverse momentum kT of the constituents of  the protons, should additionally distort the shapes of the  spectra in the low pT regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' NLO+PS predictions are  hence smaller than NLO ones at small pT ≲ 30 GeV, be-  fore becoming considerably greater at intermediate trans-  verse momenta (pT ∈ [60, 500] GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' At larger pT val-  ues, QCD radiation encoded in the real matrix elements  dominates, so that NLO+PS predictions agree with NLO  ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Same as figure 5 but for Type-III seesaw leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  This last effect can be even more evidenced by study-  ing the LO+PS curves (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Non-zero pT values are  here purely arising from shower effects, which gives rise  to a too soft spectrum in the high-pT regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In conclu-  sion, NLO+PS should be the best for describing such an  observable, while NLO (LO+PS) predictions fail at low  Pp-→Z- + @ LHC14  mz=600 GeV  10-2  [fb/GeV]  10-4  NLO  10-6  LO +PS  NLO +PS  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='5  do/doNLO  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0  0  500  1000  1500  2000  Pr [GeV]PP→Z-+ @ LHC14  ms=900 GeV  10-3  [fb/GeV]  10-5  NLO  LO+PS  10-7  NLO +PS  2  do/doNLO  0  500  1000  1500  2000  Pr [GeV]Pp→Z-Z+ @ LHC14  10  mz=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='GeV  [fb/GeV]  10-6  NLO  10-8  LO+PS  NLO +PS  2  do/doNLO  0  500  1000  1500  2000  Pr [GeV]12  (high) pT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Finally, we can note that multiplying LO+PS  predictions by an overall K-factor, as traditionally done  in many experimental and phenomenological studies, is  unjustified in the aim of an accurate signal description,  and will even yield qualitatively wrong results at high pT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  CONCLUSION  Numerous extensions of the Standard Model feature  additional leptons that carry a variety of different elec-  tric charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  They are consequently actively searched  for at collider experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In this work, we have stud-  ied the production of these extra leptons in effective  frameworks representative of the TeV-scale phenomenol-  ogy of several models featuring additional leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' By  providing FeynRules implementations and the associ-  ated UFO libraries for Type-III seesaw models and new  physics scenarios with VLLs, we complete the set of pub-  licly available models suitable for calculations relevant  for the production of new leptons at colliders beyond  the LO or LO+PS accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  For the fist time, VLL  and Type-III lepton production can now be simulated  at NLO+PS accuracy, as was already the case for other  neutrino mass models or left-right-symmetric scenarios  which both involve new non-coloured fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' The cor-  responding model files can be downloaded from https:  //feynrules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='irmp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='be/wiki/NLOModels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  We have reported the most precise calculations of to-  tal rates to date for the production of a pair of Type-III  leptons or VLLs lying in the trivial or fundamental repre-  sentations of SU(2)L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Our predictions include both NLO  QCD corrections and threshold resummation effects at  NNLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Higher-order QCD effects increase the production  rates by 25%–30%, the exact value depending on the sce-  nario and the extra lepton mass, and scale uncertainties  are reduced below 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' We have additionally investigated  the impact of these corrections on the distributions in the  invariant mass of the produced heavy lepton system and  observed a notable increase in the differential rates and  a significant reduction of the scale uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Finally, we have made use of the designed UFO models  to highlight the joint impact of NLO corrections and PS  matching on an example of observable relevant for exist-  ing searches for extra leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' We have chosen the distri-  bution in the transverse momentum of the di-lepton sys-  tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' We have illustrated how NLO+PS improves fixed-  order NLO computations at low pT , and provides a better  modelling of the physical spectra at intermediate pT val-  ues below 1 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' These calculations have been achieved  fully automatically, in the MG5aMC framework, in a  setup similar to that used in ATLAS and CMS studies as  well as in many existing phenomenological explorations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Upgrades of existing studies and searches to include NLO  corrections matched with PS should therefore be straight-  forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The authors are grateful to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Cirelli and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Soyez for  motivating discussions during the course of this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  We acknowledge support from the European Union’s  Horizon 2020 research and innovation programme (grant  agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 824093,  STRONG-2020,  EU Virtual  Access “NLOAccess”), the European Research Coun-  cil (ERC grant agreement ID 101041109, “BOSON”),  the French ANR (grants ANR-20-CE31-0015, “PrecisO-  nium”, and ANR-21-CE31-0013, “DMwithLLPatLHC”),  and the CNRS IEA (grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 205210, “GlueGraph”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Appendix A: Resummation coefficients  In this section, we provide analytical results for the  resummation coefficients appearing in (10) in the case  of a Drell-Yan-like process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Those comprise the process-  dependent term ˜g0,q¯q, as well as the universal coefficients  of the exponent gk,q¯q with k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' The latter are iden-  tical to those given in appendix A of [90] (the λ param-  eter of [90] being replaced by ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' We then provide be-  low the expression for the process-dependent coefficient  ˜g0,q¯q(M 2, µ2  F , µ2  R) that can be expanded in as as  ˜g0,q¯q(M 2, µ2  F , µ2  R) =  dˆσ(0)  q¯q  d ln M 2  �  1 +  �  k=1  ak  s(µ2  R)˜g(k)  0,q¯q  �  ,  (A1)  where  dˆσ(0)  q¯q  d ln M 2 is the Born partonic cross section differen-  tiating with respect to ln M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' The ˜g(1)  0,q¯q and ˜g(2)  0,q¯q coeffi-  cients in QCD (for nq = 5) relevant for resummation at  NNLL explicitly read  ˜g(1)  0,q¯q = −64  3  + 64  3 ζ2 − 8Lfr + 8Lqr ,  ˜g(2)  0,q¯q = −1291  9  + 64ζ2  9  + 368ζ2  2  3  + 4528ζ3  27  +  188L2  fr  3  + 4L2  qr  3  + Lfr  �1324  9  − 1888ζ2  9  + 32ζ3  3  �  + Lqr  �148  9  − 64Lfr + 416ζ2  9  − 32ζ3  3  �  ,  (A2)  with Lqr = ln M 2  µ2  R , Lfr = ln µ2  F  µ2  R and ζn being the Riemann  zeta function ζ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  In the processes (12) and (13), the di-lepton system  is produced either through an s-channel virtual photon  exchange, or through an s-channel Z-boson exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In  other words, the Born partonic cross section can be split  into three pieces, namely the square of photon-exchange  diagram, that of the Z-exchange diagram, and the inter-  ference between the two diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Mathematically, this  13  can be written as  ˆσ(0)  q¯q = ˆσ(0),γ  q¯q  + ˆσ(0),Z  q¯q  + ˆσ(0),int  q¯q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  (A3)  The photon exchange contribution depends on the elec-  tric charge of the initial-state quarks Qq, and on that of  the final-state leptons Qℓ,  ˆσ(0),γ  q¯q  = Q2  qQ2  ℓ  4πα2  9M 2  �  1 − 4m2  ℓ  M 2  �  1 + 2m2  ℓ  M 2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  (A4)  In this expression, mℓ is the mass of the produced lepton,  and it is thus respectively given by m ˜  E, mE and mΣ in  the three processes shown in (12) and (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' As all exotic  leptons produced in these processes satisfy Qℓ = −1,  ˆσ(0),γ  q¯q  is identical in all three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  The Z-boson exchange contribution and the interfer-  ence term read  ˆσ(0),Z  q¯q  = f 2  ℓ  (V 2  q + A2  q)  4  M 4  (M 2 − m2  Z)2 + Γ2  Zm2  Z  ˆσ(0),γ  q¯q  Q2q  ,  ˆσ(0),int  q¯q  = fℓQqVq  M 2(M 2 − m2  Z)  (M 2 − m2  Z)2 + Γ2  Zm2  Z  ˆσ(0),γ  q¯q  Q2q  ,  (A5)  where Vq = Iq −2s2  WQq and Aq = Iq represent the vector  and axial couplings between the initial quarks and the Z  boson respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Here, the parameters cW and sW are  the cosine and sine of the electroweak mixing angle, and  Iq is the weak isospin quantum number of the quark q,  that is thus equal to 1/2 for up-type quarks and −1/2 for  down-type quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' In addition, ΓZ denotes the width of  the Z boson, and the prefactor fℓ depends on the quan-  tum numbers of the lepton ℓ produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' It is thus given  by  fℓ =  �  �  �  �  �  �  �  �  �  −c−2  W  for  ℓ = ˜E ,  c2  W −s2  W  2s2  W c2  W  for  ℓ = E ,  s−2  W  for  ℓ = Σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  (A6)  [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Panico and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Wulzer, The Composite  Nambu-Goldstone Higgs, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 913.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Springer, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  arXiv:1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='01961 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [2] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Cacciapaglia, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Pica, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Sannino,  “Fundamental Composite Dynamics: A Review,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 877 (2020) 1–70, arXiv:2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='04914 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [3] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Cacciapaglia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Deandrea, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Sridhar, “Review  of fundamental composite dynamics,” Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' ST  231 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 7, (2022) 1221–1222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Morais, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Pasechnik, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Porod, “Grand  Unified Origin of Gauge Interactions and Families  Replication in the Standard Model,” Universe 7 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 12,  (2021) 461, arXiv:2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='04804 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Morais, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Pasechnik, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Porod, “Prospects  for new physics from gauge left-right-colour-family  grand unification hypothesis,” Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' C 80  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 12, (2020) 1162, arXiv:2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='06383 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Martin, “Extra vector-like matter and the lightest  Higgs scalar boson mass in low-energy supersymmetry,”  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' D 81 (2010) 035004, arXiv:0910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='2732  [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Endo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Hamaguchi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Iwamoto, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Yokozaki,  “Higgs Mass and Muon Anomalous Magnetic Moment  in Supersymmetric Models with Vector-Like Matters,”  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' D 84 (2011) 075017, arXiv:1108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='3071  [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [8] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Araz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Banerjee, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Frank, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Fuks, and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Goudelis, “Dark matter and collider signals in an  MSSM extension with vector-like multiplets,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' D 98 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 11, (2018) 115009, arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='07224  [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Pati and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Salam, “Lepton Number as the  Fourth Color,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' D 10 (1974) 275–289.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [Erratum: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='D 11, 703–703 (1975)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [10] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Mohapatra and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Pati, “A Natural Left-Right  Symmetry,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' D 11 (1975) 2558.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [11] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Senjanovic and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Mohapatra, “Exact Left-Right  Symmetry and Spontaneous Violation of Parity,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' D 12 (1975) 1502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [12] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Mohapatra, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Paige, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Sidhu,  “Symmetry Breaking and Naturalness of Parity  Conservation in Weak Neutral Currents in Left-Right  Symmetric Gauge Theories,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' D 17 (1978)  2462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Halverson, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Orlofsky, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Pierce, “Vectorlike  Leptons as the Tip of the Dark Matter Iceberg,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' D 90 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 1, (2014) 015002, arXiv:1403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1592  [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [14] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Minkowski, “µ → eγ at a Rate of One Out of 109  Muon Decays?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=',” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' B 67 (1977) 421–428.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [15] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Mohapatra and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Senjanovic, “Neutrino Mass  and Spontaneous Parity Nonconservation,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 44 (1980) 912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [16] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Yanagida, “Horizontal gauge symmetry and masses  of neutrinos,” Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' C 7902131 (1979) 95–99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Gell-Mann, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Ramond, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Slansky, “Complex  Spinors and Unified Theories,” Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' C 790927  (1979) 315–321, arXiv:1306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='4669 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Glashow, “The Future of Elementary Particle  Physics,” NATO Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' B 61 (1980) 687.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [19] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Shrock, “General Theory of Weak Leptonic and  Semileptonic Decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Leptonic Pseudoscalar Meson  Decays, with Associated Tests For, and Bounds on,  Neutrino Masses and Lepton Mixing,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' D 24  14  (1981) 1232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Schechter and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Valle, “Neutrino Masses in  SU(2) x U(1) Theories,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' D 22 (1980) 2227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [21] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Cai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Han, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Li, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Ruiz, “Lepton Number  Violation: Seesaw Models and Their Collider Tests,”  Front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' in Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 6 (2018) 40, arXiv:1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='02180  [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [22] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Foot, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Lew, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' He, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Joshi, “Seesaw  Neutrino Masses Induced by a Triplet of Leptons,” Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' C 44 (1989) 441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [23] ATLAS Collaboration, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Aaboud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=', “Search for  heavy Majorana or Dirac neutrinos and right-handed W  gauge bosons in final states with two charged leptons  and two jets at √s = 13 TeV with the ATLAS  detector,” JHEP 01 (2019) 016, arXiv:1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='11105  [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [24] ATLAS Collaboration, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Aaboud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=', “Search for a  right-handed gauge boson decaying into a  high-momentum heavy neutrino and a charged lepton in  pp collisions with the ATLAS detector at √s = 13  TeV,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' B 798 (2019) 134942,  arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='12679 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [25] ATLAS Collaboration, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Aad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=', “Search for  heavy neutral leptons in decays of W bosons produced  in 13 TeV pp collisions using prompt and displaced  signatures with the ATLAS detector,” JHEP 10 (2019)  265, arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='09787 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [26] ATLAS Collaboration, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Aad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=', “Search for  type-III seesaw heavy leptons in dilepton final states in  pp collisions at √s = 13 TeV with the ATLAS  detector,” Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' C 81 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 3, (2021) 218,  arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='07949 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [27] ATLAS Collaboration, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Aad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=', “Search for new  phenomena in three- or four-lepton events in pp  collisions at √s =13 TeV with the ATLAS detector,”  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' B 824 (2022) 136832, arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='00404  [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [28] ATLAS Collaboration, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Aad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=', “Search for  type-III seesaw heavy leptons in leptonic final states in  pp collisions at √s = 13 TeV with the ATLAS  detector,” arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='02039 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [29] CMS Collaboration, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Sirunyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=', “Search for  Evidence of the Type-III Seesaw Mechanism in  Multilepton Final States in Proton-Proton Collisions at  √s = 13 TeV,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 119 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 22, (2017)  221802, arXiv:1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='07962 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [30] CMS Collaboration, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Sirunyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=', “Search for  heavy neutral leptons in events with three charged  leptons in proton-proton collisions at √s = 13 TeV,”  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 120 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 22, (2018) 221801,  arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='02965 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [31] CMS Collaboration, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Sirunyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=', “Search for  a heavy right-handed W boson and a heavy neutrino in  events with two same-flavor leptons and two jets at  √s = 13 TeV,” JHEP 05 (2018) 148,  arXiv:1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='11116 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [32] CMS Collaboration, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Sirunyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=', “Search for  heavy Majorana neutrinos in same-sign dilepton  channels in proton-proton collisions at √s = 13 TeV,”  JHEP 01 (2019) 122, arXiv:1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='10905 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [33] CMS Collaboration, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Sirunyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=', “Search for  heavy neutrinos and third-generation leptoquarks in  hadronic states of two τ leptons and two jets in  proton-proton collisions at √s = 13 TeV,” JHEP 03  (2019) 170, arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='00806 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [34] CMS Collaboration, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Sirunyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=', “Search for  vector-like leptons in multilepton final states in  proton-proton collisions at √s = 13 TeV,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' D  100 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 5, (2019) 052003, arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='10853 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [35] CMS Collaboration, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Sirunyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=', “Search for  physics beyond the standard model in multilepton final  states in proton-proton collisions at √s = 13 TeV,”  JHEP 03 (2020) 051, arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='04968 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [36] CMS Collaboration, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Tumasyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=', “Search for a  right-handed W boson and a heavy neutrino in  proton-proton collisions at √s = 13 TeV,” JHEP 04  (2022) 047, arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='03949 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [37] CMS Collaboration, “Search for resonant and  nonresonant production of pairs of dijet resonances in  proton-proton collisions at √s = 13 TeV,”  arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='09997 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [38] CMS Collaboration, “Search for pair-produced  vector-like leptons in final states with third-generation  leptons and at least three b quark jets in proton-proton  collisions at √s = 13 TeV,” arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='09700  [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [39] ATLAS Collaboration, “Search for heavy neutral  leptons in decays of W bosons using a dilepton  displaced vertex in √s = 13 TeV pp collisions with the  ATLAS detector,” arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='11988 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [40] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Degrande, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Mattelaer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Ruiz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Turner,  “Fully-Automated Precision Predictions for Heavy  Neutrino Production Mechanisms at Hadron Colliders,”  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' D 94 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 5, (2016) 053002,  arXiv:1602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='06957 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [41] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Fuks, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Neundorf, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Peters, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Ruiz, and  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Saimpert, “Majorana neutrinos in same-sign  W ±W ± scattering at the LHC: Breaking the TeV  barrier,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' D 103 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 5, (2021) 055005,  arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='02547 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [42] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Mattelaer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Mitra, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Ruiz, “Automated  Neutrino Jet and Top Jet Predictions at  Next-to-Leading-Order with Parton Shower Matching in  Effective Left-Right Symmetric Models,”  arXiv:1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='08985 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [43] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Debove, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Fuks, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Klasen, “Threshold  resummation for gaugino pair production at hadron  colliders,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' B 842 (2011) 51–85,  arXiv:1005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='2909 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [44] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Fuks, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Klasen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Lamprea, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Rothering,  “Gaugino production in proton-proton collisions at a  center-of-mass energy of 8 TeV,” JHEP 10 (2012) 081,  arXiv:1207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='2159 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [45] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Fuks, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Klasen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Lamprea, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Rothering,  “Precision predictions for electroweak superpartner  production at hadron colliders with Resummino,” Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' C 73 (2013) 2480, arXiv:1304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0790 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [46] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Fiaschi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Klasen, “Higgsino and gaugino pair  production at the LHC with aNNLO+NNLL precision,”  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' D 102 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 9, (2020) 095021,  arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='02294 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [47] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Degrande, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Duhr, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Fuks, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Grellscheid,  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Mattelaer, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Reiter, “UFO - The Universal  FeynRules Output,” Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 183  (2012) 1201–1214, arXiv:1108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='2040 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [48] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Alwall, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Frederix, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Frixione, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Hirschi,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Maltoni, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Mattelaer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Shao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Stelzer,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Torrielli, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Zaro, “The automated computation  15  of tree-level and next-to-leading order differential cross  sections, and their matching to parton shower  simulations,” JHEP 07 (2014) 079, arXiv:1405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0301  [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [49] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Frederix, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Frixione, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Hirschi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Pagani, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Shao, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Zaro, “The automation of next-to-leading  order electroweak calculations,” JHEP 07 (2018) 185,  arXiv:1804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='10017 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' [Erratum: JHEP 11, 085  (2021)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [50] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Sterman, “Summation of Large Corrections to  Short Distance Hadronic Cross-Sections,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' B  281 (1987) 310–364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [51] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Catani and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Trentadue, “Resummation of the QCD  Perturbative Series for Hard Processes,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' B  327 (1989) 323–352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [52] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Vogt, “Next-to-next-to-leading logarithmic threshold  resummation for deep inelastic scattering and the  Drell-Yan process,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' B 497 (2001) 228–234,  arXiv:hep-ph/0010146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [53] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Buchkremer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Cacciapaglia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Deandrea, and  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Panizzi, “Model Independent Framework for  Searches of Top Partners,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' B 876 (2013)  376–417, arXiv:1305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='4172 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [54] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Fuks and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Shao, “QCD next-to-leading-order  predictions matched to parton showers for vector-like  quark models,” Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' C 77 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 2, (2017) 135,  arXiv:1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='04622 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [55] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Christensen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' de Aquino, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Degrande,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Duhr, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Fuks, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Herquet, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Maltoni, and  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Schumann, “A Comprehensive approach to new  physics simulations,” Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' C 71 (2011) 1541,  arXiv:0906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='2474 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [56] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Alloul, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Christensen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Degrande, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Duhr,  and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Fuks, “FeynRules 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0 - A complete toolbox for  tree-level phenomenology,” Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  185 (2014) 2250–2300, arXiv:1310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1921 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [57] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Skands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=', “SUSY Les Houches accord:  Interfacing SUSY spectrum calculators, decay packages,  and event generators,” JHEP 07 (2004) 036,  arXiv:hep-ph/0311123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [58] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Biggio and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Bonnet, “Implementation of the Type  III Seesaw Model in FeynRules/MadGraph and  Prospects for Discovery with Early LHC Data,” Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' C 72 (2012) 1899, arXiv:1107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='3463 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [59] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Li and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' He, “Neutrino Masses and Heavy  Triplet Leptons at the LHC: Testability of Type III  Seesaw,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' D 80 (2009) 093003,  arXiv:0907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='4193 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [60] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Abada, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Biggio, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Bonnet, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Gavela, and  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Hambye, “Low energy effects of neutrino masses,”  JHEP 12 (2007) 061, arXiv:0707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='4058 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [61] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Abada, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Biggio, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Bonnet, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Gavela, and  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Hambye, “mu —> e gamma and tau —> l gamma  decays in the fermion triplet seesaw model,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  D 78 (2008) 033007, arXiv:0803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0481 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [62] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Fuks, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Nemevˇsek, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Ruiz, “Doubly Charged  Higgs Boson Production at Hadron Colliders,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' D 101 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 7, (2020) 075022, arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='08975  [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [63] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Duhr and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Fuks, “A superspace module for the  FeynRules package,” Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 182  (2011) 2404–2426, arXiv:1102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='4191 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [64] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Frixione, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Fuks, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Hirschi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Mawatari, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Shao, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Sunder, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Zaro, “Automated  simulations beyond the Standard Model:  supersymmetry,” JHEP 12 (2019) 008,  arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='04898 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [65] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Degrande, “Automatic evaluation of UV and R2  terms for beyond the Standard Model Lagrangians: a  proof-of-principle,” Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 197  (2015) 239–262, arXiv:1406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='3030 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [66] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Hahn, “Generating Feynman diagrams and  amplitudes with FeynArts 3,” Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  140 (2001) 418–431, arXiv:hep-ph/0012260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [67] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Bellm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=', “Herwig 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0/Herwig++ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0 release  note,” Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' C 76 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 4, (2016) 196,  arXiv:1512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='01178 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [68] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Gleisberg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Hoeche, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Krauss, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Schonherr,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Schumann, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Siegert, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Winter, “Event  generation with SHERPA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1,” JHEP 02 (2009) 007,  arXiv:0811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='4622 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [69] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Frixione, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Kunszt, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Signer, “Three jet  cross-sections to next-to-leading order,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' B  467 (1996) 399–442, arXiv:hep-ph/9512328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [70] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Frixione, “A General approach to jet cross-sections in  QCD,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' B 507 (1997) 295–314,  arXiv:hep-ph/9706545.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [71] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Frederix, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Frixione, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Maltoni, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Stelzer,  “Automation of next-to-leading order computations in  QCD: The FKS subtraction,” JHEP 10 (2009) 003,  arXiv:0908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='4272 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [72] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Frederix, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Frixione, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Papanastasiou,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Prestel, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Torrielli, “Off-shell single-top  production at NLO matched to parton showers,” JHEP  06 (2016) 027, arXiv:1603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='01178 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [73] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Ossola, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Papadopoulos, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Pittau,  “Reducing full one-loop amplitudes to scalar integrals  at the integrand level,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' B 763 (2007)  147–169, arXiv:hep-ph/0609007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [74] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Mastrolia, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Mirabella, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Peraro, “Integrand  reduction of one-loop scattering amplitudes through  Laurent series expansion,” JHEP 06 (2012) 095,  arXiv:1203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0291 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' [Erratum: JHEP 11, 128  (2012)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [75] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Passarino and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Veltman, “One Loop  Corrections for e+ e- Annihilation Into mu+ mu- in the  Weinberg Model,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' B 160 (1979) 151–207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [76] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Davydychev, “A Simple formula for reducing  Feynman diagrams to scalar integrals,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' B  263 (1991) 107–111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [77] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Denner and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Dittmaier, “Reduction schemes for  one-loop tensor integrals,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' B 734 (2006)  62–115, arXiv:hep-ph/0509141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [78] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Hirschi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Frederix, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Frixione, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Garzelli,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Maltoni, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Pittau, “Automation of one-loop  QCD corrections,” JHEP 05 (2011) 044,  arXiv:1103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='0621 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [79] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Ossola, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Papadopoulos, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Pittau,  “CutTools: A Program implementing the OPP  reduction method to compute one-loop amplitudes,”  JHEP 03 (2008) 042, arXiv:0711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='3596 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [80] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Peraro, “Ninja: Automated Integrand Reduction via  Laurent Expansion for One-Loop Amplitudes,”  Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 185 (2014) 2771–2797,  arXiv:1403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='1229 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [81] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Hirschi and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Peraro, “Tensor integrand reduction  via Laurent expansion,” JHEP 06 (2016) 060,  arXiv:1604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='01363 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  16  [82] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Denner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Dittmaier, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Hofer, “Collier: a  fortran-based Complex One-Loop LIbrary in Extended  Regularizations,” Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 212 (2017)  220–238, arXiv:1604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='06792 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [83] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Cascioli, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Maierhofer, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Pozzorini, “Scattering  Amplitudes with Open Loops,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 108  (2012) 111601, arXiv:1111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='5206 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [84] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Frixione and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Webber, “Matching NLO QCD  computations and parton shower simulations,” JHEP  06 (2002) 029, arXiv:hep-ph/0204244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [85] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Ajjath and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Shao, “N3LO+N3LL QCD  improved Higgs pair cross sections,” arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='03914  [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [86] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Sj¨ostrand, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Ask, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Christiansen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Corke,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Desai, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Ilten, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Mrenna, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Prestel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  Rasmussen, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Skands, “An introduction to  PYTHIA 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='2,” Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 191 (2015)  159–177, arXiv:1410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='3012 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [87] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Hou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=', “New CTEQ global analysis of  quantum chromodynamics with high-precision data  from the LHC,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' D 103 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' 1, (2021) 014013,  arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='10053 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [88] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Buckley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Ferrando, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Lloyd, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Nordstr¨om,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Page, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' R¨ufenacht, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Sch¨onherr, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Watt,  “LHAPDF6: parton density access in the LHC  precision era,” Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' C 75 (2015) 132,  arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='7420 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [89] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Biggio, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Fernandez-Martinez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Filaci,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Hernandez-Garcia, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Lopez-Pavon, “Global  Bounds on the Type-III Seesaw,” JHEP 05 (2020) 022,  arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='11790 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='  [90] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Ajjath, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Chakraborty, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Das, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Mukherjee,  and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content=' Ravindran, “Resummed prediction for Higgs  boson production through bb annihilation at N3LL,”  JHEP 11 (2019) 006, arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
+page_content='03771 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfEwbs/content/2301.03640v1.pdf'}
diff --git a/adE2T4oBgHgl3EQfZwdG/content/tmp_files/2301.03867v1.pdf.txt b/adE2T4oBgHgl3EQfZwdG/content/tmp_files/2301.03867v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..8122459ea145d3122bd951bad70130cd8e1f025b
--- /dev/null
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@@ -0,0 +1,623 @@
+Sentiment-based Engagement Strategies for intuitive Human-Robot
+Interaction
+Thorsten Hempel
+a, Laslo Dinges
+b and Ayoub Al-Hamadi
+c
+Neuro-Information Technology, Faculty of Electrical Engineering and Information Technology,
+Otto von Guericke University, Magdeburg, Germany
+{thorsten.hempel, laslo.dinges, ayoub.al-hamadi}@ovgu.de
+Keywords:
+human-robot interaction, approaching strategy, sentiment estimation, emotion detection, anticipating human
+behaviors, approaching people
+Abstract:
+Emotion expressions serve as important communicative signals and are crucial cues in intuitive interactions
+between humans. Hence, it is essential to include these fundamentals in robotic behavior strategies when inter-
+acting with humans to promote mutual understanding and to reduce misjudgements. We tackle this challenge
+by detecting and using the emotional state and attention for a sentiment analysis of potential human inter-
+action partners to select well-adjusted engagement strategies. This way, we pave the way for more intuitive
+human-robot interactions, as the robot’s action conforms to the person’s mood and expectation. We propose
+four different engagement strategies with implicit and explicit communication techniques that we implement
+on a mobile robot platform for initial experiments.
+1
+Introduction
+Once utopian, robots are increasingly moving from
+industrial and laboratory settings to the real-world
+in order to assist humans in everyday tasks. In this
+process, human-robot interaction became a central
+point of interest in robotic research that investigates
+the manifold challenges in performing interactive and
+collaborative tasks. As a general principle, the robot
+needs to be acquired with the necessary skills to en-
+able intuitive interactions with arbitrary human inter-
+action partners, regardless of the corresponding task,
+the human’s intention and communication behavior.
+To this end, a fundamental objective is the anticipa-
+tion of appropriate strategies to proactively approach
+or evade people based on the situation-specific con-
+text, such as the person’s mood, attitude (M¨uller-
+Abdelrazeq et al., 2019; Elprama et al., 2016) and
+ressentiments towards robots (Naneva et al., 2020;
+Miller et al., 2021). First, it requires the determina-
+tion of people’s interest in interacting at all (Satake
+et al., 2009; Finke et al., 2005). Thereupon, the robot
+has to find an appropriate strategy for either engaging
+people to establish an interaction or to avoid a con-
+a
+https://orcid.org/0000-0002-3621-7194
+b
+https://orcid.org/0000-0003-0364-8067
+c
+https://orcid.org/0000-0002-3632-2402
+frontation if desired. In order to reach these under-
+lying decisions, the robot has to cope with a row of
+sub-tasks: estimating people’s focus of attention (Dini
+et al., 2017), predicting their mental state, and carry-
+ing out a (dis-)engagement strategy (Avelino et al.,
+2021).
+Especially, the estimation of the interaction will-
+ingness is a very challenging task, as this mental state
+tends to provide only vague social signals.
+As an
+alternative approach, many methods (Oertel et al.,
+2020) go after the detection of engagement in order
+to determine if a person already approached the robot
+and waits for the robot’s reaction. This state expresses
+itself more clearly, e.g., in voice commands (Foster
+et al., 2017), gestures and proxemics, but it under-
+mines the proactive approach and neglects situations
+where the human partner is in need of help, but too
+uncertain or not aware of the robot.
+In this work, we close the gap between the lack of
+proactive behavior and advanced mental state predic-
+tions by introducing a sentiment-based engagement
+strategy.
+By combining the focus of attention and emotions,
+we derive a sentiment analysis that allows us to select
+fine-grained behavior patterns that exceed current bi-
+nary engagement and disengagement strategies. We
+implement our approach on a mobile robot platform
+and execute our strategies using explicit and implicit
+arXiv:2301.03867v1  [cs.RO]  10 Jan 2023
+
+embodied communication.
+Our models are trained
+on minimum sized networks in order to perform our
+method on mobile systems with limited computa-
+tional resources.
+2
+Related Work
+In recent years, proxemic rules have been a popular
+tool to draw conclusions about the different states of
+human-robot interactions (Mumm and Mutlu, 2011)
+and how spatial zones can be leverages to improve
+them (Syrdal et al., 2007). (Repiso et al., 2018) pre-
+dict appropriate encounter points to achieve a natu-
+ral engagement with a group of people. Similarly,
+(Satake et al., 2009) proactively approach detected
+people to offer help. (Fischer et al., 2021) verbally
+greet people and dynamically adapt the voice volume
+based on the distance to the target person. Follow-
+ing (Kendon, 1990), (Carvalho et al., 2021) apply the
+Kendon’s greeting model to approach people while
+tracking the human mental states of the interaction us-
+ing multiple features, such as gaze, facial expressions,
+and gestures. Yet, all of these methods neglect if the
+human counterpart is actually interested and ready to
+initiate an interaction. This is problematic, as in the
+case of negative attitude, the robot’s interaction ef-
+forts can be perceived as rude and annoying (Br¨ohl
+et al., 2019). An initial attempt to address this was
+presented by (Kato et al., 2015), who only approach
+and offer help if they detect signs of attention towards
+the robot. But this still doesn’t incorporate the actual
+mental attitude of the human counterpart towards the
+interaction itself.
+The recent advances in the area of deep-learning
+opened up new possibilities for the visual analysis of
+humans, starting from general detection tasks to the
+estimation of facial micro expressions. This allows
+the aggregation of additional relevant information,
+such as emotions (Chuah and Yu, 2021; Spezialetti
+et al., 2020), to better understand the behavior and in-
+tentions of the human interaction partner. To the best
+of our knowledge, we are the first to leverage this ad-
+vancement by predicting sentiment states based on vi-
+sual emotion estimation to improve interaction strat-
+egy for robots. We summarize our main contributions
+as follows:
+• We propose an image-based sentiment analysis to
+gauge the current interaction preference of poten-
+tial human interaction partners.
+• We propose four different behavior strategies (En-
+gage, Attract, Ignore, Avoid) that comprises a
+number of different explicit and implicit commu-
+nication modalities.
+Yaw, Tilt Head
+Rotate Base
+Lifting Torso
+Speech
+Figure 1: The TIAGo robot platform with the key opera-
+tions used in our method.
+• We design ultra lightweight models to implement
+our method and integrate it on a mobile robot plat-
+form.
+• We conduct initial experiments in laboratory set-
+tings for an early evaluation.
+3
+Method
+This section explains each step of our proposed
+method for sentiment-based interaction behavior.
+The sentiment analysis is based on two main fea-
+tures: head pose and emotion estimation. The head
+pose estimation is a leading indicator to gauge the cur-
+rent visual focus of attention of the person. The emo-
+tion estimation predicts the current emotional state of
+the person. Together, it allows the assumption about
+a person that a) is seeking for interaction, b) is in-
+decisive about it or c) doesn’t want to engage with
+the robot at all. Determining each sentiment state en-
+ables to perform a dedicated robot behavior that cor-
+responds with the expected reaction from the person
+and, thus, improves an intuitive interaction. Figure 2
+illustrates an overview of our proposed system. In the
+following subsection, we will give details about each
+component of the system and its interplay.
+3.1
+Robot platform
+We use the TIAGo robot from PAL Robotics, a mobile
+service robot for indoor environments. It is equipped
+with an RGB-D camera mounted inside its head, that
+can be yawed and pitched to dynamically perceive the
+environment. For grasping and moving objects, the
+
+PAILO
+PALOrobot has a manipulator arm with a 5 finger gripper.
+The torso can be lifted to adjust the robot’s height, and
+integrated speakers allow the output of voice com-
+mands. These abilities allow the combination of ex-
+plicit and implicit communication modalities that we
+use in our method.
+• Head Following: The robot’s head aims at and fol-
+lows a moving person as an implicit signal of at-
+tentiveness without actively approaching.
+• Body Following: If a person exceeds the range of
+the robot’s moving head, the entire robot base ro-
+tates in order to keep track of the person. This
+behavior communicates an attention commitment
+and is therefore an even stronger signal as the
+head following.
+• Torso Lifting: Lifting the torso is a rather subtle
+communication cue, as it is perceived similarly to
+the social convention of standing up while greet
+someone.
+• Speech: Approaching people with speech rep-
+resents an explicit communication strategy, that
+stronger commits them to a reaction.
+Hence,
+speech should be only used if the approached per-
+son is highly expected to welcome an interaction.
+3.2
+Visual Attention
+Typical indicators for gauging the human visual at-
+tention are the gaze and, more coarsely, the head
+pose. As the gaze depends on the eyes, which takes
+only a small part in face images, it is prone to er-
+rors. Therefore, we estimate the visual attention of
+the surrounding persons based on a head pose predic-
+tion algorithm. At first, we locate faces in the image
+stream using an ultra light SSD face detector. Then,
+the face crops are further processed by the 6DRep-
+Net (Hempel et al., 2022), that uses a rotation repre-
+sentation to directly regress yaw, pitch, and roll an-
+gle of the faces. To assure real-time processing capa-
+bilities, we replace the original 6DRepNet backbone
+with the efficient MobileNetv3-Small (Howard et al.,
+2019). The new head pose prediction model is able to
+maintain 90% of its accuracy compared to the origi-
+nal model, but is downsized to only one tenth of its
+parameters count.
+3.3
+Emotion Estimation
+In order to assess the attitude towards robots as well
+as differentiated subjective states such as dislike or
+willingness to cooperate, basic emotions can be used
+as indicating features. Naturally, these are frequently
+communicated by facial expressions.
+Hence, au-
+tomatic facial expression recognition is an essen-
+tial method to generate features for Human-Robot-
+Collaboration and attention prediction.
+Previous facial expression recognition methods
+adhere to the established pipeline of face detection,
+landmark extraction, and action unit (AU). Then,
+emotions are classified using these AUs as feature
+vectors (Wegrzyn et al., 2017; Vinkemeier et al.,
+2018; Werner et al., 2017).
+However, end-to-end
+learning on a single but more comprehensive, and bet-
+ter generalizing database, often outperforms such tra-
+ditional approaches. The AffectNet database (Molla-
+hosseini et al., 2017) for example contains about 500k
+in-the-wild samples for neutral and the basic emo-
+tions happy, disgust, fear, surprise, anger, and sad-
+ness. Furthermore, it is the only database, that also
+includes ground truth of valence and arousal (VA),
+which are, in contrast to the basic emotion classes,
+continuous labels ∈ [−1,1] which can be used for re-
+gression tasks. VA is less intuitively interpretable by
+humans, but it allows differentiation between facial
+expressions of varying degrees. This is particularly
+useful for capturing modest but long-term changes
+in expressed emotion. In addition, a multitask net-
+work (simultaneous training of classification and re-
+gression) also improved the accuracy of the classifi-
+cation. Furthermore, Valence summarizes essential
+information about several emotions in one parameter
+(negative, neutral, positive), which may facilitate the
+development of heuristic behavior rules for some sce-
+narios.
+In our former work, we proposed a multitask
+network based on YOLOv3 (Dinges et al., 2021).
+For the current paper, we compared EfficientNetV2-
+S, ResNet18, and MobileNetV3-Large. Despite the
+poorer converging training loss, MobileNet, which is
+optimized for limited hardware, achieved almost the
+same accuracy on the test set as EfficientNet (devia-
+tion < 1%). In addition, we used a weighted random
+sampler to handle the imbalanced dataset and re-train
+the network on the training and test set to generate
+an application model with increased robustness to un-
+seen data.
+3.4
+Sentiment Analysis and
+Approaching Strategy
+Detecting emotional expressions combined with the
+visual focus of attention allows perceiving important
+communicative signals. As in human-human interac-
+tion, you may be more likely to greet a stranger who
+smiles at you but avoid eye contact if that stranger
+is scowling.
+We transfer these principles to deter-
+
+Engage
+Attract
+Avoid
+Ignore
+Head Pose Estimation
+Emotion estimation
+Face Detection
+Perception
+Behavior Strategy
+Sentiment Analysis
+Emotional state with
+focus of attention  
+Figure 2: Method overview of our proposed system. Detected faces are used to estimate the head pose and the corresponding
+emotions based on facial expressions. Both information is used to determine the emotional state with the corresponding visual
+focus of the person’s attention that indicates the sentiment and allows a fine-tuned behavior strategy.
+Happy
+Uncertain
+Surprise
+Neutral
+Sad
+Fear
+Disgust
+Anger
+Facing
+Averted
+ENGAGE:
+Lift torso, follow with
+head and base, and
+approach by speech
+ATTRACT:
+Lift torso and follow
+with head
+AVOID:
+Look down and avoid
+attraction  
+ 
+ 
+ 
+IGNORE:
+ 
+Follow with head
+ 
+ 
+ 
+ 
+Figure 3: Behavior strategies based on the emotional and
+attentive state.
+mine states of sentiment based on the combination
+of the different emotions and visual attention. Is a
+person showing visual signs of fear or anger, while
+he is attentive towards the robot, it indicates a nega-
+tive attitude towards the robot. In this case, a passive
+robot behavior without hectic movements can help to
+defuse the situation and to calm the person. If a per-
+son is feared, but not attentive towards the robot, the
+reason for the fear might not be the robot. Execut-
+ing a more active robot behavior to show awareness
+could be therefore a more suitable strategy. Hence,
+our sentiment analysis allows us to compartmental-
+ize the common hard binary engagement strategy into
+soft behavior. Instead of choosing between engaging
+and not engaging, we choose between four different
+behaviors: Engage, Attract, Avoid, Ignore.
+• Engage: Engaging will extend the torso of the
+robot to signal the awareness of the person’s pres-
+ence. Additionally, voice commands will greet
+the people to actively initiate an interaction. This
+strategy is called, when the attention is directed at
+the robot with clear positive emotional attitude.
+• Attract: Positive emotional attitude without atten-
+tion towards the robot leads to more passive be-
+havior. The goal is to signal the offer for inter-
+action and help without actively committing to an
+interaction. This is done by extending the torso
+to show awareness, facing the person, but without
+verbally greeting. This way, a non-verbal offer for
+support (especially in case of uncertainty and sad-
+ness) is subtly given, that can be easily accepted
+or ignored.
+• Avoid: The avoiding strategy is executed when
+strong negative emotions (Fear, Disgust, Anger)
+are facing the robot. In this case, the robot avoids
+eye contact (keeps the face outside the camera
+center) and takes a passive part. Especially in case
+of fear, the goal is to increase the feeling of safe-
+ness by avoiding unexpected motions.
+• Ignore: If the negative emotions are not directed
+at the robot, the robot mostly ignores the person,
+only following it with its head to capture changes
+in emotional or attention states.
+Figure 3 shows all combinations of emotional and at-
+tention states with corresponding behavior strategy.
+4
+Implementation
+We used the ROS1 middleware to implement our pro-
+posed solution on our robot platform, as this also
+offers a simplified integration into other ROS-based
+1https://www.ros.org/
+
+Figure 4: Sentiment analysis in a laboratory setting. The
+person is tracked using a face detector, followed by a head
+pose and emotion estimation. As the person shows atten-
+tiveness and happy, a positive sentiment towards the robot
+is expected and a proactive initiation strategy selected.
+robot platforms. Face detection, head pose and emo-
+tion estimation are executed in separate nodes, that
+forward the necessary information to perform the sen-
+timent analysis and the action controls. While the lat-
+ter is performed on the robot itself, the inference of
+the neural networks is outsourced to a separate note-
+book that is mounted on the shoulder of the robot.
+This ensures the complete mobility of the robot, while
+exploiting additional computational resources. The
+notebook has a Quadro RTX 4000 processing all
+models in parallel with the following inference times:
+• Face Detection:
+6.7 ms
+• Head Pose Estimation:
+1.4 ms
+• Emotion Estimation:
+6.3 ms
+The visual perception can be therefore performed in
+real time and is able to continuously track people in
+the surroundings.
+5
+Experiment
+We conducted an initial experiment in a laboratory
+setting to test our implemented method on the TIAGo
+robot platform. The objective of the experiment was
+to gain experience about the model’s performances in
+real application scenarios. Figure 4 shows a snapshot
+from one of our tests. The robot platform detects a
+face in the environment and predicts head pose and
+emotional state. The focus of attention is on the robot
+(indicated by the green arrow) in conjunction with ex-
+pressing happiness, so the robot initiates an interac-
+tion by following the engage strategy.
+While the models provide very robust predic-
+tions, the implementation is currently entirely based
+on single shot detections. Yet, over the duration of
+face tracking, facial expressions change intermedi-
+ately and short distractions make the head pose oc-
+casionally oscillating. This leads to abrupt changes of
+states within short time periods. Hence, to improve
+the robustness of the sentiment analysis, conclusions
+about the current state should incorporate the predic-
+tions from multiple time frames.
+6
+Conclusion
+In this paper, we present a sentiment-based method-
+ology to improve the intuitiveness and reasoning in
+human-robot interactions. Our method focuses on the
+visual perception of mental states and the focus of at-
+tention to derive a sentiment analysis for suitable in-
+teraction initiation strategies. We take also into ac-
+count the case where people are not interested in an
+engagement at all, which has been mainly neglected
+in the recent literature. To implement our method,
+we trained lightweight perception models that we in-
+tegrate on a mobile robot platform. Finally, we con-
+ducted initial experiments using a mobile robot plat-
+form to analyze the performance of our models and
+the overall system.
+In the future, we will extend our work by
+embedding our sentiment analysis in a temporal-
+probabilistic framework to improve the robustness in
+real-world scenarios.
+Here, we will also incorpo-
+rate the valence and arousal predictions to evaluate
+the emotion’s intensity and transitions between emo-
+tional states. Further enhancements offer the inclu-
+sion of the robot’s mobility to actively engage and dis-
+engage people or to add additional interaction modal-
+ities, such as following people by command, getting
+out of the way of passing people, or approaching a
+place that offers a better overview of the surround-
+ings.
+Finally, we will conduct a study with test subjects
+to evaluate our approach in terms of intuitiveness and
+sympathy.
+ACKNOWLEDGEMENTS
+This work is funded and supported by the Fed-
+eral Ministry of Education and Research of Ger-
+many (BMBF) (AutoKoWAT-3DMAt under grant Nr.
+13N16336) and German Research Foundation (DFG)
+under grants Al 638/13-1, Al 638/14-1 and Al 638/15-
+1.
+
+anger
+surprise
+happy
+TTREFERENCES
+Avelino,
+J.,
+Garcia-Marques,
+L.,
+Ventura,
+R.,
+and
+Bernardino, A. (2021). Break the ice: a survey on
+socially aware engagement for human–robot first en-
+counters.
+International Journal of Social Robotics,
+13:1851 – 1877.
+Br¨ohl, C., Nelles, J., Brandl, C., Mertens, A., and Nitsch,
+V. (2019).
+Human-robot collaboration acceptance
+model: Development and comparison for germany,
+japan, china and the usa. I. J. Social Robotics, 11:709–
+726.
+Carvalho, M., Avelino, J., Bernardino, A., Ventura, R.
+M. M., and Moreno, P. (2021). Human-robot greet-
+ing: tracking human greeting mental states and acting
+accordingly.
+2021 IEEE/RSJ International Confer-
+ence on Intelligent Robots and Systems (IROS), pages
+1935–1941.
+Chuah, S. H.-W. and Yu, J. (2021).
+The future of ser-
+vice: The power of emotion in human-robot interac-
+tion.
+Journal of Retailing and Consumer Services,
+61:102551.
+Dinges, L., Al-Hamadi, A., Hempel, T., and Al Aghbari,
+Z. (2021). Using facial action recognition to evaluate
+user perception in aggravated hrc scenarios. In 2021
+12th International Symposium on Image and Signal
+Processing and Analysis (ISPA), pages 195–199.
+Dini, A., Murko, C., Yahyanejad, S., Augsd¨orfer, U., Hof-
+baur, M., and Paletta, L. (2017). Measurement and
+prediction of situation awareness in human-robot in-
+teraction based on a framework of probabilistic atten-
+tion. In 2017 IEEE/RSJ International Conference on
+Intelligent Robots and Systems (IROS), pages 4354–
+4361.
+Elprama, S., El Makrini, I., Vanderborght, B., and Jacobs,
+A. (2016). Acceptance of collaborative robots by fac-
+tory workers: a pilot study on the role of social cues
+of anthropomorphic robots.
+Finke, M., Koay, K. L., Dautenhahn, K., Nehaniv, C., Wal-
+ters, M., and Saunders, J. (2005). Hey, i’m over here
+- how can a robot attract people’s attention? In RO-
+MAN 2005. IEEE International Workshop on Robot
+and Human Interactive Communication, 2005., pages
+7–12.
+Fischer, K., Naik, L., Langedijk, R. M., Baumann, T.,
+Jel´ınek, M., and Palinko, O. (2021). Initiating human-
+robot interactions using incremental speech adapta-
+tion. In Companion of the 2021 ACM/IEEE Interna-
+tional Conference on Human-Robot Interaction, HRI
+’21 Companion, page 421–425, New York, NY, USA.
+Association for Computing Machinery.
+Foster, M. E., Gaschler, A., and Giuliani, M. (2017). Au-
+tomatically classifying user engagement for dynamic
+multi-party human–robot interaction.
+International
+Journal of Social Robotics, 9.
+Hempel, T., Abdelrahman, A. A., and Al-Hamadi, A.
+(2022). 6d rotation representation for unconstrained
+head pose estimation.
+In 2022 IEEE International
+Conference on Image Processing (ICIP), pages 2496–
+2500.
+Howard, A., Sandler, M., Chu, G., Chen, L.-C., Chen, B.,
+Tan, M., Wang, W., Zhu, Y., Pang, R., Vasudevan, V.,
+Le, Q. V., and Adam, H. (2019). Searching for mo-
+bilenetv3. In Proceedings of the IEEE/CVF Interna-
+tional Conference on Computer Vision (ICCV).
+Kato, Y., Kanda, T., and Ishiguro, H. (2015).
+May i
+help you? - design of human-like polite approach-
+ing behavior-. In 2015 10th ACM/IEEE International
+Conference on Human-Robot Interaction (HRI), pages
+35–42.
+Kendon, A. (1990). Conducting Interaction: Patterns of Be-
+havior in Focused Encounters. Cambridge University
+Press, Cambridge, U.K.
+Miller, L., Kraus, J., Babel, F., and Baumann, M. (2021).
+More than a feeling—interrelation of trust layers in
+human-robot interaction and the role of user disposi-
+tions and state anxiety. Frontiers in Psychology, 12.
+Mollahosseini, A., Hasani, B., and Mahoor, M. H. (2017).
+Affectnet: A database for facial expression, valence,
+and arousal computing in the wild. IEEE Transactions
+on Affective Computing, 10:18–31.
+M¨uller-Abdelrazeq, S. L., Sch¨onefeld, K., Haberstroh, M.,
+and Hees, F. (2019). Interacting with Collaborative
+Robots—A Study on Attitudes and Acceptance in In-
+dustrial Contexts, pages 101–117. Springer Interna-
+tional Publishing, Cham.
+Mumm, J. and Mutlu, B. (2011). Human-robot proxemics:
+Physical and psychological distancing in human-robot
+interaction.
+In 2011 6th ACM/IEEE International
+Conference on Human-Robot Interaction (HRI), pages
+331–338.
+Naneva, S., Gou, M. S., Webb, T. L., and Prescott, T. J.
+(2020). A systematic review of attitudes, anxiety, ac-
+ceptance, and trust towards social robots.
+Interna-
+tional Journal of Social Robotics, pages 1–23.
+Oertel, C., Castellano, G., Chetouani, M., Nasir, J., Obaid,
+M., Pelachaud, C., and Peters, C. (2020). Engagement
+in human-agent interaction: An overview. Frontiers in
+Robotics and AI, 7.
+Repiso, E., Garrell, A., and Sanfeliu, A. (2018).
+Robot
+approaching and engaging people in a human-robot
+companion framework. In 2018 IEEE/RSJ Interna-
+tional Conference on Intelligent Robots and Systems
+(IROS), pages 8200–8205.
+Satake, S., Kanda, T., Glas, D. F., Imai, M., Ishiguro, H.,
+and Hagita, N. (2009). How to approach humans?-
+strategies for social robots to initiate interaction. In
+2009 4th ACM/IEEE International Conference on
+Human-Robot Interaction (HRI), pages 109–116.
+Spezialetti, M., Placidi, G., and Rossi, S. (2020). Emotion
+recognition for human-robot interaction: Recent ad-
+vances and future perspectives. Frontiers in Robotics
+and AI, 7.
+Syrdal, D. S., Lee Koay, K., Walters, M. L., and Dauten-
+hahn, K. (2007). A personalized robot companion?
+- the role of individual differences on spatial prefer-
+ences in hri scenarios. In RO-MAN 2007 - The 16th
+IEEE International Symposium on Robot and Human
+Interactive Communication, pages 1143–1148.
+
+Vinkemeier, D., Valstar, M. F., and Gratch, J. (2018). Pre-
+dicting folds in poker using action unit detectors and
+decision trees. In FG, pages 504–511.
+Wegrzyn, M., Vogt, M., Kireclioglu, B., Schneider, J., and
+Kissler, J. (2017). Mapping the emotional face. How
+individual face parts contribute to successful emotion
+recognition. PLOS ONE, 12(5):e0177239.
+Werner, P., Handrich, S., and Al-Hamadi, A. (2017). Fa-
+cial action unit intensity estimation and feature rel-
+evance visualization with random regression forests.
+In 2017 Seventh International Conference on Affective
+Computing and Intelligent Interaction (ACII), volume
+2018-Janua, pages 401–406. IEEE.
+
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+page_content='Sentiment-based Engagement Strategies for intuitive Human-Robot  Interaction  Thorsten Hempel  a, Laslo Dinges  b and Ayoub Al-Hamadi  c  Neuro-Information Technology, Faculty of Electrical Engineering and Information Technology,  Otto von Guericke University, Magdeburg, Germany  {thorsten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
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+page_content='dinges, ayoub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='al-hamadi}@ovgu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='de  Keywords:  human-robot interaction, approaching strategy, sentiment estimation, emotion detection, anticipating human  behaviors, approaching people  Abstract:  Emotion expressions serve as important communicative signals and are crucial cues in intuitive interactions  between humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Hence, it is essential to include these fundamentals in robotic behavior strategies when inter-  acting with humans to promote mutual understanding and to reduce misjudgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' We tackle this challenge  by detecting and using the emotional state and attention for a sentiment analysis of potential human inter-  action partners to select well-adjusted engagement strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' This way, we pave the way for more intuitive  human-robot interactions, as the robot’s action conforms to the person’s mood and expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' We propose  four different engagement strategies with implicit and explicit communication techniques that we implement  on a mobile robot platform for initial experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  1  Introduction  Once utopian, robots are increasingly moving from  industrial and laboratory settings to the real-world  in order to assist humans in everyday tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' In this  process, human-robot interaction became a central  point of interest in robotic research that investigates  the manifold challenges in performing interactive and  collaborative tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' As a general principle, the robot  needs to be acquired with the necessary skills to en-  able intuitive interactions with arbitrary human inter-  action partners, regardless of the corresponding task,  the human’s intention and communication behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  To this end, a fundamental objective is the anticipa-  tion of appropriate strategies to proactively approach  or evade people based on the situation-specific con-  text, such as the person’s mood, attitude (M¨uller-  Abdelrazeq et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Elprama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', 2016) and  ressentiments towards robots (Naneva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' First, it requires the determina-  tion of people’s interest in interacting at all (Satake  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Finke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Thereupon, the robot  has to find an appropriate strategy for either engaging  people to establish an interaction or to avoid a con-  a  https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='org/0000-0002-3621-7194  b  https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='org/0000-0003-0364-8067  c  https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='org/0000-0002-3632-2402  frontation if desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' In order to reach these under-  lying decisions, the robot has to cope with a row of  sub-tasks: estimating people’s focus of attention (Dini  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', 2017), predicting their mental state, and carry-  ing out a (dis-)engagement strategy (Avelino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=',  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Especially, the estimation of the interaction will-  ingness is a very challenging task, as this mental state  tends to provide only vague social signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  As an  alternative approach, many methods (Oertel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=',  2020) go after the detection of engagement in order  to determine if a person already approached the robot  and waits for the robot’s reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' This state expresses  itself more clearly, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', in voice commands (Foster  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', 2017), gestures and proxemics, but it under-  mines the proactive approach and neglects situations  where the human partner is in need of help, but too  uncertain or not aware of the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  In this work, we close the gap between the lack of  proactive behavior and advanced mental state predic-  tions by introducing a sentiment-based engagement  strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  By combining the focus of attention and emotions,  we derive a sentiment analysis that allows us to select  fine-grained behavior patterns that exceed current bi-  nary engagement and disengagement strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' We  implement our approach on a mobile robot platform  and execute our strategies using explicit and implicit  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='03867v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='RO]  10 Jan 2023  embodied communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Our models are trained  on minimum sized networks in order to perform our  method on mobile systems with limited computa-  tional resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  2  Related Work  In recent years, proxemic rules have been a popular  tool to draw conclusions about the different states of  human-robot interactions (Mumm and Mutlu, 2011)  and how spatial zones can be leverages to improve  them (Syrdal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (Repiso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', 2018) pre-  dict appropriate encounter points to achieve a natu-  ral engagement with a group of people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Similarly,  (Satake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', 2009) proactively approach detected  people to offer help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (Fischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', 2021) verbally  greet people and dynamically adapt the voice volume  based on the distance to the target person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Follow-  ing (Kendon, 1990), (Carvalho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', 2021) apply the  Kendon’s greeting model to approach people while  tracking the human mental states of the interaction us-  ing multiple features, such as gaze, facial expressions,  and gestures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Yet, all of these methods neglect if the  human counterpart is actually interested and ready to  initiate an interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' This is problematic, as in the  case of negative attitude, the robot’s interaction ef-  forts can be perceived as rude and annoying (Br¨ohl  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' An initial attempt to address this was  presented by (Kato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', 2015), who only approach  and offer help if they detect signs of attention towards  the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' But this still doesn’t incorporate the actual  mental attitude of the human counterpart towards the  interaction itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  The recent advances in the area of deep-learning  opened up new possibilities for the visual analysis of  humans, starting from general detection tasks to the  estimation of facial micro expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' This allows  the aggregation of additional relevant information,  such as emotions (Chuah and Yu, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Spezialetti  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', 2020), to better understand the behavior and in-  tentions of the human interaction partner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' To the best  of our knowledge, we are the first to leverage this ad-  vancement by predicting sentiment states based on vi-  sual emotion estimation to improve interaction strat-  egy for robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' We summarize our main contributions  as follows:  We propose an image-based sentiment analysis to  gauge the current interaction preference of poten-  tial human interaction partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  We propose four different behavior strategies (En-  gage, Attract, Ignore, Avoid) that comprises a  number of different explicit and implicit commu-  nication modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Yaw, Tilt Head  Rotate Base  Lifting Torso  Speech  Figure 1: The TIAGo robot platform with the key opera-  tions used in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  We design ultra lightweight models to implement  our method and integrate it on a mobile robot plat-  form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  We conduct initial experiments in laboratory set-  tings for an early evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  3  Method  This section explains each step of our proposed  method for sentiment-based interaction behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  The sentiment analysis is based on two main fea-  tures: head pose and emotion estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' The head  pose estimation is a leading indicator to gauge the cur-  rent visual focus of attention of the person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' The emo-  tion estimation predicts the current emotional state of  the person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Together, it allows the assumption about  a person that a) is seeking for interaction, b) is in-  decisive about it or c) doesn’t want to engage with  the robot at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Determining each sentiment state en-  ables to perform a dedicated robot behavior that cor-  responds with the expected reaction from the person  and, thus, improves an intuitive interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Figure 2  illustrates an overview of our proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' In the  following subsection, we will give details about each  component of the system and its interplay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='1  Robot platform  We use the TIAGo robot from PAL Robotics, a mobile  service robot for indoor environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' It is equipped  with an RGB-D camera mounted inside its head, that  can be yawed and pitched to dynamically perceive the  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' For grasping and moving objects, the  PAILO  PALOrobot has a manipulator arm with a 5 finger gripper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  The torso can be lifted to adjust the robot’s height, and  integrated speakers allow the output of voice com-  mands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' These abilities allow the combination of ex-  plicit and implicit communication modalities that we  use in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Head Following: The robot’s head aims at and fol-  lows a moving person as an implicit signal of at-  tentiveness without actively approaching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Body Following: If a person exceeds the range of  the robot’s moving head, the entire robot base ro-  tates in order to keep track of the person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' This  behavior communicates an attention commitment  and is therefore an even stronger signal as the  head following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Torso Lifting: Lifting the torso is a rather subtle  communication cue, as it is perceived similarly to  the social convention of standing up while greet  someone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Speech: Approaching people with speech rep-  resents an explicit communication strategy, that  stronger commits them to a reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Hence,  speech should be only used if the approached per-  son is highly expected to welcome an interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='2  Visual Attention  Typical indicators for gauging the human visual at-  tention are the gaze and, more coarsely, the head  pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' As the gaze depends on the eyes, which takes  only a small part in face images, it is prone to er-  rors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Therefore, we estimate the visual attention of  the surrounding persons based on a head pose predic-  tion algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' At first, we locate faces in the image  stream using an ultra light SSD face detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Then,  the face crops are further processed by the 6DRep-  Net (Hempel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', 2022), that uses a rotation repre-  sentation to directly regress yaw, pitch, and roll an-  gle of the faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' To assure real-time processing capa-  bilities, we replace the original 6DRepNet backbone  with the efficient MobileNetv3-Small (Howard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=',  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' The new head pose prediction model is able to  maintain 90% of its accuracy compared to the origi-  nal model, but is downsized to only one tenth of its  parameters count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='3  Emotion Estimation  In order to assess the attitude towards robots as well  as differentiated subjective states such as dislike or  willingness to cooperate, basic emotions can be used  as indicating features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Naturally, these are frequently  communicated by facial expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Hence, au-  tomatic facial expression recognition is an essen-  tial method to generate features for Human-Robot-  Collaboration and attention prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Previous facial expression recognition methods  adhere to the established pipeline of face detection,  landmark extraction, and action unit (AU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Then,  emotions are classified using these AUs as feature  vectors (Wegrzyn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Vinkemeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=',  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Werner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  However, end-to-end  learning on a single but more comprehensive, and bet-  ter generalizing database, often outperforms such tra-  ditional approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' The AffectNet database (Molla-  hosseini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', 2017) for example contains about 500k  in-the-wild samples for neutral and the basic emo-  tions happy, disgust, fear, surprise, anger, and sad-  ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Furthermore, it is the only database, that also  includes ground truth of valence and arousal (VA),  which are, in contrast to the basic emotion classes,  continuous labels ∈ [−1,1] which can be used for re-  gression tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' VA is less intuitively interpretable by  humans, but it allows differentiation between facial  expressions of varying degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' This is particularly  useful for capturing modest but long-term changes  in expressed emotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' In addition, a multitask net-  work (simultaneous training of classification and re-  gression) also improved the accuracy of the classifi-  cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Furthermore, Valence summarizes essential  information about several emotions in one parameter  (negative, neutral, positive), which may facilitate the  development of heuristic behavior rules for some sce-  narios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  In our former work, we proposed a multitask  network based on YOLOv3 (Dinges et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  For the current paper, we compared EfficientNetV2-  S, ResNet18, and MobileNetV3-Large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Despite the  poorer converging training loss, MobileNet, which is  optimized for limited hardware, achieved almost the  same accuracy on the test set as EfficientNet (devia-  tion < 1%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' In addition, we used a weighted random  sampler to handle the imbalanced dataset and re-train  the network on the training and test set to generate  an application model with increased robustness to un-  seen data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='4  Sentiment Analysis and  Approaching Strategy  Detecting emotional expressions combined with the  visual focus of attention allows perceiving important  communicative signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' As in human-human interac-  tion, you may be more likely to greet a stranger who  smiles at you but avoid eye contact if that stranger  is scowling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  We transfer these principles to deter-  Engage  Attract  Avoid  Ignore  Head Pose Estimation  Emotion estimation  Face Detection  Perception  Behavior Strategy  Sentiment Analysis  Emotional state with  focus of attention  Figure 2: Method overview of our proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Detected faces are used to estimate the head pose and the corresponding  emotions based on facial expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Both information is used to determine the emotional state with the corresponding visual  focus of the person’s attention that indicates the sentiment and allows a fine-tuned behavior strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Happy  Uncertain  Surprise  Neutral  Sad  Fear  Disgust  Anger  Facing  Averted  ENGAGE:  Lift torso, follow with  head and base, and  approach by speech  ATTRACT:  Lift torso and follow  with head  AVOID:  Look down and avoid  attraction  IGNORE:  Follow with head  Figure 3: Behavior strategies based on the emotional and  attentive state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  mine states of sentiment based on the combination  of the different emotions and visual attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Is a  person showing visual signs of fear or anger, while  he is attentive towards the robot, it indicates a nega-  tive attitude towards the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' In this case, a passive  robot behavior without hectic movements can help to  defuse the situation and to calm the person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' If a per-  son is feared, but not attentive towards the robot, the  reason for the fear might not be the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Execut-  ing a more active robot behavior to show awareness  could be therefore a more suitable strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Hence,  our sentiment analysis allows us to compartmental-  ize the common hard binary engagement strategy into  soft behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Instead of choosing between engaging  and not engaging, we choose between four different  behaviors: Engage, Attract, Avoid, Ignore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Engage: Engaging will extend the torso of the  robot to signal the awareness of the person’s pres-  ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Additionally, voice commands will greet  the people to actively initiate an interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' This  strategy is called, when the attention is directed at  the robot with clear positive emotional attitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Attract: Positive emotional attitude without atten-  tion towards the robot leads to more passive be-  havior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' The goal is to signal the offer for inter-  action and help without actively committing to an  interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' This is done by extending the torso  to show awareness, facing the person, but without  verbally greeting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' This way, a non-verbal offer for  support (especially in case of uncertainty and sad-  ness) is subtly given, that can be easily accepted  or ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Avoid: The avoiding strategy is executed when  strong negative emotions (Fear, Disgust, Anger)  are facing the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' In this case, the robot avoids  eye contact (keeps the face outside the camera  center) and takes a passive part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Especially in case  of fear, the goal is to increase the feeling of safe-  ness by avoiding unexpected motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Ignore: If the negative emotions are not directed  at the robot, the robot mostly ignores the person,  only following it with its head to capture changes  in emotional or attention states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Figure 3 shows all combinations of emotional and at-  tention states with corresponding behavior strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  4  Implementation  We used the ROS1 middleware to implement our pro-  posed solution on our robot platform, as this also  offers a simplified integration into other ROS-based  1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='ros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='org/  Figure 4: Sentiment analysis in a laboratory setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' The  person is tracked using a face detector, followed by a head  pose and emotion estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' As the person shows atten-  tiveness and happy, a positive sentiment towards the robot  is expected and a proactive initiation strategy selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  robot platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Face detection, head pose and emo-  tion estimation are executed in separate nodes, that  forward the necessary information to perform the sen-  timent analysis and the action controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' While the lat-  ter is performed on the robot itself, the inference of  the neural networks is outsourced to a separate note-  book that is mounted on the shoulder of the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  This ensures the complete mobility of the robot, while  exploiting additional computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' The  notebook has a Quadro RTX 4000 processing all  models in parallel with the following inference times:  Face Detection:  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='7 ms  Head Pose Estimation:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='4 ms  Emotion Estimation:  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='3 ms  The visual perception can be therefore performed in  real time and is able to continuously track people in  the surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  5  Experiment  We conducted an initial experiment in a laboratory  setting to test our implemented method on the TIAGo  robot platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' The objective of the experiment was  to gain experience about the model’s performances in  real application scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Figure 4 shows a snapshot  from one of our tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' The robot platform detects a  face in the environment and predicts head pose and  emotional state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' The focus of attention is on the robot  (indicated by the green arrow) in conjunction with ex-  pressing happiness, so the robot initiates an interac-  tion by following the engage strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  While the models provide very robust predic-  tions, the implementation is currently entirely based  on single shot detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Yet, over the duration of  face tracking, facial expressions change intermedi-  ately and short distractions make the head pose oc-  casionally oscillating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' This leads to abrupt changes of  states within short time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Hence, to improve  the robustness of the sentiment analysis, conclusions  about the current state should incorporate the predic-  tions from multiple time frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  6  Conclusion  In this paper, we present a sentiment-based method-  ology to improve the intuitiveness and reasoning in  human-robot interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Our method focuses on the  visual perception of mental states and the focus of at-  tention to derive a sentiment analysis for suitable in-  teraction initiation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' We take also into ac-  count the case where people are not interested in an  engagement at all, which has been mainly neglected  in the recent literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' To implement our method,  we trained lightweight perception models that we in-  tegrate on a mobile robot platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Finally, we con-  ducted initial experiments using a mobile robot plat-  form to analyze the performance of our models and  the overall system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  In the future, we will extend our work by  embedding our sentiment analysis in a temporal-  probabilistic framework to improve the robustness in  real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Here, we will also incorpo-  rate the valence and arousal predictions to evaluate  the emotion’s intensity and transitions between emo-  tional states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Further enhancements offer the inclu-  sion of the robot’s mobility to actively engage and dis-  engage people or to add additional interaction modal-  ities, such as following people by command, getting  out of the way of passing people, or approaching a  place that offers a better overview of the surround-  ings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Finally, we will conduct a study with test subjects  to evaluate our approach in terms of intuitiveness and  sympathy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  This work is funded and supported by the Fed-  eral Ministry of Education and Research of Ger-  many (BMBF) (AutoKoWAT-3DMAt under grant Nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  13N16336) and German Research Foundation (DFG)  under grants Al 638/13-1, Al 638/14-1 and Al 638/15-  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  anger  surprise  happy  TTREFERENCES  Avelino,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=',  Garcia-Marques,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=',  Ventura,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=',  and  Bernardino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Break the ice: a survey on  socially aware engagement for human–robot first en-  counters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  International Journal of Social Robotics,  13:1851 – 1877.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Br¨ohl, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Nelles, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Brandl, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Mertens, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', and Nitsch,  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Human-robot collaboration acceptance  model: Development and comparison for germany,  japan, china and the usa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Social Robotics, 11:709–  726.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Carvalho, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Avelino, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Bernardino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Ventura, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', and Moreno, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Human-robot greet-  ing: tracking human greeting mental states and acting  accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  2021 IEEE/RSJ International Confer-  ence on Intelligent Robots and Systems (IROS), pages  1935–1941.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Chuah, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' and Yu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  The future of ser-  vice: The power of emotion in human-robot interac-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Journal of Retailing and Consumer Services,  61:102551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Dinges, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Al-Hamadi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Hempel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', and Al Aghbari,  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Using facial action recognition to evaluate  user perception in aggravated hrc scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' In 2021  12th International Symposium on Image and Signal  Processing and Analysis (ISPA), pages 195–199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Dini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Murko, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Yahyanejad, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Augsd¨orfer, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Hof-  baur, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', and Paletta, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Measurement and  prediction of situation awareness in human-robot in-  teraction based on a framework of probabilistic atten-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' In 2017 IEEE/RSJ International Conference on  Intelligent Robots and Systems (IROS), pages 4354–  4361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Elprama, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', El Makrini, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Vanderborght, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', and Jacobs,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Acceptance of collaborative robots by fac-  tory workers: a pilot study on the role of social cues  of anthropomorphic robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Finke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Koay, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Dautenhahn, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Nehaniv, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Wal-  ters, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', and Saunders, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Hey, i’m over here  how can a robot attract people’s attention?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' In RO-  MAN 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' IEEE International Workshop on Robot  and Human Interactive Communication, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', pages  7–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Fischer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Naik, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Langedijk, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Baumann, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=',  Jel´ınek, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', and Palinko, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Initiating human-  robot interactions using incremental speech adapta-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' In Companion of the 2021 ACM/IEEE Interna-  tional Conference on Human-Robot Interaction, HRI  ’21 Companion, page 421–425, New York, NY, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Foster, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Gaschler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', and Giuliani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Au-  tomatically classifying user engagement for dynamic  multi-party human–robot interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  International  Journal of Social Robotics, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Hempel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Abdelrahman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', and Al-Hamadi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' 6d rotation representation for unconstrained  head pose estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  In 2022 IEEE International  Conference on Image Processing (ICIP), pages 2496–  2500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Howard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Sandler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Chu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Chen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=',  Tan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Zhu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Pang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Vasudevan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=',  Le, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', and Adam, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Searching for mo-  bilenetv3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' In Proceedings of the IEEE/CVF Interna-  tional Conference on Computer Vision (ICCV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Kato, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Kanda, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', and Ishiguro, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  May i  help you?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' - design of human-like polite approach-  ing behavior-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' In 2015 10th ACM/IEEE International  Conference on Human-Robot Interaction (HRI), pages  35–42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Kendon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Conducting Interaction: Patterns of Be-  havior in Focused Encounters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Cambridge University  Press, Cambridge, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Miller, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Kraus, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Babel, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', and Baumann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  More than a feeling—interrelation of trust layers in  human-robot interaction and the role of user disposi-  tions and state anxiety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Frontiers in Psychology, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Mollahosseini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Hasani, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', and Mahoor, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Affectnet: A database for facial expression, valence,  and arousal computing in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' IEEE Transactions  on Affective Computing, 10:18–31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  M¨uller-Abdelrazeq, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Sch¨onefeld, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Haberstroh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=',  and Hees, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Interacting with Collaborative  Robots—A Study on Attitudes and Acceptance in In-  dustrial Contexts, pages 101–117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Springer Interna-  tional Publishing, Cham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Mumm, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' and Mutlu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Human-robot proxemics:  Physical and psychological distancing in human-robot  interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  In 2011 6th ACM/IEEE International  Conference on Human-Robot Interaction (HRI), pages  331–338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Naneva, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Gou, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Webb, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', and Prescott, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' A systematic review of attitudes, anxiety, ac-  ceptance, and trust towards social robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Interna-  tional Journal of Social Robotics, pages 1–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Oertel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Castellano, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Chetouani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Nasir, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Obaid,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Pelachaud, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', and Peters, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Engagement  in human-agent interaction: An overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Frontiers in  Robotics and AI, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Repiso, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Garrell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', and Sanfeliu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Robot  approaching and engaging people in a human-robot  companion framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' In 2018 IEEE/RSJ Interna-  tional Conference on Intelligent Robots and Systems  (IROS), pages 8200–8205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Satake, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Kanda, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Glas, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Imai, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Ishiguro, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=',  and Hagita, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' How to approach humans?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='-  strategies for social robots to initiate interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' In  2009 4th ACM/IEEE International Conference on  Human-Robot Interaction (HRI), pages 109–116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Spezialetti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Placidi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', and Rossi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Emotion  recognition for human-robot interaction: Recent ad-  vances and future perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Frontiers in Robotics  and AI, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Syrdal, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Lee Koay, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Walters, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', and Dauten-  hahn, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' A personalized robot companion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  the role of individual differences on spatial prefer-  ences in hri scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' In RO-MAN 2007 - The 16th  IEEE International Symposium on Robot and Human  Interactive Communication, pages 1143–1148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Vinkemeier, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Valstar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', and Gratch, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Pre-  dicting folds in poker using action unit detectors and  decision trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' In FG, pages 504–511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Wegrzyn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Vogt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Kireclioglu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Schneider, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', and  Kissler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Mapping the emotional face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' How  individual face parts contribute to successful emotion  recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' PLOS ONE, 12(5):e0177239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  Werner, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', Handrich, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=', and Al-Hamadi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' Fa-  cial action unit intensity estimation and feature rel-  evance visualization with random regression forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content='  In 2017 Seventh International Conference on Affective  Computing and Intelligent Interaction (ACII), volume  2018-Janua, pages 401–406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfZwdG/content/2301.03867v1.pdf'}
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+arXiv:2301.13177v1  [math.NA]  30 Jan 2023
+Infinite-Variate 푳2-Approximation with Nested
+Subspace Sampling
+Kumar Harsha, Michael Gnewuch, and Marcin Wnuk
+Abstract We consider 퐿2-approximation on weighted reproducing kernel Hilbert
+spaces of functions depending on infinitely many variables. We focus on unrestricted
+linear information, admitting evaluations of arbitrary continuous linear functionals.
+We distinguish between ANOVA and non-ANOVA spaces, where, by ANOVA spaces,
+we refer to function spaces whose norms are induced by an underlying ANOVA func-
+tion decomposition. In ANOVA spaces, we prove that there is an optimal algorithm
+to solve the approximation problem using linear information. This way, we can deter-
+mine the exact polynomial convergence rate of 푛-th minimal worst-case errors. For
+non-ANOVA spaces, we also establish upper and lower error bounds. Even though
+the bounds do not match in this case, they reveal that for weights with a moderate
+decay behavior, the convergence rate of 푛-th minimal errors is strictly higher in
+ANOVA than in non-ANOVA spaces.
+1 Introduction
+We want to investigate 퐿2-approximation on weighted reproducing kernel Hilbert
+spaces (RKHSs) of functions depending on infinitely many variables. For each of
+those RKHSs, there is a family of weights and a function decomposition, which
+determine the norm on the space. With the help of the function decomposition, it
+is possible to represent an infinite-variate function as an infinite sum of functions
+depending only on finitely many variables.
+K. Harsha •M. Gnewuch •M. Wnuk
+Institute for Mathematics, Osnabrück University, Albrechtraße 28a, 49074 Osnabrück, Germany
+e-mail: kumar.harsha@uni-osnabrueck.de
+M. Gnewuch
+e-mail: michael.gnewuch@uni-osnabrueck.de
+M. Wnuk
+e-mail: marcin.wnuk@uni-osnabrueck.de
+1
+
+2
+Kumar Harsha, Michael Gnewuch, and Marcin Wnuk
+In this paper, we focus on linear algorithms that may obtain information about
+an input function 푓 by evaluating finitely many continuous linear functionals in 푓 .
+The information cost of such an algorithm is the sum of the cost of all functional
+evaluations, which, in turn, depends on the underlying function decomposition of
+the weighted RKHS we are working on. Such a situation has already been stud-
+ied in [14, 15], where the considered cost model is a generalization of the one
+introduced in [10] for information consisting only of function evaluation (so-called
+standard information). In [15], a rather specific approximation problem, called the
+G-approximationproblem, is analyzed; there, the approximationerror is measured in
+some very special Hilbert space G. In [14], 퐿2-approximation is investigated in full
+generality, apart from the case that can be reduced to the G-approximation problem.
+To describe the results from [14,15] and from this paper in a more common language,
+we distinguish two cases of weighted RKHSs: the ones whose underlying function
+decomposition is an ANOVA decomposition are called ANOVA spaces, whereas the
+remaining RKHSs are referred to as non-ANOVA spaces (for a rigorous definition,
+see Section 2.1).
+The case of ANOVA spaces was covered by Wasilkowski and Woźniakowski, who
+in [15] exactly determined the polynomial convergence rates 푟∞ of 푛-th minimal
+errors (for a definition, see Section 2.2.2). For the non-ANOVA spaces, Wasilkowski
+was able to derive in [14] a lower bound for 푟∞; with the help of auxiliary weights, he
+re-used the constructive results from [15]. He also proved a matching lower bound.
+Interestingly, it turns out that for slowly decaying weights, the convergence rate in
+the ANOVA case is superior to the one in the non-ANOVA case, while it stays the
+same if the weights decrease quickly enough.
+In this paper, we consider the nested subspace sampling cost model, which is less
+generous than the cost model in [14, 15]. For this cost model, we have derived the
+exact polynomialconvergencerate 푟∞ of 푛-th minimal errors in the ANOVA case, see
+Theorem 2. In the non-ANOVA case, we are able to provide upper and constructive
+lower bounds on 푟∞, see Propositions 1 and 2. Although these bounds do not match,
+they show that in some cases the convergence rate in the ANOVA case is higher than
+the one in the non-ANOVA case, cf. Corollary 2. The reason for that is mainly that the
+cost of an algorithm may depend on what kind of function decomposition underlies
+the function space it is defined on. This dependence, unfortunately, prevents us from
+simply using the embedding machinery developed in [5, 6, 8] to transfer results for
+ANOVA spaces directly to spaces with a differentunderlying function decomposition.
+Notation For a set u, we denote by |u| its cardinality. We use the shorthand
+[푘] := {1, . . . , 푘} for every 푘 ∈ N.
+2 Function Spaces and Algorithms
+For general results on reproducing kernel Hilbert spaces (RKHS) we refer to [1], and
+to [7] for results on the function spaces discussed below.
+
+Infinite-Variate 퐿2-Approximation with Nested Subspace Sampling
+3
+2.1 Weighted Function Spaces
+Let 퐷 ≠ ∅ be an arbitrary set, Σ a 휎-algebra on 퐷, and 휌 : Σ → [0, 1] a probability
+measure on 퐷. Let 푘 : 퐷 × 퐷 → R be a reproducing kernel on 퐷 such that
+퐻(1) ∩ 퐻(푘) = {0};
+here and in the whole paper we follow the convention to denote by 퐻(푘) the uniquely
+determined Hilbert space with reproducing kernel 푘 and its norm by ∥ · ∥퐻 (푘); in
+particular 퐻(1) is the space of all constant functions on the corresponding domain
+퐷. Put
+U := {u ⊂ N | |u| < ∞},
+where |u| refers to the cardinality of u. For u ∈ U we define 푘u : 퐷N × 퐷N → R by
+푘u(x, y) :=
+�
+푗 ∈u
+푘(푥 푗, 푦 푗)
+for all x, y ∈ 퐷N.
+Since 푘u only depends on the variables 푥 푗, 푗 ∈ u, we may alternatively view it as a
+function on 퐷u by putting
+푘u(xu, yu) := 푘u(˜x, ˜y)
+for arbitrary ˜x, ˜y ∈ 퐷N satisfying ˜xu = xu, ˜yu = yu, where xu := {푥 푗} 푗 ∈u ∈ 퐷u.
+We use an analogous convention for arbitrary functions on 퐷N that only depend on
+variables 푥 푗, 푗 ∈ u. We denote the RKHS with kernel 푘u by 퐻u := 퐻(푘u) and its
+norm by ∥ · ∥퐻u; in particular, we have 퐻∅ = 퐻(1) due to the convention that an
+empty product is equal to 1.
+For a family 훾 = {훾u}u∈U of non-negative weights, we put
+U훾 := {u ∈ U | 훾u > 0}.
+We work with product weights which are given by a decreasing sequence of weights
+1 ≥ 훾1 ≥ 훾2 ≥ ... ≥ 0 assigned to all of the variables. Then the weight (or
+importance) of an interacting set of variables u is given by 훾u = �
+푗 ∈u 훾 푗. Moreover,
+we define
+픛 :=
+�
+x ∈ 퐷N
+����
+�
+u∈U훾
+훾u푘u(x, x) < ∞
+�
+and the reproducing kernel 퐾훾 : 픛 × 픛 → R by
+퐾훾(x, y) :=
+�
+u∈U훾
+훾u푘u(x, y)
+x, y ∈ 픛.
+(1)
+We denote the corresponding RKHS by 퐻훾 := 퐻(퐾훾). 퐻훾 decomposes into an
+orthogonal sum of subspaces
+
+4
+Kumar Harsha, Michael Gnewuch, and Marcin Wnuk
+퐻훾 =
+�
+u∈U훾
+퐻(훾u푘u) =
+�
+u∈U훾
+훾u퐻u.
+(2)
+Note that for u ∈ U훾 we have that 퐻(훾u푘u) and 퐻u are equal as vector spaces, but
+their norms differ by a factor 훾−1/2
+u
+. Hence the norm in 퐻훾 is given by
+∥ 푓 ∥2
+퐻훾 =
+�
+u∈U훾
+훾−1
+u ∥ 푓u∥2
+퐻u ,
+where
+푓 =
+�
+u∈U훾
+푓u,
+푓u ∈ 퐻u,
+(3)
+is the uniquely defined function decomposition of 푓 in 퐻훾.
+We assume that
+푀 :=
+∫
+퐷
+푘(푥, 푥) d휌(푥) < ∞
+(4)
+and
+�
+u∈U훾
+훾u푀 |u| < ∞.
+(5)
+From our assumptions (4) and (5) follows that we have the continuous and compact
+embeddings 퐻(1 + 푘) ⊆ 퐿2(퐷, 휌), 퐻u ⊆ 퐿2(퐷u, 휌u) for all u ∈ U훾, and 퐻훾 ⊆
+퐿2(픛, 휌N), see [16]. In particular, the solution operator we consider
+S : 퐻훾 → 퐿2(픛, 휌N) , 푓 ↦→ 푓
+is well-defined and continuous.
+Definition 1 (ANOVA kernels) Let 푘 : 퐷 × 퐷 → R be a reproducing kernel such
+that 푘(·, 푥) ∈ 퐿2(퐷, 휌) for all 푥 ∈ 퐷. If the kernel satisfies the ANOVA condition
+∫
+퐷
+푘(푦, 푥)d휌(푦) = 0
+∀푥 ∈ 퐷,
+(6)
+then we call 푘 an ANOVA kernel. Otherwise, we call it a non-ANOVA kernel.
+Remark 1 1. It is easily seen that under the assumptions of Definition 1, the ANOVA
+condition (6) is equivalent to the condition that
+∫
+퐷 푓 (푦)휌(푦) = 0 for all 푓 ∈ 퐻(푘).
+2. Consider the reproducing kernel as in (1) based on an ANOVA kernel 푘 : 퐷×퐷 →
+R. Then for each 푓 ∈ 퐻훾, the function decomposition as in (3) is the infinite-
+variate ANOVA decomposition, see [2, Remark 2.12]. For instance, it implies that
+for each 푓 ∈ 퐻훾, for all u ∈ U and x ∈ 퐷N
+∫
+퐷
+푓u(x) d휌(푥 푗) = 0
+if 푗 ∈ 푢.
+ExamplesofANOVAspacesincludeKorobov spaces,seee.g.[12,Appendix A.1]and
+unanchored Sobolev spaces, see e.g. [12, Appendix A.2.1,A.2.3], [2, Section 5],
+
+Infinite-Variate 퐿2-Approximation with Nested Subspace Sampling
+5
+and [4, Section 5]. A notable example of non-ANOVA spaces are anchored Sobolev
+spaces, see e.g. [12, Appendix A.2.2] or [10].
+2.2 Algorithms, Cost Models, and Errors
+2.2.1 Admissible algorithms and cost
+We want to approximate 푓 ∈ 퐻훾 by algorithms using finitely many information eval-
+uations. Since our problem is linear, we may confine ourselves to linear algorithms
+using arbitrary linear information of the form
+A( 푓 ) =
+푛
+�
+푗=1
+퐿u 푗 ( 푓 ) · 푔 푗
+(7)
+for some 푛 ∈ N, 푔 푗 ∈ 퐿2(픛, 휌N), index sets u푗 ∈ U훾, and linear functionals
+퐿u 푗 ∈ 퐻∗
+훾 such that 퐿u 푗 |퐻v = 0 if v ⊈ u푗, see [13]. Owing to the Riesz representation
+theorem, we can define the linear functional 퐿u 푗 as inner product with a representer
+as
+퐿u 푗 ( 푓 ) :=
+�
+푓 , 휂u 푗
+�
+퐻훾 ,
+where 휂u 푗 ∈
+�
+v⊆u 푗
+퐻v.
+(8)
+The worst-case error of the algorithm A for 퐿2-approximation with respect to
+퐻훾 is given as
+푒(A, 퐻훾) := 푒(A, S, 퐻훾) :=
+sup
+∥ 푓 ∥퐻훾 ≤1
+∥S( 푓 ) − A( 푓 )∥퐿2 (픛,휌N).
+Let us fix a cost function $ : N → [1, ∞) which is non-decreasing. For nested
+subspace sampling (NSS) (see [5, Sect. 5.1] or [2,3,9,11]) we consider a sequence
+of nested finite index sets
+∅ = v0 ⊂ v1 ⊂ v2 ⊂ . . . ,
+(9)
+with v푖 := [푖] for 푖 ∈ N. Then the cost of any information evaluation 퐿u( 푓 ) as in (8)
+is given by
+cost(퐿u) = inf{$(|v푖|) : u ⊆ v푖}.
+2.2.2 Minimal errors and complexity
+For the results in this paper, we work with the NSS cost model unless specified
+otherwise. On that basis, we define the 푛-th minimal error for 푛 ∈ N as
+푒(푛, 퐻훾) := inf{푒(A, 퐻훾) : A as in (7) and cost(A) ≤ 푛}.
+
+6
+Kumar Harsha, Michael Gnewuch, and Marcin Wnuk
+The approximation problem is said to be strongly tractable if and only if there exist
+non-negative constants 푐, 훼 > 0 such that for all 푛 ∈ N, it holds that
+푒(푛, 퐻훾) ≤ 푐 · 푛−훼.
+In order to quantify convergence rates, we introduce the notion of decay rates. The
+polynomial decay rate of a null sequence of positive reals x := {푥 푗} 푗 ∈N is given by
+dx := decay(x) = sup
+�
+푝 ≥ 0 :
+�
+푗 ∈N
+푥1/푝
+푗
+< ∞
+�
+.
+(10)
+This leads to the definition of the polynomialconvergence rate of 푛-th minimal errors,
+c.f. [5, Sec 4],
+푟∞ := 푟∞(퐻훾) = decay({푒(푛, 퐻훾)}푛∈N),
+(11)
+which is the quantity of interest in later sections.
+We now introduce complexity concepts which also appear in tractability studies
+as another perspective on the minimal errors. Let 휀 > 0 be the error demand. The
+worst case 휀-information complexity, also referred to as 휀-complexity of function
+approximation is defined as, c.f. [15],
+comp(휀, 퐻훾) := inf
+�
+cost(A) : A as in (7) and 푒(A, 퐻훾) ≤ 휀
+�
+.
+The approximation problemis strongly tractable if and only ifthereexist non-negative
+constants 푐, 훼 > 0 such that
+comp(휀, 퐻훾) ≤ 푐 · 휀−훼.
+Consequently, the exponent of tractability is defined as
+푝∞ := 푝∞(퐻훾) = inf
+�
+푝 ≥ 0 : ∃퐶푝 > 0∀휀 ∈ (0, 1) comp(휀, 퐻훾) ≤ 퐶푝휀−푝�
+.
+(12)
+We can relate the exponent of tractability to the convergence rate of the 푛-th minimal
+error as 푟∞ = 1/푝∞.
+2.2.3 Univariate and Multivariate 푳2-Approximation
+The polynomial convergence rate of the 푛-th minimal error of the univariate approxi-
+mation problem, according to (11), is denoted by 푟1(퐻(1 + 푘)) = decay({푒(푛, 퐻(1 +
+푘))}푛∈N).
+Consider the univariate solution operator 푆 : 퐻(1 + 푘) → 퐿2(퐷, 휌), 푓 ↦→ 푓 .
+It follows from (4) that 푆 is compact. This means that the self-adjoint operator
+푊 := 푆∗ ◦ 푆 is compact and positive definite. Consequently, 푊 has a null sequence
+of positive eigenvalues 휆 := {휆 푗} 푗 ∈N with a corresponding set of eigenfunctions
+{휂 푗} 푗 ∈N that form a complete orthonormal system of 퐻(1 + 푘) satisfying
+
+Infinite-Variate 퐿2-Approximation with Nested Subspace Sampling
+7
+푊(휂 푗) = 휆 푗휂 푗.
+(13)
+Let d휆 denote the decay rate of the eigenvalues 휆 of 푊 according to (10). Then it
+is well known that the convergence rate of the 푛-th minimal errors of the univariate
+approximation problem is d휆/2, see e.g. [13].
+Let 푆u : 퐻u → 퐿2(퐷u, 휌u) be given by 푆u( 푓 ) = 푓 for all 푓 ∈ 퐻u. Owing
+to the tensor product construction of spaces 퐻u for u ∈ U훾, the eigenpairs of
+the operator 푊u := 푆∗
+u ◦ 푆u : 퐻u → 퐻u are of product form. In particular, for
+u = {푢1, 푢2, . . . , 푢푘} ∈ U with 푘 = |u|, and for all j = ( 푗1, 푗2, . . . , 푗푘) ∈ N푘, we
+define
+휆u,j :=
+푘
+�
+푖=1
+휆 푗푖
+and
+휂u,j(x) :=
+푘
+�
+푖=1
+휂 푗푖 (푥푢푖),
+using the eigenvalues and eigenfunctions of the univariate operator as in (13). Then
+푊u(휂u,j) = 휆u,j휂u,j,
+(14)
+and �휂u,i, 휂u,j
+�
+퐻u = 훿i,j for i, j ∈ N푘. Using the eigenpairs of 푊u, we can express the
+SVD of 푆u as
+푆u( 푓u) =
+�
+j∈Nu
+�
+휆u,j
+�
+푓u, 휂u,j
+�
+퐻u · ¯휂u,j,
+(15)
+where
+¯휂u,j = 푆u(휂u,j)/
+�
+휆u,j
+(16)
+form an orthonormal system in 퐿2(퐷u, 휌u). In the same way as for the spaces 퐻u,
+we now define a self-adjoint operator for the infinite-variate setting
+W := S∗ ◦ S : 퐻훾 → 퐻훾
+which is compact and positive definite.
+3 ANOVA spaces
+For notational clarity, we use 퐾A
+훾 and 퐻A
+훾 to refer to the infinite-variate ANOVA
+kernel and the corresponding function space in this section. It is well known that in
+ANOVA spaces we have, see, e.g. [2],
+⟨ 푓u, 푔v⟩퐿2 (퐷u∪v,휌u∪v) = 0
+for u, v ∈ U, u ≠ v, 푓u ∈ 퐻u, 푔v ∈ 퐻v.
+(17)
+Lemma 1 In the ANOVA setting, any linear continuous algorithm A : 퐻A
+훾 →
+퐿2(픛, 휌N) satisfying A(퐻u) ⊆ S(퐻u) = 퐻u for all u ∈ U훾 also satisfies
+∥S( 푓 ) − A( 푓 )∥2
+퐿2 (픛,휌N) =
+�
+u∈U훾
+∥푆u( 푓u) − 퐴u( 푓u)∥2
+퐿2 (퐷u,휌u) ,
+
+8
+Kumar Harsha, Michael Gnewuch, and Marcin Wnuk
+where 퐴u : 퐻u → 퐿2(퐷u, 휌u) and 퐴u( 푓 ) = A( 푓 ) for all 푓 ∈ 퐻u.
+Proof The statement follows directly from (3) and (17), using the fact that S( 푓u) =
+푆u( 푓u).
+□
+Analogous to [15], we define an approximation algorithm based on the SVD of 푆u
+from (15). Let 훿 ∈ (0, 1) and define for 푓 ∈ 퐻u
+퐴∗
+u, 훿( 푓 ) :=
+�
+j∈퐽 ( 훿,u)
+�
+휆u,j
+�
+푓 , 휂u,j
+�
+퐻u · ¯휂u,j,
+(18)
+where 퐽(훿, u) := {j : �휆u,j > 훿} and ¯휂u,j is given by (16). We want to combine all
+the sub-algorithms 퐴∗
+u, 훿 : 퐻u → 퐿2(퐷u, 휌u) for u ∈ U훾 to attack the approximation
+problem on 퐻훾. Let 휀 ∈ (0, 1) be the worst-case error threshold for the infinite-variate
+problem, and define the set
+푀훾(휀) :=
+�
+u∈U훾
+{(u, j) : j ∈ 퐽(휀/√훾u, u)}
+=
+�
+(u, j) : u ∈ U훾, j ∈ Nu, and 훾u휆u,j > 휀2�
+.
+(19)
+We now define the algorithm Aopt
+휀,훾 : 퐻A
+훾 → 퐿2(픛, 휌N) as
+Aopt
+휀,훾 ( 푓 ) :=
+�
+u∈U훾
+퐴∗
+u,휀/√훾u ( 푓u) =
+�
+(u,j) ∈푀훾 (휀)
+�
+푓 , 휂u,j
+�
+퐻u · 푆u(휂u,j).
+(20)
+Lemma 2 For any u ∈ U훾 and 훿 > 0, the algorithm 퐴∗
+u, 훿 from (18) satisfies
+���푆u − 퐴∗
+u, 훿
+��� ≤ 훿.
+Proof Let 훿 ∈ (0, 1). Then using (15) and Bessel’s inequality, we get
+��푆u( 푓 ) − 퐴∗
+u, 훿( 푓 )
+��2
+퐿2 (퐷u,휌u) =
+������
+�
+j∉퐽 ( 훿,u)
+�
+휆u,j
+� 푓 , 휂u,j
+�
+퐻u · ¯휂u,j
+������
+2
+퐿2(퐷u,휌u)
+=
+�
+j∉퐽 ( 훿,u)
+휆u,j
+���
+�
+푓 , 휂u,j
+�
+퐻u
+���
+2 �� ¯휂u,j
+��2
+퐿2(퐷u,휌u) ≤ 훿2
+�
+j∉퐽 ( 훿,u)
+���
+�
+푓 , 휂u,j
+�
+퐻u
+���
+2
+≤ 훿2 ∥ 푓 ∥2
+퐻u .
+The result now follows by taking supremum over 푓 ∈ 퐻u with ∥ 푓 ∥퐻u = 1.
+□
+The next result is reformulated from [15, Theorem 1(i)], and has a straightforward
+proof.
+Corollary 1 In the ANOVA setting, the algorithm Aopt
+휀,훾 as in (20) achieves a worst
+case error at most 휀.
+The following result was mentioned in [15] without a proof, therefore we present it
+here for the reader’s convenience.
+
+Infinite-Variate 퐿2-Approximation with Nested Subspace Sampling
+9
+Lemma 3 The operator W = S∗S admits the following decomposition over ANOVA
+spaces
+W( 푓 ) =
+�
+u∈U훾
+훾u푊u( 푓u)
+∀푓 ∈ 퐻A
+훾 , 푓 =
+�
+u∈U훾
+푓u.
+(21)
+Proof The Definition 1 and the dominated convergence theorem imply that
+(W 푓 )(y) =
+�
+W( 푓 ), 퐾훾(·, y)
+�
+퐻 A
+훾 =
+�
+S( 푓 ), S(퐾훾 (·, y))
+�
+퐿2 (픛,휌N)
+=
+∫
+픛
+퐾훾(x, y) 푓 (x)d휌N(x) =
+∫
+픛
+�
+u∈U훾
+훾u푘u(xu, yu) 푓 (x)d휌N(x)
+=
+�
+u∈U훾
+훾u
+∫
+퐷u 푘u(xu, yu) 푓u(xu)d휌u(xu) =
+�
+u∈U훾
+훾u(푊u 푓u)(yu).
+The result now follows.
+□
+Using the eigenpairs from (14) and Lemma 3, we can conclude that the eigenpairs
+of W with positive eigenvalues are
+��훾u휆u,j, 휉u,j
+� : u ∈ U훾, j ∈ Nu� ,
+(22)
+with eigenfunctions 휉u,j := √훾u휂u,j normalized in 퐻A
+훾 . We can now express the
+solution operator S using its singular values and the orthonormal system as
+S( 푓 ) =
+�
+u∈U훾
+�
+j∈Nu
+�
+푓 , 휂u,j
+�
+퐻u · S(휂u,j) =
+�
+u∈U훾
+�
+j∈Nu
+�
+훾u휆u,j
+�
+푓 , 휉u,j
+�
+퐻훾 · ¯휂u,j,
+where ¯휂u,j is as defined in (16).
+The next result is the analogue of [15, Theorem 1(iii)] where it was proven for
+unrestricted subspace sampling. It can be checked that the proof follows verbatim
+and the arguments hold irrespective of the cost model.
+Theorem 1 In the nested subspace sampling model, the cost of the algorithm Aopt
+휀,훾
+is minimal amongst all algorithms that achieve a worst case error of at most 휀.
+Using the nested sequence from (9), we set for notational convenience
+푉0 := {∅}
+and
+푉푘 := {u ∈ U훾 : 푘 ∈ u ⊆ [푘]}
+for 푘 ≥ 1.
+We knowthat Aopt
+휀,훾 from(20) is the optimal algorithmfor the approximationproblem
+in ANOVA function spaces because it achieves an 휀 worst case error as shown in
+Corollary 1, and it follows from Theorem1 that it has minimal cost as well. Therefore,
+it suffices to compute the cost of Aopt
+휀,훾 to determine the 휀-complexity. Let
+푀(휀, 푘) :=
+�
+(u, j) : u ∈ 푉푘, j ∈ Nu, and 훾u휆u,j > 휀2�
+,
+and
+푛푘 := |푀(휀, 푘)| .
+(23)
+In order to simplify the cost analysis later on, we can rewrite the optimal algorithm
+from (20) in a multilevel fashion as
+
+10
+Kumar Harsha, Michael Gnewuch, and Marcin Wnuk
+Aopt
+휀,훾 ( 푓 ) =
+푚휀
+�
+푘=1
+�
+(u,j) ∈푀 (휀,푘)
+� 푓 , 휂u,j
+�
+퐻u · S(휂u,j),
+(24)
+where 푚휀 ∈ N is the largest number 푘 such that 푛푘 > 0.
+Lemma 4 For an error threshold 휀 > 0, and with the nested subspaces as in (9), the
+optimal algorithm Aopt
+휀,훾 samples from 푚휀 levels which can be estimated as
+푚휀 ≳ 휀−2/푝1
+∀푝1 ∈ (d훾, ∞),
+and
+푚휀 ≲ 휀−2/푝2
+∀푝2 ∈ (1, d훾).
+Proof From Definition (10), we have for all 푝1 ∈ (d훾, ∞) and 푝2 ∈ (1, d훾)
+훾 푗 ≳ 푗−푝1
+and
+훾 푗 ≲ 푗−푝2.
+(25)
+Using (19), we have ({푚휀 + 1}, 1) ∉ 푀훾(휀) which means 훾푚휀+1휆1 < 휀2. Using the
+lower bound from (25), we get for some 푐1 > 0
+푐1휆1(푚휀 + 1)−푝1 ≤ 휆1훾푚휀+1 ≤ 휀2 =⇒ (푐1휆1)1/푝1휀−2/푝1 ≤ 푚휀 + 1.
+Therefore, we get
+푚휀 ≥ (푐1휆1)1/푝1휀−2/푝1 − 1,
+(26)
+and the lower bound from the lemma follows.
+To show the remaining upper bound from the lemma, we note that �{푚휀}, 1� ∈
+푀훾(휀) must hold, which is equivalent to the condition 휆1훾푚휀 > 휀2. Then using (25)
+we get for some constant 푐2 > 0
+휀2 < 휆1훾푚휀 ≤ 푐2휆1푚−푝2
+휀
+=⇒ 푚휀 ≤ (푐2휆1)1/푝2휀−2/푝2,
+(27)
+giving us the desired upper bound.
+□
+We now want to estimate the cardinality of 푀(휀, 푘) as defined in (23). Before
+we proceed, we introduce the following notation. Let (u1, j1), (u2, j2) ∈ 푀(휀) such
+that u1 ∩ u2 = ∅, |u1| = 푘1, |u2| = 푘2 and j1 ∈ N푘1, j2 ∈ N푘2. Then, we define
+(u1, j2) × (u2, j2) = (u, j) with u = u1 ∪ u2, j ∈ Nu1∪u2 such that 푗푖 = 푗1,푖 if 푖 ∈ u1,
+and 푗푖 = 푗2,푖 if 푖 ∈ u2. Moreover, we define 푀(휀, 푘) := �(푘, 푗) : 푗 ∈ N, 훾푘휆 푗 > 휀2�,
+and 푛푘 :=
+���푀(휀, 푘)
+���.
+Let d훾 denote the decay rate of {훾 푗} 푗 ∈N. We assume that �
+푗 ∈N 훾1/d훾
+푗
+< 1. Then
+from [3, Theorem 5 and Corollary 1], we have for every 휏 ∈ [1, d훾] and every
+constant 퐶휏 > 0 that
+�
+푘∈u⊆[푘]
+훾1/휏
+u
+퐶 |u|
+휏
+≍ 훾1/휏
+푘
+.
+(28)
+Lemma 5 Let 휀 > 0 and 푘 ∈ [푚휀]. Then for all 푞1 ∈ (d휆, ∞) and 푞2 ∈ (1, d휆) we
+have the following properties.
+
+Infinite-Variate 퐿2-Approximation with Nested Subspace Sampling
+11
+1. 푀(휀, 푘 + 1) = 푀(휀, 푘 + 1) ∪ �푘
+ℓ=1
+�
+푗 ∈N
+�
+({푘 + 1}, 푗) × 푀
+�
+휀
+√
+훾푘+1휆 푗 , ℓ
+��
+.
+2. |푀(휀, 푘 + 1)| =
+���푀(휀, 푘 + 1)
+��� + �푘
+ℓ=1
+�
+푗 ∈N
+����푀
+�
+휀
+√
+훾푘+1휆 푗 , ℓ
+�����.
+3.
+���푀(휀, 푘)
+��� ≳ 훾1/푞1
+푘
+휀−2/푞1.
+4.
+���푀(휀, 푘)
+��� ≲ 훾1/푞2
+푘
+휀−2/푞2.
+5. |푀(휀, 푘)| ≲ 휀−2/푞2 �
+u∈푉푘 훾1/푞2
+u
+푐(휆, 푞2) |푢|−1, where 푐(휆, 푞2) = �
+푗 ∈N 휆1/푞2
+푗
+< ∞.
+6. |푀(휀, 푘)| ≲ 휀−2/푞2훾1/푞2
+푘
+, if additionally 푞2 ∈ (1, min{d휆, d훾}).
+Proof Similar to (25), we get for all 푞1 ∈ (d휆, ∞) and 푞2 ∈ (1, d휆) that
+휆 푗 ≳ 푗−푞1
+and
+휆 푗 ≲ 푗−푞2.
+(29)
+We assume that 휀2 < 훾1휆1 ≤ 1, since otherwise the set 푀(휀) is empty.
+1. Let (u ∪ {푘 + 1}, (j, 푗푘+1)) ∈ 푀(휀, 푘 + 1) where ∅ ≠ u ⊆ [푘], j ∈ Nu and
+푗푘+1 ∈ N. Since we have product weights, the definition (23) requires that 훾u휆j >
+휀2/훾푘+1휆 푗푘+1. Let ℓ = max(u), then u ∈ 푉ℓ and
+(u, j) ∈ 푀
+�
+휀
+�훾푘+1휆 푗푘+1
+, ℓ
+�
+.
+2. Note that the right hand side in statement 1 is a union of disjoint sets.
+3. Using the lower bound from (29), and the definition of 푀(휀, 푘), we get for some
+푐1 > 0 that
+푐1훾푘(푛푘 + 1)−푞1 ≤ 휆푛푘+1훾푘 ≤ 휀2 =⇒ 푛푘 ≥ 푐1/푞1
+1
+훾1/푞1
+푘
+휀−2/푞1 − 1.
+4. Using the upper bound from (29), and the definition of 푀(휀, 푘), we get for some
+푐2 > 0 that
+휀2 < 훾푘휆푛푘 ≤ 푐2훾푘푛−푞2
+푘
+=⇒ 푛푘 ≤ 푐1/푞2
+2
+훾1/푞2
+푘
+휀−2/푞2.
+5. We prove this result by induction. For 푘 = 1, we have due to statement 4
+푛1 = |푀(휀, 1)| =
+���푀(휀, 1)
+��� ≲ 훾1/푞2
+1
+휀−2/푞2.
+We suppose now that the bound is true for 푀(휀,1), . . . , 푀(휀, 푘). Using statements
+2 and 4, we get for some 푐2 > 0
+
+12
+Kumar Harsha, Michael Gnewuch, and Marcin Wnuk
+|푀(휀, 푘 + 1)| =
+���푀(휀, 푘 + 1)
+��� +
+푘
+�
+ℓ=1
+�
+푗 ∈N
+�����푀
+�
+휀
+�훾푘+1휆 푗
+, ℓ
+������
+≤ 푐2휀−2/푞2훾1/푞2
+푘+1 + 푐2
+푘
+�
+ℓ=1
+�
+푗 ∈N
+�
+휀
+�훾푘+1휆 푗
+�−2/푞2 �
+u∈푉ℓ
+훾1/푞2
+u
+푐(휆, 푞2) |푢|−1
+= 푐2휀−2/푞2훾1/푞2
+푘+1
+�
+1 +
+푘
+�
+ℓ=1
+�
+u∈푉ℓ
+훾1/푞2
+u
+푐(휆, 푞2) |푢|−1 �
+푗 ∈N
+휆1/푞2
+푗
+�
+= 푐2휀−2/푞2
+
+훾1/푞2
+푘+1 + 훾1/푞2
+푘+1
+�
+∅≠u⊆[푘]
+훾1/푞2
+u
+푐(휆, 푞2) |푢|
+
+= 푐2휀−2/푞2 �
+u∈푉푘+1
+훾1/푞2
+u
+푐(휆, 푞2) |푢|−1.
+6. The estimate follows directly by applying (28) to the formula in statement 5.
+This completes the proof.
+□
+Theorem 2 Let $(푘) = Θ(푘푠) for some 푠 > 0. Then the optimal algorithm Aopt
+휀,훾 as
+in (20) achieves the following convergence rate using the nested subspace sampling
+cost model in ANOVA function spaces with product weights
+푟 퐴
+∞ = min
+�d휆
+2 ,
+d훾
+2(1 + 푠)
+�
+.
+Proof Let 휀 be the desired worst case error, and 푚휀 be the highest level for our
+algorithm.
+We first show 푟 퐴
+∞ ≤ min
+�
+d휆
+2 ,
+d훾
+2(1+푠)
+�
+using $(푘) = Ω(푘푠). Since 푀(휀, 푘) ⊂
+푀(휀, 푘), we get using the lower bound on
+���푀(휀, 푘)
+��� from statement 3 of Lemma 5,
+and using (25) that
+cost(Aopt
+휀,훾) ≥
+푚휀
+�
+푘=1
+���푀(휀, 푘)
+���·$(푘) ≳ 휀−2/푞1
+푚휀
+�
+푘=1
+훾1/푞1
+푘
+푘푠 ≳ 휀−2/푞1
+푚휀
+�
+푘=1
+푘푠−푝1/푞1, (30)
+where for 훿 > 0 arbitrarily small, we choose 푝1 = d훾(1 + 훿) and 푞1 = d휆(1 + 훿).
+From the last expression in (30), we may distinguish three cases.
+1. d훾/(1 + 푠) > d휆 =⇒ 푠 − 푝1/푞1 < −1: This means we have �푚휀
+푘=1 푘푠−푝1/푞1 ≳ 1.
+Consequently, as 푞1 → d휆, we get 푟 퐴
+∞ ≤ d휆/2.
+2. d훾/(1 + 푠) = d휆 =⇒ 푠 − 푝1/푞1 = −1: Here we get the estimate
+푚휀
+�
+푘=1
+푘−1 ≳ 1 + ln(푚휀) ≳ 1 + 2
+푝1
+ln(1/휀),
+
+Infinite-Variate 퐿2-Approximation with Nested Subspace Sampling
+13
+where the last relation follows from Lemma 4. Hence, we get cost(Aopt
+휀,훾) ≳
+휀−2/푞1(1 + ln(1/휀)). Since the definition of the exponent of tractability as in (12)
+ignores the logarithmic terms, we get 푟 퐴
+∞ ≤ d휆/2 as 푞1 → d휆.
+3. d훾/(1 + 푠) < d휆 =⇒ 푠 − 푝1/푞1 > −1: In this case, we can use Lemma 4 to get
+푚휀
+�
+푘=1
+푘푠−푝1/푞1 ≳ (푚휀)1−푝1/푞1+푠 ≳ 휀−2(1−푝1/푞1+푠)/푝1,
+which implies that cost(Aopt
+휀,훾) ≳ 휀−2(1+푠)/푝1. As 푝1 → d훾, we have 푟 퐴
+∞ ≤
+d훾/2(1 + 푠).
+We now show that 푟 퐴
+∞ ≥ min
+�
+푟1,
+d훾
+2(1+푠)
+�
+using $(푘) ≲ 푘푠. We use (25) and
+statement 6 from Lemma 5 to get
+cost(Aopt
+휀,훾) =
+푚휀
+�
+푘=1
+|푀(휀, 푘)| · $(푘) ≲ 휀−2/푞2
+푚휀
+�
+푘=1
+훾1/푞2
+푘
+푘푠 ≲ 휀−2/푞2
+푚휀
+�
+푘=1
+푘푠−푝2/푞2,
+(31)
+where for 훿 > 0 arbitrarily small, we choose 푝2 = d훾(1 − 훿) and 푞2 = d휆(1 − 훿). We
+analyze the last expression in (31) in three cases as in the first part of the proof.
+1. d훾/(1 + 푠) > d휆 =⇒ 푠 − 푝2/푞2 < −1: Then �푚휀
+푘=1 푘푠−푝2/푞2 ≲ 1. Therefore, we
+may choose 푞2 → d휆 to get 푟 퐴
+∞ ≥ d휆/2.
+2. d훾/(1 + 푠) = d휆
+=⇒
+푠 − 푝2/푞2 = −1: For this case, we get the estimate
+�푚휀
+푘=1 푘−1 ≲ 1 + ln(푚휀) ≲ 1 +
+2
+푝2 ln(1/휀), where we used Lemma 4 for the last
+relation. Consequently, we have cost(Aopt
+휀,훾) ≲ 휀−2/푞2(1 + ln(1/휀)). Similarly as
+in the first part of the proof, we can ignore the logarithmic terms as 푞2 → d휆 to
+get 푟 퐴
+∞ ≥ d휆/2.
+3. d훾/(1 + 푠) < d휆 =⇒ 푠 − 푝2/푞2 > −1: Due to Lemma 4 we get
+푚휀
+�
+푘=1
+푘−푝2/푞2+푠 ≲ 푚1−푝2/푞2+푠
+휀
+≲ 휀−2(1−푝2/푞2+푠)/푝2.
+Hence, we get cost(Aopt
+휀,훾) ≲ 휀−2(1+푠)/푝2 and we may choose 푝2 → d훾 to get
+푟 퐴
+∞ ≥ d훾/2(1 + 푠).
+This completes the proof.
+□
+4 Non-ANOVA spaces
+Let us now turn to the case where the underlying function decomposition of our
+Hilbert space 퐻훾 is not of ANOVA-type. We start by proving a lower bound on the
+convergence rate of 푛-th minimal errors.
+
+14
+Kumar Harsha, Michael Gnewuch, and Marcin Wnuk
+Proposition 1 Let $(푘) ≳ 푘푠 for any 푘 ∈ N and some 푠 > 0. Then the optimal
+convergence rate for the algorithms defined as in (7) satisfy in non-ANOVA function
+spaces
+푟∞ ≤ min
+�d휆
+2 , d훾 − 1
+2푠
+�
+.
+Proof The bound 푟∞ ≤ d휆/2 follows from the rate of convergence for the univariate
+problem.
+To show the other part, we adapt the proof strategy from [14, Proposition 4].
+Let A푁 be a linear algorithm using finitely many function evaluations, such that
+cost(A푁 ) ≤ 푁. We now choose 퐿 := sup {푘 : $(푘) ≤ 푁} . Due to this definition,
+푉 := [퐿] is a superset of all the indices that the algorithm A푁 relies on. Since
+we assumed that $(푘) ≳ 푘푠, there exists a constant 푐 > 0 such that 푘푠/푐푠 ≤ $(푘),
+implying 퐿 ≤ 푐푁1/푠.
+Consider a univariate function ℎ ∈ 퐻(푘) such that ∥ℎ∥퐻 (푘) = 1 and 푐1 :=
+∫
+퐷 ℎ(푥)d휌(푥) ≠ 0, cf. Remark 1. Then define 푓 ∗ ∈ 퐻훾 as
+푓 ∗(x) := 1
+휅
+�
+푗∉푉
+훾 푗ℎ(푥 푗);
+with 휅 :=
+��
+푗∉푉
+훾 푗.
+As shown in [14], A푁 ( 푓 ∗) = 0, ∥ 푓 ∥퐻훾 = 1, and also
+∥ 푓 ∗∥2
+퐿2 (픛,휌N) =
+�
+∥ℎ∥2
+퐿2 (퐷,휌) − 푐2
+1
+� �
+푗∉푉 훾2
+푗
+�
+푗∉푉 훾 푗
++ 푐2
+1
+�
+푗∉푉
+훾 푗.
+From (10), we have for any arbitrarily small 훿 > 0 that 훾 푗 ≳ 푗−(d훾+훿/4), and also
+훾 푗 ≲ 푗−(d훾−훿/2). With this, we can conclude that
+�
+푗∉푉 훾2
+푗
+�
+푗∉푉 훾 푗
+≳
+�∞
+푗=퐿+1 푗−2d훾−훿/2
+�∞
+푗=퐿+1 푗−d훾+훿/2 ≳ 퐿−d훾−훿 ≳ 푁−(d훾+훿)/푠.
+Similarly, we have that �
+푗∉푉 훾 푗 ≳ 퐿1−d훾−훿 ≳ 푁 (1−d훾−훿)/푠 . From the construction
+of 푓 ∗, we have for the error that
+푒(A푁 ; 퐻훾) =
+sup
+∥ 푓 ∥퐻훾 =1
+∥ 푓 − A푁 ( 푓 )∥퐿2 (픛,휌N) ≥ ∥ 푓 ∗∥퐿2 (픛,휌N) ≳ 푁 (1−d훾−훿)/2푠.
+This shows that 푟∞ ≤ (d훾 − 1)/2푠, and the result follows.
+□
+For non-ANOVA spaces, Lemma 3 does not hold in general, which means that
+we cannot express the eigenpairs of W : 퐻훾 → 퐻훾 in terms of the eigenpairs of 푊u.
+For the upper error bound, we need results from [14] which are stated below for the
+reader’s convenience. Let 퐶0 := sup 푓 ∈퐻 (푘)
+∥ 푓 ∥퐿2 (퐷,휌)
+∥ 푓 ∥퐻 (푘)
+< ∞.
+Lemma 6 For all 푐 ∈ (1/d훾, 1) it holds that supu∈U훾 퐶2|u|
+0
+훾1−푐
+u
+< ∞.
+
+Infinite-Variate 퐿2-Approximation with Nested Subspace Sampling
+15
+1. For
+�훾푐 = {�훾u,푐}u∈U훾,
+�훾u,푐 := 훾1−푐
+u
+,
+we define 휙 : 퐻훾 → 퐻�훾푐, 푓 ↦→ �
+u∈U훾 훾−푐/2
+u
+푓u. Then 휙 is an isometric isomor-
+phism between 퐻훾 and 퐻�훾푐. Furthermore, 퐻훾 ⊆ 퐻�훾푐.
+2. Let A be a linear continuous algorithm satisfying
+A(퐻u) ⊆ 퐿2(퐷u, 휌u),
+(32)
+for all u ∈ U훾. Then we get the upper bound
+∥S( 푓 ) − A( 푓 )∥2
+퐿2 (픛,휌N) ≤ 퐶훾
+�
+u∈U훾
+���푆u( �푓u) − 퐴u( �푓u)
+���
+2
+퐿2 (퐷u,휌u)
+(33)
+where 퐶훾 = �
+v∈U훾 훾푐
+v, 휙( 푓 ) =: �푓 = �
+u∈U훾 �푓u for all 푓 ∈ 퐻훾, and 퐴u : 퐻u →
+퐿2(퐷u, 휌u) such that 퐴u( 푓 ) = A( 푓 ) for all 푓 ∈ 퐻u.
+It is straightforward to check that 휙 defines an isometric isomorphism. 퐻훾 ⊆ 퐻�훾푐
+is due to [14, Lemma 2]. The inequality (33) has been derived in the proof of [14,
+Lemma 2] for any algorithm A satisfying (32).
+Proposition 2 For any 휀 ∈ (0, 1), define the set 푀�훾푐 (휀) as in (19) and also the
+algorithm Aopt
+휀,�훾푐 : 퐻�훾푐 → 퐿2(픛, 휌N) as defined in (20). Then the algorithm Aopt
+휀,�훾푐
+satisfies (32) and 푒(Aopt
+휀,�훾푐, S, 퐻훾) ≤ 휀. The algorithm Aopt
+휀,�훾푐 establishes that
+푟∞ ≥ min
+�d휆
+2 , d훾 − 1
+2(1 + 푠)
+�
+.
+Proof Using Lemmas 6 and 2, we get
+���S( 푓 ) − Aopt
+휀,�훾푐 ( 푓 )
+���
+2
+퐿2(픛,휌N) ≲
+�
+u∈U훾
+����푆u( �푓u) − 퐴∗
+u,휀/√
+�훾u,푐 ( �푓u)
+����
+2
+퐿2 (퐷u,휌u)
+≤ 휀2 �
+u∈U훾
+1
+�훾u,푐
+��� �푓u
+���
+2
+퐻u = 휀2 ��� �푓
+���
+2
+퐻�훾푐
+= 휀2 ∥ 푓 ∥2
+퐻훾 .
+Notice that Aopt
+휀,�훾푐 has the same cost in 퐻�훾푐 and in its subspace 퐻훾, since the
+function space decomposition in both spaces rely on the same family of subspaces
+(퐻u)u∈U훾. Let d�훾푐 denote the decay rate of product weights �훾푐. Then from statement
+5 of Lemma 5, (31), and the subsequent analysis in Theorem 2, we get
+푟∞ ≥ min
+�d휆
+2 ,
+d�훾푐
+2(1 + 푠)
+�
+.
+Finally, d�훾푐 = d훾(1 − 푐)
+푐→1/d훾
+−−−−−−→ d훾 − 1. This completes the proof.
+□
+
+16
+Kumar Harsha, Michael Gnewuch, and Marcin Wnuk
+Corollary 2 If 1 < d훾 < 1 + 푠, then the convergence rate of 퐿2-approximation in
+ANOVA spaces is strictly larger than the one in non-ANOVA spaces.
+Proof Under the given condition,the optimal convergence rate of 푛-th minimal errors
+in the ANOVA case from Theorem 2 is greater than the largest possible convergence
+rate in the non-ANOVA case from Proposition 1, i.e. (d훾 − 1)/2푠 < d훾/2(1 + 푠). □
+As we see from the analysis, the convergence rates from the upper and lower error
+bounds in the non-ANOVA spaces do not match. In future work, we will consider
+bridging the gap.
+References
+1. N. Aronszajn. Theory of reproducing kernels. Trans. Amer. Math. Soc., 68:337–404, 1950.
+2. J. Baldeaux and M. Gnewuch.
+Optimal randomized multilevel algorithms for infinite-
+dimensional integration on function spaces with ANOVA-type decomposition. SIAM J. Numer.
+Anal., 52:1128–1155, 2014.
+3. J. Dick and M. Gnewuch. Infinite-dimensional integration in weighted Hilbert spaces: anchored
+decompositions, optimal deterministic algorithms, and higher order convergence. Found. Com-
+put. Math., 14:1027–1077, 2014.
+4. J. Dick and M. Gnewuch. Optimal randomized changing dimension algorithms for infinite-
+dimensional integration on function spaces with ANOVA-type decomposition.
+J. Approx.
+Theory, 184:111–145, 2014.
+5. M. Gnewuch, M. Hefter, A. Hinrichs, and K. Ritter. Embeddings of weighted Hilbert spaces
+and applicationsto multivariate and infinite-dimensional integration. Journal of Approximation
+Theory, 222:8–39, 2017.
+6. M. Gnewuch, M. Hefter, A. Hinrichs, K. Ritter, and G. W. Wasilkowski. Embeddingsforinfinite-
+dimensional integration and 퐿2-approximation with increasing smoothness. J. Complexity,
+54:101406, 2019.
+7. M. Gnewuch, S. Mayer, and K. Ritter. On weighted Hilbert spaces and integration of functions
+of infinitely many variables. J. Complexity, 30:29–47, 2014.
+8. M. Hefter and K. Ritter.
+On embeddings of weighted tensor product Hilbert spaces.
+J.
+Complexity, 31:405–423, 2015.
+9. F. J. Hickernell, T. Müller-Gronbach, B. Niu, and K. Ritter. Multi-level Monte Carlo algorithms
+for infinite-dimensional integration on RN. J. Complexity, 26:229–254, 2010.
+10. F. Y. Kuo, I. H. Sloan, G. W. Wasilkowski, and H. Woźniakowski. Liberating the dimension.
+J. Complexity, 26:422–454, 2010.
+11. T. Müller-Gronbach and K. Ritter. Variable subspace sampling and multi-level algorithms. In
+P. L’Ecuyer and A. B. Owen, editors, Monte Carlo and Quasi-Monte Carlo Methods 2008.
+Springer, 2009.
+12. E. Novak and H. Woźniakowski. Tractability of Multivariate Problems. Vol. 1: Linear Infor-
+mation. EMS Tracts in Mathematics. European Mathematical Society (EMS), Zürich, 2008.
+13. J. F. Traub, G. W. Wasilkowski, and H. Woźniakowski. Information-Based Complexity. Aca-
+demic Press, New York, 1988.
+14. G. W. Wasilkowski. Liberating the dimension for 퐿2-approximation. J. Complexity, 28:304–
+319, 2012.
+15. G. W. Wasilkowski and H. Woźniakowski. Liberating the dimension forfunction approximation.
+J. Complexity, 27:86–110, 2011.
+16. Marcin Wnuk. A short note on compact embeddings of reproducing kernel Hilbert spaces in
+퐿2 for infinite-variate function approximation, 2022.
+
diff --git a/etFPT4oBgHgl3EQfzzXg/content/tmp_files/load_file.txt b/etFPT4oBgHgl3EQfzzXg/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..51a1d9e3eaa0ebb0042c4fb7a5034f2dc1a1cecd
--- /dev/null
+++ b/etFPT4oBgHgl3EQfzzXg/content/tmp_files/load_file.txt
@@ -0,0 +1,467 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf,len=466
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='13177v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='NA]  30 Jan 2023  Infinite-Variate 푳2-Approximation with Nested  Subspace Sampling  Kumar Harsha, Michael Gnewuch, and Marcin Wnuk  Abstract We consider 퐿2-approximation on weighted reproducing kernel Hilbert  spaces of functions depending on infinitely many variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' We focus on unrestricted  linear information, admitting evaluations of arbitrary continuous linear functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  We distinguish between ANOVA and non-ANOVA spaces, where, by ANOVA spaces,  we refer to function spaces whose norms are induced by an underlying ANOVA func-  tion decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' In ANOVA spaces, we prove that there is an optimal algorithm  to solve the approximation problem using linear information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' This way, we can deter-  mine the exact polynomial convergence rate of 푛-th minimal worst-case errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' For  non-ANOVA spaces, we also establish upper and lower error bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Even though  the bounds do not match in this case, they reveal that for weights with a moderate  decay behavior, the convergence rate of 푛-th minimal errors is strictly higher in  ANOVA than in non-ANOVA spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  1 Introduction  We want to investigate 퐿2-approximation on weighted reproducing kernel Hilbert  spaces (RKHSs) of functions depending on infinitely many variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' For each of  those RKHSs, there is a family of weights and a function decomposition, which  determine the norm on the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' With the help of the function decomposition, it  is possible to represent an infinite-variate function as an infinite sum of functions  depending only on finitely many variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Harsha •M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Gnewuch •M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Wnuk  Institute for Mathematics, Osnabrück University, Albrechtraße 28a, 49074 Osnabrück, Germany  e-mail: kumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='harsha@uni-osnabrueck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='de  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Gnewuch  e-mail: michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='gnewuch@uni-osnabrueck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='de  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Wnuk  e-mail: marcin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='wnuk@uni-osnabrueck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='de  1  2  Kumar Harsha, Michael Gnewuch, and Marcin Wnuk  In this paper, we focus on linear algorithms that may obtain information about  an input function 푓 by evaluating finitely many continuous linear functionals in 푓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  The information cost of such an algorithm is the sum of the cost of all functional  evaluations, which, in turn, depends on the underlying function decomposition of  the weighted RKHS we are working on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Such a situation has already been stud-  ied in [14, 15], where the considered cost model is a generalization of the one  introduced in [10] for information consisting only of function evaluation (so-called  standard information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' In [15], a rather specific approximation problem, called the  G-approximationproblem, is analyzed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' there, the approximationerror is measured in  some very special Hilbert space G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' In [14], 퐿2-approximation is investigated in full  generality, apart from the case that can be reduced to the G-approximation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  To describe the results from [14,15] and from this paper in a more common language,  we distinguish two cases of weighted RKHSs: the ones whose underlying function  decomposition is an ANOVA decomposition are called ANOVA spaces, whereas the  remaining RKHSs are referred to as non-ANOVA spaces (for a rigorous definition,  see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  The case of ANOVA spaces was covered by Wasilkowski and Woźniakowski, who  in [15] exactly determined the polynomial convergence rates 푟∞ of 푛-th minimal  errors (for a definition, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' For the non-ANOVA spaces, Wasilkowski  was able to derive in [14] a lower bound for 푟∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' with the help of auxiliary weights, he  re-used the constructive results from [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' He also proved a matching lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Interestingly, it turns out that for slowly decaying weights, the convergence rate in  the ANOVA case is superior to the one in the non-ANOVA case, while it stays the  same if the weights decrease quickly enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  In this paper, we consider the nested subspace sampling cost model, which is less  generous than the cost model in [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' For this cost model, we have derived the  exact polynomialconvergencerate 푟∞ of 푛-th minimal errors in the ANOVA case, see  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' In the non-ANOVA case, we are able to provide upper and constructive  lower bounds on 푟∞, see Propositions 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Although these bounds do not match,  they show that in some cases the convergence rate in the ANOVA case is higher than  the one in the non-ANOVA case, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' The reason for that is mainly that the  cost of an algorithm may depend on what kind of function decomposition underlies  the function space it is defined on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' This dependence, unfortunately, prevents us from  simply using the embedding machinery developed in [5, 6, 8] to transfer results for  ANOVA spaces directly to spaces with a differentunderlying function decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Notation For a set u, we denote by |u| its cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' We use the shorthand  [푘] := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' , 푘} for every 푘 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  2 Function Spaces and Algorithms  For general results on reproducing kernel Hilbert spaces (RKHS) we refer to [1], and  to [7] for results on the function spaces discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Infinite-Variate 퐿2-Approximation with Nested Subspace Sampling  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='1 Weighted Function Spaces  Let 퐷 ≠ ∅ be an arbitrary set, Σ a 휎-algebra on 퐷, and 휌 : Σ → [0, 1] a probability  measure on 퐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Let 푘 : 퐷 × 퐷 → R be a reproducing kernel on 퐷 such that  퐻(1) ∩ 퐻(푘) = {0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  here and in the whole paper we follow the convention to denote by 퐻(푘) the uniquely  determined Hilbert space with reproducing kernel 푘 and its norm by ∥ · ∥퐻 (푘);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' in  particular 퐻(1) is the space of all constant functions on the corresponding domain  퐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Put  U := {u ⊂ N | |u| < ∞},  where |u| refers to the cardinality of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' For u ∈ U we define 푘u : 퐷N × 퐷N → R by  푘u(x, y) :=  �  푗 ∈u  푘(푥 푗, 푦 푗)  for all x, y ∈ 퐷N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Since 푘u only depends on the variables 푥 푗, 푗 ∈ u, we may alternatively view it as a  function on 퐷u by putting  푘u(xu, yu) := 푘u(˜x, ˜y)  for arbitrary ˜x, ˜y ∈ 퐷N satisfying ˜xu = xu, ˜yu = yu, where xu := {푥 푗} 푗 ∈u ∈ 퐷u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  We use an analogous convention for arbitrary functions on 퐷N that only depend on  variables 푥 푗, 푗 ∈ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' We denote the RKHS with kernel 푘u by 퐻u := 퐻(푘u) and its  norm by ∥ · ∥퐻u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' in particular, we have 퐻∅ = 퐻(1) due to the convention that an  empty product is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  For a family 훾 = {훾u}u∈U of non-negative weights, we put  U훾 := {u ∈ U | 훾u > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  We work with product weights which are given by a decreasing sequence of weights  1 ≥ 훾1 ≥ 훾2 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' ≥ 0 assigned to all of the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Then the weight (or  importance) of an interacting set of variables u is given by 훾u = �  푗 ∈u 훾 푗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Moreover,  we define  픛 :=  �  x ∈ 퐷N  ����  �  u∈U훾  훾u푘u(x, x) < ∞  �  and the reproducing kernel 퐾훾 : 픛 × 픛 → R by  퐾훾(x, y) :=  �  u∈U훾  훾u푘u(x, y)  x, y ∈ 픛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  (1)  We denote the corresponding RKHS by 퐻훾 := 퐻(퐾훾).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' 퐻훾 decomposes into an  orthogonal sum of subspaces  4  Kumar Harsha, Michael Gnewuch, and Marcin Wnuk  퐻훾 =  �  u∈U훾  퐻(훾u푘u) =  �  u∈U훾  훾u퐻u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  (2)  Note that for u ∈ U훾 we have that 퐻(훾u푘u) and 퐻u are equal as vector spaces, but  their norms differ by a factor 훾−1/2  u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Hence the norm in 퐻훾 is given by  ∥ 푓 ∥2  퐻훾 =  �  u∈U훾  훾−1  u ∥ 푓u∥2  퐻u ,  where  푓 =  �  u∈U훾  푓u,  푓u ∈ 퐻u,  (3)  is the uniquely defined function decomposition of 푓 in 퐻훾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  We assume that  푀 :=  ∫  퐷  푘(푥, 푥) d휌(푥) < ∞  (4)  and  �  u∈U훾  훾u푀 |u| < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  (5)  From our assumptions (4) and (5) follows that we have the continuous and compact  embeddings 퐻(1 + 푘) ⊆ 퐿2(퐷, 휌), 퐻u ⊆ 퐿2(퐷u, 휌u) for all u ∈ U훾, and 퐻훾 ⊆  퐿2(픛, 휌N), see [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' In particular, the solution operator we consider  S : 퐻훾 → 퐿2(픛, 휌N) , 푓 ↦→ 푓  is well-defined and continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Definition 1 (ANOVA kernels) Let 푘 : 퐷 × 퐷 → R be a reproducing kernel such  that 푘(·, 푥) ∈ 퐿2(퐷, 휌) for all 푥 ∈ 퐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' If the kernel satisfies the ANOVA condition  ∫  퐷  푘(푦, 푥)d휌(푦) = 0  ∀푥 ∈ 퐷,  (6)  then we call 푘 an ANOVA kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Otherwise, we call it a non-ANOVA kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Remark 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' It is easily seen that under the assumptions of Definition 1, the ANOVA  condition (6) is equivalent to the condition that  ∫  퐷 푓 (푦)휌(푦) = 0 for all 푓 ∈ 퐻(푘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Consider the reproducing kernel as in (1) based on an ANOVA kernel 푘 : 퐷×퐷 →  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Then for each 푓 ∈ 퐻훾, the function decomposition as in (3) is the infinite-  variate ANOVA decomposition, see [2, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' For instance, it implies that  for each 푓 ∈ 퐻훾, for all u ∈ U and x ∈ 퐷N  ∫  퐷  푓u(x) d휌(푥 푗) = 0  if 푗 ∈ 푢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  ExamplesofANOVAspacesincludeKorobov spaces,seee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' [12,Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='1]and  unanchored Sobolev spaces, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' [12, Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='1,A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='3], [2, Section 5],  Infinite-Variate 퐿2-Approximation with Nested Subspace Sampling  5  and [4, Section 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' A notable example of non-ANOVA spaces are anchored Sobolev  spaces, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' [12, Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='2] or [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='2 Algorithms, Cost Models, and Errors  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='1 Admissible algorithms and cost  We want to approximate 푓 ∈ 퐻훾 by algorithms using finitely many information eval-  uations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Since our problem is linear, we may confine ourselves to linear algorithms  using arbitrary linear information of the form  A( 푓 ) =  푛  �  푗=1  퐿u 푗 ( 푓 ) · 푔 푗  (7)  for some 푛 ∈ N, 푔 푗 ∈ 퐿2(픛, 휌N), index sets u푗 ∈ U훾, and linear functionals  퐿u 푗 ∈ 퐻∗  훾 such that 퐿u 푗 |퐻v = 0 if v ⊈ u푗, see [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Owing to the Riesz representation  theorem, we can define the linear functional 퐿u 푗 as inner product with a representer  as  퐿u 푗 ( 푓 ) :=  �  푓 , 휂u 푗  �  퐻훾 ,  where 휂u 푗 ∈  �  v⊆u 푗  퐻v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  (8)  The worst-case error of the algorithm A for 퐿2-approximation with respect to  퐻훾 is given as  푒(A, 퐻훾) := 푒(A, S, 퐻훾) :=  sup  ∥ 푓 ∥퐻훾 ≤1  ∥S( 푓 ) − A( 푓 )∥퐿2 (픛,휌N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Let us fix a cost function $ : N → [1, ∞) which is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' For nested  subspace sampling (NSS) (see [5, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='1] or [2,3,9,11]) we consider a sequence  of nested finite index sets  ∅ = v0 ⊂ v1 ⊂ v2 ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' ,  (9)  with v푖 := [푖] for 푖 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Then the cost of any information evaluation 퐿u( 푓 ) as in (8)  is given by  cost(퐿u) = inf{$(|v푖|) : u ⊆ v푖}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='2 Minimal errors and complexity  For the results in this paper, we work with the NSS cost model unless specified  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' On that basis, we define the 푛-th minimal error for 푛 ∈ N as  푒(푛, 퐻훾) := inf{푒(A, 퐻훾) : A as in (7) and cost(A) ≤ 푛}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  6  Kumar Harsha, Michael Gnewuch, and Marcin Wnuk  The approximation problem is said to be strongly tractable if and only if there exist  non-negative constants 푐, 훼 > 0 such that for all 푛 ∈ N, it holds that  푒(푛, 퐻훾) ≤ 푐 · 푛−훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  In order to quantify convergence rates, we introduce the notion of decay rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' The  polynomial decay rate of a null sequence of positive reals x := {푥 푗} 푗 ∈N is given by  dx := decay(x) = sup  �  푝 ≥ 0 :  �  푗 ∈N  푥1/푝  푗  < ∞  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  (10)  This leads to the definition of the polynomialconvergence rate of 푛-th minimal errors,  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' [5, Sec 4],  푟∞ := 푟∞(퐻훾) = decay({푒(푛, 퐻훾)}푛∈N),  (11)  which is the quantity of interest in later sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  We now introduce complexity concepts which also appear in tractability studies  as another perspective on the minimal errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Let 휀 > 0 be the error demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' The  worst case 휀-information complexity, also referred to as 휀-complexity of function  approximation is defined as, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' [15],  comp(휀, 퐻훾) := inf  �  cost(A) : A as in (7) and 푒(A, 퐻훾) ≤ 휀  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  The approximation problemis strongly tractable if and only ifthereexist non-negative  constants 푐, 훼 > 0 such that  comp(휀, 퐻훾) ≤ 푐 · 휀−훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Consequently, the exponent of tractability is defined as  푝∞ := 푝∞(퐻훾) = inf  �  푝 ≥ 0 : ∃퐶푝 > 0∀휀 ∈ (0, 1) comp(휀, 퐻훾) ≤ 퐶푝휀−푝�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  (12)  We can relate the exponent of tractability to the convergence rate of the 푛-th minimal  error as 푟∞ = 1/푝∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='3 Univariate and Multivariate 푳2-Approximation  The polynomial convergence rate of the 푛-th minimal error of the univariate approxi-  mation problem, according to (11), is denoted by 푟1(퐻(1 + 푘)) = decay({푒(푛, 퐻(1 +  푘))}푛∈N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Consider the univariate solution operator 푆 : 퐻(1 + 푘) → 퐿2(퐷, 휌), 푓 ↦→ 푓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  It follows from (4) that 푆 is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' This means that the self-adjoint operator  푊 := 푆∗ ◦ 푆 is compact and positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Consequently, 푊 has a null sequence  of positive eigenvalues 휆 := {휆 푗} 푗 ∈N with a corresponding set of eigenfunctions  {휂 푗} 푗 ∈N that form a complete orthonormal system of 퐻(1 + 푘) satisfying  Infinite-Variate 퐿2-Approximation with Nested Subspace Sampling  7  푊(휂 푗) = 휆 푗휂 푗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  (13)  Let d휆 denote the decay rate of the eigenvalues 휆 of 푊 according to (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Then it  is well known that the convergence rate of the 푛-th minimal errors of the univariate  approximation problem is d휆/2, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Let 푆u : 퐻u → 퐿2(퐷u, 휌u) be given by 푆u( 푓 ) = 푓 for all 푓 ∈ 퐻u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Owing  to the tensor product construction of spaces 퐻u for u ∈ U훾, the eigenpairs of  the operator 푊u := 푆∗  u ◦ 푆u : 퐻u → 퐻u are of product form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' In particular, for  u = {푢1, 푢2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' , 푢푘} ∈ U with 푘 = |u|, and for all j = ( 푗1, 푗2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' , 푗푘) ∈ N푘, we  define  휆u,j :=  푘  �  푖=1  휆 푗푖  and  휂u,j(x) :=  푘  �  푖=1  휂 푗푖 (푥푢푖),  using the eigenvalues and eigenfunctions of the univariate operator as in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Then  푊u(휂u,j) = 휆u,j휂u,j,  (14)  and �휂u,i, 휂u,j  �  퐻u = 훿i,j for i, j ∈ N푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Using the eigenpairs of 푊u, we can express the  SVD of 푆u as  푆u( 푓u) =  �  j∈Nu  �  휆u,j  �  푓u, 휂u,j  �  퐻u · ¯휂u,j,  (15)  where  ¯휂u,j = 푆u(휂u,j)/  �  휆u,j  (16)  form an orthonormal system in 퐿2(퐷u, 휌u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' In the same way as for the spaces 퐻u,  we now define a self-adjoint operator for the infinite-variate setting  W := S∗ ◦ S : 퐻훾 → 퐻훾  which is compact and positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  3 ANOVA spaces  For notational clarity, we use 퐾A  훾 and 퐻A  훾 to refer to the infinite-variate ANOVA  kernel and the corresponding function space in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' It is well known that in  ANOVA spaces we have, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' [2],  ⟨ 푓u, 푔v⟩퐿2 (퐷u∪v,휌u∪v) = 0  for u, v ∈ U, u ≠ v, 푓u ∈ 퐻u, 푔v ∈ 퐻v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  (17)  Lemma 1 In the ANOVA setting, any linear continuous algorithm A : 퐻A  훾 →  퐿2(픛, 휌N) satisfying A(퐻u) ⊆ S(퐻u) = 퐻u for all u ∈ U훾 also satisfies  ∥S( 푓 ) − A( 푓 )∥2  퐿2 (픛,휌N) =  �  u∈U훾  ∥푆u( 푓u) − 퐴u( 푓u)∥2  퐿2 (퐷u,휌u) ,  8  Kumar Harsha, Michael Gnewuch, and Marcin Wnuk  where 퐴u : 퐻u → 퐿2(퐷u, 휌u) and 퐴u( 푓 ) = A( 푓 ) for all 푓 ∈ 퐻u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Proof The statement follows directly from (3) and (17), using the fact that S( 푓u) =  푆u( 푓u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  □  Analogous to [15], we define an approximation algorithm based on the SVD of 푆u  from (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Let 훿 ∈ (0, 1) and define for 푓 ∈ 퐻u  퐴∗  u, 훿( 푓 ) :=  �  j∈퐽 ( 훿,u)  �  휆u,j  �  푓 , 휂u,j  �  퐻u · ¯휂u,j,  (18)  where 퐽(훿, u) := {j : �휆u,j > 훿} and ¯휂u,j is given by (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' We want to combine all  the sub-algorithms 퐴∗  u, 훿 : 퐻u → 퐿2(퐷u, 휌u) for u ∈ U훾 to attack the approximation  problem on 퐻훾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Let 휀 ∈ (0, 1) be the worst-case error threshold for the infinite-variate  problem, and define the set  푀훾(휀) :=  �  u∈U훾  {(u, j) : j ∈ 퐽(휀/√훾u, u)}  =  �  (u, j) : u ∈ U훾, j ∈ Nu, and 훾u휆u,j > 휀2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  (19)  We now define the algorithm Aopt  휀,훾 : 퐻A  훾 → 퐿2(픛, 휌N) as  Aopt  휀,훾 ( 푓 ) :=  �  u∈U훾  퐴∗  u,휀/√훾u ( 푓u) =  �  (u,j) ∈푀훾 (휀)  �  푓 , 휂u,j  �  퐻u · 푆u(휂u,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  (20)  Lemma 2 For any u ∈ U훾 and 훿 > 0, the algorithm 퐴∗  u, 훿 from (18) satisfies  ���푆u − 퐴∗  u, 훿  ��� ≤ 훿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Proof Let 훿 ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Then using (15) and Bessel’s inequality, we get  ��푆u( 푓 ) − 퐴∗  u, 훿( 푓 )  ��2  퐿2 (퐷u,휌u) =  ������  �  j∉퐽 ( 훿,u)  �  휆u,j  � 푓 , 휂u,j  �  퐻u · ¯휂u,j  ������  2  퐿2(퐷u,휌u)  =  �  j∉퐽 ( 훿,u)  휆u,j  ���  �  푓 , 휂u,j  �  퐻u  ���  2 �� ¯휂u,j  ��2  퐿2(퐷u,휌u) ≤ 훿2  �  j∉퐽 ( 훿,u)  ���  �  푓 , 휂u,j  �  퐻u  ���  2  ≤ 훿2 ∥ 푓 ∥2  퐻u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  The result now follows by taking supremum over 푓 ∈ 퐻u with ∥ 푓 ∥퐻u = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  □  The next result is reformulated from [15, Theorem 1(i)], and has a straightforward  proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Corollary 1 In the ANOVA setting, the algorithm Aopt  휀,훾 as in (20) achieves a worst  case error at most 휀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  The following result was mentioned in [15] without a proof, therefore we present it  here for the reader’s convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Infinite-Variate 퐿2-Approximation with Nested Subspace Sampling  9  Lemma 3 The operator W = S∗S admits the following decomposition over ANOVA  spaces  W( 푓 ) =  �  u∈U훾  훾u푊u( 푓u)  ∀푓 ∈ 퐻A  훾 , 푓 =  �  u∈U훾  푓u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  (21)  Proof The Definition 1 and the dominated convergence theorem imply that  (W 푓 )(y) =  �  W( 푓 ), 퐾훾(·, y)  �  퐻 A  훾 =  �  S( 푓 ), S(퐾훾 (·, y))  �  퐿2 (픛,휌N)  =  ∫  픛  퐾훾(x, y) 푓 (x)d휌N(x) =  ∫  픛  �  u∈U훾  훾u푘u(xu, yu) 푓 (x)d휌N(x)  =  �  u∈U훾  훾u  ∫  퐷u 푘u(xu, yu) 푓u(xu)d휌u(xu) =  �  u∈U훾  훾u(푊u 푓u)(yu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  The result now follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  □  Using the eigenpairs from (14) and Lemma 3, we can conclude that the eigenpairs  of W with positive eigenvalues are  ��훾u휆u,j, 휉u,j  � : u ∈ U훾, j ∈ Nu� ,  (22)  with eigenfunctions 휉u,j := √훾u휂u,j normalized in 퐻A  훾 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' We can now express the  solution operator S using its singular values and the orthonormal system as  S( 푓 ) =  �  u∈U훾  �  j∈Nu  �  푓 , 휂u,j  �  퐻u · S(휂u,j) =  �  u∈U훾  �  j∈Nu  �  훾u휆u,j  �  푓 , 휉u,j  �  퐻훾 · ¯휂u,j,  where ¯휂u,j is as defined in (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  The next result is the analogue of [15, Theorem 1(iii)] where it was proven for  unrestricted subspace sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' It can be checked that the proof follows verbatim  and the arguments hold irrespective of the cost model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Theorem 1 In the nested subspace sampling model, the cost of the algorithm Aopt  휀,훾  is minimal amongst all algorithms that achieve a worst case error of at most 휀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Using the nested sequence from (9), we set for notational convenience  푉0 := {∅}  and  푉푘 := {u ∈ U훾 : 푘 ∈ u ⊆ [푘]}  for 푘 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  We knowthat Aopt  휀,훾 from(20) is the optimal algorithmfor the approximationproblem  in ANOVA function spaces because it achieves an 휀 worst case error as shown in  Corollary 1, and it follows from Theorem1 that it has minimal cost as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Therefore,  it suffices to compute the cost of Aopt  휀,훾 to determine the 휀-complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Let  푀(휀, 푘) :=  �  (u, j) : u ∈ 푉푘, j ∈ Nu, and 훾u휆u,j > 휀2�  ,  and  푛푘 := |푀(휀, 푘)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  (23)  In order to simplify the cost analysis later on, we can rewrite the optimal algorithm  from (20) in a multilevel fashion as  10  Kumar Harsha, Michael Gnewuch, and Marcin Wnuk  Aopt  휀,훾 ( 푓 ) =  푚휀  �  푘=1  �  (u,j) ∈푀 (휀,푘)  � 푓 , 휂u,j  �  퐻u · S(휂u,j),  (24)  where 푚휀 ∈ N is the largest number 푘 such that 푛푘 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Lemma 4 For an error threshold 휀 > 0, and with the nested subspaces as in (9), the  optimal algorithm Aopt  휀,훾 samples from 푚휀 levels which can be estimated as  푚휀 ≳ 휀−2/푝1  ∀푝1 ∈ (d훾, ∞),  and  푚휀 ≲ 휀−2/푝2  ∀푝2 ∈ (1, d훾).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Proof From Definition (10), we have for all 푝1 ∈ (d훾, ∞) and 푝2 ∈ (1, d훾)  훾 푗 ≳ 푗−푝1  and  훾 푗 ≲ 푗−푝2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  (25)  Using (19), we have ({푚휀 + 1}, 1) ∉ 푀훾(휀) which means 훾푚휀+1휆1 < 휀2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Using the  lower bound from (25), we get for some 푐1 > 0  푐1휆1(푚휀 + 1)−푝1 ≤ 휆1훾푚휀+1 ≤ 휀2 =⇒ (푐1휆1)1/푝1휀−2/푝1 ≤ 푚휀 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Therefore, we get  푚휀 ≥ (푐1휆1)1/푝1휀−2/푝1 − 1,  (26)  and the lower bound from the lemma follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  To show the remaining upper bound from the lemma, we note that �{푚휀}, 1� ∈  푀훾(휀) must hold, which is equivalent to the condition 휆1훾푚휀 > 휀2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Then using (25)  we get for some constant 푐2 > 0  휀2 < 휆1훾푚휀 ≤ 푐2휆1푚−푝2  휀  =⇒ 푚휀 ≤ (푐2휆1)1/푝2휀−2/푝2,  (27)  giving us the desired upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  □  We now want to estimate the cardinality of 푀(휀, 푘) as defined in (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Before  we proceed, we introduce the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Let (u1, j1), (u2, j2) ∈ 푀(휀) such  that u1 ∩ u2 = ∅, |u1| = 푘1, |u2| = 푘2 and j1 ∈ N푘1, j2 ∈ N푘2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Then, we define  (u1, j2) × (u2, j2) = (u, j) with u = u1 ∪ u2, j ∈ Nu1∪u2 such that 푗푖 = 푗1,푖 if 푖 ∈ u1,  and 푗푖 = 푗2,푖 if 푖 ∈ u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Moreover, we define 푀(휀, 푘) := �(푘, 푗) : 푗 ∈ N, 훾푘휆 푗 > 휀2�,  and 푛푘 :=  ���푀(휀, 푘)  ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Let d훾 denote the decay rate of {훾 푗} 푗 ∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' We assume that �  푗 ∈N 훾1/d훾  푗  < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Then  from [3, Theorem 5 and Corollary 1], we have for every 휏 ∈ [1, d훾] and every  constant 퐶휏 > 0 that  �  푘∈u⊆[푘]  훾1/휏  u  퐶 |u|  휏  ≍ 훾1/휏  푘  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  (28)  Lemma 5 Let 휀 > 0 and 푘 ∈ [푚휀].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Then for all 푞1 ∈ (d휆, ∞) and 푞2 ∈ (1, d휆) we  have the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Infinite-Variate 퐿2-Approximation with Nested Subspace Sampling  11  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' 푀(휀, 푘 + 1) = 푀(휀, 푘 + 1) ∪ �푘  ℓ=1  �  푗 ∈N  �  ({푘 + 1}, 푗) × 푀  �  휀  √  훾푘+1휆 푗 , ℓ  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' |푀(휀, 푘 + 1)| =  ���푀(휀, 푘 + 1)  ��� + �푘  ℓ=1  �  푗 ∈N  ����푀  �  휀  √  훾푘+1휆 푗 , ℓ  �����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  ���푀(휀, 푘)  ��� ≳ 훾1/푞1  푘  휀−2/푞1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  ���푀(휀, 푘)  ��� ≲ 훾1/푞2  푘  휀−2/푞2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' |푀(휀, 푘)| ≲ 휀−2/푞2 �  u∈푉푘 훾1/푞2  u  푐(휆, 푞2) |푢|−1, where 푐(휆, 푞2) = �  푗 ∈N 휆1/푞2  푗  < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' |푀(휀, 푘)| ≲ 휀−2/푞2훾1/푞2  푘  , if additionally 푞2 ∈ (1, min{d휆, d훾}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Proof Similar to (25), we get for all 푞1 ∈ (d휆, ∞) and 푞2 ∈ (1, d휆) that  휆 푗 ≳ 푗−푞1  and  휆 푗 ≲ 푗−푞2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  (29)  We assume that 휀2 < 훾1휆1 ≤ 1, since otherwise the set 푀(휀) is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Let (u ∪ {푘 + 1}, (j, 푗푘+1)) ∈ 푀(휀, 푘 + 1) where ∅ ≠ u ⊆ [푘], j ∈ Nu and  푗푘+1 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Since we have product weights, the definition (23) requires that 훾u휆j >  휀2/훾푘+1휆 푗푘+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Let ℓ = max(u), then u ∈ 푉ℓ and  (u, j) ∈ 푀  �  휀  �훾푘+1휆 푗푘+1  , ℓ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Note that the right hand side in statement 1 is a union of disjoint sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Using the lower bound from (29), and the definition of 푀(휀, 푘), we get for some  푐1 > 0 that  푐1훾푘(푛푘 + 1)−푞1 ≤ 휆푛푘+1훾푘 ≤ 휀2 =⇒ 푛푘 ≥ 푐1/푞1  1  훾1/푞1  푘  휀−2/푞1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Using the upper bound from (29), and the definition of 푀(휀, 푘), we get for some  푐2 > 0 that  휀2 < 훾푘휆푛푘 ≤ 푐2훾푘푛−푞2  푘  =⇒ 푛푘 ≤ 푐1/푞2  2  훾1/푞2  푘  휀−2/푞2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' We prove this result by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' For 푘 = 1, we have due to statement 4  푛1 = |푀(휀, 1)| =  ���푀(휀, 1)  ��� ≲ 훾1/푞2  1  휀−2/푞2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  We suppose now that the bound is true for 푀(휀,1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' , 푀(휀, 푘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Using statements  2 and 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' we get for some 푐2 > 0  12  Kumar Harsha,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Michael Gnewuch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' and Marcin Wnuk  |푀(휀,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' 푘 + 1)| =  ���푀(휀,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' 푘 + 1)  ��� +  푘  �  ℓ=1  �  푗 ∈N  �����푀  �  휀  �훾푘+1휆 푗  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' ℓ  ������  ≤ 푐2휀−2/푞2훾1/푞2  푘+1 + 푐2  푘  �  ℓ=1  �  푗 ∈N  �  휀  �훾푘+1휆 푗  �−2/푞2 �  u∈푉ℓ  훾1/푞2  u  푐(휆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' 푞2) |푢|−1  = 푐2휀−2/푞2훾1/푞2  푘+1  �  1 +  푘  �  ℓ=1  �  u∈푉ℓ  훾1/푞2  u  푐(휆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' 푞2) |푢|−1 �  푗 ∈N  휆1/푞2  푗  �  = 푐2휀−2/푞2  \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  훾1/푞2  푘+1 + 훾1/푞2  푘+1  �  ∅≠u⊆[푘]  훾1/푞2  u  푐(휆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' 푞2) |푢|  \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  = 푐2휀−2/푞2 �  u∈푉푘+1  훾1/푞2  u  푐(휆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' 푞2) |푢|−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' The estimate follows directly by applying (28) to the formula in statement 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  □  Theorem 2 Let $(푘) = Θ(푘푠) for some 푠 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Then the optimal algorithm Aopt  휀,훾 as  in (20) achieves the following convergence rate using the nested subspace sampling  cost model in ANOVA function spaces with product weights  푟 퐴  ∞ = min  �d휆  2 ,  d훾  2(1 + 푠)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Proof Let 휀 be the desired worst case error, and 푚휀 be the highest level for our  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  We first show 푟 퐴  ∞ ≤ min  �  d휆  2 ,  d훾  2(1+푠)  �  using $(푘) = Ω(푘푠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Since 푀(휀, 푘) ⊂  푀(휀, 푘), we get using the lower bound on  ���푀(휀, 푘)  ��� from statement 3 of Lemma 5,  and using (25) that  cost(Aopt  휀,훾) ≥  푚휀  �  푘=1  ���푀(휀, 푘)  ���·$(푘) ≳ 휀−2/푞1  푚휀  �  푘=1  훾1/푞1  푘  푘푠 ≳ 휀−2/푞1  푚휀  �  푘=1  푘푠−푝1/푞1, (30)  where for 훿 > 0 arbitrarily small, we choose 푝1 = d훾(1 + 훿) and 푞1 = d휆(1 + 훿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  From the last expression in (30), we may distinguish three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' d훾/(1 + 푠) > d휆 =⇒ 푠 − 푝1/푞1 < −1: This means we have �푚휀  푘=1 푘푠−푝1/푞1 ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Consequently, as 푞1 → d휆, we get 푟 퐴  ∞ ≤ d휆/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' d훾/(1 + 푠) = d휆 =⇒ 푠 − 푝1/푞1 = −1: Here we get the estimate  푚휀  �  푘=1  푘−1 ≳ 1 + ln(푚휀) ≳ 1 + 2  푝1  ln(1/휀),  Infinite-Variate 퐿2-Approximation with Nested Subspace Sampling  13  where the last relation follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Hence, we get cost(Aopt  휀,훾) ≳  휀−2/푞1(1 + ln(1/휀)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Since the definition of the exponent of tractability as in (12)  ignores the logarithmic terms, we get 푟 퐴  ∞ ≤ d휆/2 as 푞1 → d휆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' d훾/(1 + 푠) < d휆 =⇒ 푠 − 푝1/푞1 > −1: In this case, we can use Lemma 4 to get  푚휀  �  푘=1  푘푠−푝1/푞1 ≳ (푚휀)1−푝1/푞1+푠 ≳ 휀−2(1−푝1/푞1+푠)/푝1,  which implies that cost(Aopt  휀,훾) ≳ 휀−2(1+푠)/푝1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' As 푝1 → d훾, we have 푟 퐴  ∞ ≤  d훾/2(1 + 푠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  We now show that 푟 퐴  ∞ ≥ min  �  푟1,  d훾  2(1+푠)  �  using $(푘) ≲ 푘푠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' We use (25) and  statement 6 from Lemma 5 to get  cost(Aopt  휀,훾) =  푚휀  �  푘=1  |푀(휀, 푘)| · $(푘) ≲ 휀−2/푞2  푚휀  �  푘=1  훾1/푞2  푘  푘푠 ≲ 휀−2/푞2  푚휀  �  푘=1  푘푠−푝2/푞2,  (31)  where for 훿 > 0 arbitrarily small, we choose 푝2 = d훾(1 − 훿) and 푞2 = d휆(1 − 훿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' We  analyze the last expression in (31) in three cases as in the first part of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' d훾/(1 + 푠) > d휆 =⇒ 푠 − 푝2/푞2 < −1: Then �푚휀  푘=1 푘푠−푝2/푞2 ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Therefore, we  may choose 푞2 → d휆 to get 푟 퐴  ∞ ≥ d휆/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' d훾/(1 + 푠) = d휆  =⇒  푠 − 푝2/푞2 = −1: For this case, we get the estimate  �푚휀  푘=1 푘−1 ≲ 1 + ln(푚휀) ≲ 1 +  2  푝2 ln(1/휀), where we used Lemma 4 for the last  relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Consequently, we have cost(Aopt  휀,훾) ≲ 휀−2/푞2(1 + ln(1/휀)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Similarly as  in the first part of the proof, we can ignore the logarithmic terms as 푞2 → d휆 to  get 푟 퐴  ∞ ≥ d휆/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' d훾/(1 + 푠) < d휆 =⇒ 푠 − 푝2/푞2 > −1: Due to Lemma 4 we get  푚휀  �  푘=1  푘−푝2/푞2+푠 ≲ 푚1−푝2/푞2+푠  휀  ≲ 휀−2(1−푝2/푞2+푠)/푝2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Hence, we get cost(Aopt  휀,훾) ≲ 휀−2(1+푠)/푝2 and we may choose 푝2 → d훾 to get  푟 퐴  ∞ ≥ d훾/2(1 + 푠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  □  4 Non-ANOVA spaces  Let us now turn to the case where the underlying function decomposition of our  Hilbert space 퐻훾 is not of ANOVA-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' We start by proving a lower bound on the  convergence rate of 푛-th minimal errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  14  Kumar Harsha, Michael Gnewuch, and Marcin Wnuk  Proposition 1 Let $(푘) ≳ 푘푠 for any 푘 ∈ N and some 푠 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Then the optimal  convergence rate for the algorithms defined as in (7) satisfy in non-ANOVA function  spaces  푟∞ ≤ min  �d휆  2 , d훾 − 1  2푠  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Proof The bound 푟∞ ≤ d휆/2 follows from the rate of convergence for the univariate  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  To show the other part, we adapt the proof strategy from [14, Proposition 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Let A푁 be a linear algorithm using finitely many function evaluations, such that  cost(A푁 ) ≤ 푁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' We now choose 퐿 := sup {푘 : $(푘) ≤ 푁} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Due to this definition,  푉 := [퐿] is a superset of all the indices that the algorithm A푁 relies on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Since  we assumed that $(푘) ≳ 푘푠, there exists a constant 푐 > 0 such that 푘푠/푐푠 ≤ $(푘),  implying 퐿 ≤ 푐푁1/푠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Consider a univariate function ℎ ∈ 퐻(푘) such that ∥ℎ∥퐻 (푘) = 1 and 푐1 :=  ∫  퐷 ℎ(푥)d휌(푥) ≠ 0, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Then define 푓 ∗ ∈ 퐻훾 as  푓 ∗(x) := 1  휅  �  푗∉푉  훾 푗ℎ(푥 푗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  with 휅 :=  ��  푗∉푉  훾 푗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  As shown in [14], A푁 ( 푓 ∗) = 0, ∥ 푓 ∥퐻훾 = 1, and also  ∥ 푓 ∗∥2  퐿2 (픛,휌N) =  �  ∥ℎ∥2  퐿2 (퐷,휌) − 푐2  1  � �  푗∉푉 훾2  푗  �  푗∉푉 훾 푗  + 푐2  1  �  푗∉푉  훾 푗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  From (10), we have for any arbitrarily small 훿 > 0 that 훾 푗 ≳ 푗−(d훾+훿/4), and also  훾 푗 ≲ 푗−(d훾−훿/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' With this, we can conclude that  �  푗∉푉 훾2  푗  �  푗∉푉 훾 푗  ≳  �∞  푗=퐿+1 푗−2d훾−훿/2  �∞  푗=퐿+1 푗−d훾+훿/2 ≳ 퐿−d훾−훿 ≳ 푁−(d훾+훿)/푠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Similarly, we have that �  푗∉푉 훾 푗 ≳ 퐿1−d훾−훿 ≳ 푁 (1−d훾−훿)/푠 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' From the construction  of 푓 ∗, we have for the error that  푒(A푁 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' 퐻훾) =  sup  ∥ 푓 ∥퐻훾 =1  ∥ 푓 − A푁 ( 푓 )∥퐿2 (픛,휌N) ≥ ∥ 푓 ∗∥퐿2 (픛,휌N) ≳ 푁 (1−d훾−훿)/2푠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  This shows that 푟∞ ≤ (d훾 − 1)/2푠, and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  □  For non-ANOVA spaces, Lemma 3 does not hold in general, which means that  we cannot express the eigenpairs of W : 퐻훾 → 퐻훾 in terms of the eigenpairs of 푊u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  For the upper error bound, we need results from [14] which are stated below for the  reader’s convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Let 퐶0 := sup 푓 ∈퐻 (푘)  ∥ 푓 ∥퐿2 (퐷,휌)  ∥ 푓 ∥퐻 (푘)  < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Lemma 6 For all 푐 ∈ (1/d훾, 1) it holds that supu∈U훾 퐶2|u|  0  훾1−푐  u  < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Infinite-Variate 퐿2-Approximation with Nested Subspace Sampling  15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' For  �훾푐 = {�훾u,푐}u∈U훾,  �훾u,푐 := 훾1−푐  u  ,  we define 휙 : 퐻훾 → 퐻�훾푐, 푓 ↦→ �  u∈U훾 훾−푐/2  u  푓u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Then 휙 is an isometric isomor-  phism between 퐻훾 and 퐻�훾푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Furthermore, 퐻훾 ⊆ 퐻�훾푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Let A be a linear continuous algorithm satisfying  A(퐻u) ⊆ 퐿2(퐷u, 휌u),  (32)  for all u ∈ U훾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Then we get the upper bound  ∥S( 푓 ) − A( 푓 )∥2  퐿2 (픛,휌N) ≤ 퐶훾  �  u∈U훾  ���푆u( �푓u) − 퐴u( �푓u)  ���  2  퐿2 (퐷u,휌u)  (33)  where 퐶훾 = �  v∈U훾 훾푐  v, 휙( 푓 ) =: �푓 = �  u∈U훾 �푓u for all 푓 ∈ 퐻훾, and 퐴u : 퐻u →  퐿2(퐷u, 휌u) such that 퐴u( 푓 ) = A( 푓 ) for all 푓 ∈ 퐻u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  It is straightforward to check that 휙 defines an isometric isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' 퐻훾 ⊆ 퐻�훾푐  is due to [14, Lemma 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' The inequality (33) has been derived in the proof of [14,  Lemma 2] for any algorithm A satisfying (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Proposition 2 For any 휀 ∈ (0, 1), define the set 푀�훾푐 (휀) as in (19) and also the  algorithm Aopt  휀,�훾푐 : 퐻�훾푐 → 퐿2(픛, 휌N) as defined in (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Then the algorithm Aopt  휀,�훾푐  satisfies (32) and 푒(Aopt  휀,�훾푐, S, 퐻훾) ≤ 휀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' The algorithm Aopt  휀,�훾푐 establishes that  푟∞ ≥ min  �d휆  2 , d훾 − 1  2(1 + 푠)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Proof Using Lemmas 6 and 2, we get  ���S( 푓 ) − Aopt  휀,�훾푐 ( 푓 )  ���  2  퐿2(픛,휌N) ≲  �  u∈U훾  ����푆u( �푓u) − 퐴∗  u,휀/√  �훾u,푐 ( �푓u)  ����  2  퐿2 (퐷u,휌u)  ≤ 휀2 �  u∈U훾  1  �훾u,푐  ��� �푓u  ���  2  퐻u = 휀2 ��� �푓  ���  2  퐻�훾푐  = 휀2 ∥ 푓 ∥2  퐻훾 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Notice that Aopt  휀,�훾푐 has the same cost in 퐻�훾푐 and in its subspace 퐻훾, since the  function space decomposition in both spaces rely on the same family of subspaces  (퐻u)u∈U훾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Let d�훾푐 denote the decay rate of product weights �훾푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Then from statement  5 of Lemma 5, (31), and the subsequent analysis in Theorem 2, we get  푟∞ ≥ min  �d휆  2 ,  d�훾푐  2(1 + 푠)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Finally, d�훾푐 = d훾(1 − 푐)  푐→1/d훾  −−−−−−→ d훾 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  □  16  Kumar Harsha, Michael Gnewuch, and Marcin Wnuk  Corollary 2 If 1 < d훾 < 1 + 푠, then the convergence rate of 퐿2-approximation in  ANOVA spaces is strictly larger than the one in non-ANOVA spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Proof Under the given condition,the optimal convergence rate of 푛-th minimal errors  in the ANOVA case from Theorem 2 is greater than the largest possible convergence  rate in the non-ANOVA case from Proposition 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' (d훾 − 1)/2푠 < d훾/2(1 + 푠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' □  As we see from the analysis, the convergence rates from the upper and lower error  bounds in the non-ANOVA spaces do not match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' In future work, we will consider  bridging the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Aronszajn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Theory of reproducing kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=', 68:337–404, 1950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Baldeaux and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Gnewuch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Optimal randomized multilevel algorithms for infinite-  dimensional integration on function spaces with ANOVA-type decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=', 52:1128–1155, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Dick and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Gnewuch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Infinite-dimensional integration in weighted Hilbert spaces: anchored  decompositions, optimal deterministic algorithms, and higher order convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Com-  put.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=', 14:1027–1077, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Dick and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Gnewuch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Optimal randomized changing dimension algorithms for infinite-  dimensional integration on function spaces with ANOVA-type decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Theory, 184:111–145, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Gnewuch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Hefter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Hinrichs, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Ritter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Embeddings of weighted Hilbert spaces  and applicationsto multivariate and infinite-dimensional integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Journal of Approximation  Theory, 222:8–39, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Gnewuch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Hefter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Hinrichs, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Ritter, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Wasilkowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Embeddingsforinfinite-  dimensional integration and 퐿2-approximation with increasing smoothness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Complexity,  54:101406, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Gnewuch, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Mayer, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Ritter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' On weighted Hilbert spaces and integration of functions  of infinitely many variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Complexity, 30:29–47, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Hefter and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Ritter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  On embeddings of weighted tensor product Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Complexity, 31:405–423, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Hickernell, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Müller-Gronbach, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Niu, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Ritter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Multi-level Monte Carlo algorithms  for infinite-dimensional integration on RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Complexity, 26:229–254, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Kuo, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Sloan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Wasilkowski, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Woźniakowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Liberating the dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Complexity, 26:422–454, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Müller-Gronbach and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Ritter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Variable subspace sampling and multi-level algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' In  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' L’Ecuyer and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Owen, editors, Monte Carlo and Quasi-Monte Carlo Methods 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  Springer, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Novak and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Woźniakowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Tractability of Multivariate Problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' 1: Linear Infor-  mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' EMS Tracts in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' European Mathematical Society (EMS), Zürich, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Traub, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Wasilkowski, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Woźniakowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Information-Based Complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Aca-  demic Press, New York, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Wasilkowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Liberating the dimension for 퐿2-approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Complexity, 28:304–  319, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Wasilkowski and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Woźniakowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Liberating the dimension forfunction approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Complexity, 27:86–110, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content='  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' Marcin Wnuk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
+page_content=' A short note on compact embeddings of reproducing kernel Hilbert spaces in  퐿2 for infinite-variate function approximation, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFPT4oBgHgl3EQfzzXg/content/2301.13177v1.pdf'}
diff --git a/f9E4T4oBgHgl3EQfqw0f/content/tmp_files/2301.05202v1.pdf.txt b/f9E4T4oBgHgl3EQfqw0f/content/tmp_files/2301.05202v1.pdf.txt
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+Progress towards the 1/2-Conjecture for the domination game
+Julien Portier∗
+Leo Versteegen †
+Abstract
+The domination game is played on a graph G by two players, Dominator and Staller,
+who alternate in selecting vertices until each vertex in the graph G is contained in the closed
+neighbourhood of the set of selected vertices. Dominator’s aim is to reach this state in as few
+moves as possible, whereas Staller wants the game to last as long as possible. In this paper, we
+prove that if G has n vertices and minimum degree at least 2, then Dominator has a strategy
+to finish the domination game on G within 10n/17+1/17 moves, thus making progress towards
+a conjecture by Bujtás, Iršič and Klavžar.
+1
+Introduction
+The domination game is an optimization game for two players introduced by Brešar, Klavžar and
+Rall in [1] and is played on a graph G without isolated vertices. In this game, the two players,
+Dominator (D) and Staller (S), alternately select vertices of G.
+A vertex is considered to be
+dominated, if itself or one of its neighbors were selected by either player, and both players must
+always select a vertex that increases the number of dominated vertices. The game ends once all
+vertices of G are dominated. Within the bound of these rules, Dominator wishes to minimize the
+number of moves played during the game, while Staller wants to achieve the opposite. We define
+the game domination number γg(G) of G to be the number of moves the game takes to finish if
+Dominator moves first and both players play optimally.
+Let n be the number of vertices of G. Of course, one should expect that the game lasts longer
+the larger n is, and so it is a natural objective for research to bound γg in terms of n. Specifically,
+Kinnersley, West and Zamani conjectured [8] that γg(G) ≤ 3n/5 for all graphs without isolated
+vertices, and the second author has recently confirmed their conjecture [11].
+Theorem 1.1. For any graph G on n vertices without isolated vertices, γg(G) ≤ 3n/5.
+The bound 3n/5 in Theorem 1.1 is best possible, as one may see by considering the path on five
+vertices. What is more, if we think of the path P5 as a single vertex to which we have appended
+two paths of length two, we can generalise this example by taking any graph H and appending
+two paths of length two to each of its vertices. The resulting graph G has 5|V (H)| vertices and
+satisfies γg(G) = 3|V (H)|. One can generalise this further (see [6, 11]), but one property that
+all the resulting extremal graphs share is that they have many leaves, i.e., vertices of degree one.
+There is an extremal graph without a leaf, namely the cycle on five vertices, but in many ways, the
+∗jp899@cam.ac.uk, Department of Pure Mathematics and Mathematical Statistics (DPMMS), University of Cam-
+bridge, Wilberforce Road, Cambridge, CB3 0WA, United Kingdom
+†lvv23@cam.ac.uk, Department of Pure Mathematics and Mathematical Statistics (DPMMS), University of Cam-
+bridge, Wilberforce Road, Cambridge, CB3 0WA, United Kingdom
+1
+arXiv:2301.05202v1  [math.CO]  12 Jan 2023
+
+five-cycle is a degenerate example that only works due to its small size, and unlike the P5, it does
+not lend itself to any sort of generalisation to larger graphs. With this caveat in mind, it is natural
+to ask whether Theorem 1.1 can be improved for the class of graphs without leaves. In [4], Bujtás,
+Iršič and Klavžar made the following two conjectures in this direction.
+Conjecture 1.2. There exists a constant c < 3/5 such that every graph G on n ≥ 6 vertices and
+minimum degree at least 2 satisfies γg(G) ≤ cn.
+Conjecture 1.3. Let G be a graph with δ(G) ≥ 2. Then γg(G) ≤ ⌈ 1
+2n⌉.
+Of course, Conjecture 1.3 is a straightforward strengthening of Conjecture 1.2, as it essentially
+sets the undetermined constant c from Conjecture 1.2 to 1/2, which is natural choice for two reasons.
+Firstly, it is simply the least constant for which there are not any counterexamples known already.
+Secondly, if we follow the heuristic notion that γg should generally decrease if we add more edges to
+the graph under consideration, then we should believe that unions of vertex-disjoint cycles should
+pose the worst case when proving an upper bound on γg for graphs with minimum degree 2. But
+for a cycle C, it is known [9] that γg(C) ≤ ⌈ 1
+2n⌉.
+In 2016, long before Theorem 1.1 was established in full, Henning and Kinnersley proved [5]
+that Theorem 1.1 holds for graphs of minimum degree 2, which is the reason that Conjecture 1.2
+asks for a constant c < 3/5. To the best of our knowledge, no improvement on the constant has
+been made since, whence Conjecture 1.2 is still open. In this paper, we prove the following.
+Theorem 1.4. For every graph G with minimum degree at least 2, we have γg(G) ≤ 10
+17n + 1
+17.
+Since 10n/17 + 1/17 < 3n/5 for all n > 5, it is easy to see that Theorem 1.4 confirms Conjec-
+ture 1.2.
+2
+The proof
+2.1
+The main idea of the proof
+To prove Theorem 1.4, we employ the discharging method introduced by Bujtás in [2]. The core
+concept of this method is that at any given time during the game, every vertex has a color, which
+has in turn a certain number of points associated with it. As the game progresses, the sum π of
+all points associated to vertices decreases until it reaches zero. If we can prove that the rate of
+decrease per move is sufficiently large, then that will allows us to bound the total number of moves
+in terms of the total number of points at the start of the game, which in turn only depends on the
+number of vertices n in the graph G under consideration.
+Concretely, we use the colours white, blue, orange and red. Let t be the number of moves that
+have been played so far and let A be the set of vertices that have been selected by either player up
+to and including move t. The colour of a vertex is determined by the following rules.
+• A vertex is white if and only if it is not dominated by A (i.e, is not itself contained in A and
+not adjacent to a vertex in A).
+• A vertex is red if and only if it and all its neighbours are dominated by A.
+• Let W be the set of white vertices as determined by A. An isolated vertex in the induced
+subgraph G[W] is called a single, while a component of exactly two vertices in G[W] is called
+a pair. An individual member of a pair is called a double. A vertex is orange if and only if
+it is dominated by A, has exactly one white neighbour, and this neighbour is a single or a
+double.
+2
+
+• All other vertices are blue.
+The number of points πv(t) associated with a vertex v after move t, depends only on the color
+of v after move t. Namely, πv(t) is 20, 10, 7, or 0 if v is white, blue, orange, or red, respectively.
+With this we define the total number of points after move t as π(t) = �
+v∈V (G) πv(t).
+At the start of the game, all vertices are white, which means that π(0) = 20n, and the game
+ends after move T if all vertices are red, which is to say that π(T) = 0. Therefore, if D can play in
+a way such that we have π(T) ≤ 20n − 34T, we would know that T ≤ 10n/17. One can show that
+this is the case as long as n > 5, but doing so requires a disproportional amount of additional effort
+relative to the only very slightly worse bound π(T) ≤ 10n/17 + 1/17. To show the latter bound,
+we divide the game for the purpose of Dominator’s strategy into two phases. The first phase lasts
+as long as D can play in a way that guarantees the game to arrive at a point where t moves have
+been played, π(t) ≤ π(0) − 34t, and D is to move next.
+Once this phase is over, one can conclude that all the remaining white vertices lie in very specific
+configurations. The problem with these configurations is that they lead to a form of zugzwang,
+where neither player would like to make the first move. A good example for such a configuration is
+a cycle on five white vertices. Indeed, it is an easy exercise to show that if the domination game is
+played on a C5 and D makes the first move, then the game will take three moves, whereas it will
+only take two moves if S goes first. Happily, if the five-cycle is part of a larger graph G, it is S who
+is in zugzwang after those three moves have been played, and there is no way for them to get out
+of it for the rest of the game. Thus, the idea is that D can afford to reduce π by less than 34 on
+average for a few moves in exchange laying the burden of the initiative on S.
+2.2
+The active phase
+Before we reveal the strategy that D should play, we first make a general observation about the
+point system.
+Claim 2.1. If the game is not over after t moves, then π(t) − π(t + 1) ≥ 20.
+Proof. Suppose that the vertex v is played on move t + 1. If v was blue before move t + 1, then
+v has a neighbour u which was white then. After move t + 1 on the other hand, v is red and u is
+not white anymore, so that π(t) − π(t + 1) ≥ 10 + (20 − 10) = 20. If v was coloured orange, then
+v has a neighbour u which was a double or a single before move t + 1. In particular, u was white
+and has at most one white neighbour, whence it will be orange or red after move t + 1. It follows
+that π(t) − π(t + 1) ≥ 7 + (20 − 7) = 20. Finally, if v was white itself, π(t) − π(t + 1) ≥ 20 because
+v is red after move t + 1.
+We say that the game is in a stable state after move t, if π(0) − π(t) ≥ 34t, and at least one of
+the following holds.
+• Either, t is even, i.e., D is to move next, or
+• there are no white vertices left, i.e., the game is over.
+Whenever the game is not over but in a stable state, it must be Dominator’s turn. Suppose that
+the game is in a stable state after move t. If D has a strategy that allows him to advance the game,
+possibly over the course of several moves, to another stable state after the current one, D plays
+according to this strategy. Letting t1 denote the smallest integer so that D does not have such a
+strategy after move t1, we can make a series of powerful observations about the state of the game
+after move t1.
+3
+
+For example, we know that there can be no vertex v such that π(t1) − π(t1 + 1) ≥ 48 if D
+plays v on move t1 + 1. Indeed, if the game is over after D plays v, the game has reached another
+stable state after move t1 + 1. If it is not, Claim 2.1 tells us that any subsequent move by S will
+reduce π further by at least 20 so that π(t1) − π(t1 + 2) ≥ 68 = 2 · 34, which means the game will
+reach a stable state since t1 + 2 is even. Likewise, there can never be a strategy for D to play that
+guarantees that π(t1) − π(t1 + 2) ≥ 68 or π(t1) − π(t1 + 3) ≥ 116, even if it begins with a move
+that decreases π by less than 48. In the proofs of all of the following claims, the assumption that
+the claim does not hold can be used to provide D with a strategy that leads to one of these three
+outcomes and thus yields a contradiction.
+Claim 2.2. No white vertex has three white neighbours and no blue vertex has four white neighbours.
+Proof. Suppose towards a contradiction that there is a white vertex x with at least three white
+neighbours after t1 moves. D may play this vertex, and since all neighbours of x are now dominated,
+each of them has at least 10 points less associated with them than before. Furthermore, x is now
+red and has 0 point associated with it so that π(t1) − π(t1 + 1) ≥ 50. Similarly if a vertex v is blue
+and has four white neighbours, D can also reduce π by at least 50 by playing v.
+Let H∗ be the subgraph of G in which we remove every connected component that is a four-
+or five-cycle and has only white vertices. Let further H be the subgraph of H∗ that is induced by
+the set of white vertices after t1 moves. By Claim 2.2, H has no vertex of degree three or higher,
+whence H must be a union of disjoint paths and cycles.
+Claim 2.3. H contains no path xyz such that x has no white neighbour other than y and z.
+Proof. If such a path existed, then D could simply play y to decrease π by 50.
+Claim 2.4. H contains no cycle on nine or more vertices.
+Proof. Suppose that k ≥ 9 and that x1x2 . . . xkx1 is a cycle in H. D may play x5, thus decreasing
+π by 46. If S replies by playing a vertex z that is adjacent to both a vertex in {x1, x2, x3} and
+{x7, x8, x9}, then π decreases further by at least 30, so we may assume without loss of generality
+that y has no neighbour in {x1, x2, x3}. In this case, D can play x2 next, which decreases π by 60
+since both x3 and x4 become red. By Claim 2.1, we have π(t1) − π(t1 + 3) ≥ 126.
+Claim 2.5. H contains no cycle on eight or seven vertices.
+Proof. Suppose that x1x2 . . . x8x1 is a cycle in H. D can play x1, which decreases π by 40, and
+if the vertex z that S plays in response is adjacent to two white vertices, then π decreases by at
+least 30. We may therefore assume that z has only one white neighbour before Staller’s move and
+thus without loss of generality that z is not adjacent to either of x3 and x4. If z is not adjacent to
+x5 either, then D can play x4 on move t1 + 3, which decreases π by another 60 points, whence we
+would have π(t1) − π(t1 + 3) ≥ 120. On the other hand, if z is adjacent to x5, then both x5 and x8
+will be orange after D plays x4 on move t + 3, whence π(t1) − π(t1 + 3) ≥ 116.
+The proof that H cannot have a cycle on seven vertices is very similar and we leave it as an
+exercise.
+Claim 2.6. H contains no cycle on six vertices.
+Proof. Suppose that x1 . . . x6x1 is a cycle in H. D can play x1, and if after Staller’s move any of the
+vertices x3, x4 and x5 is still white, D can play x4 next and we have π(t1)−π(t1+3) ≥ 120. Of course,
+if after move t1+2, none of x3, x4 and x5 are white anymore, we even have π(t1)−π(t1+2) ≥ 120.
+4
+
+Claim 2.7. H contains no cycle on four vertices.
+Proof. Suppose that x1 . . . x4x1 is a cycle in H. Because H is a subgraph of H∗, one of the four
+vertices, x1 say, must have another neighbour u, which must be blue. D can play x3, which makes
+x3 red as well as x2 and x4 orange. We distinguish which colour x1 and u have after S played
+some vertex v on move t1 + 2. If x1 is red, then x2 and x4 are red as well, and we must have
+π(t1) − π(t1 + 2) ≥ 80. If x1 is not red, then it must be white, and thus u cannot be red either. In
+particular, u ̸= v. If u is orange, we can use the fact v is now red, and that v must have been white
+or had a white neighbour before move t1 + 2 to see that π(t1) − π(t1 + 2) ≥ 69.
+This leaves the possibility that u is still blue after move t1 +2. In this case, u must have another
+white neighbour y besides x1, and D can play u after which u, x1, x2 and x4 must be red and y is
+not white anymore so that π(t1) − π(t1 + 3) ≥ 120 by Claim 2.1.
+We know now that H must consist only of isolated vertices, isolated edges and five-cycles, but
+the structure of H can be even more rigidly determined. Indeed, we will show that H does in
+fact not contain any five-cycles or isolated vertices and that the edges of H lie in highly specific
+configurations when viewed in G. In the following proofs, it is understood that whenever we play
+a vertex that is adjacent to a double, the double is coloured orange or red but never blue.
+Claim 2.8. If two vertices x1, y1 ∈ H are part of (not necessarily distinct) five-cycles, they cannot
+have a common blue neighbour.
+Proof. Suppose that there are two five-cycles x1 . . . x5x1 and y1 . . . y5y1 in H and a vertex u that
+is adjacent to both x1 and y1. If y1 = xi for some i ̸= 1, there must be some j ∈ [5] such that
+{x1, xi} ⊂ {xj} ∪ N(xj). Therefore, by Claim 2.2, u will become red or orange after D plays xj so
+that π decreases by at least 49. Let us therefore assume that the two cycles are disjoint.
+Suppose now that u has a third white neighbour z, noting that u cannot have a fourth white
+neighbour by Claim 2.2. D can play u which reduces π by at least 40. If S replies by playing a
+vertex that has a neighbour in {x2, x3, x4, x5} and {y2, y3, y4, y5}, then π(t1) − π(t1 + 2) ≥ 70 and
+we are done. Otherwise, we can assume without loss of generality that {x2, x3, x4, x5} are all still
+white after move t1 +2, and D can play x3 which colours x2 and x3 red as well as x1 and x4 orange,
+thus reducing π by at least 56 so that π(t1) − π(t1 + 3) ≥ 116.
+We are left with the possibility that x1 and y1 are the only white neighbours of u. In this case,
+D may play x1, which reduces π by 46. If S replies by playing a vertex that is adjacent to at least
+two of the vertices {x3, x4, y1, y2, y5}, they reduce π further by at least 30. Should S play a vertex
+that does not dominate either of x3 or x4, D can play x3 next so that π(t1) − π(t1 + 3) ≥ 120.
+Lastly, if S plays a vertex that does not dominate either of y1, y2 and y5, D can play y1 next so that
+π(t1) − π(t1 + 3) ≥ 122.
+Claim 2.9. No (blue) vertex has three white neighbours.
+Proof. Let u be a blue vertex with three white neighbours x1, y and z. If all three of them are
+singles or doubles, D can play u to decrease π by at least 49. Let us therefore assume that x1 is
+part of a five-cycle x1 . . . x5x1 in H. By Claim 2.8, y and z must be singles or doubles. If y is a
+single, D can still play u to reduce π by at least 53 so let us assume that y is adjacent to another
+double y′, which must have itself a second neighbour, which we denote by v. How D should proceed
+depends on the number and type of the white neighbours of v.
+Suppose first that v has only one white neighbour, D can play y to reduce π by 47. If S replies
+by playing a vertex that dominates x1 or at least two white vertices overall, π decreases further by
+5
+
+at least 23 and we are done. Otherwise, D can play one of x1, x2 or x5 next to reduce π by at least
+49.
+Suppose next that v has exactly two white neighbours and that one of them (the one that is
+not y′), which we call x′
+1, is part of a five-cycle x′
+1 . . . x′
+5x′
+1 of white vertices. In this scenario, D can
+reduce π by 49 by playing x′
+1.
+If neither of the above assumptions are true, we know by Claim 2.8 that v is adjacent to a single
+or double z′ ̸= y′. If z′ = y, D can play v to reduce π by 50 right away. Else, D can play u on move
+t1 +1 to reduce π by at least 46. As usual, we may assume that the vertex that S plays in response,
+dominates only one white vertex. We can also assume that this white vertex is either y′ or z′ as
+otherwise D could subsequently reduce π by at least 50 by playing v. In particular, x2, x3 and x4
+are still white after move t1 + 2, which means that D can play x3 on move t1 + 3, thus reducing π
+by at least 56.
+Claim 2.10. H contains no cycle on five vertices.
+Proof. Suppose that x1 . . . x5x1 is a cycle in H. We know by definition of H∗ and H that one of
+the vertices in our cycle, x1 say, must have another neighbour u, which must be blue. By Claim 2.2
+and Claim 2.9, u can have zero or one further white neighbours. In either case, D can play x1,
+which reduces π by at least 49.
+Claim 2.11. No double is adjacent to an orange vertex.
+Proof. Suppose xy is a pair and that x is adjacent to an orange vertex u. Since no vertex has
+degree 1, y must have another neighbour v, which cannot be white and must be distinct from u.
+Furthermore, v cannot have three white neighbours by Claim 2.9 and if D plays x, then π will
+decrease by at least 50.
+Claim 2.12. No single is adjacent to both a blue and an orange vertex.
+Proof. By playing the blue neighbour of the single, D would decrease π by at least 50.
+Claim 2.13. No blue vertex is adjacent to both doubles of a pair.
+Proof. By playing such a blue vertex, D would decrease π by at least 50.
+Claim 2.14. No double is adjacent to two blue vertices.
+Proof. By playing the double, D would decrease π by at least 49.
+Claim 2.15. No (blue) vertex is adjacent to two singles.
+Proof. By playing such a blue vertex, D would decrease π by at least 56.
+Claim 2.16. No single has a blue neighbour.
+Proof. Suppose towards a contradiction that there exists a single x with a blue neighbour u. Since
+the degree of x is at least 2, x must have another neighbour v, and by Claim 2.12, v must be blue.
+By Claim 2.15, u and v must both be adjacent to a double, which we call y and z respectively.
+Assume first that y and z are not adjacent. In this case, D can first play y which decreases π
+by 46. If S replies by playing a vertex that is adjacent to x or z, that will decrease π by at least
+23 and we are done. In any other case, D may play v next to decrease π by at least 50, so that
+π(t1) − π(t1 + 3) ≥ 116.
+6
+
+If y and z are adjacent, D can still decrease π by 46 by playing y. By Claim 2.14 and Claim 2.11,
+y and z have no other neighbours than each other and u and v respectively. Therefore, Claim 2.11
+is still valid after this move by D. Hence, if S plays an orange vertex, then π will decrease by at
+least 34 due to Claim 2.12. If S plays a single, all of its orange neighbours become red and all of
+its blue neighbours become orange so that π reduces by at least 26. If S plays a blue vertex, π
+decreases by at least 36 and if S plays a double, π decreases by 46. In any of these cases, we have
+π(t1) − π(t1 + 2) ≥ 72.
+One can conclude from Claim 2.11 and Claim 2.16 that S would have no good move if it was
+their turn rather than Dominator’s. Often, this will allow D to bring S into a zugzwang of sorts.
+The next claim formalizes this.
+Claim 2.17. Suppose that after move t ≥ t1, no double is adjacent to an orange vertex and no
+single is adjacent to both a blue and an orange vertex, then move t + 1 will decrease π by at least
+34.
+Proof. We denote the vertex played on move t + 1 by x. If x is a double, π decreases by 46, and
+if x is blue, π decreases by 36. If x is a single, it must have at least two orange neighbours by
+Claim 2.16 and π decreases by at least 34. The same holds if x is orange itself.
+As a simple consequence of Claim 2.17, we obtain the following.
+Claim 2.18. No orange vertex and no single exists.
+Proof. If an orange vertex existed, D could reduce π by 34 by playing this vertex without introducing
+a new orange or blue vertex, whence Claim 2.17 would be applicable with t = t1 + 1 (recall
+Claim 2.11).
+Since no orange vertex exists, no single can exist either by Claim 2.16.
+Another consequence of Claim 2.17 is that the isolated four-cycles we removed in our definition
+of H∗ do not actually pose a problem.
+Claim 2.19. No isolated cycle on four vertices contained in G has only white vertices after move
+t1.
+Proof. In any such cycle, D could just play any vertex to reduce π by 46, after which Claim 2.17 is
+applicable.
+Claim 2.20. Every pair xy must lie inside a six-cycle uxyvy′x′u such that x′ and y′ are white.
+Furthermore, the six-cycle is unique up to relabelling of its constituent vertices.
+Proof. To see that any six-cycle as in the claim must be unique, note that by Claim 2.9, u has no
+other white neighbours than x and x′, and v has no white neighbours other than y and y′.
+Let now xy be a pair. By Claim 2.11 and Claim 2.14, x and y have both exactly one blue
+neighbour, which we denote by u and v respectively. These in turn must have exactly one more
+white neighbour each, which we call x′ and y′ respectively. By Claim 2.18, both x′ and y′ are
+doubles, by Claim 2.13, y′ ̸= x and x′ ̸= y and by Claim 2.14, x′ ̸= y′.
+If x′ and y′ were adjacent, there would be nothing left to show, so we may suppose towards a
+contradiction that they are not. In this situation, D may play x to decrease π by 46. We can argue
+as in the proof of Claim 2.18 that if S plays any vertex other than u or v, π will decrease by at least
+34. If S does play one of u and v, without loss of generality the former, then π decreases by 20 and
+D can play y′ next, thus decreasing π by at least 50 so that we have π(t1) − π(t1 + 3) ≥ 116.
+7
+
+2.3
+The reactive phase
+We now know that after t1, all remaining white vertices in G have degree 2 and lie either in an
+isolated five-cycle of white vertices or in a six-cycle as described in Claim 2.20. Let k be the number
+of configurations of either type. The following claim asserts that D can rapidly end the game from
+here on out.
+Claim 2.21. D can play in a way such that the game ends after at most t1 + 2k + 1 moves.
+Proof. D can ensure that S plays at most one vertex in any of the k cycles, from which the claim
+follows. Indeed, whenever S plays a vertex x in one of the remaining cycles C, and there are white
+vertices left in C, D can make all of them red by playing a vertex that is opposite to x in C. If
+there are no white vertices left in C after Staller’s move as well as on move t1 + 1, D can simply
+play an arbitrary vertex.
+2.4
+Putting everything together
+Proof of Theorem 1.4. We have π(0) = 20n, and by definition of t1, π(t1) ≤ π(0) − 34t1. Fur-
+thermore, since each of the k remaining five- and six-cycles carries 100 points, we know that
+π(t1) = 100k. Therefore, we obtain 100k ≤ 20n − 34t1 or
+t1 ≤ 10
+17n − 50
+17k.
+(1)
+If k = 0, then the game is over after move t1, and there is nothing left to show. Otherwise, we
+apply Claim 2.21 to see that D can conclude the game after t1 + 2k + 1 moves, which by (1) is at
+most 10n/17 + 1/17.
+3
+Concluding remarks
+Before we discuss in how far the proof of Theorem 1.4 is amenable to further improvement, we
+point out that instead of letting D reduce π greedily as fast as possible, one can extract a more
+explicitly formulated strategy for D from the Claims 2.2 through 2.21. Indeed, whenever it is Dom-
+inator’s turn, one looks for the first one among these claims that is not yet satisfied, and lets D
+play according to the instructions in the proof of that claim. If all of the claims are satisfied, then
+one can move to play according to the explicit strategy of the reactive phase.
+We decided to keep the argument relatively short and simple with the bound 10
+17n + 1
+17, but we
+believe that with a more careful case analysis the method used in this paper could yield a better
+constant C < 10
+17 in Theorem 1.4. However, it seems that a substantially different approach would
+be required to improve the bound beyond
+7
+13n, as the example G of multiple disjoint copies of the
+graph G13 in Figure 1 shows.
+According to the strategy that the Claims 2.2 to 2.21 describe on G, D will start by playing the
+vertex x1 or y1 in one of the copies of G13, to which S can reply by playing x2 or y2 in the same
+copy. After all of those four vertices have been played in every copy of G13, three more vertices will
+have to be played in each of them. Overall, 7 out of 13 vertices will be played in every instance of
+G13 over the course of the game.
+We will now make some remarks on conjectures that are related to Conjecture 1.3. Firstly, the
+proof of Theorem 1.4 can be adapted to yield progress on the following conjecture by Rall, which
+was first mentioned in [7].
+8
+
+x1
+x2
+y1
+y2
+Figure 1: The graph G13. After D plays the vertices with encircled in blue, and S plays the vertices
+encircled in red, the vertices are coloured as shown here.
+Conjecture 3.1. If G has a Hamiltonian path, then γg(G) ≤ ⌈ 1
+2n⌉.
+Indeed, we can show the following as an easy corollary to the proof of Theorem 1.4.
+Theorem 3.2. If G has a Hamiltonian path, then γg(G) ≤ ⌈ 10
+17n⌉.
+Sketch proof. The worst case is that G has two leaves l1 and l2. In this situation, D can play the
+owner of l1 in the first move to decrease π by 50. If l2 is not red before Dominator’s next move, its
+unique neighbour cannot be red either, so playing it decreases π by at least 27. Therefore, we have
+π(0) − π(4) ≥ 117. If l2 becomes red before the second move, we have π(0) − π(2) ≥ 60. In either
+case, it is easy to check that if D plays as in the proof of Theorem 1.4 from there on out, the game
+will last at most 10n/17 + 21/34 ≤ ⌈ 10
+17n⌉ moves.
+The next conjecture was made by Bujtás, Iršič and Klavžar in [4] as a relaxation of Conjec-
+ture 1.3.
+Conjecture 3.3. There exists a universal constant C such that for every graph G with δ(G) ≥ 2,
+we have γg(G) ≤ 1
+2n + C.
+It turns out however, that Conjecture 1.3 and Conjecture 3.3 are almost equivalent. To see
+this, we consider a variant of the domination game, which can be seen as a generalisation of most
+other variants that have been studied. The transversal game was defined in [3] as a game played
+on a hypergraph H, in which the two players, Edge-hitter and Staller, alternately select a vertex
+from H, with the rule that each newly selected vertex must hit, i.e., be contained in, at least one
+edge that does not intersect the set of previously selected vertices. The game ends when the set of
+selected vertices becomes a transversal in H, i.e., when every edge of H intersects the set of selected
+vertices. Edge-hitter aims to end the game in as few moves as possible, whereas Staller wants the
+game to last as long as possible. The game transversal number τg(H) of H is the number of moves
+played if Edge-hitter starts the game and both players play optimally. In [10], the authors proved
+the following result.
+Lemma 3.4. Let C be a set of hypergraphs closed under taking multiple disjoint copies of a hyper-
+graph in C and suppose that there exists c > 0 such that for every hypergraph H ∈ C on n vertices,
+τg(H) ≤ (c + o(1))n. Then we also have τg(H) ≤ cn + 1 for every H ∈ C on n vertices.
+9
+
+The closed neighbourhood hypergraph of a graph G is defined as the hypergraph HG with
+vertex set V (HG) = V (G) and hyperedge set E(HG) = {NG[x]|x ∈ V (G)} consisting of the closed
+neighbourhoods of vertices in G. As remarked in [3], the domination game played on a graph G
+can be seen as a special instance of the transversal game being played on the closed neighbourhood
+hypergraph HG of G. By taking C = {HG : G is a graph of minimum degree at least 2} and c =
+1/2, we obtain the following as a direct corollary of Lemma 3.4.
+Corollary 3.5. Suppose there exists a strategy for D to finish the domination game on each graph
+G of minimum degree at least 2 in at most 1
+2(1+o(1))n moves. Then there also exists a strategy for
+D to finish the domination game on each graph G of minimum degree at least 2 in at most 1
+2n + 1
+moves.
+Thus, if Conjecture 3.3 is true, we can actually choose C = 1.
+References
+[1] B. Brešar, S. Klavžar, and D. F. Rall. Domination game and an imagination strategy. SIAM
+Journal on Discrete Mathematics, 24(3):979–991, 2010.
+[2] C. Bujtás. Domination game on forests. Discrete Math., 338(12):2220–2228, 2015. ISSN 0012-
+365X. doi: 10.1016/j.disc.2015.05.022. URL https://doi.org/10.1016/j.disc.2015.05.
+022.
+[3] C. Bujtás, M. A. Henning, and Z. Tuza. Transversal game on hypergraphs and the 34-conjecture
+on the total domination game. SIAM Journal on Discrete Mathematics, 30(3):1830–1847, 2016.
+[4] C. Bujtás, V. Iršič, and S. Klavžar. 1/2-conjectures on the domination game and claw-free
+graphs. European Journal of Combinatorics, 101:103467, 2022.
+[5] M. A. Henning and W. B. Kinnersley. Domination game: A proof of the 3/5-conjecture for
+graphs with minimum degree at least two. SIAM Journal on Discrete Mathematics, 30(1):
+20–35, 2016.
+[6] M. A. Henning and C. Löwenstein. Domination game: extremal families for the 3/5-conjecture
+for forests. Discuss. Math. Graph Theory, 37(2):369–381, 2017. ISSN 1234-3099. doi: 10.7151/
+dmgt.1931. URL https://doi.org/10.7151/dmgt.1931.
+[7] T. James, S. Klavžar, and A. Vijayakumar. The domination game on split graphs. Bulletin of
+the Australian Mathematical Society, 99(2):327–337, 2019.
+[8] W. B. Kinnersley, D. B. West, and R. Zamani.
+Extremal problems for game domination
+number. SIAM Journal on Discrete Mathematics, 27(4):2090–2107, 2013.
+[9] G. Košmrlj.
+Domination game on paths and cycles.
+Ars Math. Contemp., 13(1):125–136,
+2017. ISSN 1855-3966. doi: 10.26493/1855-3974.891.e93. URL https://doi.org/10.26493/
+1855-3974.891.e93.
+[10] J. Portier and L. Versteegen. A proof of the 3/4 conjecture for the total domination game.
+arXiv preprint arXiv:2211.16432, 2022.
+[11] L. Versteegen.
+A proof of the 3/5-conjecture in the domination game.
+arXiv preprint
+arXiv:2212.04527, 2022.
+10
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf,len=559
+page_content='Progress towards the 1/2-Conjecture for the domination game  Julien Portier∗  Leo Versteegen †  Abstract  The domination game is played on a graph G by two players, Dominator and Staller,  who alternate in selecting vertices until each vertex in the graph G is contained in the closed  neighbourhood of the set of selected vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Dominator’s aim is to reach this state in as few  moves as possible, whereas Staller wants the game to last as long as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' In this paper, we  prove that if G has n vertices and minimum degree at least 2, then Dominator has a strategy  to finish the domination game on G within 10n/17+1/17 moves, thus making progress towards  a conjecture by Bujtás, Iršič and Klavžar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  1  Introduction  The domination game is an optimization game for two players introduced by Brešar, Klavžar and  Rall in [1] and is played on a graph G without isolated vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' In this game, the two players,  Dominator (D) and Staller (S), alternately select vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  A vertex is considered to be  dominated, if itself or one of its neighbors were selected by either player, and both players must  always select a vertex that increases the number of dominated vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' The game ends once all  vertices of G are dominated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Within the bound of these rules, Dominator wishes to minimize the  number of moves played during the game, while Staller wants to achieve the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' We define  the game domination number γg(G) of G to be the number of moves the game takes to finish if  Dominator moves first and both players play optimally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Let n be the number of vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Of course, one should expect that the game lasts longer  the larger n is, and so it is a natural objective for research to bound γg in terms of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Specifically,  Kinnersley, West and Zamani conjectured [8] that γg(G) ≤ 3n/5 for all graphs without isolated  vertices, and the second author has recently confirmed their conjecture [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' For any graph G on n vertices without isolated vertices, γg(G) ≤ 3n/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  The bound 3n/5 in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='1 is best possible, as one may see by considering the path on five  vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' What is more, if we think of the path P5 as a single vertex to which we have appended  two paths of length two, we can generalise this example by taking any graph H and appending  two paths of length two to each of its vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' The resulting graph G has 5|V (H)| vertices and  satisfies γg(G) = 3|V (H)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' One can generalise this further (see [6, 11]), but one property that  all the resulting extremal graphs share is that they have many leaves, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=', vertices of degree one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  There is an extremal graph without a leaf, namely the cycle on five vertices, but in many ways, the  ∗jp899@cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='uk, Department of Pure Mathematics and Mathematical Statistics (DPMMS), University of Cam-  bridge, Wilberforce Road, Cambridge, CB3 0WA, United Kingdom  †lvv23@cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='uk, Department of Pure Mathematics and Mathematical Statistics (DPMMS), University of Cam-  bridge, Wilberforce Road, Cambridge, CB3 0WA, United Kingdom  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='05202v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='CO]  12 Jan 2023  five-cycle is a degenerate example that only works due to its small size, and unlike the P5, it does  not lend itself to any sort of generalisation to larger graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' With this caveat in mind, it is natural  to ask whether Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='1 can be improved for the class of graphs without leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' In [4], Bujtás,  Iršič and Klavžar made the following two conjectures in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' There exists a constant c < 3/5 such that every graph G on n ≥ 6 vertices and  minimum degree at least 2 satisfies γg(G) ≤ cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Let G be a graph with δ(G) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Then γg(G) ≤ ⌈ 1  2n⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Of course, Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='3 is a straightforward strengthening of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='2, as it essentially  sets the undetermined constant c from Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='2 to 1/2, which is natural choice for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Firstly, it is simply the least constant for which there are not any counterexamples known already.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Secondly, if we follow the heuristic notion that γg should generally decrease if we add more edges to  the graph under consideration, then we should believe that unions of vertex-disjoint cycles should  pose the worst case when proving an upper bound on γg for graphs with minimum degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' But  for a cycle C, it is known [9] that γg(C) ≤ ⌈ 1  2n⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  In 2016, long before Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='1 was established in full, Henning and Kinnersley proved [5]  that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='1 holds for graphs of minimum degree 2, which is the reason that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='2  asks for a constant c < 3/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' To the best of our knowledge, no improvement on the constant has  been made since, whence Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='2 is still open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' In this paper, we prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' For every graph G with minimum degree at least 2, we have γg(G) ≤ 10  17n + 1  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Since 10n/17 + 1/17 < 3n/5 for all n > 5, it is easy to see that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='4 confirms Conjec-  ture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  2  The proof  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='1  The main idea of the proof  To prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='4, we employ the discharging method introduced by Bujtás in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' The core  concept of this method is that at any given time during the game, every vertex has a color, which  has in turn a certain number of points associated with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' As the game progresses, the sum π of  all points associated to vertices decreases until it reaches zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If we can prove that the rate of  decrease per move is sufficiently large, then that will allows us to bound the total number of moves  in terms of the total number of points at the start of the game, which in turn only depends on the  number of vertices n in the graph G under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Concretely, we use the colours white, blue, orange and red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Let t be the number of moves that  have been played so far and let A be the set of vertices that have been selected by either player up  to and including move t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' The colour of a vertex is determined by the following rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  A vertex is white if and only if it is not dominated by A (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='e, is not itself contained in A and  not adjacent to a vertex in A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  A vertex is red if and only if it and all its neighbours are dominated by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Let W be the set of white vertices as determined by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' An isolated vertex in the induced  subgraph G[W] is called a single, while a component of exactly two vertices in G[W] is called  a pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' An individual member of a pair is called a double.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' A vertex is orange if and only if  it is dominated by A, has exactly one white neighbour, and this neighbour is a single or a  double.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  2  All other vertices are blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  The number of points πv(t) associated with a vertex v after move t, depends only on the color  of v after move t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Namely, πv(t) is 20, 10, 7, or 0 if v is white, blue, orange, or red, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  With this we define the total number of points after move t as π(t) = �  v∈V (G) πv(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  At the start of the game, all vertices are white, which means that π(0) = 20n, and the game  ends after move T if all vertices are red, which is to say that π(T) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Therefore, if D can play in  a way such that we have π(T) ≤ 20n − 34T, we would know that T ≤ 10n/17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' One can show that  this is the case as long as n > 5, but doing so requires a disproportional amount of additional effort  relative to the only very slightly worse bound π(T) ≤ 10n/17 + 1/17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' To show the latter bound,  we divide the game for the purpose of Dominator’s strategy into two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' The first phase lasts  as long as D can play in a way that guarantees the game to arrive at a point where t moves have  been played, π(t) ≤ π(0) − 34t, and D is to move next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Once this phase is over, one can conclude that all the remaining white vertices lie in very specific  configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' The problem with these configurations is that they lead to a form of zugzwang,  where neither player would like to make the first move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' A good example for such a configuration is  a cycle on five white vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Indeed, it is an easy exercise to show that if the domination game is  played on a C5 and D makes the first move, then the game will take three moves, whereas it will  only take two moves if S goes first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Happily, if the five-cycle is part of a larger graph G, it is S who  is in zugzwang after those three moves have been played, and there is no way for them to get out  of it for the rest of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Thus, the idea is that D can afford to reduce π by less than 34 on  average for a few moves in exchange laying the burden of the initiative on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='2  The active phase  Before we reveal the strategy that D should play, we first make a general observation about the  point system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If the game is not over after t moves, then π(t) − π(t + 1) ≥ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Suppose that the vertex v is played on move t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If v was blue before move t + 1, then  v has a neighbour u which was white then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' After move t + 1 on the other hand, v is red and u is  not white anymore, so that π(t) − π(t + 1) ≥ 10 + (20 − 10) = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If v was coloured orange, then  v has a neighbour u which was a double or a single before move t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' In particular, u was white  and has at most one white neighbour, whence it will be orange or red after move t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' It follows  that π(t) − π(t + 1) ≥ 7 + (20 − 7) = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Finally, if v was white itself, π(t) − π(t + 1) ≥ 20 because  v is red after move t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  We say that the game is in a stable state after move t, if π(0) − π(t) ≥ 34t, and at least one of  the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Either, t is even, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=', D is to move next, or  there are no white vertices left, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=', the game is over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Whenever the game is not over but in a stable state, it must be Dominator’s turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Suppose that  the game is in a stable state after move t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If D has a strategy that allows him to advance the game,  possibly over the course of several moves, to another stable state after the current one, D plays  according to this strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Letting t1 denote the smallest integer so that D does not have such a  strategy after move t1, we can make a series of powerful observations about the state of the game  after move t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  3  For example, we know that there can be no vertex v such that π(t1) − π(t1 + 1) ≥ 48 if D  plays v on move t1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Indeed, if the game is over after D plays v, the game has reached another  stable state after move t1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If it is not, Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='1 tells us that any subsequent move by S will  reduce π further by at least 20 so that π(t1) − π(t1 + 2) ≥ 68 = 2 · 34, which means the game will  reach a stable state since t1 + 2 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Likewise, there can never be a strategy for D to play that  guarantees that π(t1) − π(t1 + 2) ≥ 68 or π(t1) − π(t1 + 3) ≥ 116, even if it begins with a move  that decreases π by less than 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' In the proofs of all of the following claims, the assumption that  the claim does not hold can be used to provide D with a strategy that leads to one of these three  outcomes and thus yields a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' No white vertex has three white neighbours and no blue vertex has four white neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Suppose towards a contradiction that there is a white vertex x with at least three white  neighbours after t1 moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' D may play this vertex, and since all neighbours of x are now dominated,  each of them has at least 10 points less associated with them than before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Furthermore, x is now  red and has 0 point associated with it so that π(t1) − π(t1 + 1) ≥ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Similarly if a vertex v is blue  and has four white neighbours, D can also reduce π by at least 50 by playing v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Let H∗ be the subgraph of G in which we remove every connected component that is a four-  or five-cycle and has only white vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Let further H be the subgraph of H∗ that is induced by  the set of white vertices after t1 moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' By Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='2, H has no vertex of degree three or higher,  whence H must be a union of disjoint paths and cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' H contains no path xyz such that x has no white neighbour other than y and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If such a path existed, then D could simply play y to decrease π by 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' H contains no cycle on nine or more vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Suppose that k ≥ 9 and that x1x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' xkx1 is a cycle in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' D may play x5, thus decreasing  π by 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If S replies by playing a vertex z that is adjacent to both a vertex in {x1, x2, x3} and  {x7, x8, x9}, then π decreases further by at least 30, so we may assume without loss of generality  that y has no neighbour in {x1, x2, x3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' In this case, D can play x2 next, which decreases π by 60  since both x3 and x4 become red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' By Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='1, we have π(t1) − π(t1 + 3) ≥ 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' H contains no cycle on eight or seven vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Suppose that x1x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' x8x1 is a cycle in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' D can play x1, which decreases π by 40, and  if the vertex z that S plays in response is adjacent to two white vertices, then π decreases by at  least 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' We may therefore assume that z has only one white neighbour before Staller’s move and  thus without loss of generality that z is not adjacent to either of x3 and x4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If z is not adjacent to  x5 either, then D can play x4 on move t1 + 3, which decreases π by another 60 points, whence we  would have π(t1) − π(t1 + 3) ≥ 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' On the other hand, if z is adjacent to x5, then both x5 and x8  will be orange after D plays x4 on move t + 3, whence π(t1) − π(t1 + 3) ≥ 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  The proof that H cannot have a cycle on seven vertices is very similar and we leave it as an  exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' H contains no cycle on six vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Suppose that x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' x6x1 is a cycle in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' D can play x1, and if after Staller’s move any of the  vertices x3, x4 and x5 is still white, D can play x4 next and we have π(t1)−π(t1+3) ≥ 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Of course,  if after move t1+2, none of x3, x4 and x5 are white anymore, we even have π(t1)−π(t1+2) ≥ 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  4  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' H contains no cycle on four vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Suppose that x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' x4x1 is a cycle in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Because H is a subgraph of H∗, one of the four  vertices, x1 say, must have another neighbour u, which must be blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' D can play x3, which makes  x3 red as well as x2 and x4 orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' We distinguish which colour x1 and u have after S played  some vertex v on move t1 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If x1 is red, then x2 and x4 are red as well, and we must have  π(t1) − π(t1 + 2) ≥ 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If x1 is not red, then it must be white, and thus u cannot be red either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' In  particular, u ̸= v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If u is orange, we can use the fact v is now red, and that v must have been white  or had a white neighbour before move t1 + 2 to see that π(t1) − π(t1 + 2) ≥ 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  This leaves the possibility that u is still blue after move t1 +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' In this case, u must have another  white neighbour y besides x1, and D can play u after which u, x1, x2 and x4 must be red and y is  not white anymore so that π(t1) − π(t1 + 3) ≥ 120 by Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  We know now that H must consist only of isolated vertices, isolated edges and five-cycles, but  the structure of H can be even more rigidly determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Indeed, we will show that H does in  fact not contain any five-cycles or isolated vertices and that the edges of H lie in highly specific  configurations when viewed in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' In the following proofs, it is understood that whenever we play  a vertex that is adjacent to a double, the double is coloured orange or red but never blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If two vertices x1, y1 ∈ H are part of (not necessarily distinct) five-cycles, they cannot  have a common blue neighbour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Suppose that there are two five-cycles x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' x5x1 and y1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' y5y1 in H and a vertex u that  is adjacent to both x1 and y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If y1 = xi for some i ̸= 1, there must be some j ∈ [5] such that  {x1, xi} ⊂ {xj} ∪ N(xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Therefore, by Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='2, u will become red or orange after D plays xj so  that π decreases by at least 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Let us therefore assume that the two cycles are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Suppose now that u has a third white neighbour z, noting that u cannot have a fourth white  neighbour by Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' D can play u which reduces π by at least 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If S replies by playing a  vertex that has a neighbour in {x2, x3, x4, x5} and {y2, y3, y4, y5}, then π(t1) − π(t1 + 2) ≥ 70 and  we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Otherwise, we can assume without loss of generality that {x2, x3, x4, x5} are all still  white after move t1 +2, and D can play x3 which colours x2 and x3 red as well as x1 and x4 orange,  thus reducing π by at least 56 so that π(t1) − π(t1 + 3) ≥ 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  We are left with the possibility that x1 and y1 are the only white neighbours of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' In this case,  D may play x1, which reduces π by 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If S replies by playing a vertex that is adjacent to at least  two of the vertices {x3, x4, y1, y2, y5}, they reduce π further by at least 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Should S play a vertex  that does not dominate either of x3 or x4, D can play x3 next so that π(t1) − π(t1 + 3) ≥ 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Lastly, if S plays a vertex that does not dominate either of y1, y2 and y5, D can play y1 next so that  π(t1) − π(t1 + 3) ≥ 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' No (blue) vertex has three white neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Let u be a blue vertex with three white neighbours x1, y and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If all three of them are  singles or doubles, D can play u to decrease π by at least 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Let us therefore assume that x1 is  part of a five-cycle x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' x5x1 in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' By Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='8, y and z must be singles or doubles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If y is a  single, D can still play u to reduce π by at least 53 so let us assume that y is adjacent to another  double y′, which must have itself a second neighbour, which we denote by v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' How D should proceed  depends on the number and type of the white neighbours of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Suppose first that v has only one white neighbour, D can play y to reduce π by 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If S replies  by playing a vertex that dominates x1 or at least two white vertices overall, π decreases further by  5  at least 23 and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Otherwise, D can play one of x1, x2 or x5 next to reduce π by at least  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Suppose next that v has exactly two white neighbours and that one of them (the one that is  not y′), which we call x′  1, is part of a five-cycle x′  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' x′  5x′  1 of white vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' In this scenario, D can  reduce π by 49 by playing x′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  If neither of the above assumptions are true, we know by Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='8 that v is adjacent to a single  or double z′ ̸= y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If z′ = y, D can play v to reduce π by 50 right away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Else, D can play u on move  t1 +1 to reduce π by at least 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' As usual, we may assume that the vertex that S plays in response,  dominates only one white vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' We can also assume that this white vertex is either y′ or z′ as  otherwise D could subsequently reduce π by at least 50 by playing v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' In particular, x2, x3 and x4  are still white after move t1 + 2, which means that D can play x3 on move t1 + 3, thus reducing π  by at least 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' H contains no cycle on five vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Suppose that x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' x5x1 is a cycle in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' We know by definition of H∗ and H that one of  the vertices in our cycle, x1 say, must have another neighbour u, which must be blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' By Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='2  and Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='9, u can have zero or one further white neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' In either case, D can play x1,  which reduces π by at least 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' No double is adjacent to an orange vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Suppose xy is a pair and that x is adjacent to an orange vertex u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Since no vertex has  degree 1, y must have another neighbour v, which cannot be white and must be distinct from u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Furthermore, v cannot have three white neighbours by Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='9 and if D plays x, then π will  decrease by at least 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' No single is adjacent to both a blue and an orange vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' By playing the blue neighbour of the single, D would decrease π by at least 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' No blue vertex is adjacent to both doubles of a pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' By playing such a blue vertex, D would decrease π by at least 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' No double is adjacent to two blue vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' By playing the double, D would decrease π by at least 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' No (blue) vertex is adjacent to two singles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' By playing such a blue vertex, D would decrease π by at least 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' No single has a blue neighbour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Suppose towards a contradiction that there exists a single x with a blue neighbour u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Since  the degree of x is at least 2, x must have another neighbour v, and by Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='12, v must be blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  By Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='15, u and v must both be adjacent to a double, which we call y and z respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Assume first that y and z are not adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' In this case, D can first play y which decreases π  by 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If S replies by playing a vertex that is adjacent to x or z, that will decrease π by at least  23 and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' In any other case, D may play v next to decrease π by at least 50, so that  π(t1) − π(t1 + 3) ≥ 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  6  If y and z are adjacent, D can still decrease π by 46 by playing y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' By Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='14 and Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='11,  y and z have no other neighbours than each other and u and v respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Therefore, Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='11  is still valid after this move by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Hence, if S plays an orange vertex, then π will decrease by at  least 34 due to Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If S plays a single, all of its orange neighbours become red and all of  its blue neighbours become orange so that π reduces by at least 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If S plays a blue vertex, π  decreases by at least 36 and if S plays a double, π decreases by 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' In any of these cases, we have  π(t1) − π(t1 + 2) ≥ 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  One can conclude from Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='11 and Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='16 that S would have no good move if it was  their turn rather than Dominator’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Often, this will allow D to bring S into a zugzwang of sorts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  The next claim formalizes this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Suppose that after move t ≥ t1, no double is adjacent to an orange vertex and no  single is adjacent to both a blue and an orange vertex, then move t + 1 will decrease π by at least  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' We denote the vertex played on move t + 1 by x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If x is a double, π decreases by 46, and  if x is blue, π decreases by 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If x is a single, it must have at least two orange neighbours by  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='16 and π decreases by at least 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' The same holds if x is orange itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  As a simple consequence of Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='17, we obtain the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' No orange vertex and no single exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If an orange vertex existed, D could reduce π by 34 by playing this vertex without introducing  a new orange or blue vertex, whence Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='17 would be applicable with t = t1 + 1 (recall  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Since no orange vertex exists, no single can exist either by Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Another consequence of Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='17 is that the isolated four-cycles we removed in our definition  of H∗ do not actually pose a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' No isolated cycle on four vertices contained in G has only white vertices after move  t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' In any such cycle, D could just play any vertex to reduce π by 46, after which Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='17 is  applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Every pair xy must lie inside a six-cycle uxyvy′x′u such that x′ and y′ are white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Furthermore, the six-cycle is unique up to relabelling of its constituent vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' To see that any six-cycle as in the claim must be unique, note that by Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='9, u has no  other white neighbours than x and x′, and v has no white neighbours other than y and y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Let now xy be a pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' By Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='11 and Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='14, x and y have both exactly one blue  neighbour, which we denote by u and v respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' These in turn must have exactly one more  white neighbour each, which we call x′ and y′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' By Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='18, both x′ and y′ are  doubles, by Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='13, y′ ̸= x and x′ ̸= y and by Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='14, x′ ̸= y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  If x′ and y′ were adjacent, there would be nothing left to show, so we may suppose towards a  contradiction that they are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' In this situation, D may play x to decrease π by 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' We can argue  as in the proof of Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='18 that if S plays any vertex other than u or v, π will decrease by at least  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If S does play one of u and v, without loss of generality the former, then π decreases by 20 and  D can play y′ next, thus decreasing π by at least 50 so that we have π(t1) − π(t1 + 3) ≥ 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='3  The reactive phase  We now know that after t1, all remaining white vertices in G have degree 2 and lie either in an  isolated five-cycle of white vertices or in a six-cycle as described in Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Let k be the number  of configurations of either type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' The following claim asserts that D can rapidly end the game from  here on out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' D can play in a way such that the game ends after at most t1 + 2k + 1 moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' D can ensure that S plays at most one vertex in any of the k cycles, from which the claim  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Indeed, whenever S plays a vertex x in one of the remaining cycles C, and there are white  vertices left in C, D can make all of them red by playing a vertex that is opposite to x in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If  there are no white vertices left in C after Staller’s move as well as on move t1 + 1, D can simply  play an arbitrary vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='4  Putting everything together  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' We have π(0) = 20n, and by definition of t1, π(t1) ≤ π(0) − 34t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Fur-  thermore, since each of the k remaining five- and six-cycles carries 100 points, we know that  π(t1) = 100k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Therefore, we obtain 100k ≤ 20n − 34t1 or  t1 ≤ 10  17n − 50  17k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  (1)  If k = 0, then the game is over after move t1, and there is nothing left to show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Otherwise, we  apply Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='21 to see that D can conclude the game after t1 + 2k + 1 moves, which by (1) is at  most 10n/17 + 1/17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  3  Concluding remarks  Before we discuss in how far the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='4 is amenable to further improvement, we  point out that instead of letting D reduce π greedily as fast as possible, one can extract a more  explicitly formulated strategy for D from the Claims 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='2 through 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Indeed, whenever it is Dom-  inator’s turn, one looks for the first one among these claims that is not yet satisfied, and lets D  play according to the instructions in the proof of that claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If all of the claims are satisfied, then  one can move to play according to the explicit strategy of the reactive phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  We decided to keep the argument relatively short and simple with the bound 10  17n + 1  17, but we  believe that with a more careful case analysis the method used in this paper could yield a better  constant C < 10  17 in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' However, it seems that a substantially different approach would  be required to improve the bound beyond  7  13n, as the example G of multiple disjoint copies of the  graph G13 in Figure 1 shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  According to the strategy that the Claims 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='2 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='21 describe on G, D will start by playing the  vertex x1 or y1 in one of the copies of G13, to which S can reply by playing x2 or y2 in the same  copy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' After all of those four vertices have been played in every copy of G13, three more vertices will  have to be played in each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Overall, 7 out of 13 vertices will be played in every instance of  G13 over the course of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  We will now make some remarks on conjectures that are related to Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Firstly, the  proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='4 can be adapted to yield progress on the following conjecture by Rall, which  was first mentioned in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  8  x1  x2  y1  y2  Figure 1: The graph G13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' After D plays the vertices with encircled in blue, and S plays the vertices  encircled in red, the vertices are coloured as shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If G has a Hamiltonian path, then γg(G) ≤ ⌈ 1  2n⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Indeed, we can show the following as an easy corollary to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If G has a Hamiltonian path, then γg(G) ≤ ⌈ 10  17n⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Sketch proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' The worst case is that G has two leaves l1 and l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' In this situation, D can play the  owner of l1 in the first move to decrease π by 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If l2 is not red before Dominator’s next move, its  unique neighbour cannot be red either, so playing it decreases π by at least 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Therefore, we have  π(0) − π(4) ≥ 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' If l2 becomes red before the second move, we have π(0) − π(2) ≥ 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' In either  case, it is easy to check that if D plays as in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='4 from there on out, the game  will last at most 10n/17 + 21/34 ≤ ⌈ 10  17n⌉ moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  The next conjecture was made by Bujtás, Iršič and Klavžar in [4] as a relaxation of Conjec-  ture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' There exists a universal constant C such that for every graph G with δ(G) ≥ 2,  we have γg(G) ≤ 1  2n + C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  It turns out however, that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='3 and Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='3 are almost equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' To see  this, we consider a variant of the domination game, which can be seen as a generalisation of most  other variants that have been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' The transversal game was defined in [3] as a game played  on a hypergraph H, in which the two players, Edge-hitter and Staller, alternately select a vertex  from H, with the rule that each newly selected vertex must hit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=', be contained in, at least one  edge that does not intersect the set of previously selected vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' The game ends when the set of  selected vertices becomes a transversal in H, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=', when every edge of H intersects the set of selected  vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Edge-hitter aims to end the game in as few moves as possible, whereas Staller wants the  game to last as long as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' The game transversal number τg(H) of H is the number of moves  played if Edge-hitter starts the game and both players play optimally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' In [10], the authors proved  the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Let C be a set of hypergraphs closed under taking multiple disjoint copies of a hyper-  graph in C and suppose that there exists c > 0 such that for every hypergraph H ∈ C on n vertices,  τg(H) ≤ (c + o(1))n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Then we also have τg(H) ≤ cn + 1 for every H ∈ C on n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  9  The closed neighbourhood hypergraph of a graph G is defined as the hypergraph HG with  vertex set V (HG) = V (G) and hyperedge set E(HG) = {NG[x]|x ∈ V (G)} consisting of the closed  neighbourhoods of vertices in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' As remarked in [3], the domination game played on a graph G  can be seen as a special instance of the transversal game being played on the closed neighbourhood  hypergraph HG of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' By taking C = {HG : G is a graph of minimum degree at least 2} and c =  1/2, we obtain the following as a direct corollary of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Suppose there exists a strategy for D to finish the domination game on each graph  G of minimum degree at least 2 in at most 1  2(1+o(1))n moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Then there also exists a strategy for  D to finish the domination game on each graph G of minimum degree at least 2 in at most 1  2n + 1  moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Thus, if Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='3 is true, we can actually choose C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  References  [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Brešar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Klavžar, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Rall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Domination game and an imagination strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' SIAM  Journal on Discrete Mathematics, 24(3):979–991, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Bujtás.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Domination game on forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=', 338(12):2220–2228, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' ISSN 0012-  365X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  [3] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Bujtás, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Henning, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Tuza.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Transversal game on hypergraphs and the 34-conjecture  on the total domination game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' SIAM Journal on Discrete Mathematics, 30(3):1830–1847, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  [4] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Bujtás, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Iršič, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Klavžar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' 1/2-conjectures on the domination game and claw-free  graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' European Journal of Combinatorics, 101:103467, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Henning and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Kinnersley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Domination game: A proof of the 3/5-conjecture for  graphs with minimum degree at least two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' SIAM Journal on Discrete Mathematics, 30(1):  20–35, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Henning and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Löwenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Domination game: extremal families for the 3/5-conjecture  for forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Discuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Graph Theory, 37(2):369–381, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' ISSN 1234-3099.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='7151/  dmgt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='1931.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='7151/dmgt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='1931.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  [7] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' James, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Klavžar, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Vijayakumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' The domination game on split graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Bulletin of  the Australian Mathematical Society, 99(2):327–337, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  [8] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Kinnersley, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' West, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Zamani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Extremal problems for game domination  number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' SIAM Journal on Discrete Mathematics, 27(4):2090–2107, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  [9] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Košmrlj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Domination game on paths and cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  Ars Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Contemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=', 13(1):125–136,  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' ISSN 1855-3966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='26493/1855-3974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='891.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='e93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='26493/  1855-3974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='891.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='e93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  [10] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Portier and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Versteegen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' A proof of the 3/4 conjecture for the total domination game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  arXiv preprint arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='16432, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  [11] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content=' Versteegen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  A proof of the 3/5-conjecture in the domination game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  arXiv preprint  arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='04527, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
+page_content='  10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E4T4oBgHgl3EQfqw0f/content/2301.05202v1.pdf'}
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+A Voigt laser operating on 87Rb 780 nm transition
+Zijie Liu,1, 2, a) Xiaolei Guan,3, a) Xiaomin Qin,1 Zhiyang Wang,1 Hangbo Shi,1 Jia Zhang,1 Jianxiang Miao,1
+Tiantian Shi,1, b) Anhong Dang,2, b) and Jingbiao Chen1
+1)State Key Laboratory of Advanced Optical Communication Systems and Networks, Institute of Quantum Electronics,
+School of Electronics, Peking University, Beijing 100871, China
+2)State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Electronics, Peking University,
+Beijing 100871, China
+3)State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications,
+Beijing 100876, China
+(*Electronic mail: ahdang@pku.edu.cn)
+(*Electronic mail: tts@pku.edu.cn)
+(Dated: 6 January 2023)
+We report the development of laser systems- a “Voigt laser” - using a Voigt anomalous dispersion optical filter as the
+frequency-selective element, working at the wavelength of 780 nm of 87Rb-D2 resonance line. Compared with Faraday
+anomalous dispersion optical filter, the Voigt anomalous dispersion optical filter can generate a stronger and more
+uniform magnetic field with a compact size of magnet, and obtains a transmission spectrum with narrower linewidth
+and more stable lineprofile. In this case, the frequency stability of the Voigt laser reaches 5×10−9 at the averaging time
+of 200 s, and the wavelength fluctuation of 8-hours free operation is ± 0.1 pm. Besides, the Voigt laser has greater
+immunity to diode current than the Faraday laser, with a wavelength fluctuation of ± 0.5 pm in the current range from
+73 mA to 150 mA. Finally, the Voigt laser frequency can be controlled by the cell temperature of the Voigt optical filter,
+which is expected to achieve a frequency detuning of 20 GHz. Consequently, the Voigt laser, whose frequency could
+correspond to the atomic transition frequency by tuning the cell temperature, obtains good robustness to the current
+and temperature fluctuation of laser diode, and could realize a compact optical standard for precise measurement once
+stabilized by modulation transfer spectroscopy.
+Diode laser with narrow linewidth plays an essential role
+in the study of atomic physics, and is used in an extremely
+broad range of applications from laser cooling to thermal va-
+por experiments1. In the usual case, the frequency-selective
+element of a traditional laser could be formed with a spatial
+grating2,3 or a Fabry–Pérot (FP) etalon4. The frequency of
+these lasers is sensitive to their diode current and temperature,
+requiring an extremely high precision temperature and cur-
+rent control. Besides, the laser frequency needs to be finely
+tuned to the atomic transition line, to achieve frequency sta-
+bilization using saturated absorption spectroscopy5, polariza-
+tion spectroscopy6, modulation transfer spectrum7,8, etc. In
+this case, there will be a risk of mode hopping, resulting in the
+laser frequency far away from the locking point, which is not
+conducive to long-term continuous work.
+Faraday laser comprises a Faraday anomalous dispersion
+optical filter (FADOF) as a frequency-selective element, could
+solve the above problem9–14.
+FADOF has attracted much
+attention since its birth15, owing to its narrow bandwidth
+and high signal-to-noise ratio (SNR)16–21.
+Since 2011,
+FADOF is applied as a frequency-selective element, reveal-
+ing a stable "Faraday laser" immune to the diode current and
+temperature9.
+Importantly, the wavelength of the Faraday
+laser is limited to the narrow-band transmission window of
+FADOF, which is around the atomic transition line. There-
+a)These authors contributed to the work eqully and should be regarded as co-
+first authors
+b)Authors to whom correspondence should be addressed:tts@pku.edu.cn and
+ahdang@pku.edu.cn
+fore, the Faraday laser could easily achieve the frequency sta-
+bilization once it is turned on, and could stably operate for a
+long term without artificial adjustment22. In recent years, the
+Faraday laser has been formally proposed to different struc-
+tures and atoms, achieving better frequency stability and anti-
+interference ability13,14,23–25.
+Nevertheless, the axial mag-
+netic field in the Faraday laser is generally generated by two
+magnets aside the cell along the optical axis, which greatly
+increases the axial length of FADOF and limits the mini-
+mum cavity length of Faraday laser24. In another method, the
+strength and uniformity of the magnetic field generated by the
+magnet around the cell are relatively small, which limits the
+further improvement of the laser stability13.
+Unlike the Faraday effect, the Voigt effect rotates the po-
+larization direction of the incident laser by applying a radial
+magnetic field26. Therefore, the Voigt anomalous dispersion
+optical filter (VADOF) could construct a stronger and more
+homogeneous magnetic field with less volume than FADOF,
+by placing two magnets close together in the radial direction
+of the cell27–29. Then, a stable transmission spectrum with a
+narrower linewidth is obtained. In this case, a compact "Voigt
+laser" could be constructed, using a VADOF for frequency-
+selection. Like the Faraday laser, the Voigt laser could realize
+compact optical frequency standard, using modulation trans-
+fer spectroscopy (MTS) for frequency stabilization22,30.
+In this work, we propose and prove a Voigt laser, which
+uses 87Rb VADOF as frequency-selective element and semi-
+conductor diode laser as gain medium. This Voigt laser op-
+erated freely for many times during half a year, and its wave-
+length fluctuation is ± 0.5 pm with an instability better than
+2 × 10−8 for 48 hours. Compared with the Faraday laser, the
+arXiv:2301.01614v1  [physics.optics]  4 Jan 2023
+
+2
+Voigt laser has a better frequency stability in a smaller vol-
+ume. Besides, the wavelength fluctuation of the Voigt laser
+is ± 0.5 pm in the diode current range from 73 mA to 150
+mA, presenting the Voigt laser’s immunity towards the pa-
+rameters of the diode laser . Meanwhile, we investigate the
+influence of the cell temperature on the Voigt laser, revealing
+a tunable laser wavelength selected by the cell temperature.
+Therefore, the frequency of this Voigt laser can corresponds
+to the atomic transition frequency by adjusting the cell tem-
+perature, and then can be stabilized using MTS technology to
+realize a compact optical frequency standard. Furthermore,
+the Voigt laser could use different gain mediums and atoms to
+improve its performance, which paves the way for basic sci-
+entific research (such as atomic clock, atomic gravimeter, and
+atomic interferometer, etc.) and industrial electronic equip-
+ment (precision optical measurement, spectral inspection).
+FIG. 1.
+Experimental setup of the VADOF transmission spec-
+trum. The green dash refers to the saturated absorption spectroscopy
+(SAS). HWP, half-wave plate; PBS, polarization beam splitter; PD,
+photon detector; The probe laser is split into two laser beams (laser
+a and laser b) by a HWP and a PBS.
+VADOF comprises a 87Rb atomic vapor cell with length of
+L = 30 mm, a set of permanent magnets, a temperature control
+system, and two polarized beam splitters (PBSs), as presented
+in Fig. 1. The vacuum cell and magnets are stuck in a desired
+Teflon thermal insulation structure. The anti-reflection coated
+cell has a transmission of 95% for λ = 780 nm laser. The tem-
+perature of the 87Rb cell is controlled by an electric heating
+element that allows to maintain the cell temperature within
+0.1°C. The radial magnetic field range from 200 Gs to 3500
+Gs, which reach a maximum with two 15 mm-thick NdFeB
+permanent magnets aside the cell. For a better Voigt effect,
+there is a 45° angle between the magnetic field direction and
+the laser polarization direction28. The two PBSs are placed on
+either side of the cell. The polarization direction of the PBS
+behind the cell can be rotated, so that the two PBSs could be
+orthogonal or parallel.
+We use a homemade 780 nm interference-filter external
+cavity diode laser (IF-ECDL) to probe the transmission spec-
+trum of VADOF, of which the laser beam diameter is 2 mm.
+The probe laser is split into two laser beams (laser a and laser
+b) by a half wavelength plate (HWP) and a PBS. We probe
+the saturated absorption spectroscopy (SAS) of 87Rb as a fre-
+quency reference using laser a, as the green dash in Fig.
+1
+shows. Laser b goes through a HWP to control the laser inten-
+sity before probing the transmission spectrum of VADOF. The
+above spectrums are simultaneously presented in an oscillo-
+scope, which connects with two photodetectors (PD, Thorlabs
+PDA8A2).
+FIG. 2. The saturated absorption spectroscopy of 87Rb-D2-line and
+the transmission spectrum of VADOF in different temperature.The
+diagram at the top shows F = 1 and F = 2 transition of 87Rb-D2-line
+in SAS. a, b ,c , d present the transmission spectrum at the cell tem-
+perature of 110°C, 95°C, 85°C, and 75°C, respectively. Peak 1, 2, 3,
+4, and 5 are typical transmission peaks in the transmission spectrum.
+The probe laser frequency detuning ranges from -30 GHz to 25 GHz.
+Before experimentally probing the spectrum of VADOF, we
+analyze its major influencing factors using a simulation tool31.
+The results show that we can reveal a spectrum with high
+transmission and narrow linewidth, when radial magnetic field
+B = 3500 Gs. Then, we investigate the transmission spectrum
+of VADOF in different temperatures at B = 3500 Gs, as shown
+in Fig. 2. The diagram at the top shows SAS of 87Rb-D2-line.
+The frequency interval between F = 1 and F = 2 transition is
+6.834 GHz, and we set the frequency at the peak of F = 2
+transition as the detuning origin.
+The probe laser frequency tuning ranges from -30 GHz to
+25 GHz, where we can almost observe the whole transmission
+spectrum. Limited by the tuning range, The spectrum above
+25 GHz is missed. However, the transmission spectrum above
+25 GHz is suggested to monotonically decrease to 0 (the same
+
+1 -HWP
+2 - PBS
+3 - PD
+4 - 87Rb cell
+5 - VADOF
+3
+2Saturated spectrum
+Absorb(a. u.)
+0.4
+F=1
+F=2
+0.2
+100
+2
+(a)
+110
+80
+60
+40
+20
+100
+95°C
+80
+(b)
+60
+40
+20
+100
+85°
+80
+(c)
+60
+e
+40
+M MJ
+20
+100
+75
+80
+(d)
+2
+60
+40
+20
+0
+-30
+-25
+-20
+-15
+-10
+5
+10
+15
+20
+25
+-5
+0
+30
+Freguency detuning(GHz3
+trend as the left side of the transmission spectrum), accord-
+ing to the theoretical prediction. Compared with conventional
+FADOF, VADOF requires higher temperatures to reveal high
+transmission. Therefore, the Voigt laser is hard to produce
+once the cell temperature is lower than 60°C. When the cell
+temperature is in the range of 60 - 80°C, the peak 1 in the
+transmission spectrum obtains the highest transmission, see-
+ing Fig. 2d. In this case, the transmission peak 1, 2, and 3 will
+rise with the cell temperature. Peaks 2 and 3 become the high-
+est transmission peak at 85°C and 95°C, respectively, as pre-
+sented in Fig. 2c and Fig. 2b. VADOF reveals a ‘wing’ spec-
+tral profile and peak 4 stands out as the highest, as the temper-
+ature rises to 110°C in Fig. 2a. Then, the "wing" peak 5 will
+become the highest and the spectral profile won’t change with
+the rising temperature. However, the linewidth of peak 5 is
+relatively wide and its center frequency is sensitive to the cell
+temperature, which is not suitable for Voigt laser. Therefore,
+the highest transmission peak gradually jumps from peak 1 to
+peak 4 when the temperature increases from 60°C to 110°C.
+Based on this, we predict that the output laser wavelength of
+the Voigt laser also undergoes corresponding jumps when the
+temperature increases.
+FIG. 3. The experiment setup of the Voigt laser. HWP, half-wave
+plate; PBS, polarization beam splitter; HR, high-reflection mirror;
+PZT, piezoelectric ceramic tube; FP, Scanning Fabry–Pérot Interfer-
+ometers (Thorlabs, SA210-5B); λ, optical wavelength meter (Bristol
+671A)
+The experimental setup of the Voigt laser is schematically
+shown in the green dash of Fig.
+3. The Voigt laser with
+the length of optical cavity of 20 cm is composed of a 87Rb
+VADOF, an anti-reflection coated laser diode, aspheric lens, a
+high-reflection mirror (HR) and a piezoelectric ceramic tube
+(PZT). After collimated by an aspheric lens, the beam diam-
+eter of the ARLD’s emitted light is enlarged to 2 mm. Then,
+the collimated light is filtered by the Voigt optical filter and
+fed back into the ARLD by HR. The reflectivity of HR is
+99.9% at the wavelength of 780 nm, and is fixed on the PZT
+for finely detuning the cavity length. The laser beam outputs
+from the second PBS. A HWP and a PBS are comprised to
+split the output laser, for scanning FP spectrum and sensing
+laser wavelength, respectively. The temperature of 87Rb cell
+could range from 20°C to 200°C and the radial magnetic field
+is between 200 Gs and 3500 Gs.
+We firstly investigate the cell temperature’s effect on the
+Voigt laser, presented in Fig.
+4. The radial magnetic field
+is selected as 3500 Gs for a high transmission peak. The LD
+current and temperature are 90 mA and 22.08°C, respectively.
+The output laser wavelength jumps with the increasing cell
+temperature, as we predicted in the discussion of Fig. 2. Once
+the temperature is stabilized at an exact value, laser with a sta-
+ble wavelength will be emitted. When the temperature varies
+between 60 - 80°C, the output laser wavelength is maintained
+at 780.256 nm (near 60 – 75°C) or 780.254 nm (near 80°C),
+corresponding to the two longitudinal modes of peak 1 in Fig.
+2. Continuing to increase the cell temperature, the laser wave-
+length will be detected as 780.241 nm (near 85°C) and then
+jump to 780.265 nm (near 90-100°C), referring to peak 2 and
+peak 3 in Fig. 2, respectively. When the temperature is near
+105°C, the transmission of peaks in Fig. 2 are close to each
+other and the output laser wavelength fluctuates widely. In
+this case, the laser wavelength will level out at 780.276 nm,
+corresponding to peak 4 in Fig. 2, if the cell temperature rises
+to 110°C. Then, the output laser will be red-shifted and the
+laser wavelength will rise with the increasing cell temperature.
+Subsequently, the Voigt laser with an EOM could achieve a
+wide frequency tuning range of 20 GHz.
+FIG. 4. The wavelength of the Voigt laser at different cell tempera-
+ture.
+Based on the above investigation, we further study the char-
+acteristics of the Voigt laser at different LD currents, seeing
+Fig. 5. The cell temperature is set to 75°C. The blue square
+in Fig.
+5a shows the laser wavelength at different LD cur-
+rents ranging from 73 mA to 150 mA. The wavelength of the
+output laser is near 780.255 nm and the wavelength fluctua-
+tion is about ± 0.5 pm. Furthermore, the Voigt laser maintains
+a single longitudinal mode when the LD current ranges from
+73 mA to 150 mA, as shown in the insert of Fig.
+5a. For
+comparison, the LD current range of Faraday laser without
+mode-hopping is 74 – 130 mA and the wavelength fluctuation
+is ± 2 pm13. Therefore, the Voigt laser has stronger immu-
+nity to current drift. In addition, our preliminary experimental
+
+1 - Diode
+2 - Aspheric lens
+3 - PBS
+4 - 87Rb cell
+5 - HWP
+6 - HR&PZT
+FP
+3
+3780.280
+B = 3500 Gs
+Current = 90 mA
+780.275
+Diode temperature = 22.08°C
+780.270
+Wavelength(nm)
+780.265
+X
+780.260
+X
+780.255
+X
+780.250
+XXXXXXX
+780.245
+X
+780.240
+780.235
+60
+70
+80
+90
+100
+110
+120
+Cell temperature (C)4
+results show that the Voigt laser is also immune to the diode
+temperature.
+FIG. 5. The characteristics of Voigt laser. a laser wavelength versus
+LD current and the result of Scanning Fabry–Pérot Interferometers.
+b Allan deviation of laser frequency for 8-hour (blue solid) and 48-
+hour (red solid) operation. The raw data is shown in the inset ( the
+blue dotted curve for 8-hour operation and the red dotted curve for
+48-hour operation), indicating the wavelength variation during oper-
+ation.
+With the LD current set as 150 mA, we investigate the fre-
+quency instability for a long-term free operation. We record
+the wavelength of the output laser using a high-precision op-
+tical wavelength meter (Bristol 671A), at a sampling rate of
+1 Hz. As shown in the insert of Fig.
+5b, the wavelength
+fluctuation is ± 0.1 pm and ± 0.5 pm for 8-hour and 48-hour
+operation, respectively. However, the wavelength fluctuation
+is ± 2 pm for 48-hour operation in Faraday laser13. We then
+analyze this data set using the standard approach of an Al-
+lan deviation, obtaining the long-term instability of the Voigt
+laser, as presented in Fig.
+5b. For a 8-hour free-running,
+the frequency instability of a Voigt laser can reach 5 × 10−9
+as the average time is up to 200 s, which is on the order of
+the Faraday laser using cavity locking technology12. When
+the operation time raises to 48 hours, the frequency instability
+can still reach 2 × 10−8, even if affected by the external vi-
+bration and the ambient temperature variation for over 10 de-
+grees. Consequently, the Voigt laser, with excellent frequency
+stability and immunity to the diode laser current and temper-
+ature, could contribute to the development of the compact op-
+tical frequency standard for the application of atomic clock,
+atomic gravimeter and outdoor precision measurement exper-
+iment.
+In Conclusion, this letter put forward a novel implementa-
+tion scheme for diode laser, named as a "Voigt laser", using
+the Voigt effect for frequency selection. The Voigt laser ob-
+tains a frequency instability of 5×10−9 and a strong immunity
+to the temperature and current of the laser diode. Moreover,
+the cell temperature characteristics of the Voigt laser in a large
+magnetic field are explored. Results show that the Voigt laser
+wavelength could be selected by adjusting the cell tempera-
+ture and the frequency variation range is 20 GHz. Therefore,
+the compact Voigt laser, corresponding to the atomic transi-
+tion line by adjusting the cell temperature, can serve for the
+atomic clocks, atomic gravimeters, and atomic gyroscopes,
+owing to to its high frequency stability and stable long-term
+operation. Importantly, the Voigt laser using different gain
+mediums and atoms, could be applied to active optical clock32
+and cold atom optical clock in the following study, promoting
+the development of optical frequency standards.
+ACKNOWLEDGMENTS
+The work was was funded by the National Natural Science
+Foundation of China (NSFC) (91436210), China Postdoctoral
+Science Foundation (BX2021020), Wenzhou Major Science
+and Technology Innovation Key Project (ZG2020046), and
+Wenzhou Key Scientific and Technological Innovation R&D
+Project (2019ZG0029).
+DATA AVAILABILITY STATEMENT
+The data that support the findings of this study are available
+from the corresponding author upon reasonable request.
+1E. C. Cook, P. J. Martin, T. L. Brown-Heft, J. C. Garman, and D. A. Steck,
+Review of Scientific Instruments 83, 043101 (2012).
+2K. Liu and M. G. Littman, Optics letters 6, 117 (1981).
+3R. Wyatt and W. Devlin, Electronics Letters 19, 110 (1983).
+4B. Bernacki, P. Hemmer, S. Smith, and S. Ezekiel, Optics letters 13, 725
+(1988).
+5A. L. Schawlow, Reviews of Modern Physics 54, 697 (1982).
+6C. Pearman, C. Adams, S. Cox, P. Griffin, D. Smith, and I. Hughes, Journal
+of Physics B: Atomic, Molecular and Optical Physics 35, 5141 (2002).
+7K. L. Corwin, Z.-T. Lu, C. F. Hand, R. J. Epstein, and C. E. Wieman,
+Applied Optics 37, 3295 (1998).
+8D. McCarron, S. King, and S. Cornish, Measurement science and technol-
+ogy 19, 105601 (2008).
+9X. Miao, L. Yin, W. Zhuang, B. Luo, A. Dang, J. Chen,
+and H. Guo,
+Review of Scientific Instruments 82, 086106 (2011).
+10Z. Tao, Y. Hong, B. Luo, J. Chen, and H. Guo, Optics Letters 40, 4348
+(2015).
+11Z. Tao, X. Zhang, D. Pan, M. Chen, C. Zhu, and J. Chen, Journal of Physics
+B: Atomic, Molecular and Optical Physics 49, 13LT01 (2016).
+12J. Keaveney, W. J. Hamlyn, C. S. Adams, and I. G. Hughes, Review of
+Scientific Instruments 87, 095111 (2016).
+13P. Chang, Y. Chen, H. Shang, X. Guan, H. Guo, J. Chen, and B. Luo,
+Applied Physics B 125, 1 (2019).
+
+780.256
+a
+780.254
+3.5
+3.0
+()
+2.5
+2.0
+1.5
+780.252
+1.0
+0.5
+0.0
+0.5
+0
+10
+20
+Detuning (GHz)
+780.250
+60
+70
+80
+90
+100.110
+120
+130　140
+150
+¥160
+Current (mA)
+b
+8 h
+(nm)
+780.25600
+48 h
+5525
+10-7
+Allan Deviation
+780.25450
+10.20...30
+40
+Time (h)
+780.2557
+wu)
+780.2
+length
+780
+Wavel
+780.
+54
+8
+2
+4
+6
+Time (h)
+101
+102
+103
+104
+100
+Average time (s)5
+14T. Shi, X. Guan, P. Chang, J. Miao, D. Pan, B. Luo, H. Guo, and J. Chen,
+IEEE Photonics Journal 12, 1 (2020).
+15Y. Ohman, Stockholms Observatoriums Annaler 19, 9 (1956).
+16I. Gerhardt, Optics letters 43, 5295 (2018).
+17L. Ling and G. Bi, Optics Letters 39, 3324 (2014).
+18M. A. Zentile, D. J. Whiting, J. Keaveney, C. S. Adams, and I. G. Hughes,
+Optics Letters 40, 2000 (2015).
+19J. Menders, K. Benson, S. Bloom, C. Liu, and E. Korevaar, Optics letters
+16, 846 (1991).
+20J. Menders, P. Searcy, K. Roff, and E. Korevaar, Optics letters 17, 1388
+(1992).
+21D. Pan, X. Xue, H. Shang, B. Luo, J. Chen, and H. Guo, Scientific Reports
+6, 1 (2016).
+22H. Shi, P. Chang, Z. Wang, Z. Liu, T. Shi, and J. Chen, IEEE Photonics
+Journal (2022).
+23Z. Tao, Y. Hong, B. Luo, J. Chen, and H. Guo, Optics Letters 40, 4348
+(2015).
+24H. Tang, H. Zhao, R. Wang, L. Li, Z. Yang, H. Wang, W. Yang, K. Han,
+and X. Xu, Optics Express 29, 38728 (2021).
+25P. Chang, H. Peng, S. Zhang, Z. Chen, B. Luo, J. Chen,
+and H. Guo,
+Scientific Reports 7, 1 (2017).
+26J. Menders, P. Searcy, K. Roff, and E. Korevaar, Optics letters 17, 1388
+(1992).
+27F. S. Ponciano-Ojeda, F. D. Logue, and I. G. Hughes, Journal of Physics
+B: Atomic, Molecular and Optical Physics 54, 015401 (2020).
+28J. Keaveney, F. S. Ponciano-Ojeda, S. M. Rieche, M. J. Raine, D. P. Hamp-
+shire,
+and I. G. Hughes, Journal of Physics B: Atomic, Molecular and
+Optical Physics 52, 055003 (2019).
+29K. Muroo, T. Matsunobe, Y. Shishido, Y. Tukubo,
+and M. Yamamoto,
+JOSA B 11, 409 (1994).
+30J. Miao, T. Shi, J. Zhang, and J. Chen, Physical Review Applied 18, 024034
+(2022).
+31J. Keaveney, C. S. Adams, and I. G. Hughes, Computer Physics Commu-
+nications 224, 311 (2018).
+32W. Zhuang and J. Chen, Optics letters 39, 6339 (2014).
+
diff --git a/j9AzT4oBgHgl3EQfpf2j/content/tmp_files/load_file.txt b/j9AzT4oBgHgl3EQfpf2j/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..8602f2ec2322d5ce2abc56053ba44c9de1ed4c38
--- /dev/null
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf,len=451
+page_content='A Voigt laser operating on 87Rb 780 nm transition  Zijie Liu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' a) Xiaolei Guan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' a) Xiaomin Qin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='1 Zhiyang Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='1 Hangbo Shi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='1 Jia Zhang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='1 Jianxiang Miao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='1  Tiantian Shi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' b) Anhong Dang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' b) and Jingbiao Chen1  1)State Key Laboratory of Advanced Optical Communication Systems and Networks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Institute of Quantum Electronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  School of Electronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Peking University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Beijing 100871,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' China  2)State Key Laboratory of Advanced Optical Communication Systems and Networks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' School of Electronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Peking University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  Beijing 100871,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' China  3)State Key Laboratory of Information Photonics and Optical Communications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Beijing University of Posts and Telecommunications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  Beijing 100876,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' China  (*Electronic mail: ahdang@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='cn)  (*Electronic mail: tts@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='cn)  (Dated: 6 January 2023)  We report the development of laser systems- a “Voigt laser” - using a Voigt anomalous dispersion optical filter as the  frequency-selective element, working at the wavelength of 780 nm of 87Rb-D2 resonance line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Compared with Faraday  anomalous dispersion optical filter, the Voigt anomalous dispersion optical filter can generate a stronger and more  uniform magnetic field with a compact size of magnet, and obtains a transmission spectrum with narrower linewidth  and more stable lineprofile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' In this case, the frequency stability of the Voigt laser reaches 5×10−9 at the averaging time  of 200 s, and the wavelength fluctuation of 8-hours free operation is ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='1 pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Besides, the Voigt laser has greater  immunity to diode current than the Faraday laser, with a wavelength fluctuation of ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='5 pm in the current range from  73 mA to 150 mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Finally, the Voigt laser frequency can be controlled by the cell temperature of the Voigt optical filter,  which is expected to achieve a frequency detuning of 20 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Consequently, the Voigt laser, whose frequency could  correspond to the atomic transition frequency by tuning the cell temperature, obtains good robustness to the current  and temperature fluctuation of laser diode, and could realize a compact optical standard for precise measurement once  stabilized by modulation transfer spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  Diode laser with narrow linewidth plays an essential role  in the study of atomic physics, and is used in an extremely  broad range of applications from laser cooling to thermal va-  por experiments1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' In the usual case, the frequency-selective  element of a traditional laser could be formed with a spatial  grating2,3 or a Fabry–Pérot (FP) etalon4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' The frequency of  these lasers is sensitive to their diode current and temperature,  requiring an extremely high precision temperature and cur-  rent control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Besides, the laser frequency needs to be finely  tuned to the atomic transition line, to achieve frequency sta-  bilization using saturated absorption spectroscopy5, polariza-  tion spectroscopy6, modulation transfer spectrum7,8, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' In  this case, there will be a risk of mode hopping, resulting in the  laser frequency far away from the locking point, which is not  conducive to long-term continuous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  Faraday laser comprises a Faraday anomalous dispersion  optical filter (FADOF) as a frequency-selective element, could  solve the above problem9–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  FADOF has attracted much  attention since its birth15, owing to its narrow bandwidth  and high signal-to-noise ratio (SNR)16–21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  Since 2011,  FADOF is applied as a frequency-selective element, reveal-  ing a stable "Faraday laser" immune to the diode current and  temperature9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  Importantly, the wavelength of the Faraday  laser is limited to the narrow-band transmission window of  FADOF, which is around the atomic transition line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' There-  a)These authors contributed to the work eqully and should be regarded as co-  first authors  b)Authors to whom correspondence should be addressed:tts@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='cn and  ahdang@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='cn  fore, the Faraday laser could easily achieve the frequency sta-  bilization once it is turned on, and could stably operate for a  long term without artificial adjustment22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' In recent years, the  Faraday laser has been formally proposed to different struc-  tures and atoms, achieving better frequency stability and anti-  interference ability13,14,23–25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  Nevertheless, the axial mag-  netic field in the Faraday laser is generally generated by two  magnets aside the cell along the optical axis, which greatly  increases the axial length of FADOF and limits the mini-  mum cavity length of Faraday laser24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' In another method, the  strength and uniformity of the magnetic field generated by the  magnet around the cell are relatively small, which limits the  further improvement of the laser stability13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  Unlike the Faraday effect, the Voigt effect rotates the po-  larization direction of the incident laser by applying a radial  magnetic field26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Therefore, the Voigt anomalous dispersion  optical filter (VADOF) could construct a stronger and more  homogeneous magnetic field with less volume than FADOF,  by placing two magnets close together in the radial direction  of the cell27–29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Then, a stable transmission spectrum with a  narrower linewidth is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' In this case, a compact "Voigt  laser" could be constructed, using a VADOF for frequency-  selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Like the Faraday laser, the Voigt laser could realize  compact optical frequency standard, using modulation trans-  fer spectroscopy (MTS) for frequency stabilization22,30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  In this work, we propose and prove a Voigt laser, which  uses 87Rb VADOF as frequency-selective element and semi-  conductor diode laser as gain medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' This Voigt laser op-  erated freely for many times during half a year, and its wave-  length fluctuation is ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='5 pm with an instability better than  2 × 10−8 for 48 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Compared with the Faraday laser, the  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='01614v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='optics]  4 Jan 2023  2  Voigt laser has a better frequency stability in a smaller vol-  ume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Besides, the wavelength fluctuation of the Voigt laser  is ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='5 pm in the diode current range from 73 mA to 150  mA, presenting the Voigt laser’s immunity towards the pa-  rameters of the diode laser .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Meanwhile, we investigate the  influence of the cell temperature on the Voigt laser, revealing  a tunable laser wavelength selected by the cell temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  Therefore, the frequency of this Voigt laser can corresponds  to the atomic transition frequency by adjusting the cell tem-  perature, and then can be stabilized using MTS technology to  realize a compact optical frequency standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Furthermore,  the Voigt laser could use different gain mediums and atoms to  improve its performance, which paves the way for basic sci-  entific research (such as atomic clock, atomic gravimeter, and  atomic interferometer, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=') and industrial electronic equip-  ment (precision optical measurement, spectral inspection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  Experimental setup of the VADOF transmission spec-  trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' The green dash refers to the saturated absorption spectroscopy  (SAS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' HWP, half-wave plate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' PBS, polarization beam splitter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' PD,  photon detector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' The probe laser is split into two laser beams (laser  a and laser b) by a HWP and a PBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  VADOF comprises a 87Rb atomic vapor cell with length of  L = 30 mm, a set of permanent magnets, a temperature control  system, and two polarized beam splitters (PBSs), as presented  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' The vacuum cell and magnets are stuck in a desired  Teflon thermal insulation structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' The anti-reflection coated  cell has a transmission of 95% for λ = 780 nm laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' The tem-  perature of the 87Rb cell is controlled by an electric heating  element that allows to maintain the cell temperature within  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='1°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' The radial magnetic field range from 200 Gs to 3500  Gs, which reach a maximum with two 15 mm-thick NdFeB  permanent magnets aside the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' For a better Voigt effect,  there is a 45° angle between the magnetic field direction and  the laser polarization direction28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' The two PBSs are placed on  either side of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' The polarization direction of the PBS  behind the cell can be rotated, so that the two PBSs could be  orthogonal or parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  We use a homemade 780 nm interference-filter external  cavity diode laser (IF-ECDL) to probe the transmission spec-  trum of VADOF, of which the laser beam diameter is 2 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  The probe laser is split into two laser beams (laser a and laser  b) by a half wavelength plate (HWP) and a PBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' We probe  the saturated absorption spectroscopy (SAS) of 87Rb as a fre-  quency reference using laser a, as the green dash in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  1  shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Laser b goes through a HWP to control the laser inten-  sity before probing the transmission spectrum of VADOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' The  above spectrums are simultaneously presented in an oscillo-  scope, which connects with two photodetectors (PD, Thorlabs  PDA8A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' The saturated absorption spectroscopy of 87Rb-D2-line and  the transmission spectrum of VADOF in different temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='The  diagram at the top shows F = 1 and F = 2 transition of 87Rb-D2-line  in SAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' a, b ,c , d present the transmission spectrum at the cell tem-  perature of 110°C, 95°C, 85°C, and 75°C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Peak 1, 2, 3,  4, and 5 are typical transmission peaks in the transmission spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  The probe laser frequency detuning ranges from -30 GHz to 25 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  Before experimentally probing the spectrum of VADOF, we  analyze its major influencing factors using a simulation tool31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  The results show that we can reveal a spectrum with high  transmission and narrow linewidth, when radial magnetic field  B = 3500 Gs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Then, we investigate the transmission spectrum  of VADOF in different temperatures at B = 3500 Gs, as shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' The diagram at the top shows SAS of 87Rb-D2-line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  The frequency interval between F = 1 and F = 2 transition is  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='834 GHz, and we set the frequency at the peak of F = 2  transition as the detuning origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  The probe laser frequency tuning ranges from -30 GHz to  25 GHz, where we can almost observe the whole transmission  spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Limited by the tuning range, The spectrum above  25 GHz is missed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' However, the transmission spectrum above  25 GHz is suggested to monotonically decrease to 0 (the same  1 -HWP  2 - PBS  3 - PD  4 - 87Rb cell  5 - VADOF  3  2Saturated spectrum  Absorb(a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=')  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='4  F=1  F=2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='2  100  2  (a)  110  80  60  40  20  100  95°C  80  (b)  60  40  20  100  85°  80  (c)  60  e  40  M MJ  20  100  75  80  (d)  2  60  40  20  0  30  25  20  15  10  5  10  15  20  25  5  0  30  Freguency detuning(GHz3  trend as the left side of the transmission spectrum), accord-  ing to the theoretical prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Compared with conventional  FADOF, VADOF requires higher temperatures to reveal high  transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Therefore, the Voigt laser is hard to produce  once the cell temperature is lower than 60°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' When the cell  temperature is in the range of 60 - 80°C, the peak 1 in the  transmission spectrum obtains the highest transmission, see-  ing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' In this case, the transmission peak 1, 2, and 3 will  rise with the cell temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Peaks 2 and 3 become the high-  est transmission peak at 85°C and 95°C, respectively, as pre-  sented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' 2c and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' VADOF reveals a ‘wing’ spec-  tral profile and peak 4 stands out as the highest, as the temper-  ature rises to 110°C in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Then, the "wing" peak 5 will  become the highest and the spectral profile won’t change with  the rising temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' However, the linewidth of peak 5 is  relatively wide and its center frequency is sensitive to the cell  temperature, which is not suitable for Voigt laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Therefore,  the highest transmission peak gradually jumps from peak 1 to  peak 4 when the temperature increases from 60°C to 110°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  Based on this, we predict that the output laser wavelength of  the Voigt laser also undergoes corresponding jumps when the  temperature increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' The experiment setup of the Voigt laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' HWP, half-wave  plate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' PBS, polarization beam splitter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' HR, high-reflection mirror;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  PZT, piezoelectric ceramic tube;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' FP, Scanning Fabry–Pérot Interfer-  ometers (Thorlabs, SA210-5B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' λ, optical wavelength meter (Bristol  671A)  The experimental setup of the Voigt laser is schematically  shown in the green dash of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' The Voigt laser with  the length of optical cavity of 20 cm is composed of a 87Rb  VADOF, an anti-reflection coated laser diode, aspheric lens, a  high-reflection mirror (HR) and a piezoelectric ceramic tube  (PZT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' After collimated by an aspheric lens, the beam diam-  eter of the ARLD’s emitted light is enlarged to 2 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Then,  the collimated light is filtered by the Voigt optical filter and  fed back into the ARLD by HR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' The reflectivity of HR is  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='9% at the wavelength of 780 nm, and is fixed on the PZT  for finely detuning the cavity length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' The laser beam outputs  from the second PBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' A HWP and a PBS are comprised to  split the output laser, for scanning FP spectrum and sensing  laser wavelength, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' The temperature of 87Rb cell  could range from 20°C to 200°C and the radial magnetic field  is between 200 Gs and 3500 Gs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  We firstly investigate the cell temperature’s effect on the  Voigt laser, presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' The radial magnetic field  is selected as 3500 Gs for a high transmission peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' The LD  current and temperature are 90 mA and 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='08°C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  The output laser wavelength jumps with the increasing cell  temperature, as we predicted in the discussion of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Once  the temperature is stabilized at an exact value, laser with a sta-  ble wavelength will be emitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' When the temperature varies  between 60 - 80°C, the output laser wavelength is maintained  at 780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='256 nm (near 60 – 75°C) or 780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='254 nm (near 80°C),  corresponding to the two longitudinal modes of peak 1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Continuing to increase the cell temperature, the laser wave-  length will be detected as 780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='241 nm (near 85°C) and then  jump to 780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='265 nm (near 90-100°C), referring to peak 2 and  peak 3 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' When the temperature is near  105°C, the transmission of peaks in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' 2 are close to each  other and the output laser wavelength fluctuates widely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' In  this case, the laser wavelength will level out at 780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='276 nm,  corresponding to peak 4 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' 2, if the cell temperature rises  to 110°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Then, the output laser will be red-shifted and the  laser wavelength will rise with the increasing cell temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  Subsequently, the Voigt laser with an EOM could achieve a  wide frequency tuning range of 20 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' The wavelength of the Voigt laser at different cell tempera-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  Based on the above investigation, we further study the char-  acteristics of the Voigt laser at different LD currents, seeing  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' The cell temperature is set to 75°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' The blue square  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  5a shows the laser wavelength at different LD cur-  rents ranging from 73 mA to 150 mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' The wavelength of the  output laser is near 780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='255 nm and the wavelength fluctua-  tion is about ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='5 pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Furthermore, the Voigt laser maintains  a single longitudinal mode when the LD current ranges from  73 mA to 150 mA, as shown in the insert of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' For  comparison, the LD current range of Faraday laser without  mode-hopping is 74 – 130 mA and the wavelength fluctuation  is ± 2 pm13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Therefore, the Voigt laser has stronger immu-  nity to current drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' In addition, our preliminary experimental  1 - Diode  2 - Aspheric lens  3 - PBS  4 - 87Rb cell  5 - HWP  6 - HR&PZT  FP  3  3780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='280  B = 3500 Gs  Current = 90 mA  780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='275  Diode temperature = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='08°C  780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='270  Wavelength(nm)  780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='265  X  780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='260  X  780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='255  X  780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='250  XXXXXXX  780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='245  X  780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='240  780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='235  60  70  80  90  100  110  120  Cell temperature (C)4  results show that the Voigt laser is also immune to the diode  temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' The characteristics of Voigt laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' a laser wavelength versus  LD current and the result of Scanning Fabry–Pérot Interferometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  b Allan deviation of laser frequency for 8-hour (blue solid) and 48-  hour (red solid) operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' The raw data is shown in the inset ( the  blue dotted curve for 8-hour operation and the red dotted curve for  48-hour operation), indicating the wavelength variation during oper-  ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  With the LD current set as 150 mA, we investigate the fre-  quency instability for a long-term free operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' We record  the wavelength of the output laser using a high-precision op-  tical wavelength meter (Bristol 671A), at a sampling rate of  1 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' As shown in the insert of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  5b, the wavelength  fluctuation is ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='1 pm and ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='5 pm for 8-hour and 48-hour  operation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' However, the wavelength fluctuation  is ± 2 pm for 48-hour operation in Faraday laser13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' We then  analyze this data set using the standard approach of an Al-  lan deviation, obtaining the long-term instability of the Voigt  laser, as presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' For a 8-hour free-running,  the frequency instability of a Voigt laser can reach 5 × 10−9  as the average time is up to 200 s, which is on the order of  the Faraday laser using cavity locking technology12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' When  the operation time raises to 48 hours, the frequency instability  can still reach 2 × 10−8, even if affected by the external vi-  bration and the ambient temperature variation for over 10 de-  grees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Consequently, the Voigt laser, with excellent frequency  stability and immunity to the diode laser current and temper-  ature, could contribute to the development of the compact op-  tical frequency standard for the application of atomic clock,  atomic gravimeter and outdoor precision measurement exper-  iment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  In Conclusion, this letter put forward a novel implementa-  tion scheme for diode laser, named as a "Voigt laser", using  the Voigt effect for frequency selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' The Voigt laser ob-  tains a frequency instability of 5×10−9 and a strong immunity  to the temperature and current of the laser diode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Moreover,  the cell temperature characteristics of the Voigt laser in a large  magnetic field are explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Results show that the Voigt laser  wavelength could be selected by adjusting the cell tempera-  ture and the frequency variation range is 20 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Therefore,  the compact Voigt laser, corresponding to the atomic transi-  tion line by adjusting the cell temperature, can serve for the  atomic clocks, atomic gravimeters, and atomic gyroscopes,  owing to to its high frequency stability and stable long-term  operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Importantly, the Voigt laser using different gain  mediums and atoms, could be applied to active optical clock32  and cold atom optical clock in the following study, promoting  the development of optical frequency standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The work was was funded by the National Natural Science  Foundation of China (NSFC) (91436210), China Postdoctoral  Science Foundation (BX2021020), Wenzhou Major Science  and Technology Innovation Key Project (ZG2020046), and  Wenzhou Key Scientific and Technological Innovation R&D  Project (2019ZG0029).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  DATA AVAILABILITY STATEMENT  The data that support the findings of this study are available  from the corresponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  1E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Cook, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Martin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Brown-Heft, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Garman, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Steck,  Review of Scientific Instruments 83, 043101 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  2K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Liu and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Littman, Optics letters 6, 117 (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
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+page_content=' Devlin, Electronics Letters 19, 110 (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  4B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Bernacki, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Hemmer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Smith, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Ezekiel, Optics letters 13, 725  (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
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+page_content=' Smith, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Hughes, Journal  of Physics B: Atomic, Molecular and Optical Physics 35, 5141 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
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+page_content=' Epstein, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
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+page_content=' Wieman,  Applied Optics 37, 3295 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
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+page_content=' Adams, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Hughes,  Optics Letters 40, 2000 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  19J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Menders, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Benson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Bloom, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Liu, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Korevaar, Optics letters  16, 846 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  20J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Menders, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Searcy, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Roff, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Korevaar, Optics letters 17, 1388  (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  21D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Pan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Xue, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Shang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Luo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Chen, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Guo, Scientific Reports  6, 1 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  22H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Shi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Chang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Liu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Shi, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Chen, IEEE Photonics  Journal (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  23Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Tao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Hong, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Luo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Chen, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Guo, Optics Letters 40, 4348  (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  24H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Tang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Zhao, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Yang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Yang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Han,  and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Xu, Optics Express 29, 38728 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  25P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Chang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Peng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Chen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Luo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Chen,  and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Guo,  Scientific Reports 7, 1 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  26J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Menders, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Searcy, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Roff, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Korevaar, Optics letters 17, 1388  (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  27F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Ponciano-Ojeda, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Logue, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Hughes, Journal of Physics  B: Atomic, Molecular and Optical Physics 54, 015401 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  28J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Keaveney, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Ponciano-Ojeda, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Rieche, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Raine, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Hamp-  shire,  and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Hughes, Journal of Physics B: Atomic, Molecular and  Optical Physics 52, 055003 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  29K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Muroo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Matsunobe, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Shishido, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Tukubo,  and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Yamamoto,  JOSA B 11, 409 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  30J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Miao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Zhang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Chen, Physical Review Applied 18, 024034  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  31J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Keaveney, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Adams, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Hughes, Computer Physics Commu-  nications 224, 311 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content='  32W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Zhuang and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
+page_content=' Chen, Optics letters 39, 6339 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfpf2j/content/2301.01614v1.pdf'}
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+THE INFLUENCES OF COLOR AND SHAPE FEATURES
+IN VISUAL CONTRASTIVE LEARNING
+Xiaoqi Zhuang
+xiaoqizhuang@outlook.com
+ABSTRACT
+In the field of visual representation learning, performance of contrastive learning has been catching
+up with the supervised method which is commonly a classification convolutional neural network.
+However, most of the research work focuses on improving the accuracy of downstream tasks such as
+image classification and object detection. For visual contrastive learning, the influences of individual
+image features (e.g., color and shape) to model performance remain ambiguous.
+This paper investigates such influences by designing various ablation experiments, the results of which
+are evaluated by specifically designed metrics. While these metrics are not invented by us, we first use
+them in the field of representation evaluation. Specifically, we assess the contribution of two primary
+image features (i.e., color and shape) in a quantitative way. Experimental results show that compared
+with supervised representations, contrastive representations tend to cluster with objects of similar
+color in the representation space, and contain less shape information than supervised representations.
+Finally, we discuss that the current data augmentation is responsible for these results. We believe that
+exploring an unsupervised augmentation method that can strongly change the shape information of
+objects is an effective research direction that can improve unsupervised visual representations.
+Keywords Contrastive Learning · Representation Learning
+1
+Introduction
+Unsupervised visual representation learning, specifically contrastive learning, has seen rapid development in recent
+years [1, 2, 3]. These methods aim to train a deep neural network encoder as a feature extractor by minimizing the
+distance of similar objects and maximizing the distance of dissimilar ones. Due to the limit of unsupervised learning,
+the ground truth of each image is only itself. Therefore, the mainstream methods usually apply different types of pretext
+tasks. [4] proposed a pretext task called instance discrimination, which applies different image transformations on each
+image to get a pair of positive samples and define all other images as negative samples. Previous works proposed 3
+key components to make such training framework succeed: deep neural networks based image transformation [2], a
+large size of negative samples, and feature consistency[1, 2]. [2] showed that adding color distortion and cropping
+into the transformation pipeline significantly improve accuracy, which is because simple image augmentation such as
+rotation and flipping could not largely change the distribution. Large negative samples, which is because the positive
+samples are only the two augmentations from a single image, the contrasting learning loss function [5] always has to
+use thousands of negative samples to project images into the representation space. Feature consistency, [1, 2] proposed
+that the representation features should be compared into the same representation space to avoid the biased contrastive
+loss from the updated encoders after backpropagation. These trivial works are the basis for contrastive learning, which
+achieved around 74% accuracy in Image-Net datasets.
+Representation learning has always used improving the accuracy of downstream tasks as an indicator to improve the
+backbone of the model. However, we believe that the accuracy of a single downstream task such as image classification
+is not sufficient for the evaluation of representation quality. Therefore, We developed metrics for representation
+models to explicitly quantify visual representations by evaluating their color and shape information. At the same time,
+supervised methods have always been considered the upper limit of unsupervised methods. We also want to find out the
+gap between them, thus to clarify the improvement ideas of unsupervised methods. Therefore, we used ResNet-18 [6]
+to train three unsupervised models and one supervised model for experiments.
+arXiv:2301.12459v1  [cs.CV]  29 Jan 2023
+
+In this paper, we did these experiments. Firstly, we try to find the gap between supervised and unsupervised models.
+We get wrong predictions from unsupervised models with image classification tasks. We found that nearly half of
+them were the same predictions for three unsupervised models. These images can be considered common faults of
+unsupervised methods, and the parts of these images where the supervised model can make correct predictions are
+considered as gaps. Secondly, in terms of color information, we found that these images and their false predictions
+are strongly correlated with color. We use the earth mover’s distance [7] metric to demonstrate that unsupervised
+representations are color biased within the representation space. For shape information, we generated a CIFAR-10 [8]
+silhouette dataset by a pretrained segmentation model. We verified that the accuracy of the unsupervised model on this
+dataset is also lower than that of the supervised model. We recomputed the accuracy of the image classification task
+with color and shape-distorted images, and the results show that unsupervised models are more robust than supervised
+models. Finally, we conclude that there is no distribution change method strongly related to the image shape among the
+data augmentation methods, which is the main reason for the gap with the supervised model.
+2
+Related Work
+Unsupervised contrastive visual learning
+Due to the limitation of an unsupervised task, the ground truth of an image is just itself, leading that it being hard to
+design the learning progress. [4] proposed a pretext task called Instance Discrimination to annotate images by taking
+them as positive and negative samples. The core concept is comparing the similarity between two images that are
+transformed by a single image. Therefore, we can train the model by reducing the distance of generated image pairs and
+extending the distance of other images. Such a learning framework requires two important keys [2]: large negative
+samples, and feature consistency. [1, 9] proposed a first-in-first-out queue to store large negative samples, and keep
+feature consistency by updating a momentum encoder slowly. [2] achieved both keys by training the model within a
+large batch size, which is straightforward but hard to reproduce . [2] proved that adding color jitter as well as cropping
+into image transformation will largely improve the model. [2] added a linear layer called "Projection Layer" after the
+backbone model, which can help raise the accuracy by nearly 7%. Such modifications and the training details in [4]
+have been the fixed set of contrastive learning. [10, 3] proposed a contrastive learning framework without negative
+samples. They prove that only comparing the similarity of two positive images can achieve nearly the same accuracy as
+the state-of-the-art models without large batch size as well as momentum encoders.
+Improvements on unsupervised representation learning
+A lot of work has been done to improve the basic methods mentioned above. [11, 12, 13] want to get high-quality
+positive and negative sample pairs for contrastive learning. They selected positive and negative samples instead of
+random selection by using similarity measures during training. [14] used the heat map of feature representations to
+obtain an approximate bounding box to execute an accurate "CROP" operation. [15, 16] suggests that inappropriate
+data augmentations will harm image information. They use enumeration to select the right data augmentations for
+each image for contrastive training. [17, 18, 19] argue that instance discrimination can only provide instance-level
+representation information. Their methods can provide high-level semantic information for unsupervised learning by
+adding clustering methods.
+Experiments of visual representation
+Researchers have done many experiments on representation bias. [20] used pure texture and shape image datasets to
+illustrate that the CNN model trained on Image-Net is texture biased. [21] explores the effects of contrastive learning in
+terms of data volume, data domain, data quality, and task granularity. [22] designed a specific representation learning
+model for shape, color, and texture, which help them analysis the specific contribution of each attributes when the
+model is referenced.
+In this paper, our work focuses on analyzing color and shape information between representations learned by contrastive
+and supervised methods. We then look for potential gaps by comparing them with supervised representations. After
+that, we believe that such gaps will instruct future work to improve unsupervised models.
+2
+
+MoCo
+SimCLR
+SimSiam
+Supervised
+200 eps
+81.86%
+83.96%
+80.41%
+95.18%
+800 eps
+88.14%
+87.78%
+88.43%
+improve
+6.28%
+3.82%
+8.02%
+Table 1: Accuracy on different contrastive learning backbone. All contrastive methods do not have major differences on
+accuracy. It also shows that the accuracy of the model increases slowly after 200 epochs, which increases by less than
+10% after training 600 epochs.
+3
+Methods
+3.1
+Dataset
+We perform experiments on a popular small image dataset: CIFAR-10 [8], which consists of 60000 32x32 color images
+in 10 classes, with 6000 images per class. We know that the experiment results from such a small dataset may be biased.
+However, all these methods [1, 2, 3] show the same problems in this dataset. We believe these results are valuable to
+discuss.
+3.2
+Training details
+Unsupervised backbone
+We trained three mainstream unsupervised models based on the ResNet-18 [6] backbone, which are: MoCo V2 [9],
+SimCLR V2 [9], and SimSiam [3]. We use the linear evaluation method as the metric, that is, training these unsupervised
+methods on the trainset in 800 epochs firstly, extracting and freezing the backbone encoder, adding an extra MLP to
+train a classifier on the trainset in 200 epochs, and then evaluating the accuracy of the classifier on the test set. Due to
+the low resolution of images in CIFAR-10, we modified the Standard ResNet-18 backbone. We delete "Gaussian blur"
+in the transformation compose, replace the 7 × 7 convolution kernel as the 3 × 3 kernel and delete the "stride" and
+"MaxPool2d". The hyperparameters of the temperature τ, learning rate and scheduler are the same as [4].
+Supervised backbone
+We also trained a supervised model based on ResNet-18 as the object of comparison. The model structure is largely the
+same as the above-unsupervised models. We add a classification layer and used labels to train the model.
+Linear Evaluation Accuracy
+The evaluation results are shown in Table 1. In terms of linear evaluation, three contrastive models achieved around
+80% accuracy after 200 epochs and 88% after 800 epochs. By contrast, the accuracy of the supervised ResNet-18 is
+95.18% within only 200 epochs.
+4
+Experiments
+We now describe our experiments in the following order. Firstly, we found the gap between contrastive models and
+the supervised model, and then proposed our assumptions based on such a gap. Secondly, we implemented specific
+experiments to verify these assumptions. Finally, we discussed the possible reasons for such issues on contrastive
+representation.
+4.1
+Gap Definition
+Table 1 shows that the accuracy difference between the three backbones in the CIFAR-10 test set is less than 1%, which
+arouses our interest in whether they do the same predictions. The results show that the intersection between false
+predictions has 581 images (nearly 49%), the intersection between false predictions with the same predictions has
+384 images (nearly 32%). This indicates that although these backbone models are trained in different methods, the
+representation features are similar in some areas as long as they are trained by instance discrimination, which also
+means that they do have the same problems.
+3
+
+Figure 1: The same wrong predictions by contrastive methods. These images are easy to be identified for both human
+beings and the supervised model. However, all models based on contrastive learning make wrong predictions, which
+these predictions, the text over the image, are probably made by the color information. For example, the plane on the
+top-left corner is predicted as a horse, where the only same semantic of these two objects is the color: brown.
+MoCo
+SimCLR
+SimSiam
+Supervised
+56.51%
+62.00%
+60.16%
+77.89%
+Table 2: kNN-4 Accuracy on GAP-265 between MoCo, SimCLR, SimSiam and supervised ResNet-18
+However, the supervised model can make 265 right predictions in these 384 images. Due to the low resolution and
+wrong labeling reasons in the CIFAR-10 dataset, we can take the supervised model as the upper limit of the classification
+performance. Therefore, such 265 images can be regarded as the knowledge gap between the supervised model and
+unsupervised models. We will call these images GAP-265 in the following content.
+Figure 1 is shown as a subset of these images. These images are easy to be identified for both human beings and
+supervised model. However, all models based on contrastive learning make wrong predictions, and these predictions
+are likely to be made based on their color information. For example, the plane on the top-left corner is predicted as a
+horse, where the only same semantic of these two objects is the color: brown.
+Such phenomenon inspires us to propose two assumptions:
+• Contrastive representation space tends to make images with similar colors closer.
+• Contrastive features contain less shape information than supervised features.
+4.2
+Feature Influence
+We use Linear Evaluation to compare different models, and thus we should prove that our hypothesis should not be
+affected by the extra linear layers. In other words, features embedded by the contrastive encoder are the reason for
+the wrong predictions in GAP-265. Therefore, we perform k-nearest neighbor (kNN) classification on GAP-265 to
+eliminate such concerns. Table 2 shows that kNN-4 accuracy of models by unsupervised methods are also largely lower
+than the one by the supervised method. Figure 2 shows an example of different nearest neighbors for the brown plane
+between MoCo and the supervised model. The nearest neighbors in MoCo are other categories, while the only similar
+field is color. By contrast, all nearest neighbors in the supervised model are the same categories but different colors.
+4.3
+Color Biased Detection
+To prove our first assumption, contrastive representation space tends to make images with similar colors closer, we
+use Earth Mover’s Distance(EMD) [7], which is also called Wasserstein Distance, as a metric to evaluate the color
+similarity between the query image and its top-4 nearest neighbors. Algorithm 1 illustrates the pseudo-code that how
+we use EMD to evaluate image similarity on color dimension. First, we convert the BGR image into its HSV form that
+conforms to human vision. Second, we calculate the color histogram of the image in the hue dimension, and convert it
+4
+
+horse
+ship
+cat
+dog
+dog
+-
+deer
+cat
+bird
+airplane
+birdFigure 2: Two examples of the false prediction and their top-4 nearest neighbours. The target image are not in the right
+regions, and they are in the region which their color are similar.
+Model
+MoCo
+SimCLR
+SimSiam
+Supervised
+EMD
+26.97
+28.58
+28.85
+33.20
+Table 3: The Earth Mover’s Distance in GAP-265 between contrasitve and supervised models.
+into the signature form[7] which includes index information. Finally, We calculate EMD between the object image and
+the signatures of its nearest four neighbors to compare whether the image is close to the image of similar color in the
+representation space. More specifically, the smaller the EMD value between the query and its neighbours, the more
+similar they are in terms of color. Table 3 shows the mean value of EMD in GAP-265 between all backbones. It is
+clear that the metrics of backbones learned by contrastive methods are near and smaller than the metric of the model by
+the supervised method. Figure 2 is also a clear example that the same query will have different nearest neighbours in
+different models, while contrastive features are clustered by color and supervised features are clustered by their labels.
+Algorithm 1 Earth Mover’s Distance on Image Color Similarity
+Input:a test image ti , model M, the overall test set test, neighbor number k
+Output:the mean EMD between ti and its nearest K neighbors
+1: function image2signature(image)
+2:
+bgr = CV2.imread(image)
+3:
+hsv = CV2.cvtColor(bgr)
+▷ convert BGR to HSV
+4:
+histHUE = CV2.calcHist(hsv, channel=0)
+▷ only calculate histogram on HUE
+5:
+sign = hist2signature(histHUE)
+▷ convert histogram to signature
+6:
+return sign
+7: end function
+8: disti = []
+▷ create an empty list to store EMD values
+9: fi, fall = M(ti), M(test)
+▷ extract features by model
+10: neighborsi = nearestNeighbors(fi, fall, k)
+▷ get the nearest K neighbors of ti
+11: signi = image2signature(ti)
+12: for n ∈ neighborsi do
+13:
+signn = image2signature(n)
+14:
+emdn = cv2.EMD(signi, signn, L2)
+15:
+dist.append(emdn)
+16: end for
+Output:mean(disti)
+4.4
+Shape Biased Detection
+To prove our second assumption, contrastive features contain less shape information than supervised features, we
+created a silhouette dataset based on CIFAR-10. As CIFAR-10 does not have a public silhouette dataset, We use a
+pretrained semantic segmentation model U2-Net [23] to generate silhouette images. Due to the low resolution and
+no fine tuning, some generated images are of low quality. We manually selected 261 high-quality silhouette images
+5
+
+d=0.000
+d=0.233
+d=0.334
+d=0.345
+d=0.345
+supervised
+d=0.000
+d=0.757
+d=0.765
+d=0.790
+d=0.806
+MoCoFigure 3: Examples of the manually selected silhouette images from CIFAR-10.
+Model
+MoCo
+SimCLR
+SimSiam
+Supervised
+Acc
+55.17%
+46.7%
+56.7%
+60.91%
+Table 4: Accuracy between different models on silhouette CIFAR-10 dataset.
+to ensure that human beings are capable of making the right predictions only by their contour. Figure 3 shows some
+examples. Table 4 shows that the supervised model achieves the highest accuracy among all models. Such accuracy
+results to some extent could show that supervised features have more shape information than contrastive features.
+4.5
+Distorted Image Evaluation
+The results of the above two experiments raise questions about the robustness of contrastive representations. Therefore,
+we did different distortions on the test set and reevaluate the accuracy. More specifically, we apply "ColorJitter",
+"HorizontalFilp", and "RandomRotation" respectively to the testset. Table 5 shows that all contrastive models have a
+lower decrease of accuracy in "ColorJitter" and "GrayScale" than supervised models, and there is no obvious difference
+in "Flip". The results indicate that contrastive representation is more robust than supervised representation for the
+knowledge that it has gained.
+5
+Discussion
+As noted in Section 1, researchers have been working hard to improve unsupervised learning, and the quantitative
+improvement is usually compared with the accuracy of supervised learning. To discover fine-grained gaps between
+unsupervised and supervised learning, we proposed GAP-265, a subset of CIFAR-10 which unsupervised models make
+the same wrong predictions while the supervised model makes the right predictions. Most of these wrong predictions
+are due to the color of the object. Section 4.3 illustrate that contrastive representation space is more dependent on color
+than supervised representation space. We also constructed a silhouette dataset based on CIFAR-10 which contains
+261 high-quality images. Section 4.4 illustrate that supervised model are more shape-aware than unsupervised models.
+Section 4.5 illustrate that unsupervised models are more robust than the supervised model.
+In our opinion, the reason why such phenomena appear is because of the data augmentation. Unsupervised models
+inevitably have to apply stronger augmentations such as "ColorJitter" and "GrayScale" than supervised models due to
+the annotation limitation. As a result, for these distorted images, unsupervised models are more robust than supervised
+models because that’s how they were trained. However, this also leads to the problem of possible color dependence
+of the representation spaces trained by contrastive learning. By contrast, current augmentation methods do not have
+such a strong transformation of image shape as "ColorJitter" transforms image color. The more difficult thing is that it
+is unsupervised learning. We can change the color distribution of the image by adjusting the contrast and brightness.
+However, the shape distribution of the image is difficult to make changes on a 2D image without relying on labels. For
+model
+baseline
+color
+flip
+gray
+MoCo
+88.14%
+-1.22%
++0.16%
+-3.35%
+SimCLR
+87.78%
+-1.15%
+-0.42%
+-4.40%
+SimSiam
+88.43%
+-0.43%
++0.10%
+-3.63%
+Supervised
+95.18%
+-2.2%
++0.22%
+-5.39%
+Table 5: Accuracy change amplitude of each model for distorted images compared to the original baseline. All
+contrastive models have lower decrease of accuracy in "ColorJitter" and "GrayScale" than supervised models, and there
+is no obvious difference in "Flip".
+6
+
+example, it is easy to change the color of the object in one image, but it is obviously impossible to obtain images from
+all angles of the object through such one image.
+6
+Conclusion
+In summary, we provided evidence that representation space based on contrastive learning today relies on color
+information more and contains less shape information compared with space based on supervised learning. In addition,
+we also indicate that contrastive space are more robust than supervised space on distortion attacks. We discussed that
+data augmentation is the cause of these problems. We hope that these experiments and conclusions can make researchers
+focus on how to add shape knowledge in the future unsupervised learning and add color as well as shape metrics to the
+evaluation metrics of unsupervised tasks.
+References
+[1] Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick. Momentum contrast for unsupervised visual
+representation learning. arXiv preprint arXiv:1911.05722, 2019.
+[2] Chen Ting, Kornblith Simon, Norouzi Mohammad, and Hinton Geoffrey. A simple framework for contrastive
+learning of visual representations. arXiv preprint arXiv:2002.05709, 2020.
+[3] Chen Xinlei and He Kaiming. Exploring simple siamese representation learning. In Proceedings of the IEEE/CVF
+Conference on Computer Vision and Pattern Recognition (CVPR), pages 15750–15758, June 2021.
+[4] Wu Zhirong, Xiong Yuanjun, Stella X Yu, and Lin Dahua. Unsupervised feature learning via non-parametric
+instance discrimination. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition,
+2018.
+[5] Aaron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predictive coding.
+arXiv preprint arXiv:1807.03748, 2018.
+[6] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. arXiv
+preprint arXiv:1512.03385, 2015.
+[7] Rubner Y., Tomasi C., and Guibas L. J. The earth mover’s distance as a metric for image retrieval. International
+Journal of Computer Vision, 40, 2000.
+[8] Alex Krizhevsky, Vinod Nair, and Geoffrey Hinton. Cifar-10 (canadian institute for advanced research). 2009.
+[9] Xinlei Chen, Haoqi Fan, Ross Girshick, and Kaiming He. Improved baselines with momentum contrastive
+learning. arXiv preprint arXiv:2003.04297, 2020.
+[10] Jean-Bastien Grill, Florian Strub, Florent Altché, Corentin Tallec, Pierre H. Richemond, Elena Buchatskaya,
+Carl Doersch, Bernardo Avila Pires, Zhaohan Daniel Guo, Mohammad Gheshlaghi Azar, Bilal Piot, Koray
+Kavukcuoglu, Rémi Munos, and Michal Valko. Bootstrap your own latent: A new approach to self-supervised
+learning, 2020.
+[11] Huynh Tri, Simon Kornblith, Walter Matthew R., Maire Michael, and Khademi Maryam. Boosting contrastive self-
+supervised learning with false negative cancellation. Proceedings of the IEEE Winter Conference on Applications
+of Computer Vision (WACV), 2022.
+[12] Wang Feng, Liu Huaping, Guo Di, and Fuchun Sun.
+Unsupervised representation learning by invariance
+propagation. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin, editors, Advances in Neural
+Information Processing Systems, volume 33, pages 3510–3520. Curran Associates, Inc., 2020.
+[13] Robinson Joshua, Chuang Ching-Yao, Sra Suvrit, and Jegelka Stefanie. Contrastive learning with hard negative
+samples. International Conference on Learning Representations, 2021.
+[14] Peng Xiangyu, Wang Kai, Zhu Zheng, and You Yang. Crafting better contrastive views for siamese representation
+learning. arXiv preprint arXiv:2202.03278, 2022.
+[15] Tete Xiao, Xiaolong Wang, Alexei A. Efros, and Trevor Darrell. What should not be contrastive in contrastive
+learning. ICLR, 2021.
+[16] Dangovski Rumen, Jing Li, Loh Charlotte, Han Seungwook, Srivastava Akash, Cheung Brian, Agrawal Pulkit,
+and Soljaˇci´c Marin. Equivariant contrastive learning. arXiv preprint arXiv:2111.00899, 2021.
+[17] Caron Mathilde, Misra Ishan, Mairal Julien, Goyal Priya, Bojanowski Piotr, and Armand Joulin. Unsupervised
+learning of visual features by contrasting cluster assignments. 2020.
+7
+
+[18] Guo Yuanfan, Xu Minghao, Li Jiawen, Ni Bingbing, Zhu Xuanyu, Sun Zhenbang, and Xu Yi. Hcsc: Hierarchical
+contrastive selective coding. arXiv preprint arXiv:2202.00455, 2022.
+[19] Junnan Li, Pan Zhou, Caiming Xiong, and Steven C.H. Hoi. Prototypical contrastive learning of unsupervised
+representations. In ICLR, 2021.
+[20] Robert Geirhos, Patricia Rubisch, Claudio Michaelis, Matthias Bethge, Felix A. Wichmann, and Wieland Brendel.
+Imagenet-trained cnns are biased towards texture; increasing shape bias improves accuracy and robustness. ICLR,
+2019.
+[21] Elijah Cole, Xuan Yang, Kimberly Wilber, Oisin Mac Aodha, and Serge Belongie. When does contrastive visual
+representation learning work? CVPR, 2022.
+[22] Yunhao Ge, Yao Xiao, Zhi Xu, Xingrui Wang, and Laurent Itti. Contributions of shape, texture, and color in visual
+recognition. ECCV, 2022.
+[23] Qin Xuebin, Zhang Zichen, Huang Chenyang, Dehghan Masood, Zaiane Osmar, and Jagersand Martin. U2-net:
+Going deeper with nested u-structure for salient object detection. volume 106, page 107404, 2020.
+8
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf,len=323
+page_content='THE INFLUENCES OF COLOR AND SHAPE FEATURES  IN VISUAL CONTRASTIVE LEARNING  Xiaoqi Zhuang  xiaoqizhuang@outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='com  ABSTRACT  In the field of visual representation learning, performance of contrastive learning has been catching  up with the supervised method which is commonly a classification convolutional neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  However, most of the research work focuses on improving the accuracy of downstream tasks such as  image classification and object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' For visual contrastive learning, the influences of individual  image features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=', color and shape) to model performance remain ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  This paper investigates such influences by designing various ablation experiments, the results of which  are evaluated by specifically designed metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' While these metrics are not invented by us, we first use  them in the field of representation evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Specifically, we assess the contribution of two primary  image features (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=', color and shape) in a quantitative way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Experimental results show that compared  with supervised representations, contrastive representations tend to cluster with objects of similar  color in the representation space, and contain less shape information than supervised representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  Finally, we discuss that the current data augmentation is responsible for these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' We believe that  exploring an unsupervised augmentation method that can strongly change the shape information of  objects is an effective research direction that can improve unsupervised visual representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  Keywords Contrastive Learning · Representation Learning  1  Introduction  Unsupervised visual representation learning, specifically contrastive learning, has seen rapid development in recent  years [1, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' These methods aim to train a deep neural network encoder as a feature extractor by minimizing the  distance of similar objects and maximizing the distance of dissimilar ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Due to the limit of unsupervised learning,  the ground truth of each image is only itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Therefore, the mainstream methods usually apply different types of pretext  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' [4] proposed a pretext task called instance discrimination, which applies different image transformations on each  image to get a pair of positive samples and define all other images as negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Previous works proposed 3  key components to make such training framework succeed: deep neural networks based image transformation [2], a  large size of negative samples, and feature consistency[1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' [2] showed that adding color distortion and cropping  into the transformation pipeline significantly improve accuracy, which is because simple image augmentation such as  rotation and flipping could not largely change the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Large negative samples, which is because the positive  samples are only the two augmentations from a single image, the contrasting learning loss function [5] always has to  use thousands of negative samples to project images into the representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Feature consistency, [1, 2] proposed  that the representation features should be compared into the same representation space to avoid the biased contrastive  loss from the updated encoders after backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' These trivial works are the basis for contrastive learning, which  achieved around 74% accuracy in Image-Net datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  Representation learning has always used improving the accuracy of downstream tasks as an indicator to improve the  backbone of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' However, we believe that the accuracy of a single downstream task such as image classification  is not sufficient for the evaluation of representation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Therefore, We developed metrics for representation  models to explicitly quantify visual representations by evaluating their color and shape information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' At the same time,  supervised methods have always been considered the upper limit of unsupervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' We also want to find out the  gap between them, thus to clarify the improvement ideas of unsupervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Therefore, we used ResNet-18 [6]  to train three unsupervised models and one supervised model for experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='12459v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='CV]  29 Jan 2023  In this paper, we did these experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Firstly, we try to find the gap between supervised and unsupervised models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  We get wrong predictions from unsupervised models with image classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' We found that nearly half of  them were the same predictions for three unsupervised models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' These images can be considered common faults of  unsupervised methods, and the parts of these images where the supervised model can make correct predictions are  considered as gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Secondly, in terms of color information, we found that these images and their false predictions  are strongly correlated with color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' We use the earth mover’s distance [7] metric to demonstrate that unsupervised  representations are color biased within the representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' For shape information, we generated a CIFAR-10 [8]  silhouette dataset by a pretrained segmentation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' We verified that the accuracy of the unsupervised model on this  dataset is also lower than that of the supervised model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' We recomputed the accuracy of the image classification task  with color and shape-distorted images, and the results show that unsupervised models are more robust than supervised  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Finally, we conclude that there is no distribution change method strongly related to the image shape among the  data augmentation methods, which is the main reason for the gap with the supervised model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  2  Related Work  Unsupervised contrastive visual learning  Due to the limitation of an unsupervised task, the ground truth of an image is just itself, leading that it being hard to  design the learning progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' [4] proposed a pretext task called Instance Discrimination to annotate images by taking  them as positive and negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' The core concept is comparing the similarity between two images that are  transformed by a single image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Therefore, we can train the model by reducing the distance of generated image pairs and  extending the distance of other images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Such a learning framework requires two important keys [2]: large negative  samples, and feature consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' [1, 9] proposed a first-in-first-out queue to store large negative samples, and keep  feature consistency by updating a momentum encoder slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' [2] achieved both keys by training the model within a  large batch size, which is straightforward but hard to reproduce .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' [2] proved that adding color jitter as well as cropping  into image transformation will largely improve the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' [2] added a linear layer called "Projection Layer" after the  backbone model, which can help raise the accuracy by nearly 7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Such modifications and the training details in [4]  have been the fixed set of contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' [10, 3] proposed a contrastive learning framework without negative  samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' They prove that only comparing the similarity of two positive images can achieve nearly the same accuracy as  the state-of-the-art models without large batch size as well as momentum encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  Improvements on unsupervised representation learning  A lot of work has been done to improve the basic methods mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' [11, 12, 13] want to get high-quality  positive and negative sample pairs for contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' They selected positive and negative samples instead of  random selection by using similarity measures during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' [14] used the heat map of feature representations to  obtain an approximate bounding box to execute an accurate "CROP" operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' [15, 16] suggests that inappropriate  data augmentations will harm image information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' They use enumeration to select the right data augmentations for  each image for contrastive training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' [17, 18, 19] argue that instance discrimination can only provide instance-level  representation information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Their methods can provide high-level semantic information for unsupervised learning by  adding clustering methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  Experiments of visual representation  Researchers have done many experiments on representation bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' [20] used pure texture and shape image datasets to  illustrate that the CNN model trained on Image-Net is texture biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' [21] explores the effects of contrastive learning in  terms of data volume, data domain, data quality, and task granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' [22] designed a specific representation learning  model for shape, color, and texture, which help them analysis the specific contribution of each attributes when the  model is referenced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  In this paper, our work focuses on analyzing color and shape information between representations learned by contrastive  and supervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' We then look for potential gaps by comparing them with supervised representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' After  that, we believe that such gaps will instruct future work to improve unsupervised models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  2  MoCo  SimCLR  SimSiam  Supervised  200 eps  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='86%  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='96%  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='41%  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='18%  800 eps  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='14%  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='78%  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='43%  improve  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='28%  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='82%  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='02%  Table 1: Accuracy on different contrastive learning backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' All contrastive methods do not have major differences on  accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' It also shows that the accuracy of the model increases slowly after 200 epochs, which increases by less than  10% after training 600 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  3  Methods  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='1  Dataset  We perform experiments on a popular small image dataset: CIFAR-10 [8], which consists of 60000 32x32 color images  in 10 classes, with 6000 images per class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' We know that the experiment results from such a small dataset may be biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  However, all these methods [1, 2, 3] show the same problems in this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' We believe these results are valuable to  discuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='2  Training details  Unsupervised backbone  We trained three mainstream unsupervised models based on the ResNet-18 [6] backbone, which are: MoCo V2 [9],  SimCLR V2 [9], and SimSiam [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' We use the linear evaluation method as the metric, that is, training these unsupervised  methods on the trainset in 800 epochs firstly, extracting and freezing the backbone encoder, adding an extra MLP to  train a classifier on the trainset in 200 epochs, and then evaluating the accuracy of the classifier on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Due to  the low resolution of images in CIFAR-10, we modified the Standard ResNet-18 backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' We delete "Gaussian blur"  in the transformation compose, replace the 7 × 7 convolution kernel as the 3 × 3 kernel and delete the "stride" and  "MaxPool2d".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' The hyperparameters of the temperature τ, learning rate and scheduler are the same as [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  Supervised backbone  We also trained a supervised model based on ResNet-18 as the object of comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' The model structure is largely the  same as the above-unsupervised models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' We add a classification layer and used labels to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  Linear Evaluation Accuracy  The evaluation results are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' In terms of linear evaluation, three contrastive models achieved around  80% accuracy after 200 epochs and 88% after 800 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' By contrast, the accuracy of the supervised ResNet-18 is  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='18% within only 200 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  4  Experiments  We now describe our experiments in the following order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Firstly, we found the gap between contrastive models and  the supervised model, and then proposed our assumptions based on such a gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Secondly, we implemented specific  experiments to verify these assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Finally, we discussed the possible reasons for such issues on contrastive  representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='1  Gap Definition  Table 1 shows that the accuracy difference between the three backbones in the CIFAR-10 test set is less than 1%, which  arouses our interest in whether they do the same predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' The results show that the intersection between false  predictions has 581 images (nearly 49%), the intersection between false predictions with the same predictions has  384 images (nearly 32%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' This indicates that although these backbone models are trained in different methods, the  representation features are similar in some areas as long as they are trained by instance discrimination, which also  means that they do have the same problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  3  Figure 1: The same wrong predictions by contrastive methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' These images are easy to be identified for both human  beings and the supervised model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' However, all models based on contrastive learning make wrong predictions, which  these predictions, the text over the image, are probably made by the color information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' For example, the plane on the  top-left corner is predicted as a horse, where the only same semantic of these two objects is the color: brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  MoCo  SimCLR  SimSiam  Supervised  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='51%  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='00%  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='16%  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='89%  Table 2: kNN-4 Accuracy on GAP-265 between MoCo, SimCLR, SimSiam and supervised ResNet-18  However, the supervised model can make 265 right predictions in these 384 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Due to the low resolution and  wrong labeling reasons in the CIFAR-10 dataset, we can take the supervised model as the upper limit of the classification  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Therefore, such 265 images can be regarded as the knowledge gap between the supervised model and  unsupervised models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' We will call these images GAP-265 in the following content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  Figure 1 is shown as a subset of these images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' These images are easy to be identified for both human beings and  supervised model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' However, all models based on contrastive learning make wrong predictions, and these predictions  are likely to be made based on their color information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' For example, the plane on the top-left corner is predicted as a  horse, where the only same semantic of these two objects is the color: brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  Such phenomenon inspires us to propose two assumptions:  Contrastive representation space tends to make images with similar colors closer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  Contrastive features contain less shape information than supervised features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='2  Feature Influence  We use Linear Evaluation to compare different models, and thus we should prove that our hypothesis should not be  affected by the extra linear layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' In other words, features embedded by the contrastive encoder are the reason for  the wrong predictions in GAP-265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Therefore, we perform k-nearest neighbor (kNN) classification on GAP-265 to  eliminate such concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Table 2 shows that kNN-4 accuracy of models by unsupervised methods are also largely lower  than the one by the supervised method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Figure 2 shows an example of different nearest neighbors for the brown plane  between MoCo and the supervised model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' The nearest neighbors in MoCo are other categories, while the only similar  field is color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' By contrast, all nearest neighbors in the supervised model are the same categories but different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='3  Color Biased Detection  To prove our first assumption, contrastive representation space tends to make images with similar colors closer, we  use Earth Mover’s Distance(EMD) [7], which is also called Wasserstein Distance, as a metric to evaluate the color  similarity between the query image and its top-4 nearest neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Algorithm 1 illustrates the pseudo-code that how  we use EMD to evaluate image similarity on color dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' First, we convert the BGR image into its HSV form that  conforms to human vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Second, we calculate the color histogram of the image in the hue dimension, and convert it  4  horse  ship  cat  dog  dog deer  cat  bird  airplane  birdFigure 2: Two examples of the false prediction and their top-4 nearest neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' The target image are not in the right  regions, and they are in the region which their color are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  Model  MoCo  SimCLR  SimSiam  Supervised  EMD  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='97  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='58  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='85  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='20  Table 3: The Earth Mover’s Distance in GAP-265 between contrasitve and supervised models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  into the signature form[7] which includes index information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Finally, We calculate EMD between the object image and  the signatures of its nearest four neighbors to compare whether the image is close to the image of similar color in the  representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' More specifically, the smaller the EMD value between the query and its neighbours, the more  similar they are in terms of color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Table 3 shows the mean value of EMD in GAP-265 between all backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' It is  clear that the metrics of backbones learned by contrastive methods are near and smaller than the metric of the model by  the supervised method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Figure 2 is also a clear example that the same query will have different nearest neighbours in  different models, while contrastive features are clustered by color and supervised features are clustered by their labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  Algorithm 1 Earth Mover’s Distance on Image Color Similarity  Input:a test image ti , model M, the overall test set test, neighbor number k  Output:the mean EMD between ti and its nearest K neighbors  1: function image2signature(image)  2:  bgr = CV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='imread(image)  3:  hsv = CV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='cvtColor(bgr)  ▷ convert BGR to HSV  4:  histHUE = CV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='calcHist(hsv, channel=0)  ▷ only calculate histogram on HUE  5:  sign = hist2signature(histHUE)  ▷ convert histogram to signature  6:  return sign  7: end function  8: disti = []  ▷ create an empty list to store EMD values  9: fi, fall = M(ti), M(test)  ▷ extract features by model  10: neighborsi = nearestNeighbors(fi, fall, k)  ▷ get the nearest K neighbors of ti  11: signi = image2signature(ti)  12: for n ∈ neighborsi do  13:  signn = image2signature(n)  14:  emdn = cv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='EMD(signi, signn, L2)  15:  dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='append(emdn)  16: end for  Output:mean(disti)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='4  Shape Biased Detection  To prove our second assumption, contrastive features contain less shape information than supervised features, we  created a silhouette dataset based on CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' As CIFAR-10 does not have a public silhouette dataset, We use a  pretrained semantic segmentation model U2-Net [23] to generate silhouette images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Due to the low resolution and  no fine tuning, some generated images are of low quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' We manually selected 261 high-quality silhouette images  5  d=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='000  d=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='233  d=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='334  d=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='345  d=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='345  supervised  d=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='000  d=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='757  d=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='765  d=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='790  d=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='806  MoCoFigure 3: Examples of the manually selected silhouette images from CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  Model  MoCo  SimCLR  SimSiam  Supervised  Acc  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='17%  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='7%  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='7%  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='91%  Table 4: Accuracy between different models on silhouette CIFAR-10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  to ensure that human beings are capable of making the right predictions only by their contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Figure 3 shows some  examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Table 4 shows that the supervised model achieves the highest accuracy among all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Such accuracy  results to some extent could show that supervised features have more shape information than contrastive features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='5  Distorted Image Evaluation  The results of the above two experiments raise questions about the robustness of contrastive representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Therefore,  we did different distortions on the test set and reevaluate the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' More specifically, we apply "ColorJitter",  "HorizontalFilp", and "RandomRotation" respectively to the testset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Table 5 shows that all contrastive models have a  lower decrease of accuracy in "ColorJitter" and "GrayScale" than supervised models, and there is no obvious difference  in "Flip".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' The results indicate that contrastive representation is more robust than supervised representation for the  knowledge that it has gained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  5  Discussion  As noted in Section 1, researchers have been working hard to improve unsupervised learning, and the quantitative  improvement is usually compared with the accuracy of supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' To discover fine-grained gaps between  unsupervised and supervised learning, we proposed GAP-265, a subset of CIFAR-10 which unsupervised models make  the same wrong predictions while the supervised model makes the right predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Most of these wrong predictions  are due to the color of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='3 illustrate that contrastive representation space is more dependent on color  than supervised representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' We also constructed a silhouette dataset based on CIFAR-10 which contains  261 high-quality images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='4 illustrate that supervised model are more shape-aware than unsupervised models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='5 illustrate that unsupervised models are more robust than the supervised model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  In our opinion, the reason why such phenomena appear is because of the data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Unsupervised models  inevitably have to apply stronger augmentations such as "ColorJitter" and "GrayScale" than supervised models due to  the annotation limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' As a result, for these distorted images, unsupervised models are more robust than supervised  models because that’s how they were trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' However, this also leads to the problem of possible color dependence  of the representation spaces trained by contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' By contrast, current augmentation methods do not have  such a strong transformation of image shape as "ColorJitter" transforms image color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' The more difficult thing is that it  is unsupervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' We can change the color distribution of the image by adjusting the contrast and brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  However, the shape distribution of the image is difficult to make changes on a 2D image without relying on labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' For  model  baseline  color  flip  gray  MoCo  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='14%  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='22%  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='16%  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='35%  SimCLR  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='78%  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='15%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='42%  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='40%  SimSiam  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='43%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='43%  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='10%  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='63%  Supervised  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='18%  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='2%  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='22%  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='39%  Table 5: Accuracy change amplitude of each model for distorted images compared to the original baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' All  contrastive models have lower decrease of accuracy in "ColorJitter" and "GrayScale" than supervised models, and there  is no obvious difference in "Flip".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  6  example, it is easy to change the color of the object in one image, but it is obviously impossible to obtain images from  all angles of the object through such one image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  6  Conclusion  In summary, we provided evidence that representation space based on contrastive learning today relies on color  information more and contains less shape information compared with space based on supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' In addition,  we also indicate that contrastive space are more robust than supervised space on distortion attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' We discussed that  data augmentation is the cause of these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' We hope that these experiments and conclusions can make researchers  focus on how to add shape knowledge in the future unsupervised learning and add color as well as shape metrics to the  evaluation metrics of unsupervised tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  References  [1] Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Momentum contrast for unsupervised visual  representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' arXiv preprint arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='05722, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  [2] Chen Ting, Kornblith Simon, Norouzi Mohammad, and Hinton Geoffrey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' A simple framework for contrastive  learning of visual representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' arXiv preprint arXiv:2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='05709, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  [3] Chen Xinlei and He Kaiming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Exploring simple siamese representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' In Proceedings of the IEEE/CVF  Conference on Computer Vision and Pattern Recognition (CVPR), pages 15750–15758, June 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  [4] Wu Zhirong, Xiong Yuanjun, Stella X Yu, and Lin Dahua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Unsupervised feature learning via non-parametric  instance discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition,  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  [5] Aaron van den Oord, Yazhe Li, and Oriol Vinyals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Representation learning with contrastive predictive coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  arXiv preprint arXiv:1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='03748, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  [6] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Deep residual learning for image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' arXiv  preprint arXiv:1512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='03385, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  [7] Rubner Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=', Tomasi C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=', and Guibas L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' The earth mover’s distance as a metric for image retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' International  Journal of Computer Vision, 40, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  [8] Alex Krizhevsky, Vinod Nair, and Geoffrey Hinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Cifar-10 (canadian institute for advanced research).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  [9] Xinlei Chen, Haoqi Fan, Ross Girshick, and Kaiming He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Improved baselines with momentum contrastive  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' arXiv preprint arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='04297, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  [10] Jean-Bastien Grill, Florian Strub, Florent Altché, Corentin Tallec, Pierre H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Richemond, Elena Buchatskaya,  Carl Doersch, Bernardo Avila Pires, Zhaohan Daniel Guo, Mohammad Gheshlaghi Azar, Bilal Piot, Koray  Kavukcuoglu, Rémi Munos, and Michal Valko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Bootstrap your own latent: A new approach to self-supervised  learning, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  [11] Huynh Tri, Simon Kornblith, Walter Matthew R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=', Maire Michael, and Khademi Maryam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Boosting contrastive self-  supervised learning with false negative cancellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Proceedings of the IEEE Winter Conference on Applications  of Computer Vision (WACV), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  [12] Wang Feng, Liu Huaping, Guo Di, and Fuchun Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  Unsupervised representation learning by invariance  propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' In H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Larochelle, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Ranzato, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Hadsell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Balcan, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Lin, editors, Advances in Neural  Information Processing Systems, volume 33, pages 3510–3520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  [13] Robinson Joshua, Chuang Ching-Yao, Sra Suvrit, and Jegelka Stefanie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Contrastive learning with hard negative  samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' International Conference on Learning Representations, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  [14] Peng Xiangyu, Wang Kai, Zhu Zheng, and You Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Crafting better contrastive views for siamese representation  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' arXiv preprint arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='03278, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  [15] Tete Xiao, Xiaolong Wang, Alexei A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Efros, and Trevor Darrell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' What should not be contrastive in contrastive  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' ICLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  [16] Dangovski Rumen, Jing Li, Loh Charlotte, Han Seungwook, Srivastava Akash, Cheung Brian, Agrawal Pulkit,  and Soljaˇci´c Marin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Equivariant contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' arXiv preprint arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='00899, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  [17] Caron Mathilde, Misra Ishan, Mairal Julien, Goyal Priya, Bojanowski Piotr, and Armand Joulin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Unsupervised  learning of visual features by contrasting cluster assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  7  [18] Guo Yuanfan, Xu Minghao, Li Jiawen, Ni Bingbing, Zhu Xuanyu, Sun Zhenbang, and Xu Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Hcsc: Hierarchical  contrastive selective coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' arXiv preprint arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='00455, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  [19] Junnan Li, Pan Zhou, Caiming Xiong, and Steven C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Hoi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Prototypical contrastive learning of unsupervised  representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' In ICLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  [20] Robert Geirhos, Patricia Rubisch, Claudio Michaelis, Matthias Bethge, Felix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Wichmann, and Wieland Brendel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  Imagenet-trained cnns are biased towards texture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' increasing shape bias improves accuracy and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' ICLR,  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  [21] Elijah Cole, Xuan Yang, Kimberly Wilber, Oisin Mac Aodha, and Serge Belongie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' When does contrastive visual  representation learning work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' CVPR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  [22] Yunhao Ge, Yao Xiao, Zhi Xu, Xingrui Wang, and Laurent Itti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' Contributions of shape, texture, and color in visual  recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' ECCV, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  [23] Qin Xuebin, Zhang Zichen, Huang Chenyang, Dehghan Masood, Zaiane Osmar, and Jagersand Martin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' U2-net:  Going deeper with nested u-structure for salient object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content=' volume 106, page 107404, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
+page_content='  8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FMT4oBgHgl3EQf6TFG/content/2301.12459v1.pdf'}
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+arXiv:2301.03235v1  [physics.atom-ph]  9 Jan 2023
+Deciphering Core, Valence and Double-Core-Polarization Contributions to Parity
+Violating Amplitudes in 133Cs using Different Methods
+a,bA. Chakraborty and aB. K. Sahoo∗
+aAtomic, Molecular and Optical Physics Division,
+Physical Research Laboratory, Navrangpura, Ahmedabad 380009, India and
+bIndian Institute of Technology Gandhinagar, Palaj, Gandhinagar 382355, India
+(Dated: January 10, 2023)
+As a prerequisite of probing physics beyond the Standard Model (BSM) of particle physics, it
+is imperative to perform calculation of parity violating electric dipole (E1P V ) amplitudes within
+0.5% accuracy in atomic systems. Latest high precision calculations of E1P V of the 6s 2S → 7s 2S
+transition in 133Cs by three different groups claim achieving its accuracy below 0.5%, but such
+claims contradict on the view that their final values differ by 1%.
+One of the major issues in
+these calculations is the opposite signs among the core correlation contribution from different works
+leading to 200% difference in its value.
+In a review [Rev.
+Mod.
+Phys.
+90, 025008 (2018)], a
+Letter [Phys. Rev. D 103, L111303 (2021)] and a Comment [Phys. Rev. D 105, 018301 (2022)],
+reliability of all these calculations are strongly contended. We unearthed here the underlying reason
+for getting sign discrepancies in various works by investigating how different electron correlation
+effects are encapsulated through the undertaken methods in the above works to determine E1P V .
+Detailed discussions presented in this work would help in guiding theoretical studies to improve
+accuracy of E1P V in atomic systems to probe BSM physics.
+I.
+INTRODUCTION
+Atomic parity violation (APV) has implications for ex-
+ploring physics beyond the Standard Model (SM) of par-
+ticle physics [1–3]. The neutral current weak interactions
+due to exchange of the Z0 bosons between electrons and
+nucleus in atomic systems lead to APV [4]. APV stud-
+ies can offer a fundamental quantity known as nuclear
+weak charge QW , from which model independent val-
+ues of electron-up and electron-down quarks coupling co-
+efficients can be inferred [4, 5]. Hence, any deviations
+of these values from the SM can be used to probe new
+physics or to complement some of the findings of the
+Large Hadron Collider facility when it is upgraded at the
+higher TeV energy scale. The nuclear spin-independent
+(NSI) component of APV has been measured to an ac-
+curacy of 0.35% in the 6s 2S1/2 − 7s 2S1/2 transition in
+133Cs [6]. Advanced experimental techniques have been
+proposed recently to improve accuracy of the above mea-
+surement, as well as carrying out similar measurement
+in the 6s 2S1/2 − 5d 2D3/2 transition of 133Cs [7, 8]. In
+order to extract QW value from these measurements, it is
+imperative to perform calculation of the parity violating
+electric dipole (E1P V ) amplitudes of the corresponding
+transitions very precisely (arguably less than 0.5%).
+There have been a long history of performing calcula-
+tions of the E1P V amplitude for the 6s 2S1/2 − 7s 2S1/2
+transition in 133Cs by employing different state-of-the-
+art relativistic atomic many-body theories at different
+levels of approximation. Among the early calculations,
+Dzuba et al had employed the time-dependent Hartree-
+Fock (TDHF) method [9, 10], while Martensson had
+∗ bijaya@prl.res.in
+applied the combined coupled-perturbed Dirac-Hartree-
+Fock (CPDF) method and random-phase approximation
+(RPA), together referred as CPDF-RPA method [11],
+to investigate the roles of core-polarization effects to
+E1P V . Both these methods are technically equivalent,
+but Martensson had also provided results at the interme-
+diate levels using approximations at the Dirac-Hartree-
+Fock (DHF), CPDF and RPA methods as well as list-
+ing contributions from double-core-polarization (DCP)
+effects explicitly.
+Later, Blundell et al had employed
+a linearized version relativistic coupled-cluster method
+in the singles and doubles excitation approximation (SD
+method) to estimate the E1P V amplitude of the above
+transition [12]. However, they had adopted a sum-over-
+states approach in which matrix elements of the electric
+dipole (E1) operator and APV interaction Hamiltonian
+were evaluated for the transitions involving np 2P1/2 in-
+termediate states (called as “Main” contribution) with
+the principal quantum number n = 6 − 9. This method
+also utilized the calculated E1 matrix elements and mag-
+netic dipole hyperfine structure constants to estimate
+the uncertainty of E1P V . Uncertainties from the ener-
+gies were removed by considering the experimental en-
+ergies, while contributions from core orbitals (referred
+as “Core” contribution henceforth) and higher np 2P1/2
+intermediate states (hereafter called as “Tail” contribu-
+tion) were estimated using lower-order methods. Follow-
+ing this work, Dzuba et al improved their calculation
+of TDHF method by incorporating correlation contribu-
+tions through the Br¨uckner orbitals (BO) and referred
+the approach as RPA+BO method [13].
+Higher-order
+contributions from the Breit, lower-order QED and neu-
+tron skin effects were added subsequently through dif-
+ferent works to claim for more precise E1P V value for
+extracting the BSM physics [14–20].
+It is worth men-
+tioning here that all these higher-order effects were es-
+
+2
+timated through different types of many-body methods
+and without accounting correlations among themselves.
+Soon after these theoretical results, relativistic coupled-
+cluster (RCC) theory with singles and doubles approx-
+imation (RCCSD method) was employed to treat both
+the electromagnetic and weak interactions on an equal
+footing [21–23].
+Moreover, it also treated correlations
+among the Main, Core and Tail contributions to E1P V
+but its accuracy was an concern due to its ab initio nature
+and use of a relatively smaller basis size in the available
+computational resources at that time.
+A decade ago, Porsev et al made further improve-
+ment in the sum-over-states result of Blundell et al
+by considering contributions from the non-linear terms
+from RCCSD method to their SD method as well as
+adding valence triple excitations (CCSDvT method) [24].
+Their claimed accuracy to the E1P V amplitude of the
+6s 2S1/2 − 7s 2S1/2 transition in 133Cs was about 0.27%.
+However, the Core and Tail contributions were still es-
+timated using a blend of many-body methods without
+explicitly stating which physical effects were taken into
+account for their evaluations. We refer these two contri-
+butions together as X-factor in this work. In an attempt
+to improve the calculated E1P V value further, Dzuba et
+al estimated the X-factor contributions using their TDHF
+approach but omitting the DCP contributions [25] in the
+similar line to their earlier works [9, 10]. This calculation
+showed an opposite sign of Core contribution than that
+was reported by Porsev et al. In 2013, Roberts reported
+the DCP contribution separately [26] and the result was
+slightly different than the value of Martenssion [11]. The
+opposite sign of the Core contribution of Dzuba et al
+with Porcev et al was criticized in two papers [27, 28],
+which prompted for carrying out further investigation
+on different correlation contributions to E1P V from the
+first-principle approach. In 2021, Sahoo et al improved
+their calculation of the above E1P V amplitude by imple-
+menting the singles, doubles and triples approximation
+to both the unperturbed and perturbed wave functions
+(RCCSDT method) and using a much bigger set of basis
+functions [29]. They also used the same V N−1 poten-
+tial as in the previous cases and presented the Core and
+Valence (Main and Tail together) contributions explic-
+itly. As per the convention adopted in this approach, the
+Core contribution agreed with the earlier RCCSD result
+[21–23], and was close to the reported value of Blundell
+et al [12] and Porsev et al [24]. In a recent Comment,
+Roberts and Ginges have argued in favour of opposite
+sign of Core contribution than other findings by giving
+intermediate results of their RPA+BO method [30]. In
+another work, Tan el al estimated combined Core and
+Tail contributions to the E1P V amplitude of the above
+transition using mixed-parity orbitals through RPA [31]
+and support the value reported in Ref. [24]. It is, there-
+fore, abundantly clear that it is necessary to find out the
+issue of sign problem with the Core contribution to the
+above E1P V amplitude. More importantly, the basis of
+dividing net E1P V result into Core, Main, Tail, DCP,
+etc.
+contributions in an approach should be properly
+defined and missing physical effects in a method com-
+pared to others need to be well understood when a mix-
+ture of methods are used to estimate these contributions
+piece-wise. Any misinterpretation or misrepresentation
+of these contributions can have repercussion effect when
+they are used to infer beyond the SM (BSM) physics.
+The present work is devoted to addressing the afore-
+mentioned sign issue with the Core contribution among
+various works, bringing to notice the shortcomings of the
+sum-over-states approach, and explaining the reason for
+the coincidental agreement between the core contribu-
+tions of Porsev et al. [24] and Sahoo et al. [29]. We start
+with various procedures that can be adopted through a
+general many-body method to evaluate E1P V amplitudes
+in atomic systems and demonstrate how the definition
+of Core contribution can vary from one procedure to an-
+other. With the help of lower-order many-body methods,
+we find out the missing contributions in a typical sum-
+over-states approach to estimate E1P V . We then analyze
+results from different methods to learn about how and
+to what extent these missing effects are incorporated in
+the previous calculations. We also make a similar anal-
+ysis for the E1P V amplitude of the 6s 2S1/2 − 5d 2D3/2
+transition in 133Cs and compare it with the result of the
+6s 2S1/2 − 7s 2S1/2 transition. There are two important
+reasons for quoting result of the S−D3/2 transition of Cs
+here. First, our analysis would be helpful to estimate its
+E1P V amplitude more precisely which is required for the
+ongoing experiments [7, 8]. Second, we intend to address
+a comment [30, 32] on why sign for Core contribution to
+the S − D3/2 transitions from different methods agree in
+contrast to the S − S transitions.
+II.
+THEORY
+The short range effective Lagrangian corresponding to
+the vector- -axial-vector neutral weak current interaction
+of an electron with up- and down-quarks in an atomic
+system can be given by [4, 33–35]
+Leq = GF
+√
+2
+�
+u,d
+�
+C1u ¯ψuγµψu + C1d ¯ψdγµψd
+� ¯ψeγµγ5ψe
+= GF
+√
+2
+�
+n
+C1n ¯ψnγµψn ¯ψeγµγ5ψe,
+(1)
+where GF = 1.16632×10−5 GeV−2 is the Fermi constant,
+sums u, d and n stand for up-quark, down-quark and nu-
+cleons respectively, and C1i with i = u, d and n repre-
+sent coupling coefficients of the interaction of an electron
+with up-quark, down-quark and nucleons (protons (Pn)
+and neutrons (Nn)) respectively.
+Adding them coher-
+ently and taking the non-relativistic approximation for
+nucleons, the temporal component can give the nuclear-
+spin-independent APV interaction Hamiltonian as
+HNSI
+PV = − GF
+2
+√
+2QW γ5ρ(r),
+(2)
+
+3
+(a)
+(b)
+(c)
+(d)
+f
+f
+f
+f
+i
+i
+i
+i
+p
+p
+a
+a
+D
+HW
+D
+HW
+HW
+HW
+D
+D
+FIG. 1. Goldstone diagrams representing Core (a and b) and
+Valence (c and d) contributions to E1P V in the V N−1 DHF
+potential of an one valence atomic system. Here, double ar-
+rows represent initial (i) and final (f) valence orbitals, single
+arrows going down (a) means occupied orbitals and single ar-
+rows going up (p) means virtual orbitals. The operators D
+and HW are shown in curly line and a line with bullet point
+respectively.
+where ρ(r) is the averaged nuclear density and QW =
+2[ZC1P n + (A − Z)C1Nn] is known as the nuclear weak
+charge with Z and A representing for atomic and mass
+numbers respectively.
+It is obvious that QW
+is a
+model dependent quantity.
+Thus, the difference of its
+actual value from the SM can provide signatures of
+physics supporting other models.
+In the SM, C1u =
+1
+2
+�
+1 − 8
+3 sin2 θSM
+W
+�
+and C1d = − 1
+2
+�
+1 − 4
+3 sin2 θSM
+W
+�
+[4, 33–
+35].
+This follows C1Nn = 2C1d + C1u = −1/2 and
+C1P n = 2C1u + C1d = (1 − 4 sin2 θSM
+W )/2 ≈ 0.04, which
+are accurate up to 1%. Hence, it is necessary to evaluate
+model independent QW in any atomic system within this
+accuracy in order to infer physics beyond the SM.
+III.
+EVALUATION PROCEDURES OF E1P V
+In the presence of APV, the net atomic Hamiltonian
+is given by
+Hat = H + HNSI
+PV
+= H + GF HW ,
+(3)
+where H contains contributions from electromagnetic in-
+teractions and HW is defined in order to treat GF as
+a small parameter to include contributions from HNSI
+PV
+perturbatively through a many-body method. Using the
+wave functions of Hat, we determine E1P V of a transition
+between states |Ψi⟩ and |Ψf⟩ as
+E1P V =
+⟨Ψf|D|Ψi⟩
+�
+⟨Ψf|Ψf⟩⟨Ψi|Ψi⟩
+,
+(4)
+where D is the E1 operator. In the earlier calculations
+of E1P V amplitude in 133Cs, atomic wave functions were
+determined by using the V N−1 potential with N is the
+number of electrons of the atom. Choice of this poten-
+tial is convenient to produce both the ground and excited
+states of Cs atom using the Fock-space formalism. We
+adopt the same formalism here, so that description of
+different correlation effects and comparison of results are
+consistent to each other in all the considered works in-
+cluding in the definition of the Core contribution.
+Following a typical approach in the many-body prob-
+lems, we define a suitable mean-field Hamiltonian H0 to
+replace the exact Hamiltonian H to obtain a set of ap-
+proximated solutions
+H0|Φk⟩ = Ek|Φk⟩,
+(5)
+where subscript k is used to identify different states.
+Since [5p6] is the common closed-shell configuration in
+the states of our interest in the present work, we ob-
+tain the solution of this closed-core first. We consider
+the DHF method to obtain its mean-field wave function
+|Φ0⟩. Then, the |Φk⟩ wave functions are obtained as
+|Φv⟩ = a†
+v|Φ0⟩.
+(6)
+Starting with |Φv⟩, we can express the exact wave func-
+tion of the state using Bloch’s prescription as [36]
+|Ψv⟩ = Ωv|Φv⟩,
+(7)
+where Ωv is known as the wave operator. In the V N−1
+potential approximation, we first solve electron correla-
+tion effects among electrons from the core orbitals of |Φ0⟩.
+Then, correlation effects involving electron from the va-
+lence orbital are included. Accordingly Ωv is divided into
+two parts
+Ωv = Ω0 + Ωv,
+(8)
+where Ω0 represents wave operator accounting correla-
+tions of electrons only from the core orbitals while Ωv
+takes care of correlations of electrons from all orbitals
+including the valence orbital.
+In a given many-body
+method, we can solve amplitudes of the above wave op-
+erators using the equations
+[Ω0, H0]P0 = UresΩ0P0
+(9)
+and
+[Ωv, H0]Pv = Ures(Ω0 + Ωv)Pv
+−ΩvPvUres(Ω0 + Ωv)Pv,
+(10)
+where Pn = |Φn⟩⟨Φn| and Qn = 1 − Pn with n ≡ 0, v,
+and Ures = H − H0 is known as the residual interaction
+that contributes to the amplitudes of Ω0 and Ωv.
+In
+fact, the energy of the |Ψv⟩ state can be evaluated as the
+expectation value of the effective Hamiltonian
+Heff = PvUres(Ω0 + Ωv)Pv
+(11)
+with respect to the reference state |Φv⟩.
+It should be
+noted that energy of the state is given by
+Ev = ⟨Φv|Heff|Φv⟩
+(12)
+For the choice of V N potential in the generation of single
+particle orbitals, amplitudes of the Ωv operator can be
+estimated as
+[Ωv, H0]Pv = Ures(Ω0 + Ωv)Pv.
+(13)
+
+4
+It means that Ev that gives the energy of the state does
+not appear in the wave function determining equation for
+the case of V N potential. Thus, the core-valence interac-
+tion effects in the construction of DHF potential in case
+of V N−1 is partly compensated through the wave oper-
+ator amplitude determining equation through the extra
+term with energy. If any method utilizes the V N−1 po-
+tential without taking into account the above-mentioned
+extra term, it can be termed as an improper theory.
+Both |Ψi⟩ and |Ψf⟩ can be determined by solving the
+equation-of-motion for Hat in the above formalism. How-
+ever, parity cannot be treated as a good quantum number
+for Hat. As a consequence, it will relax one degree of free-
+dom in describing atomic states for which computations
+of amplitudes for the Ω0 and Ωv operators will increase
+by many folds. Compared to H, the strength of HNSI
+PV
+in Hat is smaller by 1012 order. Thus, it is important
+that contributions from H is accounted as much as pos-
+sible in the determination of the above wave functions
+with the available computational resources and only the
+first-order effect due to HNSI
+PV can be accounted. Anyway,
+inclusion of higher-order effects from HNSI
+PV will not serve
+for any purpose in our study as they will be much smaller
+then our interest. Thus, we express atomic wave function
+|Ψv⟩ of a general state with valence orbital v as
+|Ψv⟩ = |Ψ(0)
+v ⟩ + GF |Ψ(1)
+v ⟩,
+(14)
+where |Ψ(0)
+v ⟩ is the zeroth-order wave function containing
+contributions only from H while |Ψ(1)
+v ⟩ includes one-order
+contribution from HW with respect to |Ψ(0)
+v ⟩. Substitut-
+ing Eq. (14) in Eq. (4) and keeping finite terms up to
+first-order in GF , we get
+E1P V ≃ GF
+�
+⟨Ψ(0)
+f |D|Ψ(1)
+i ⟩
+Nif
++
+⟨Ψ(1)
+f |D|Ψ(0)
+i ⟩
+Nif
+�
+, (15)
+where normalization factor Nif = �NfNi with Nv =
+⟨Ψ(0)
+v |Ψ(0)
+v ⟩. Presenting results in this paper GF is ab-
+sorbed with the E1P V value, so it does not appear ex-
+plicitly in our calculation. It can be noted that contri-
+bution from the first term is referred as the initial per-
+turbed state contribution whereas contribution from the
+second term is referred as the final perturbed state con-
+tribution in the above expression during the discussion of
+results later. In order to treat both the electromagnetic
+and weak interaction Hamiltonians on an equal footing
+in a many-body method and consider correlations among
+them, solutions of the unperturbed and first-order per-
+turbed wave functions should satisfy
+H|Ψ(0)
+v ⟩ = E(0)
+v |Ψ(0)
+v ⟩
+(16)
+and
+(H − E(0)
+v )|Ψ(1)
+v ⟩ = (E(1)
+v
+− HW )|Ψ(0)
+v ⟩,
+(17)
+respectively, where E(1)
+v
+= 0 for odd-parity interaction
+operators.
+Using the wave operator formalism, we can express
+|Ψ(0)
+v ⟩ = Ωv(0)|Φv⟩
+= (Ω(0)
+0
++ Ω(0)
+v )|Φv⟩
+(18)
+and
+|Ψ(1)
+v ⟩ = Ωv(1)|Φv⟩
+= (Ω(1)
+0
++ Ω(1)
+v )|Φv⟩.
+(19)
+Substituting the wave operators, Eq.
+(15) can be ex-
+pressed as
+E1P V =
+⟨Φf|(Ω(0)
+0
++ Ω(0)
+f )†D(Ω(1)
+0
++ Ω(1)
+i )|Φi⟩
+Nif
++
+⟨Φf|(Ω(1)
+0
++ Ω(1)
+f )†D(Ω(1)
+0
++ Ω(1)
+i )|Φi⟩
+Nif
+=
+⟨Φ0|af(Ω(0)
+0
++ Ω(0)
+f )†D(Ω(1)
+0
++ Ω(1)
+i )a†
+i|Φ0⟩
+Nif
++
+⟨Φ0|af(Ω(1)
+0
++ Ω(1)
+f )†D(Ω(1)
+0
++ Ω(1)
+i )a†
+i|Φ0⟩
+Nif
+= ⟨Φ0|af[Ω(0)†
+0
+DΩ(1)
+0
++ Ω(1)†
+0
+DΩ(0)
+0 ]a†
+i|Φ0⟩
+Nif
++
+⟨Φ0|af[Ω(0)†
+f
+DΩ(1)
+0
++ Ω(1)†
+f
+DΩ(0)
+0 ]a†
+i|Φ0⟩
+Nif
++ ⟨Φ0|af[Ω(0)†
+0
+DΩ(1)
+i
++ Ω(1)†
+0
+DΩ(0)
+i ]a†
+i|Φ0⟩
+Nif
++
+⟨Φ0|af[Ω(0)†
+f
+DΩ(1)
+i
++ Ω(1)†
+f
+DΩ(0)
+i ]a†
+i|Φ0⟩
+Nif
+. (20)
+In the above expression,contribution from the first term is
+referred as “Core” correlation contribution while the rest
+is termed as “Valence” correlation contribution in this
+paper. Analogously, contributions to Nv = ⟨Φ0|av(Ω(0)
+0 +
+Ω(0)
+v )†(Ω(0)
+0
++ Ω(0)
+v )a†
+v|Φ0⟩ can also be divided into two
+parts.
+A.
+Sum-over-states approach
+In the sum-over-states approach, the first-order wave
+function of a general state can be expressed as
+|Ψ(1)
+v ⟩ =
+�
+I̸=v
+|Ψ(0)
+I ⟩ ⟨Ψ(0)
+I |HW |Ψ(0)
+v ⟩
+Nv(E(0)
+v
+− E(0)
+I )
+,
+(21)
+where |Ψ(0)
+I ⟩ are the zeroth-order intermediate states and
+E(0)
+n
+is the unperturbed energy of the nth level. Thus, Eq.
+
+5
+(i)
+(ii)
+(iii)
+(iv)
+(v)
+(vi)
+(vii)
+(viii)
+(ix)
+(x)
+(xi)
+(xii)
+(xiii)
+(xiv)
+(xiv)
+(xvi)
+(xvii)
+(xviii)
+(xix)
+(xx)
+(xxi)
+(xxii)
+(xxiii)
+(xxiv)
+(xxv)
+(xxvi)
+(xxvii)
+(xxviii)
+(xxix)
+(xxx)
+(xxxi)
+(xxxii)
+(xxxiii)
+(xxxiv)
+(xxxv)
+(xxxvi)
+(xxxvii)
+(xxxviii)
+(xxxix)
+(xL)
+FIG. 2. A few important electron correlation contributing di-
+agrams to E1P V in the RMBPT(3) method. The dotted lines
+represent the atomic Hamiltonian.
+Diagrams (i)-(viii) give
+Core contributions in the RMBPT(3)w method, but they cor-
+respond to Valence contributions in the RMBPT(3)d method.
+Diagrams (ix)-(xvi) follow other way around.
+(15) can be written as
+E1P V =
+�
+I̸=i
+⟨Ψ(0)
+f |D|Ψ(0)
+I ⟩⟨Ψ(0)
+I |HW |Ψ(0)
+i ⟩
+Nif(E(0)
+i
+− E(0)
+I )
++
+�
+I̸=f
+⟨Ψ(0)
+f |HW |Ψ(0)
+I ⟩⟨Ψ(0)
+I |D|Ψ(0)
+i ⟩
+Nif(E(0)
+f
+− E(0)
+I )
+. (22)
+In the above expression of E1P V , correlations among
+the H and HW that appear through Eq. (17) are omit-
+ted. Secondly, there could be conflict with the definitions
+of using Core, Main and Tail contributions to E1P V with
+the definitions used in various first-principle based calcu-
+lations.
+This is explicitly demonstrated later how for-
+mula given by Eq. (20) can be altered to redefine Core
+and Valence contributions. To understand definitions of
+Core, Main and Tail contributions used in Ref. [24], we
+follow the work of Blundell et al [12] where these terms
+were used for the first time in the context of estimating
+E1P V . Division of the total E1P V value in this calcula-
+tion was made as “Main”, “Core” and “Tail” contribu-
+tions based on the mere assumption that |Ψ(0)
+i ⟩, |Ψ(0)
+f ⟩
+and |Ψ(0)
+I ⟩ are represented by only single Slater determi-
+nants like in the DHF method. Thus, the intermediate
+states |Ψ(0)
+I ⟩ are considered to have only the np 2P1/2
+configurations. In such assumption, the Core (C), Main
+(V ) and Tail (T ) contributions to the E1P V amplitude of
+the 6s 2S1/2−7s 2S1/2 transition in 133Cs were estimated
+as
+E1P V (C) =
+�
+n≤5
+⟨7S1/2|D|nP1/2⟩⟨nP1/2|HW |6S1/2⟩
+(E6S1/2 − EnP1/2)
++
+�
+n≤5
+⟨7S1/2|HW |nP1/2⟩⟨nP1/2|D|6S1/2⟩
+(E7S1/2 − EnP1/2)
+(23)
+E1P V (V ) =
+�
+n=6−9
+⟨7S1/2|D|nP1/2⟩⟨nP1/2|HW |6S1/2⟩
+(E6S1/2 − EnP1/2)
++
+�
+n=6−9
+⟨7S1/2|HW |nP1/2⟩⟨nP1/2|D|6S1/2⟩
+(E7S1/2 − EnP1/2)
+(24)
+and
+E1P V (T ) =
+�
+n≥10
+⟨7S1/2|D|nP1/2⟩⟨nP1/2|HW |6S1/2⟩
+(E6S1/2 − EnP1/2)
++
+�
+n≥10
+⟨7S1/2|HW |nP1/2⟩⟨nP1/2|D|6S1/2⟩
+(E7S1/2 − EnP1/2)
+,(25)
+respectively. However, wave functions of multi-electron
+atomic systems are determined through a many-body
+method by expressing as a linear combination of many
+Slater determinants which can differ by either single or
+multiple entries of rows or columns. As a result, con-
+tributions from cross-terms involving other Slater de-
+terminants, e.g.
+excited configurations 5p56s7s of the
+intermediate states with respect to both the 6s 2S1/2
+and 7s 2S1/2 states, cannot appear through the above
+
+6
+(i)
+(ii)
+(iii)
+(iv)
+(v)
+(vi)
+(vii)
+(viii)
+(ix)
+(x)
+(xi)
+(xii)
+...
+FIG. 3.
+A few representative DCP diagrams from the
+RMBPT(3) method.
+All-order forms of diagrams from (i)
+to (viii) appear in the CPDF-RPA method, but the rest are
+not. These missing contributions appear in the RCC theory
+to all-orders.
+breakup. One of such contributions is referred to as DCP
+effects which arise through the CPDF-RPA (or TDHF)
+method as described by Martensson [11] and Roberts
+[26].
+There are other contributions that could come
+through effects that are neither part of BO contributions
+nor CPDF-RPA method. However those effects appear
+through the first-principle approach of the RCC method
+employed by Sahoo et al [32] as shown later part of this
+paper. These contributions are not small and demands
+for an appropriate many-body method to account their
+contributions at the par with the np 2P1/2 intermediate
+states. This argument can be understood better with the
+following explanations.
+In the 133Cs atom, the low-lying excited states have a
+common core [5p6] and differ by only a valence orbital.
+Thus, the DHF wave functions of these states can be
+expressed as |ΦI⟩ = a†
+I|Φ0⟩ and the exact wave functions
+can be defined as
+|Ψ(0)
+I ⟩ = ΩI(0)|ΦI⟩
+= (Ω(0)
+0
++ Ω(0)
+I )|ΦI⟩,
+(26)
+where ΩI(0) and Ω(0)
+I
+are the total and valence corre-
+lation contributing wave operators respectively.
+Using
+these wave operators, we can express Eq. (22) as
+E1P V =
+�
+I̸=i
+⟨Φ0|af(Ω(0)
+0
++ Ω(0)
+f )†D(Ω(0)
+0
++ Ω(0)
+I )a†
+I|Φ0⟩
+Nif
+× ⟨Φ0|aI(Ω(0)
+0
++ Ω(0)
+I )†HW (Ω(0)
+0
++ Ω(0)
+i )a†
+i|Φ0⟩
+(E(0)
+i
+− E(0)
+I )
++
+�
+I̸=f
+⟨Φ0|af(Ω(0)
+0
++ Ω(0)
+f )†HW (Ω(0)
+0
++ Ω(0)
+I )a†
+I|Φ0⟩
+Nif
+× ⟨Φ0|aI(Ω(0)
+0
++ Ω(0)
+I )†D(Ω(0)
+0
++ Ω(0)
+i )a†
+i|Φ0⟩
+(E(0)
+f
+− E(0)
+I )
+. (27)
+Since wave operators from the initial, final and inter-
+mediate states include linear combinations of configu-
+rations describing one-hole–one-particle, two-hole–two-
+particle etc. excitations, it is obvious that a sum-over-
+states approach cannot include contributions from the
+higher-level excited configurations contributing to the in-
+termediate states.
+B.
+First-principle approach
+It is evident that it is imperative to determine the
+E1P V amplitudes in atomic systems using the first prin-
+ciple approaches that account contributions from all pos-
+sible intermediate configurations. This can be done using
+either Eq. (15) or Eq. (20). It is desirable to solve both
+Eqs.
+(16) and (17) in the former case, while Bloch’s
+equations for the unperturbed and perturbed wave op-
+erators need to be solved for the later approach. The
+amplitude solving Bloch’s equations for the unperturbed
+operators Ω(0)
+0
+and Ω(0)
+v
+are similar to that are given by
+Eqs. (9) and (10), respectively. The Bloch’s equations
+for the first-order perturbed wave operators can be given
+by
+[Ω(1)
+0 , H0]P0 = (HW Ω(0)
+0
++ UresΩ(1)
+0 )P0
+(28)
+and
+[Ω(1)
+v , H0]Pv = [HW (Ω(0)
+0
++ Ω(0)
+v ) + Ures(Ω(1)
+0
++ Ω(1)
+v )Pv
+−Ω(1)
+v E(0)
+v .
+(29)
+As mentioned in Introduction, several all-order meth-
+ods in the CPDF, RPA, CPDF-RPA/TDHF and RCC
+theory frameworks are employed to determine the E1P V
+amplitudes in 133Cs. Here, we attempt to formulate all
+these methods using wave operators so that in the end it
+would be easier for us to make one-to-one relations among
+various contributions arising through these methods. Es-
+pecially, such exercise is going to be useful in explaining
+the reason for which there is a sign difference between
+the Core contributions to E1P V from the TDHF method
+of Dzuba et al [25] and RCC method employed by Sahoo
+et al [29].
+
+7
+=
++
++
++
++
+a
+(a)
+(b)
+(c)
+(d)
+(e)
+p
++
++... ...
+(f)
+ΩCPDF
+0
++
+(g)
+FIG. 4. Goldstone diagrams contributing to amplitude deter-
+mining equation of ΩCP DF
+0
+. Through iterative scheme these
+effects are included to all-orders in the CPDF method.
+We can rewrite Eq. (15) as
+E1P V =
+⟨Ψ(0)
+f |HW |˜Ψ(1)
+i ⟩
+Nif
++
+⟨˜Ψ(1)
+f |HW |Ψ(0)
+i ⟩
+Nif
+. (30)
+This can be equivalently expressed by either
+E1P V =
+⟨Ψ(0)
+f |D|Ψ(1)
+i ⟩
+Nif
++
+⟨Ψ(0)
+f |HW |˜Ψ(1)
+i ⟩
+Nif
+(31)
+or
+E1P V =
+⟨˜Ψ(1)
+f |HW |Ψ(0)
+i ⟩
+Nif
++
+⟨Ψ(1)
+f |D|Ψ(0)
+i ⟩
+Nif
+.
+(32)
+In the above expressions, we define
+|˜Ψ(1)
+i ⟩ =
+�
+I̸=f
+|Ψ(0)
+I ⟩
+⟨Ψ(0)
+I |D|Ψ(0)
+i ⟩
+(E(0)
+i
+− E(0)
+I
++ ω)
+(33)
+and
+|˜Ψ(1)
+f ⟩ =
+�
+I̸=i
+|Ψ(0)
+I ⟩
+⟨Ψ(0)
+I |D|Ψ(0)
+f ⟩
+(E(0)
+f
+− E(0)
+I
+− ω)
+(34)
+with ω = E(0)
+f
+− E(0)
+i
+is the excitation energy between
+the initial and final states. It implies that mathemati-
+cally Eqs. (15), (30), (31) and (32) are equal in an ex-
+act many-body method. Thus, any of these expressions
+can be used in the determination of the E1P V ampli-
+tude. We shall demonstrate later that the CPDF, RPA,
+CPDF-RPA and RCC methods are using different formu-
+las as mentioned above. So it is important to understand
+their relations and classifications of individual contribu-
+tions through the above methods. Since approximations
+made to the unperturbed and perturbed wave functions
+are not at the same level in these methods, it is obvious to
+guess that results from these methods can be very differ-
+ent unless electron correlation effects in an atomic system
+are negligibly small. It is also not clear whether classifi-
+cations of Core and Tail contributions in these methods
+are uniquely defined or not.
+To understand the above points, lets find out the Core
+contributions from Eqs. (15) and (30) by expressing the
+perturbed wave function due to the D operator in terms
+of wave operators as
+|˜Ψ(1)
+v ⟩ = (˜Ω(1)
+0
++ ˜Ω(1)
+v )|Φv⟩.
+(35)
+With this, the Core contributing terms in both HW and
+D perturbing approaches are given by
+E1P V (C) = ⟨Φ0|af[Ω(0)†
+0
+DΩ(1)
+0
++ Ω(1)†
+0
+DΩ(0)
+0 ]a†
+i|Φ0⟩
+Nif
+(36)
+and
+E1P V (C) = ⟨Φ0|af[˜Ω(1)†
+0
+HW Ω(0)
+0
++ Ω(0)†
+0
+HW ˜Ω(1)
+0 ]a†
+i|Φ0⟩
+Nif
+.(37)
+After a careful analysis it can be shown that the Core con-
+tributions arising through the wave operators Ω(1)
+0
+and
+˜Ω(1)
+0
+can be different. Similar arguments also hold for the
+Tail contributions arising through the perturbed valence
+operators Ω(1)
+v
+and ˜Ω(1)
+v . To get better inside of this argu-
+ment, we can rewrite the sum-over-states formula given
+by Eq. (27) as
+E1P V =
+�
+I̸=i
+⟨Φ0|afΩ(0)†
+0
+DΩ(0)
+0 Ω(0)†
+0
+HW Ω(0)
+0 a†
+i|Φ0⟩
+Nif(E(0)
+f
+− E(0)
+I
+− ω)
++
+�
+I̸=i
+⟨Φ0|afΩ(0)†
+f
+DΩ(0)
+0 Ω(0)†
+0
+HW Ω(0)
+i a†
+i|Φ0⟩
+Nif(E(0)
+f
+− E(0)
+I
+− ω)
++
+�
+I̸=i
+⟨Φ0|afΩ(0)†
+0
+DΩ(0)
+0 Ω(0)†
+0
+HW Ω(0)
+i a†
+i|Φ0⟩
+Nif(E(0)
+f
+− E(0)
+I
+− ω)
++
+�
+I̸=i
+⟨Φ0|afΩ(0)†
+f
+DΩ(0)
+0 Ω(0)†
+0
+HW Ω(0)
+0 a†
+i|Φ0⟩
+Nif(E(0)
+f
+− E(0)
+I
+− ω)
++
+�
+I̸=i
+⟨Φ0|afΩ(0)†
+0
+DΩ(0)
+I Ω(0)†
+I
+HW Ω(0)
+0 a†
+i|Φ0⟩
+Nif(E(0)
+f
+− E(0)
+I
+− ω)
++
+�
+I̸=i
+⟨Φ0|afΩ(0)†
+f
+DΩ(0)
+I Ω(0)†
+I
+HW Ω(0)
+i a†
+i|Φ0⟩
+Nif(E(0)
+f
+− E(0)
+I
+− ω)
++
+�
+I̸=i
+⟨Φ0|afΩ(0)†
+0
+DΩ(0)
+I Ω(0)†
+I
+HW Ω(0)
+i a†
+i|Φ0⟩
+Nif(E(0)
+f
+− E(0)
+I
+− ω)
++
+�
+I̸=i
+⟨Φ0|afΩ(0)†
+f
+DΩ(0)
+I Ω(0)†
+I
+HW Ω(0)
+0 a†
+i|Φ0⟩
+Nif(E(0)
+f
+− E(0)
+I
+− ω)
++
+�
+I̸=i
+⟨Φ0|afΩ(0)†
+0
+DΩ(0)
+I Ω(0)†
+0
+HW Ω(0)
+0 a†
+i|Φ0⟩
+Nif(E(0)
+f
+− E(0)
+I
+− ω)
++
+�
+I̸=i
+⟨Φ0|afΩ(0)†
+f
+DΩ(0)
+I Ω(0)†
+0
+HW Ω(0)
+i a†
+i|Φ0⟩
+Nif(E(0)
+f
+− E(0)
+I
+− ω)
++
+�
+I̸=i
+⟨Φ0|afΩ(0)†
+0
+DΩ(0)
+I Ω(0)†
+0
+HW Ω(0)
+i a†
+i|Φ0⟩
+Nif(E(0)
+f
+− E(0)
+I
+− ω)
+
+8
++
+�
+I̸=i
+⟨Φ0|afΩ(0)†
+f
+DΩ(0)
+I Ω(0)†
+0
+HW Ω(0)
+0 a†
+i|Φ0⟩
+Nif(E(0)
+f
+− E(0)
+I
+− ω)
++
+�
+I̸=i
+⟨Φ0|afΩ(0)†
+0
+DΩ(0)
+0 Ω(0)†
+I
+HW Ω(0)
+0 a†
+i|Φ0⟩
+Nif(E(0)
+f
+− E(0)
+I
+− ω)
++
+�
+I̸=i
+⟨Φ0|afΩ(0)†
+f
+DΩ(0)
+0 Ω(0)†
+I
+HW Ω(0)
+i a†
+i|Φ0⟩
+Nif(E(0)
+f
+− E(0)
+I
+− ω)
++
+�
+I̸=i
+⟨Φ0|afΩ(0)†
+0
+DΩ(0)
+0 Ω(0)†
+I
+HW Ω(0)
+i a†
+i|Φ0⟩
+Nif(E(0)
+f
+− E(0)
+I
+− ω)
++
+�
+I̸=i
+⟨Φ0|afΩ(0)†
+f
+DΩ(0)
+0 Ω(0)†
+I
+HW Ω(0)
+0 a†
+i|Φ0⟩
+Nif(E(0)
+f
+− E(0)
+I
+− ω)
++ {HW ⇔ D}
+(38)
+Both Eqs. (27) and Eq. (38) are equal but they are
+given differently. These equations are nothing but the
+expanded forms of Eqs. (20) and (27) respectively. How-
+ever, different terms are rearranged to place them under
+the categories of Core and Valence contributing terms
+in the respective formulas. Thus, we may now outline
+findings from the above discussions as follows
+1. It is important to note that in the evaluation of
+E1P V both the HW and D operators can be treated
+symmetrically. Thus, in an approximated method
+where correlation effects through both these opera-
+tors are not incorporated equivalently, distinctions
+of “Core” and “Valence” contributions to E1P V
+cannot be defined uniquely.
+2. As a consequence of the above point, estimating
+both the “Core” and “Valence” contributions using
+a blend of many-body methods could mislead the
+final result.
+3. Numerical stability to the calculation of E1P V can
+be verified by evaluating expressions given by Eqs.
+(15), (30) and (31) simultaneously though it can be
+a strenuous procedure.
+4. Scaling wave functions for estimating a part of con-
+tribution or using experimental value of ω in a
+method in which both the HW and D operators
+are not treated symmetrically may not always im-
+ply that the result is improved rather it could in-
+troduce further uncertainty to the calculation.
+The last point mentioned above can be understood as
+discussed below. Lets use the experimental value for ω
+(shown as ωex) to define the first-order perturbed wave
+functions due to the D operator
+|˜Ψ(1)
+i ⟩ =
+�
+I̸=f
+|Ψ(0)
+I ⟩
+⟨Ψ(0)
+I |D|Ψ(0)
+i ⟩
+(E(0)
+i
+− E(0)
+I
++ ωex)
+(39)
+=
++
++
++
++
+p
+(a)
+(b)
+(c)
+(d)
+(e)
+v
++
++... ...
+(f)
+ΩCPDF
+v
+(g)
+FIG. 5. Diagrams denoting amplitude solving equation for
+ΩCP DF
+v
+. These core-polarization effects are included to all-
+orders in the CPDF method.
+and
+|˜Ψ(1)
+f ⟩ =
+�
+I̸=i
+|Ψ(0)
+I ⟩
+⟨Ψ(0)
+I |D|Ψ(0)
+f ⟩
+(E(0)
+f
+− E(0)
+I
+− ωex)
+.
+(40)
+Substituting these wave functions in Eq. (22), the sum-
+over-states expression for E1P V can be given by
+E1P V =
+�
+I̸=i
+⟨Ψ(0)
+f |D|Ψ(0)
+I ⟩⟨Ψ(0)
+I |HW |Ψ(0)
+i ⟩
+Nif(E(0)
+i
+− E(0)
+I
+− δω)
++
+�
+I̸=f
+⟨Ψ(0)
+f |HW |Ψ(0)
+I ⟩⟨Ψ(0)
+I |D|Ψ(0)
+i ⟩
+Nif(E(0)
+f
+− E(0)
+I
++ δω)
+, (41)
+where δω = ωex − ω, with ω being the theoretical value,
+cannot be zero when ω is obtained using a particular
+many-body method. As can be seen, introduction of ωex
+value affects contributions from the initial and final per-
+turbed terms differently leading to inconsistency in the
+evaluation of E1P V than the ab initio calculation. This
+can be better evident from the following inequalities in
+an approximated many-body method
+E1P V =
+⟨Ψ(0)
+f |D|Ψ(1)
+i ⟩
+Nif
++
+⟨Ψ(1)
+f |D|Ψ(0)
+i ⟩
+Nif
+̸=
+⟨Ψ(0)
+f |HW |˜Ψ(1)
+i ⟩
+Nif
++
+⟨˜Ψ(1)
+f |HW |Ψ(0)
+i ⟩
+Nif
+̸=
+⟨Ψ(0)
+f |D|Ψ(1)
+i ⟩
+Nif
++
+⟨Ψ(0)
+f |HW |˜Ψ(1)
+i ⟩
+Nif
+̸=
+⟨˜Ψ(1)
+f |HW |Ψ(0)
+i ⟩
+Nif
++
+⟨Ψ(1)
+f |D|Ψ(0)
+i ⟩
+Nif
+.
+(42)
+The above inequalities are the result of introduction of
+δω in the first-order perturbed wave function due to D
+after the substitution of ωex in place of ω.
+IV.
+MANY-BODY METHODS OF E1P V
+The main objective in the APV study is to obtain the
+E1P V amplitude within sub-one percent accuracy from
+
+9
+the atomic many-body theory perspective.
+In view of
+this, it is imperative to evaluate both the zeroth-order
+and first-order atomic wave functions in Eq. (15) very
+accurately by employing a powerful relativistic atomic
+many-body method. Owing to complications in account-
+ing for various contributions, the entire calculation is
+usually performed in several steps. The majority of the
+contribution from H arises from electron correlation ef-
+fects due to Coulomb interactions in the presence of APV
+interactions, while corrections from the Breit and QED
+interactions are added separately.
+It would be neces-
+sary to include all these interactions through a com-
+mon many-body theory in order to consider correlations
+among themselves as well.
+Since corrections from the
+Breit and QED interactions to E1P V are very small, their
+reported estimations are found to be almost consistent
+to each other by various works [14–20, 32]. Thus, we fo-
+cus here mainly on the discussions considering the Dirac-
+Coulomb (DC) interaction Hamiltonian in the determi-
+nation of unperturbed wave functions. Again, our aim in
+the present paper is to demonstrate how to achieve high-
+accuracy calculations of the E1P V amplitudes in 133Cs
+by understanding about the roles of Core correlations to
+these quantities and identifying contributions that can
+arise through a particular many-body method but can be
+missed by another method. To explain these points ex-
+plicitly, we discuss calculations of the E1P V amplitudes
+in 133Cs using the following methods
+1. Relativistic
+Many-body
+Perturbation
+Theory
+(RMBPT):
+This
+method
+is
+employed
+in
+the
+Rayleigh-Schr¨odinger perturbation theory frame-
+work to fathom about significance of various
+physical effects to E1P V arising at the lower-order
+level and to learn how they propagate in the
+all-order perturbative methods.
+2. CPDF method:
+This method is employed in or-
+der to reproduce previous results reported by other
+groups using the same method using our basis func-
+tions that are later considered in the RCC calcula-
+tions.
+3. RPA: This method is employed for the same pur-
+pose as the above and show importance of coore-
+lation contributions arising through this method
+which are missing in the CPDF method.
+4. CPDF-RPA method:
+This method is employed
+again for the above purpose as well as understand-
+ing the reason for which Core contribution from
+Ref. [25] has a different sign than other works.
+5. RCC method: This method is employed both in
+the sum-over-states and first-principle approaches.
+Differences in both the results are further com-
+pared with the values that are included through
+the CPDF-RPA method.
+(a)
+(b)
+(c)
+(d)
+f
+f
+f
+f
+i
+i
+i
+i
+p
+p
+a
+a
+ΩCPDF
+0
+ΩCPDF
+v
+ΩCPDF†
+0
+D
+D
+D
+D
+ΩCPDF†
+v
+FIG. 6.
+Property diagrams of the CPDF method.
+These
+diagrams are similar to the DHF method, but the HW op-
+erators of the DHF diagrams are replaced by ΩCP DF
+0
+and
+ΩCP DF
+v
+. Expansion of ΩCP DF
+0
+and ΩCP DF
+v
+in terms of am-
+plitude diagrams shown in Figs. 4 and 5 can reveal how the
+DHF contributions and lower-order core-polarization effects
+of the RMBPT(3) method are incorporated within the CPDF
+method.
+The atomic Hamiltonian H with DC approximation
+can be expressed as sum of one-body and two-body op-
+erators
+H =
+�
+i
+�
+c⃗αD · ⃗pi + (β − 1)c2 + Vnuc(ri)
+�
++
+�
+i,j>i
+1
+rij
+=
+�
+i
+h(ri) +
+�
+i,j>i
+g(rij),
+(43)
+where c is the speed of light, ⃗αD and β are the Dirac
+matrices, ⃗p is the single particle momentum operator,
+and �
+i,j
+1
+rij represents the Coulomb potential between
+the electrons located at the ith and jth positions. The
+entire Hamiltonian can be divided into one-body part
+(�
+i h(ri)) and two-body part (�
+i,j>i g(rij)) for conve-
+nience. Using the DHF method, we can obtain the single
+particle orbitals using the modified DHF Hamiltonian as
+F = �
+i f(ri) = �
+i(hi + u(0)
+i ) and residual interaction
+as Vres = H − F in the DC approximation. Hence
+f|i⟩ = ǫi|i⟩
+(44)
+with the single particle energies ǫi giving unperturbed
+DHF energy E0
+=
+�Nc
+b
+ǫb and the DHF potential
+UDHF = �
+i u0(ri) is given by
+u0(ri)|i⟩ =
+Nc
+�
+b
+[⟨b|g|b⟩|i⟩ − ⟨b|g|i⟩|b⟩]
+(45)
+for b summing over all occupied-orbitals Nc. In the DHF
+method, both Eqs. (44) and (45) are solved iteratively to
+obtain self-consistent solutions. It can be followed from
+the above expression that for the determinant expressed
+by |Φk⟩ = a†
+k|Φ0⟩, the DHF energy is given by Ek = E0 +
+ǫk. Similarly, DHF energies of the excited configurations
+are given by Ep,q,···
+a,b,··· = E0 + ǫp + ǫp + · · · − ǫa − ǫb − · · ·
+and {p, q, · · · } denoting for the occupied and unoccupied
+orbitals respectively. We discuss below E1P V expressions
+
+10
+=
++
++
++
++
+a
+(a)
+(b)
+(c)
+(d)
+(e)
+p
++
+(f)
++... ...
+ΩRPA
+0
++
+(g)
+FIG. 7. Graphical representation of ΩRP A
+0
+and its expansion
+in terms of lower-order RMBPT(3) method.
+using the DHF method and different many-body methods
+that are employed to account for the electron correlation
+effects due to Vres.
+A.
+DHF method
+Using wave functions from the DHF method, we can
+evaluate the E1P V amplitude in the mean-field approach
+as
+E1P V = ⟨Φf|D|Φ(1)
+i ⟩ + ⟨Φ(1)
+f |D|Φi⟩,
+(46)
+where |Φ(1)
+n=i,f⟩ is the first-order perturbed wave func-
+tion with respect to |Φn=i,f⟩. We can express these wave
+function as
+|Φ(1)
+n ⟩ =
+�
+I
+|ΦI⟩⟨ΦI|HW |Φn⟩
+En − EI
+,
+(47)
+where |ΦI⟩ are the intermediate states with mean-field
+energies EI. Substituting this expression above, it yields
+E1P V =
+�
+I̸=i
+⟨Φf|D|ΦI⟩⟨ΦI|HW |Φi⟩
+Ei − EI
++
+�
+I̸=f
+⟨Φf|HW |ΦI⟩⟨ΦI|D|Φi⟩
+Ef − EI
+.
+(48)
+Using HW = �
+i hw(ri) and D = �
+i d(ri), and following
+the Slater-Condon rules, it gives
+E1P V =
+�
+a
+⟨f|d|a⟩⟨a|hw|i⟩
+ǫi − ǫa
++
+�
+a
+⟨f|hw|a⟩⟨a|d|i⟩
+ǫf − ǫa
++
+�
+p̸=i
+⟨f|d|p⟩⟨p|hw|i⟩
+ǫi − ǫp
++
+�
+p̸=f
+⟨f|hw|p⟩⟨p|d|i⟩
+ǫf − ǫp
+, (49)
+where |k = a, p⟩ denotes kth single particle DHF orbital
+with energy ǫk for the occupied orbitals denoted by a and
+virtual orbitals denoted by p. Contributions arising from
+the first two terms of the above expression are referred
+to as the lowest-order Core contributions while contri-
+butions from the later two terms are said to be Valence
+=
++
++
++
++
+p
+(a)
+(b)
+(c)
+(d)
+(e)
+v
++
++... ...
+(f)
+(ΩRPA
+v
+)
++
+(g)
+FIG. 8. Graphical representation of ΩRP A
+v
+and its expansion
+in terms of lower-order RMBPT(3) method.
+contributions that include the lowest-orders to both Main
+and Tail parts.
+In terms of wave operators, the DHF expression for
+E1P V can be given by
+E1P V = ⟨Φf|DΩi(0,1)|Φi⟩ + ⟨Φf|Ωf(0,1)†D|Φi⟩, (50)
+where Ωv(0,1) = Ω(0,1)
+0
++ Ω(0,1)
+v
+.
+Further, Ω(0,1)
+0
+=
+�
+a,p
+⟨Φp
+a|HW |Φ0⟩
+E0−Ep
+a
+a†
+paa = �
+a,p
+⟨p|hw|a⟩
+ǫa−ǫp a†
+paa ≡ �
+a,p Ωp
+a
+and Ω(0,1)
+v
+= �
+p
+⟨Φp
+v|HW |Φv⟩
+Ev−Ep
+v
+a†
+pav
+=
+⟨p|hw|v⟩
+ǫv−ǫp a†
+paa
+≡
+�
+p Ωp
+v for the intermediate states |Φp
+a⟩ with energies
+Ep
+a = E0+ǫp−ǫa and |Φp
+v⟩ with energies Ep
+v = E0+ǫp with
+respect to the |Φ0⟩ and |Φv⟩ respectively. Representing
+the wave operators in terms of the Goldstone diagrams,
+we show the Core and Valence contributions to E1P V in
+Fig. 1. Figs. 1(a) and (b) correspond to Core contribut-
+ing terms, while Figs. 1(c) and (d) correspond to Valence
+contributing terms here.
+B.
+RMBPT method
+We employ the RMBPT method in the Rayleigh-
+Schr¨odinger approach and estimate contributions only up
+to third-order of perturbation (RMBPT(3) method) by
+considering two orders of Vres and one-order of HW ; i.e.
+the net Hamiltonian is expressed as
+Hat = F + λ1Vres + λ2HW ,
+(51)
+where λ1 and λ2 are arbitrary parameters introduced to
+count orders of Vres and HW in the calculation. Here, we
+can calculate either matrix element of D after perturb-
+ing wave functions by HW or matrix element of HW after
+perturbing wave functions by D. We adopt here both ap-
+proaches for two reasons. First, it can help us to identify
+the lower-order contributions terms to the CPDF and
+RPA methods so that their inclusion through the RCC
+method can be easily understood. Second, it would be
+interesting to see classification of the Core and Valence
+contributions in both the approaches of the RMBPT
+method. In both the approaches, the unperturbed wave
+
+11
+operators in the RMBPT method can be given by
+Ω(0)
+n
+=
+�
+k
+Ω(k,0)
+n
+.
+(52)
+Similarly, the first-order perturbed wave operators can
+be denoted by
+Ω(1)
+n
+=
+�
+k
+Ω(k,1)
+n
+,
+(53)
+where subscript n = 0, v stands for core or valence oper-
+ators and superscript k denotes for order of Vres. Ampli-
+tudes of the wave operators in the RMBPT method are
+determined by
+[Ω(k,0)
+0
+, F]P0 = Q0VresΩ(k−1,0)
+0
+P0
+(54)
+and
+[Ω(k,0)
+v
+, F]Pv = QvVres(Ω(k−1,0)
+0
++ Ω(k−1,0)
+v
+)Pv
+−
+k−1
+�
+m=1
+Ω(k−m,0)
+v
+E(m,0)
+v
+,
+(55)
+where E(m,0)
+v
+is the mth-order perturbed energy of
+the total energy E(0)
+v
+= �∞
+m=1 E(m,0)
+v
+and is evalu-
+ated by using kth-order effective Hamiltonian H(m,0)
+eff
+=
+PvVres(Ω(m−1,0)
+0
++Ω(m−1,0)
+v
+)Pv as part of the net effective
+Hamiltonian H(0)
+eff = �∞
+m=1 H(m,0)
+eff
+.
+Amplitudes of the perturbed wave operators due to
+HW can be evaluated by
+[Ω(k,1)
+0
+, F]P0 = Q0HwΩ(k,0)
+0
+P0 + Q0VresΩ(k−1,1)
+0
+P0 (56)
+and
+[Ω(k,1)
+v
+, F]Pv = QvVres(Ω(k−1,1)
+0
++ Ω(k−1,1)
+v
+)Pv
++QvHw(Ω(k,0)
+0
++ Ω(k,0)
+v
+)Pv
+−
+k−1
+�
+m=1
+Ω(k−m,1)
+v
+E(m,0)
+v
+.
+(57)
+In the above expression, the lower-order unperturbed
+wave operators are given by Ω(0,0)
+n
+= 1 and Ω(1,0)
+n
+= 0.
+For the case of considering D as perturbation, Eqs. (56)
+and (57) can be again used to solve amplitudes of the
+˜Ω(1)
+0 , ˜Ω(1)
+i
+and ˜Ω(1)
+f
+operators in the RMBPT methods
+by replacing Ωp
+a/v by Ωp+
+a/v =
+⟨p|d|a⟩
+ǫa/v−ǫp−ωa†
+paa/v and Ωp†
+a/v
+by complex conjugate of Ωp−
+a/v =
+⟨p|d|v⟩
+ǫa/v−ǫp+ωa†
+paa/v. This
+follows, the nth-order E1P V expression as
+E1P V =
+1
+�n−1
+l=0 N l
+if
+�
+n
+�
+k=0
+�
+⟨Φf|(Ω(n−k,0)
+0
++ Ω(n−k,0)
+f
+)†D
+(Ω(k,1)
+0
++ Ω(k,1)
+i
+)
+�
+|Φi⟩ +
+n
+�
+k=0
+�
+⟨Φf|(Ω(k,1)
+0
++Ω(k,1)
+f
+)†D(Ω(n−k,0)
+0
++ Ω(n−k,0)
+i
+�
+|Φi⟩)
+�
+,
+(58)
+(a)
+(b)
+(c)
+(d)
+f
+f
+f
+f
+i
+i
+i
+i
+p
+p
+a
+a
+ΩRPA
+0
+ΩRPA
+v
+ΩRPA†
+0
+ΩRPA†
+v
+HW
+HW
+HW
+HW
+FIG. 9. Property contributing diagrams of the RPA. These
+diagrams are similar to the CPDF diagrams, but the core-
+polarization effects are included through the D operator in-
+stead of HW . Expansion of the ΩRP A
+0
+and ΩRP A
+v
+in terms
+of Figs. 7 and 8 can show that the RPA property diagrams
+include the DHF contributions and core-polarization effects
+that are distinctly different than the CPDF method.
+where HW
+is considered in the perturbation with
+N k
+if
+=
+[(�
+l⟨Φf|(Ω(k−l,0)
+0
++
+Ω(k−l,0)
+f
+)†(Ω(l,0)
+0
++
+Ω(l,0)
+f
+)|Φf⟩)(�
+m⟨Φi|(Ω(k−m,0)
+0
++ Ω(k−m,0)
+i
+)†(Ω(m,0)
+0
++
+Ω(m,0)
+i
+)|Φi⟩)]1/2.
+In the case of D as perturbation, it yields
+E1P V =
+1
+�n−1
+l=0 N l
+if
+�
+n
+�
+k=0
+�
+⟨Φf|(Ω(n−k,0)
+0
++ Ω(n−k,0)
+f
+)†HW
+(˜Ω(k,1)
+0
++ ˜Ω(k,1)
+i
+)
+�
+|Φi⟩ +
+n
+�
+k=0
+�
+⟨Φf|(˜Ω(k,1)
+0
++˜Ω(k,1)
+f
+)†HW (Ω(n−k,0)
+0
++ Ω(n−k,0)
+i
+�
+|Φi⟩)
+�
+.
+(59)
+In Fig. 2, we show the important correlation contribut-
+ing Goldstone diagrams to E1P V arising through the
+RMBPT(3) method. It should be noted that the lowest-
+order diagrams of the RMBPT(3) method are same as
+the diagrams corresponding to the DHF method and
+they are not shown here. Since both Eqs. (58) and (59)
+are equivalent at given level of approximation, the Gold-
+stone diagrams are identical in both the cases.
+Thus,
+the Core and Valence contributions arising through both
+the expressions can be distinguished and quoted sepa-
+rately by adopting the definitions of the respective wave
+operators.
+This would help us identifying lower-order
+Core and Valence correlation contributions to the CPDF,
+RPA, CPDF-RPA and RCC methods that are going to be
+discussed next. In order to distinguish results while pre-
+senting from both the approaches, we use the notations
+RMBPT(3)w and RMBPT(3)d in place of RMBPT(3) for
+the cases with HW as the perturbation and with D as the
+perturbation respectively.
+C.
+CPDF Method
+In the CPDF method, it is possible to extend the E1P V
+calculation to all-orders in a very simple manner by ex-
+
+12
+tending the DHF expression and with much less com-
+putational effort compared to the RMBPT(3) method.
+This can be derived by starting with the DHF expres-
+sion, given by
+E1P V = ⟨f|d|i(1)⟩ + ⟨f (1)|d|i⟩,
+(60)
+where
+|k(1)⟩ =
+�
+I̸=k
+|I⟩⟨I|hw|k⟩
+ǫk − ǫI
+.
+(61)
+In the CPDF method, the first-order perturbed sin-
+gle particle orbital |k(1)⟩ is obtained by including core-
+polarization effects due to Vres to all-orders (denoted by
+|kP V ⟩) by defining an effective potential, similar to u(0)
+i ,
+in the presence of hw. To arrive at this expression, we
+consider the net Hamiltonian Hint to define the modified
+single particle DHF Hamiltonian f P V
+i
+= fi + λ2hw and
+potential as
+f P V
+i
+|˜i⟩ = ˜ǫi|˜i⟩
+(62)
+and
+˜ui =
+Nc
+�
+b
+�
+⟨˜b|gbi|˜b⟩|˜i⟩ − ⟨˜b|gbi|˜i⟩|˜b⟩
+�
+,
+(63)
+where the tilde symbol denotes solution for Hint in place
+of H. Now expanding |˜i⟩ = |i⟩ + λ2|iP V ⟩ + O(λ2
+2) from
+Eq. (62) and retaining terms that are linear in λ2, we
+can get
+(fi − ǫi)|iP V ⟩ = −hw|i⟩ − uP V
+i
+|i⟩,
+(64)
+where
+uP V
+i
+|i⟩ =
+Nc
+�
+b
+�
+⟨b|g|b⟩|iP V ⟩ − ⟨b|g|iP V ⟩|b⟩
++⟨bP V |g|b⟩|i⟩ − ⟨bP V |g|i⟩|b⟩
+�
+.
+(65)
+Like Eqs.
+(44) and (45), both Eqs.
+(64) and (65)
+are also solved iteratively to obtain the self-consistent
+solutions to account for core-polarization effects to all-
+orders. Using the above APV modified orbitals, E1P V
+can be evaluated as
+E1P V = ⟨f|d|iP V ⟩ + ⟨f P V |d|i⟩.
+(66)
+To make one-to-one comparison between contributions
+arising through the CPDF method and other many-body
+methods including lower-order terms of the RMBPT(3)
+method, we can present the CPDF expression for E1P V
+using the wave operators as
+E1P V = ⟨Φf|DΩi,CP DF |Φi⟩ + ⟨Φf|Ωf,CP DF †D|Φi⟩,(67)
+where
+Ωv,CP DF
+=
+ΩCP DF
+0
++
+ΩCP DF
+v
+=
+�∞
+k=1
+��
+a,p Ω(k,1)
+a,p
++ �
+p Ω(k,1)
+v,p
+�
+with
+subscripts
+0
+and v stand for operators responsible for including core
+=
++
++
++
++
++
++
++
++
++
++
++
++
++
++... ...
+(i)
+(ii)
+(iii)
+(iv)
+(v)
+(vi)
+(vii)
+(viii)
+(ix)
+(x)
+(xi)
+(xii)
+(xiii)
+(xiv)
+ΩCPDF+
+(xv)
+(xvi)
++
++
+FIG. 10. Goldstone diagrams to the amplitude solving equa-
+tions of the ΩCP DF ± operator that gives rise to DCP con-
+tributions in the CPDF-RPA method.
+Diagrams from (ix)
+to (xvi) along with their exchanges are coming due to the
+implementation of the orthogonalization condition.
+and valence correlations, and superscript k denotes order
+of Vres. Amplitudes of these operators are given by
+Ω(k,1)
+a,p
+= Ωp
+a +
+�
+b,q
+�[⟨pb|g|aq⟩ − ⟨pb|g|qa⟩]
+ǫa − ǫp
+Ω(k−1,1)
+b,q
++Ω(k−1,1)†
+b,q
+[⟨pq|g|ab⟩ − ⟨pq|g|ba⟩]
+ǫa − ǫp
+�
+(68)
+and
+Ω(k,1)
+v,p
+= Ωp
+v +
+�
+b,q
+�[⟨pb|g|vq⟩ − ⟨pb|g|qv⟩]
+ǫv − ǫp
+Ω(k−1,1)
+b,q
++Ω(k−1,1)†
+b,q
+[⟨pq|g|vb⟩ − ⟨pq|g|bv⟩]
+ǫv − ǫp
+�
+,
+(69)
+where the sums over a, b and p, q represent occupied and
+virtual operators respectively. To compute amplitude of
+the above operators, we set Ω(0,1)
+a,p
+≈ Ωp
+a and Ω(0,1)
+v,p
+≈ Ωp
+v
+in the beginning to initiate the iteration procedure from
+k = 1. As can be followed here that only the effective
+singly excited configurations are contributing through
+the Ωv,CP DF operators. Thus, it completely misses out
+pair-correlation contributions.
+The Goldstone diagrams that contribute to the ampli-
+tudes of ΩCP DF
+0
+are shown in Fig. 4. Similarly, the Gold-
+stone diagrams contributing to the amplitudes of ΩCP DF
+v
+are shown in Fig.
+5.
+Using these operators, we show
+the final Goldstone diagrams that contribute to E1P V
+in Fig. 6. By analyzing these diagrams in terms of the
+Goldstone diagrams shown in Figs. 4 and 5, it is easy to
+follow how the core-polarization effects are included to
+all-orders through the CPDF method. Again, compar-
+ing those diagrams with the diagrams from the DHF and
+
+13
+ΩCPDF†
+0
+ΩRPA†
+0
+ΩCPDF
+v
+ΩRPA
+v
+ΩCPDF+
+(i)
+(ii)
+(iii)
+(iv)
+(v)
+(vi)
+(vii)
+(viii)
+(ix)
+(x)
+(xi)
+(xii)
+(xiii)
+(xiv)
+(xv)
+(xvi)
+...
+D
+HW
+FIG. 11. Goldstone diagrams contributing to the E1P V am-
+plitude in the CPDF-RPA method.
+Comparing these dia-
+grams with the diagrams of the RMBPT(3) method, can be
+understood by expanding the ΩCP DF
+0/v
+, ΩRP A
+0/v
+and ΩCP DF ±
+operators, it can be followed that the CPDF-RPA method
+does not include contributions from pair-correlation effects
+and some of the DCP effects representing diagrams.
+RMBPT(3) methods the relations among them can be
+easily understood. One can also realize here that number
+of Goldstone diagrams appear in the CPDF method are
+very less than the RMBPT(3) method. Thus, the com-
+putational efforts in the CPDF method are much smaller
+compared to the RMBPT(3) method though it is an all-
+order theory.
+D.
+RPA
+From Eq.
+(66), it can be followed that the CPDF
+method includes correlation effects in the first-order wave
+functions through the HW operator only. Thus, it com-
+pletely misses out correlation effects in the unperturbed
+state that can arise through the D operator. It can also
+be noted that the CPDF method is formulated based
+on Eq. (15). Therefore proceeding with a similar man-
+ner based on Eq.
+(30), it can lead to capturing core-
+polarization effects through the D operator and the RPA
+is formulated exactly on the same line.
+To derive the RPA expression, we consider the net
+Hamiltonian H±
+int = H + λ3D ∓ ω to define the modi-
+fied single particle DHF Hamiltonian f ±
+i = fi + λ3d ∓ ω
+and potential as
+f ±
+i |˜i±⟩ = ˜ǫ±
+i |˜i±⟩
+(70)
+=
++
++
++
++
+T(0)
+1
+T(0)
+2
+=
++
++
++
++
+(ii)
+(iii)
+(iv)
+(i)
+(ii)
+(iii)
+(v)
+(vi)
++... ...
+(i)
+(v)
++... ...
++
+(iv)
+T(0)
+2
+FIG. 12.
+Goldstone diagrams demonstrating breakdown of
+the T (0) operators in terms of lower-order perturbative exci-
+tations.
+and
+˜u±
+i =
+Nc
+�
+b
+�
+⟨˜b±|g|˜b±⟩|˜i±⟩ − ⟨˜b±|g|˜i±⟩|˜b±⟩
+�
+,
+(71)
+respectively, where the superscript ± denotes solution for
+H±
+int in place of H. Now expanding |i±⟩ = |i⟩ + λ3|i±⟩ +
+O(λ2
+3) and retaining terms that are linear in λ3, we can
+get
+(fi − ǫi ∓ ω)|i±⟩ = −d|i⟩ − u±
+i |i⟩,
+(72)
+where
+u±
+i |i⟩ =
+Nc
+�
+b
+�
+⟨b|g|b⟩|i±⟩ − ⟨b|g|i±⟩|b⟩
++⟨b∓|g|b⟩|i⟩ − ⟨b∓|g|i⟩|b⟩
+�
+.
+(73)
+Here also both Eqs. (72) and (73) are solved iteratively
+to obtain the self-consistent solutions to account for core-
+polarization effects to all-orders. Using the above D op-
+erator modified orbitals, E1P V can be evaluated as
+E1P V = ⟨Φf|HW Ωi,+|Φi⟩ + ⟨Φf|Ωf,−†HW |Φi⟩, (74)
+where
+Ωv,±
+=
+Ω±
+0
++
+Ω±
+v
+=
+�∞
+k=1
+��
+a,p Ω±(k,1)
+a,p
++ �
+p Ω±(k,1)
+v,p
+�
+with
+subscripts
+0 and v stand for operators responsible for including
+Core and Valence correlations, and superscript k denotes
+order of Vres. Amplitudes of these operators are given
+by
+Ω±(k,1)
+a,p
+= Ωp±
+a
++
+�
+b,q
+�[⟨pb|g|aq⟩ − ⟨pb|g|qa⟩]
+ǫa − ǫp ± ω
+Ω±(k−1,1)
+b,q
++Ω∓(k−1,1)†
+b,q
+[⟨pq|g|ab⟩ − ⟨pq|g|ba⟩]
+ǫa − ǫp ± ω
+�
+(75)
+
+14
+=
++
++
++... ...
+=
++
++
++... ...
+(i)
+S(0)
+1v
+(ii)
+(iii)
+S(0)
+2v
+(i)
+(ii)
+(iii)
+FIG. 13.
+Goldstone diagrams demonstrating breakdown of
+the S(0)
+v
+operators in terms of lower-order perturbative exci-
+tations.
+and
+Ω±(k,1)
+v,p
+= Ωp±
+v
++
+�
+b,q
+�[⟨pb|g|vq⟩ − ⟨pb|g|qv⟩]
+ǫv − ǫp ± ω
+Ω±(k−1,1)
+b,q
++Ω∓(k−1,1)†
+b,q
+[⟨pq|g|vb⟩ − ⟨pq|g|bv⟩]
+ǫv − ǫp ± ω
+�
+,
+(76)
+where we assume Ω±(0,1)
+a,p
+≈ Ωp±
+a
+and Ω±(0,1)
+v,p
+≈ Ωp±
+v
+initially for the iteration procedure. We also intend to
+mention here is that in Eqs.
+(75) and (76) , ω value
+can be used from the experiment while in the ab ini-
+tio framework it is taken from the DHF method. Fol-
+lowing the explanation in the previous sub-section, it is
+obvious that the RPA wave operators will pick up core-
+polarization correlations through the D operator to all-
+orders. Again based on the classification adopted in this
+work, the Goldstone diagrams contributing to the ampli-
+tude determining equation for Core operator are shown
+in Fig. 7, while the diagrams contributing to the ampli-
+tudes of the Valence operator are shown in Fig. 8.
+Using the above operators, we show the final Goldstone
+diagrams that contribute to E1P V of RPA in Fig. 9. By
+analyzing these diagrams in terms of the Goldstone di-
+agrams shown in Figs. 7 and 8, it can be followed how
+the core-polarization effects are included through D to
+all-orders through the RPA. Again, comparing these dia-
+grams with the diagrams from the DHF and RMBPT(3)
+methods the relations among both the methods can be
+understood. Though the number of Goldstone diagrams
+appear in the RPA and the CPDF method are same, but
+amplitudes in the CPDF method are expected to con-
+verge faster than the RPA owing to strong correlations
+arising through D than HW . It can be noticed here that
+the Core correlations (without DHF contributions) aris-
+ing in the RPA are distinctly different than that appear
+via the CPDF method.
+=
++
++
++
++
+(i)
+(iii)
+(iv)
+(v)
++... ...
+(ii)
+T(1)
+1
+=
+T(1)
+2
+T (1)
+1
+(i)
++
+(iii)
+T (0)
+2
++
++... ...
+T (0)
+2
+(iv)
++
+(v)
+T (0)
+2
+(ii)
+T (1)
+1
+(vi)
++
+T (1)
+1
+FIG. 14.
+Goldstone diagrams demonstrating breakdown of
+the T (1) operators in terms of lower-order perturbative exci-
+tations.
+E.
+CPDF-RPA/TDHF method
+As realized above, the CPDF method and the RPA
+include core-polarization effects only through the first-
+order perturbed wave functions but the unperturbed
+wave functions and energies in both the cases are used
+from the DHF method.
+In order to achieve core-
+polarization effects through both the states it is necessary
+to include the HW and D operators in the perturbation.
+The simplest approach for doing so would be to add both
+the CPDF and RPA results and remove repeated appear-
+ance of the DHF value from one of the approaches. How-
+ever, such an approach would omit correlations among
+both the HW and D operators which may not be negli-
+gible.
+Keeping in view of the above, we define the total
+Hamiltonian as
+Ht = H + λ2HW + λ3D
+≡ Hint + λ3D.
+(77)
+Treating both the HW and D operators perturbatively,
+the exact atomic wave function (|Ψv⟩) of Ht can be ex-
+pressed as
+|Ψv⟩ = |Ψ(0,0)
+v
+⟩ + λ2|Ψ(1,0)
+v
+⟩ + λ3|Ψ(0,1)
+v
+⟩
++λ2λ3|Ψ(1,1)
+v
+⟩ + · · · .
+(78)
+As denoted earlier, |Ψ(m,n)
+v
+⟩ represents consideration of
+m orders of Hw and n orders of D in the atomic wave
+function |Ψv⟩ of H. In the wave operator formalism, it
+is given by
+Ωv|Φv⟩ = Ω(0,0)
+v
+|Φv⟩ + λ2Ω(1,0)
+v
+|Φv⟩ + λ3Ω(0,1)
+v
+|Φv⟩
++λ2λ3Ω(1,1)
+v
+|Φv⟩ + · · · ,
+(79)
+
+15
+=
++
++
++
++
+S(1)
+1v
+(i)
+(ii)
+(iii)
+(iv)
+=
++
++
++
+S(1)
+2v
+(i)
+(ii)
+(iii)
+(iv)
++... ...
++
+(v)
+S(0)
+1v
+(vi)
++... ...
+(v)
++
+FIG. 15.
+Goldstone diagrams demonstrating breakdown of
+the S(1)
+v
+operators in terms of lower-order perturbative exci-
+tations.
+where superscripts denote the same meaning as above. It
+can be noted that each Ω(m,n)
+v
+component will have parts
+carrying Core and Valence correlations separately.
+In this case, we can determine the E1P V amplitude as
+the transition amplitude of O ≡ λ2Hw + λ3D between
+the initial perturbed state to the final unperturbed state
+or between the initial unperturbed state to the final per-
+turbed state (see Eqs. (31) and (32)). i.e.
+E1P V = ⟨Ψ(0,0)
+f
+|Ψ(1,1)
+i
+⟩ + ⟨Ψ(0,0)
+f
+|D|Ψ(1,0)
+i
+⟩
++⟨Ψ(0,0)
+f
+|HW |Ψ(0,1)
+i
+⟩
+= ⟨Φf|Ω(0,0)†
+f
+Ω(1,1)
+i
+|Φi⟩ + ⟨Φf|Ω(0,0)†
+f
+DΩ(1,0)
+i
+|Φi⟩
++ ⟨Φf|Ω(0,0)†
+f
+HW Ω(0,1)
+i
+|Φi⟩
+(80)
+or
+E1P V = ⟨Ψ(1,1)
+f
+|Ψ(0,0)
+i
+⟩ + ⟨Ψ(1,0)
+f
+|D|Ψ(0,0)
+i
+⟩
++⟨Ψ(0,1)
+f
+|HW |Ψ(0,0)
+i
+⟩
+= ⟨Φf|Ω(1,1)†
+f
+Ω(0,0)
+i
+|Φi⟩ + ⟨Φf|Ω(1,0)†
+f
+DΩ(0,0)
+i
+|Φi⟩
++ ⟨Φf|Ω(0,1)†
+f
+HW Ω(0,0)
+i
+|Φi⟩,
+(81)
+keeping terms that are of the order of λ2λ3. Note that
+both HW and D operators are treated on an equal foot-
+ing in this approach. Thus, definitions of both Core and
+Valence contributions to E1P V will be identical for both
+Eqs. (80) and (81). Also, it would be prudent to use
+both the expressions in an approximated method to ver-
+ify numerical uncertainty to the final result. However, if
+ωex (some earlier studies have done it through the scaling
+D
+T(1)
+1
+T(1)†
+1
+S(1)
+1i
+S(1)†
+1f
+S(1)
+2i
+S(1)†
+2f
+S(0)
+2i
+S(1)†
+1f
+S(0)†
+1f
+S(1)
+1i
+S(0)†
+2f
+T(1)
+1
+S(0)
+1i
+S(1)†
+2f
+S(0)
+2i
+D
+(ii)
+(i)
+(iii)
+(iv)
+(v)
+(vi)
+(vii)
+(viii)
+(ix)
+(x)
+S(0)†
+2f
+S(1)
+1i
+S(0)
+2i
+S(0)†
+1f
+T(1)†
+1
+(xi)
+(xii)
+(xiii)
+S(1)†
+2f
+S(0)
+2i
+S(0)†
+2f
+S(1)
+2i
+(xiv)
+T(1)†
+1
+S(1)
+2i
+(xv)
+T(1)
+1
+S(1)†
+2f
+(xvi)
+... ...
+FIG. 16. A few important E1P V evaluating diagrams in the
+RCCSD method. Diagrams shown as DT (1) and its complex
+conjugate c.c.) term are the dominant Core contributing ef-
+fects that include all the core-polarization effects of the CPDF
+method, pair-correlation effects of the RMBPT(3) method
+and their correlations. Similarly, diagrams shown as DS(1)
+1i
+and S(1)†
+1f D contain Valence contributing core-polarization ef-
+fects of the CPDF method, pair-correlation effects from the
+RMBPT(3) method and their correlations. DS(1)
+2i and S(1)†
+2f D
+representing diagrams contain Core contributions from the
+RPA and DCP contributions from the CPDF-RPA method
+as well as that are absent in this method but appears in the
+RMBPT(3) method. Non-linear RCCSD terms incorporate
+higher-order correlation effects of the above terms and ac-
+count correlations among core-polarization, pair-correlation
+and their intercombinations.
+procedure) is used then the results from both these equa-
+tions may not agree to each other due to inconsistencies
+in the treatment of the intermediate states through these
+equations.
+The modified single particle Hamiltonian for the cor-
+responding Hamiltonian Ht = Hint + λ3D in the CPDF-
+RPA method can be written as f P V ±
+i
+= f P V
+i
++ λ3d ∓ ω.
+It follows
+f P V ±
+i
+|i⟩ = ǫi|i⟩
+(82)
+and
+ui =
+Nc
+�
+b
+�
+⟨b|g|b⟩|i⟩ − ⟨b|g|i⟩|b⟩
+�
+,
+(83)
+where the bar symbol denotes solution for Ht. By ex-
+panding, we get |i⟩ = |iP V ⟩ + λ3|iP V ±⟩ + O(λ2
+3). It gives
+(f P V
+i
+− ǫP V
+i
+∓ ω)|iP V ±⟩ = −d|iP V ⟩ − uP V (1)
+i
+|iP V ⟩,(84)
+
+16
+where
+uP V (1)
+i
+|iP V ⟩ =
+Nc
+�
+b
+�
+⟨bP V |g|bP V ⟩|iP V ±⟩
+−⟨bP V |g|iP V ±⟩|bP V ⟩
++⟨bP V |g|bP V ⟩|iP V ±⟩
+−⟨bP V |g|iP V ±⟩|bP V ⟩
+�
+.
+(85)
+Further expanding Eqs.
+(84) and (85), and retaining
+terms of the order of λ2λ3 we get
+(fi − ǫi ∓ ω)|iP V ±⟩ = −d|iP V ⟩ − u±
+i |iP V ⟩ − hw|i±⟩
+−uP V
+i
+|i±⟩ − uP V ±
+i
+|i⟩,
+(86)
+where
+uP V ±
+i
+|i⟩ =
+Nc
+�
+b
+�
+⟨b∓|g|bP V ⟩|i⟩
+−⟨b∓|g|i⟩|bP V ⟩
++⟨bP V |g|b±⟩|i⟩
+−⟨bP V |g|i⟩|b±⟩
++⟨b|g|bP V ±⟩|i⟩
+−⟨b|g|i⟩|bP V ±⟩
++⟨bP V ∓|g|b⟩|i⟩
+−⟨bP V ∓|g|i⟩|b⟩
+�
+.
+(87)
+It can be further noted that in the CPDF method the
+perturbed core DHF orbital (|aP V ⟩) is orthogonal to the
+unperturbed core orbital (|a⟩) and the same is also true
+in the RPA. i.e. ⟨a|aP V ⟩ = 0 and ⟨a|a±⟩ = 0. However,
+⟨a|aP V ±⟩ ̸= 0 in the CPDF-RPA method. This necessi-
+tates to use the orthogonalized core orbitals (|ao±⟩) by
+imposing the condition
+|ao±⟩ = |aP V ±⟩ −
+�
+b
+|b⟩⟨b|aP V ±⟩.
+(88)
+In Fig. 10, we show the Goldstone diagrams contribut-
+ing to the determination of the |aP V ±⟩ and also the extra
+diagrams that are subtracted to obtain |ao±⟩. It can be
+understood below that it is not required to obtain the
+modified orbitals for the virtual orbitals in the CPDF-
+RPA method to estimate the E1P V amplitude, so we do
+not show Goldstone diagrams contributing to the ampli-
+tudes of valence operator here.
+Using the following expressions in the formula given by
+Eq. (80), we can write
+E1P V = ⟨f|d + u+
+i |iP V ⟩ + ⟨f|hw + uP V
+i
+|i+⟩
++⟨f|uP V +
+i
+|i⟩.
+(89)
+Similarly, using the formula given by Eq. (81) we can get
+E1P V = ⟨f P V |d + u+
+i |i⟩ + ⟨f −|hw + uP V
+i
+|i⟩
++⟨f|uP V −
+f
+|i⟩
+= ⟨f P V |d + u+
+i |i⟩ + ⟨f −|hw + uP V
+i
+|i⟩
++⟨f|uP V +
+i
+|i⟩.
+(90)
+TABLE I. E1P V values, in units 10−11i(−Qw/Nn)|e|a0, of
+the 6s 2S1/2 − 7s 2S1/2 and 6s 2S1/2 − 5d 2D3/2 transitions
+in 133Cs from DC Hamiltonian reported by various works.
+Methods shown as ‘Sum-over’ and ‘Mixed’ are obtained us-
+ing sum-over-states approach and mixed many-body methods
+respectively. Results shown in bold fonts are claimed to be
+within 0.5% accuracy.
+Method
+This work Others
+This work Others
+6s 2S1/2 − 7s 2S1/2
+6s 2S1/2 − 5d 2D3/2
+DHF
+0.7375
+0.736 [11]
+−2.3933
+RMBPT(3)w 1.0902
+−2.4639
+CPDF
+0.9226
+0.924 [11]
+−2.7989
+RPA
+0.7094
+0.707 [11]
+−2.2362
+CPDF-RPA∗ 0.8876
+0.8914 [25]
+−3.1665
+−3.70 [37]
+0.8923† [25]
+CPDF-RPA
+0.8844
+0.886 [11]
+−3.0281
+−3.80 [26]
+0.907 [26]
+RCCSD
+0.8964
+0.8961 [29]
+−3.5641
+−3.210[38]
+RCCSDT
+0.8967 [29]
+Sum-over
+0.9053 [24]
+−3.76 [13]
+0.8998† [24]
+Mixed-states
+0.8967 [25]
+−3.62 [13]
+0.8938† [25]
+0.9083‡ [25]
+†Note: Scaled value.
+‡Scaled value + borrowed contribution from Ref. [24].
+In the wave operator form, Eq. (89) can be given by
+E1P V = ⟨Φf|DΩi,CP DF |Φi⟩ + ⟨Φf|HwΩi,+|Φi⟩
++⟨Φf|ΩCP DF +|Φi⟩.
+(91)
+From (90), we can write
+E1P V = ⟨Φf|Ωf,CP DF †D|Φi⟩ + ⟨Φf|Ω−†Hw|Φi⟩
++⟨Φf|Ωf,CP DF −|Φi⟩
+= ⟨Φf|Ωf,CP DF †D|Φi⟩ + ⟨Φf|Ωf,−†Hw|Φi⟩
++⟨Φf|ΩCP DF +|Φi⟩.
+(92)
+In the above expressions, we define
+ΩCP DF + =
+�
+i,j
+(⟨f|u+
+i |iP V ⟩ + ⟨f|uP V
+i
+|i+⟩
++⟨f|uP V +
+i
+|i⟩)a†
+jai
+(93)
+and
+ΩCP DF − =
+�
+i,j
+(⟨f P V |u−
+f |i⟩ + ⟨f −|uP V
+f
+|i⟩
++⟨f|uP V −
+f
+|i⟩)a†
+jai
+(94)
+It is also worth noting that some of the works in the lit-
+erature do not consider contribution from ⟨f|uP V +
+i
+|i⟩ in
+the CPDF-RPA method, and their contributions are sep-
+arately quoted as ‘DCP’ effects. In Fig. 11, we show the
+
+17
+TABLE II. Reduced E1 (in a.u.) and HW (in units of 10−11i(−Qw/Nn)|e|a0) matrix elements from the RCCSD method, and
+the excitation energies (in cm−1) of the low-lying states of 133Cs that are used to estimate the ‘Main’ contribution of E1P V for
+the 6S − 7S transition. E1 matrix elements and energies are compared with the available most precise experimental values.
+Transition
+E1 amplitude
+Excitation Energy
+HW amplitude
+This work
+Experiment
+This work
+Experiment [45]
+6P1/2-6S
+4.5487
+4.5097(74) [41]
+−11243.93
+−11178.27
+−1.2541
+7P1/2-6S
+0.3006
+0.2825(20) [42]
+−21838.93
+−21765.35
+−0.7135
+8P1/2-6S
+0.0914
+−25787.48
+−25708.83
+−0.4808
+9P1/2-6S
+−0.0388
+−27735.96
+−27637.00
+0.3471
+6P1/2-7S
+−4.2500
+4.233(22)[43]
+7352.53
+7357.26
+0.6067
+7P1/2-7S
+10.2967
+10.308(15)[44]
+−3242.47
+−3229.82
+0.3445
+8P1/2-7S
+0.9492
+−7191.02
+−7173.31
+0.2320
+9P1/2-7S
+−0.3867
+−9139.50
+−9101.47
+−0.1674
+diagrams that are contributing to E1P V in the CPDF-
+RPA method icluding the DCP effect.
+Now compar-
+ing diagrams from Fig. 11 and the diagrams from the
+RMBPT(3) method shown in Fig. 2, it can be shown that
+some of the Core and Valence correlation diagrams of the
+RMBPT method are appearing as the Core diagrams of
+the CPDF-RPA method (so also for Valence contribu-
+tions).
+This, therefore, clearly demonstrates that the
+definitions of Core and Valence correlation contributions
+to E1P V are not unique and their classifications differ
+based on the approach adopted in a many-body method.
+Among the approximated methods where various physi-
+cal effects or correlation effects through both the HW and
+D operators are not included on an equal footing, it may
+not be possible to make an one-to-one comparative anal-
+ysis among contributions arising through the Core and
+Valence correlations. In such a scenario, it is suggestive
+to compare only the final results from different methods.
+The advantages of using the CPDF-RPA method are
+that this method includes core-polarization effects to all-
+orders, treats both the HW and D operators on an equal
+footing (which means the results would remain invari-
+ant in what-so-ever order either HW or D is included
+in Ht along with H), and gives DCP effects that are
+not present in the CPDF method or RPA. However, it
+still misses out many non-core-polarization effects includ-
+ing pair-correlation contributions and correlations among
+core-polarization, and pair-correlation effects in the de-
+termination of E1P V . Again, orthogonalization of per-
+turbed occupied orbitals is incorporated by hand while
+the approach does not take care of them in a natural man-
+ner. We would like to mention that some of the earlier
+calculations using the CPDF-RPA method neglected con-
+tributions from the DCP effects (see Ref. [26]). Results
+from such as an approximation are denoted as CPDF-
+RPA* method.
+In fact, omission of some of the DCP
+contributions in this method is just due to a similar rea-
+son. Moreover, the CPDF, RPA and CPDF-RPA meth-
+ods cannot be derived using Bloch’s prescription though
+we have expressed them using wave operators just to
+make one-to-one connection of these methods with the
+TABLE III. Estimated ‘Main’ contributions to the E1P V val-
+ues, in units 10−11i(−Qw/Nn)|e|a0, of the 6s 2S1/2 −7s 2S1/2
+transition in 133Cs using matrix elements involving np 2P1/2
+intermediate states in the sum-over-states approach.
+Four
+cases are being considered: (a) ab initio result in which cal-
+culated values from Table II are used; (b) replacing calculated
+E1 matrix elements by their experimental values; (c) retaining
+calculated E1 matrix elements and using experimental ener-
+gies; and (d) using experimental values for both E1 matrix
+elements and energies.
+Approach
+⟨7S|D|6SP NC⟩
+⟨7SP NC|D|6S⟩
+Total
+(a)
+−0.4461
+1.3171
+0.8710
+(b)
+−0.4373
+1.3121
+0.8748
+(c)
+−0.4522
+1.3156
+0.8634
+(d)
+−0.4434
+1.3106
+0.8672
+RMBPT method. Thus, these methods are simply the
+extension of the DHF method and cannot take into ac-
+count the effects that may have been neglected in gen-
+erating the single particle orbitals. For example, the ef-
+fects of V N, V N−1, V N−2 etc.
+potentials used in the
+determination of wave functions through the Bloch equa-
+tion are taken care through an effective Hamiltonian like
+Heff = PvVresΩ(0)
+v Pv that appears in solving amplitude
+of Ω(0)
+v . However, the wave operator amplitude solving
+equations in the CPDF, RPA and CPDF-RPA methods
+remain to be the same. Thus, the valence electron inter-
+action neglected in the construction of DHF potential (in-
+active valence orbital) is not amended through the above
+methods like in the RMBPT method. All these short-
+comings of the CPDF-RPA method will be adequately
+addressed by the RCC method.
+F.
+RCC method
+The RCC method both in the non-relativistic and its
+counter relativistic forms have been extensively consid-
+ered these days to include electron correlation effects
+
+18
+in the determination of properties of different types of
+many-body systems accurately such as nuclear, atomic,
+molecular and solid state systems.
+This is the reason
+for which these days it is commonly referred to as the
+gold standard of many-body theory. Compared to the
+CPDF, RPA and CPDF-RPA methods, implementation
+and computational efforts in the RCC method are ex-
+tensively complex and expensive. It can account correla-
+tions through both the HW and D operators to all-orders,
+as well take care other shortcomings of the CPDF-RPA
+method.
+All CPDF-RPA effects along with other ef-
+fects like pair-correlations, inter-correlations among core-
+polarization and pair-correlation effects, corrections due
+to choice of V N−1 DHF potential approximation etc. are
+sub-summed within our RCC method. This theory was
+already implemented and results using the method were
+already reported for Cs [21, 23], Ba+ [21], Yb+ [39] and
+Ra+ [40]. Here, we consider this theory in the singles
+and doubles approximation (RCCSD method) only to
+demonstrate how it captures correlations of previously
+mentioned methods including appearance of orthogonal-
+ization of perturbed core orbitals in a natural fashion, all-
+order pair-correlations, additional DCP effects and nor-
+malization of wave functions etc. compared to a mixed
+many-body method. Since all these effects are present
+within the RCC theory and the wave functions are ob-
+tained through iterative scheme, all of these effects are
+inter-correlated. Incorporation of additional effects from
+the Breit and QED interactions can also be treated in the
+similar fashion, if their corresponding interaction poten-
+tial terms are added in the atomic Hamiltonian. Going
+beyond the RCCSD method approximation in the RCC
+theory such as the RCCSDT method, means it can cap-
+ture even higher-order non core-polarization effects and
+further inter-correlations among core-polarization and
+pair-correlation effects that are beyond the reach of the
+mixed many-body methods employed earlier to estimate
+the E1P V amplitudes. As shown in Ref. [29] though the
+energies, E1 matrix elements and magnetic dipole hy-
+perfine structure constants from both the RCCSD and
+RCCSDT methods were quite significant, the difference
+in the E1P V values from both these two methods was
+rather small.
+This suggests that consideration of the
+RCCSD method would be sufficient enough to address
+the earlier mentioned concerns.
+In the RCC theory framework, the exact wave function
+of an atomic state can be given by
+|Ψv⟩ = eS|Φv⟩,
+(95)
+where S is an excitation operator carrying out excitations
+of electrons out of core orbitals from the DHF wave func-
+tion |Φv⟩ to generate the excited state configurations due
+to Vres. In other words, these excitation configurations
+can be thought of as contributions taken from each or-
+der of corrections to the wave function from the RMBPT
+method to construct an all-order form. We can further
+define
+S = T + Sv,
+(96)
+TABLE IV. Core and Valence correlation contributions to
+the E1P V values, in units 10−11i(−Qw/Nn)|e|a0, for the
+6s 2S1/2−7s 2S1/2 and 6s 2S1/2−5d 2D3/2 transitions in 133Cs
+from the RMBPT(3)w and RMBPT(3)d approaches. Results
+(a) considering Heff effect due to V N−1 potential and (b)
+using DHF orbital energies are shown for comparison.
+Approach
+RMBPT(3)w
+RMBPT(3)d
+Core
+Valence
+Core
+Valence
+6s 2S1/2 − 7s 2S1/2
+(a)
+−0.00205
+1.09220
+−0.00003
+1.24290
+(b)
+−0.00206
+0.43938
+−0.00031
+0.43763
+6s 2S1/2 − 5d 2D3/2
+(a)
+−0.16391
+−2.30000
+−0.12208
+−2.55427
+(b)
+−0.18267
+−3.97004
+−0.12513
+−4.02758
+in order to distinguish excitations of electrons among the
+core orbitals, denoted by T , and excitations of electrons
+from valence orbital or valence orbital along with core
+orbitals, denoted by Sv in the V N−1 framework of con-
+structing the DHF wave function |Φv⟩ = a†
+v|Φ0⟩. Accord-
+ingly we can write
+|Ψv⟩ = eT +Sv|Φv⟩
+= eT {1 + Sv} |Φv⟩.
+(97)
+Here eSv = 1 + Sv is the exact form for the atomic states
+having one valence orbital v. In the RCCSD method, the
+excitation operators are denoted as
+T = T1 + T2
+(98)
+and
+Sv = S1v + S2v,
+(99)
+where subscripts 1 and 2 stand for the singles and doubles
+excitations respectively.
+In the wave operator form given by Eqs. (7) and (8),
+it corresponds to
+Ω0 = eT
+(100)
+and
+Ωv = eT Sv.
+(101)
+Following the Bloch’s equations given by Eqs. (9) and
+(10) in general form, amplitudes of the T and Sv excita-
+tion operators are obtained by
+⟨Φ∗
+0|(HeT )l|Φ0⟩ = 0
+(102)
+and
+⟨Φ∗
+v|[(HeT )l − Ev]Sv + (HeT )l|Φv⟩ = −⟨Φ∗
+v|(HeT )l|Φv⟩,
+(103)
+where subscript l denotes for the linked terms, bra states
+with superscript ∗ means excited states with respect to
+the respective DHF ket states appear in the equations.
+
+19
+TABLE V. Ab initio contributions to Core and Valence parts
+of the E1P V values, in units 10−11i(−Qw/Nn)|e|a0, for the
+6s 2S1/2−7s 2S1/2 and 6s 2S1/2−5d 2D3/2 transitions in 133Cs
+from different methods considered in this work using the DC
+Hamiltonian. Available results from previous calculations are
+also given for comparison.
+Method
+This work
+Others
+Core
+Valence
+Core
+Valence
+6s 2S1/2 − 7s 2S1/2
+DHF
+−0.00173
+0.73923
+−0.00174[30]
+RMBPT(3)w −0.00205
+1.09220
+RMBPT(3)d −0.00003
+1.24290
+CPDF
+−0.00199
+0.92454
+−0.00201 [30]
+RPA
+0.00028
+0.70912
+CPDF-RPA∗
+0.00169
+0.88591
+0.00170 [30]
+CPDF-RPA
+0.00169
+0.88267
+RCCSD
+−0.00197
+0.89840
+−0.0019 [29] 0.8980 [29]
+6s 2S1/2 − 5d 2D3/2
+DHF
+−0.11684 −2.27646
+RMBPT(3)w −0.16391 −2.3000
+RMBPT(3)d −0.12208 −2.55427
+CPDF
+−0.19122 −2.60768
+RPA
+−0.12037 −2.11585
+CPDF-RPA∗ −0.20786 −2.95860
+CPDF-RPA
+−0.20786 −2.82021
+RCCSD
+−0.14745 −3.41667
+In the ab initio procedure, the energy of the |Ψv⟩ is de-
+termined by calculating expectation value of the effective
+Hamiltonian i.e.
+Ev = ⟨Φv|Heff|Φv⟩
+= ⟨Φv|Pv(HeT )l {1 + Sv} Pv|Φv⟩,
+(104)
+with respect to |Φv⟩. As can be noticed Ev is a function
+of Sv and Sv itself depends on Ev. Thus, the non-linear
+Eqs. (103) and (104) are solved iteratively to obtain am-
+plitudes of Ωv. As pointed out earlier, appearance of Ev
+in the determination of Sv amplitudes is a consequence
+of using orbitals from V N−1 potential. It is also possible
+to obtain amplitudes of the RCC operators by substitut-
+ing experimental energy in the semi-empirical approach.
+Also, one can scale the Sv amplitudes by multiplying
+by a suitable parameter λ to obtain the calculated Ev
+value matching with the experimental energy in the sim-
+ilar spirit of the scaling procedures adopted in Refs. [24]
+and [25].
+To evaluate E1P V , we need to express the RCC opera-
+tors in terms of both the unperturbed and first-order per-
+turbed operators. As explained in Sec. III, we have three
+different options to obtain the E1P V amplitude in the
+RCC theory framework. First, by adopting the approach
+similar to the CPDF method, in which HW is considered
+as external perturbation and matrix element of D can be
+determined. Second, considering D as the external per-
+turbative operator as in the RPA. Third and the most
+effective approach would be along the line of the CPDF-
+RPA method, in which, both the HW and D operators
+can be treated as external perturbations. The implemen-
+tation of the third approach would be more challenging
+and computationally very expensive as it will demand to
+store amplitudes of four different types of perturbed RCC
+operators instead of storing only one type of perturbed
+amplitudes in the first case and two types in the second
+case.
+Among the first two approaches, computational
+efforts are almost similar but implementation-wise con-
+sidering HW as perturbation will be more natural and
+is easier to deal with its angular momentum couplings
+owing to its scalar form. Moreover, amplitudes of the
+perturbed operators due to HW will converge faster than
+when D is treated as perturbation. Again, it is possible
+to use experimental energies in the first approach to ob-
+tain semi-empirical results in case it is required while it
+is a problem in the second case owing to the fact that is
+already discussed earlier. From this view, we adopt the
+first approach to estimate the E1P V value.
+We expand the T and Sv operators by treating HW
+as the perturbation to separate out the solutions for the
+unperturbed and the first-order wave functions by ex-
+pressing
+T = T (0) + λ2T (1)
+(105)
+and
+Sv = S(0)
+v
++ λ2S(1)
+v ,
+(106)
+where the superscript meanings are same as specified ear-
+lier. This yields
+|Ψ(0)
+v ⟩ = (Ω(0)
+0
++ Ω(0)
+v )|Φv⟩
+(107)
+and
+|Ψ(0)
+v ⟩ = (Ω(1)
+0
++ Ω(1)
+v )|Φv⟩
+(108)
+with the definitions Ω(0)
+0
+= eT (0), Ω(1)
+0
+= eT (0)T (1),
+Ω(0)
+v
+= eT (0)S(0)
+v
+and Ω(1)
+v
+= eT (0) �
+T (1)S(0)
+v
++ S(1)
+v
+�
+. The
+unperturbed operator amplitudes are obtained by solv-
+ing the usual RCC theory equations as mentioned above.
+The first-order perturbed RCC operator amplitudes are
+determined as
+⟨Φ∗
+0|(HeT (0))lT (1)|Φ0⟩ = −⟨Φ∗
+0|(HW eT (0))l|Φ0⟩(109)
+and
+⟨Φ∗
+v|[(HeT (0))l − E(0)
+v ]S(1)
+v |Φv⟩ = −⟨Φ∗
+v|[(HW eT (0))l
++(HeT (0))lT (1)]{1 + S(0)
+v }|Φv⟩ (110)
+As can be seen, the exact calculated energy also enters
+into the amplitude determining equation of S(1)
+v
+because
+of the V N−1 potential.
+This is one of the advantages
+
+20
+of the RCC method over the CPDF-RPA method.
+In
+Figs.
+12,13, 14, 15 we show some of the important
+Goldstone diagrams contributing to the T (0)
+2
+, T (0)
+1
+, S(0)
+1v ,
+T (1)
+2
+,T (1)
+1
+,S(0)
+2v , S(1)
+1v and S(1)
+2v amplitudes. These diagrams
+can be compared with the CPDF-RPA wave operator am-
+plitude determining diagrams in order to understand how
+they are embedded within the RCC operators irrespec-
+tive of the fact that denominators in the RCC method
+will contain the exact energy of the state instead of the
+DHF energy in the CPDF-RPA method.
+The E1P V expression between the states |Ψi⟩ and |Ψf⟩
+in the RCC theory is given by
+E1P V =
+⟨Φf|{S(1)
+f
++ (S(0)†
+f
++ 1)T (1)†} ¯D{1 + S(0)
+i
+}|Φi⟩
+⟨Φf|{S(0)†
+f
++ 1} ¯N{1 + S(0)
+i
+}|Φi⟩
++
+⟨Φf|{S(0)†
+f
++ 1} ¯D{T (1)(1 + S(0)
+i
+) + S(1)
+i
+}|Φi⟩
+⟨Φf|{S(0)†
+f
++ 1} ¯N{1 + S(0)
+i
+}|Φi⟩
+,
+(111)
+where ¯D = eT (0)+DeT (0) and ¯N = eT (0)+eT (0). Unlike the
+CPDF, RPA and CPDF-RPA methods, normalization
+factors appear explicitly in the RCC expression. Using
+the wave operator notations, one can easily identify which
+RCC terms contribute to the Core and Valence correla-
+tions in the evaluation of E1P V . It means basically, any
+term is connected either with the Sn=i,f (0/1) operators or
+with their conjugate operators will be a part of the Va-
+lence correlation otherwise they will be a part of the Core
+correlation. It can be further clarified that the definitions
+of Core and Valence correlation contributions to E1P V in
+our RCC theory are in the line of the RMBPT(3)w and
+CPDF methods, and different than the RPA and CPDF-
+RPA methods.
+In Fig.
+16 we show a few important
+contributing Goldstone diagrams from the RCC method
+to Core and Valence correlations. Also, for better under-
+standing, the Goldstone diagrams of the RCCSD opera-
+tors are further demonstrated as the sum of lower-order
+Goldstone diagrams of the RMBPT(3) method.
+From
+these relations, it can be followed that the RCC method
+includes correlation effects from core-polarization, pair-
+correlation, and DCP to all orders.
+It is also obvious
+from the above diagrams that orthogonalization to core
+orbitals and extra DCP contributions are also appearing
+in a natural manner in our RCC theory. Moreover, corre-
+lations among all these effects are implicitly present due
+to the fact that singles and doubles excitation amplitude
+equations are coupled through many non-linear terms in
+the RCC theory.
+V.
+RESULTS & DISCUSSIONS
+We present the calculated values of E1P V
+of the
+6S − 7S and 6S − 5D3/2 transitions in
+133Cs from
+the DHF, RMBPT(3), CPDF, RPA, CPDF-RPA and
+RCCSD methods.
+As mentioned in Introduction, the
+main intention of carrying out this study is to demon-
+TABLE VI. Comparison of contributions from the initial and
+final perturbed states to E1P V of the 6s 2S1/2 − 7s 2S1/2
+transition of 133Cs, in units 10−11i(−Qw/Nn)|e|a0, at differ-
+ent levels of approximation between the present work and that
+are reported in Refs. [11, 30].
+Method
+⟨7SP NC|D|6S⟩
+⟨7S|D|6SP NC⟩
+Ours
+Ref. [11]
+Ours
+Ref. [11]
+Total contribution
+DHF
+1.01168
+1.010
+−0.27418
+−0.274
+CPDF
+1.26664
+1.267
+−0.34409
+−0.344
+RPA
+1.02557
+1.023
+−0.31617
+−0.316
+CPDF-RPA*
+1.27910
+1.279
+−0.39150
+−0.391
+Ours
+Ref. [30]
+Ours
+Ref. [30]
+Core contribution
+DHF
+−0.02638 −0.02645
+0.02465
+0.02472
+CPDF
+−0.04298 −0.04319
+0.04099
+0.04119
+RPA
+−0.03536
+0.03564
+CPDF-RPA* −0.05794 −0.05822
+0.05963
+0.05992
+Valence contribution
+DHF
+1.03806
+−0.29883
+CPDF
+1.30962
+−0.38508
+RPA
+1.06094
+0.70912
+CPDF-RPA*
+1.33704
+−0.45113
+strate similarities and differences among various con-
+tributions to E1P V through the above methods.
+This
+would be useful in addressing the issue of the sign for the
+Core correlation contributions to the E1P V value of the
+6S − 7S transition in 133Cs that are reported differently
+by various groups [24, 25, 29]. Moreover, this exercise
+would be useful in understanding the missing contribu-
+tions in a method compared to others that are under con-
+sideration here so that accuracy of the earlier reported
+results in an atomic system, including Cs, obtained us-
+ing a particular method can be further improved. We
+have allowed correlations from all the occupied orbitals
+and allowed excitations of electrons from a given set of
+virtual orbitals in all the considered many-body methods
+to make comparative analysis of results from them. In
+order to show that we have taken a sufficiently large set
+basis functions, we validate our calculations by compar-
+ing our E1P V values from the DHF, CPDF, RPA and
+CPDF-RPA methods with the earlier reported values by
+Martensson [11].
+The reason for presenting the E1P V
+value of the 6S − 5D3/2 transition in 133Cs is to answer
+to a comment by Roberts and Ginges in Ref. [30], where
+they argue about the agreement of the sign of Core contri-
+bution to the E1P V value of a S − D transition reported
+earlier using the RCCSD method [40] while observing a
+sign difference for the 6S − 7S transition in 133Cs.
+In Table I, we present the E1P V values of the 6S − 7S
+and 6S −5D3/2 transitions in 133Cs using the DC Hamil-
+
+21
+tonian from a number of methods including the DHF
+method in order to understand the importance of corre-
+lation effects in their determination and to demonstrate
+that choice of a method matters a lot for their rigourous
+inclusion. The values shown in bold fonts in this table
+are claimed to be accurate within 0.5% by the earlier
+works. A careful look into these results reveal that some
+of them differ by 1% from each other, which suggests
+there could be issues with the estimation of accuracy in
+these calculations that needs to be investigated. Results
+from sum-over-states approach, given as ‘Sum-over’ in
+the table, uses scaled E1 matrix elements and energies
+from the CCSDvT method to estimate the Main con-
+tribution of E1P V for the 6S − 7S transition while the
+X-factor is obtained using a blend of many-body meth-
+ods [24]. In Ref. [25], the same ‘Main’ contribution is
+utilized but Core and Tail contributions to the X-factor
+are estimated using the CPDF-RPA* method (denoted
+as only RPA in the original paper). Pair-correlation ef-
+fects to these estimations were estimated using the BO-
+correlation method. Result from these RPA+BO meth-
+ods is given under ‘Mixed-states’ in the above table.
+Thus large discrepancy seen in both the results from the
+above table come from the X-factors estimated in Refs
+[24, 25]. If the total X-factors had agreed between two
+works but individual contributions would have differed,
+then the difference in the results could have been at-
+tributed to distribution of contributions under the Core
+and Valence correlations in the considered approaches in
+both the works. From the significant differences seen be-
+tween the X-factors from the Mixed-states approach and
+our RCCSD method in both the 6S −7S and 6S −5D3/2
+transitions, it does not support such distributions.
+To understand the reasons for significant discrepancies
+seen in the X-factors from various works, we first analyze
+the Main contribution to the 6S − 7S transition by us-
+ing the calculated properties from our RCCSD method
+in the sum-over-states approach. We used the E1 ma-
+trix elements and energies from our calculations as well
+as from experiments to show the differences. In Table II,
+we present the HW matrix elements, E1 matrix elements
+and energies obtained using the DC Hamiltonian in the
+RCCSD method. The calculated E1 matrix elements and
+energies are also compared with the experimental values
+[41–45] in the same table. Using these values we esti-
+mate the Main contributions to E1P V of the 6S − 7S
+transition and they are given in Table III. Results (a)
+from ab initio calculations, (b) using experimental E1
+values with calculated energies, (c) calculated E1 values
+with experimental energies and (d) using experimental
+values for both the E1 matrix elements and energies are
+given separately. This analysis shows that result from
+(b) is larger than (a), but results from (c) and (d) are
+lower than (a).
+It means that accuracy of energies in
+a given method affect the results more than E1 matrix
+elements. Later we demonstrate explicitly that it intro-
+duces error to E1P V estimation when we use experimen-
+tal energy only of the initial or final state through the
+first-principle calculations.
+Differences between the ab
+initio and semi-empirical calculations can be minimised
+by including contributions from the triples, quadruples
+etc. higher-level excitations of the RCC method. In Ref.
+[29], Sahoo et al have demonstrated difference between
+the ab initio calculations of E1P V of the 6S − 7S tran-
+sition using the RCCSD and RCCSDT methods is very
+small. They further showed that Core contributions are
+almost same in both the methods, and the agreement of
+the E1P V values from both the methods was the result
+of opposite trends of correlation effects in the evaluation
+of the HW matrix elements than the E1 matrix elements
+and energies. A similar trend was anticipated from the
+Tail contribution. Subtracting the ab initio value of Main
+from the final RCCSD result, we find the X-factor to
+E1P V of the 6S −7S transition as 0.0254, against 0.0175
+and 0.0256 of Refs. [24] and [25] respectively in units of
+10−11i(−Qw/Nn)|e|a0. It means that there is a large dif-
+ference between the X-factor of Ref. [24] and our work,
+whereas these values almost agree between Ref. [25] and
+the present work. Since there is a sign difference between
+the Core contribution from Ref.
+[25] and the RCCSD
+value of Ref. [29], the above analysis suggests that the
+sign difference is solely due to different definitions used
+for the Core contribution in both the works.
+In order to explain how definition of Core contribution
+changes depending upon the choice of an approach to
+estimate E1P V , we present the Core and Valence contri-
+butions separately to the 6S − 7S and 6S − 5D3/2 tran-
+sitions from both the RMBPT(3)w and RMBPT(3)d ap-
+proaches in Table IV. Just for the sake of demonstrating
+how appearance of Heff in the wave function determining
+equation to choice of V N−1 modify the result, we present
+RMBPT(3) results considering effect of Heff (given re-
+sults as (a) in the above table) and replacing it with
+DHF energy as the case of the CPDF-RPA method (cor-
+responding results are given under (b) in the above ta-
+ble). As can be seen, the Core and Valence contributions
+from both the RMBPT(3)w and RMBPT(3)d approaches
+are coming out differently whereas the final results from
+both the methods are almost close to each other. It can
+also be realized that changes in the results for both the
+transitions are enormous when Heff is considered in the
+wave function solving equation than otherwise.
+To further figure out about the mismatch in the X-
+factors from various works, we present the Core and Va-
+lence contributions to the E1P V values separately for
+both the 6S − 7S and 6S − 5D3/2 transitions arising
+through the first-principle calculations in Table V. As
+can be seen from the table, signs of Core contributions
+to E1P V of both the transitions from the DHF method
+and many-body methods at a given level of approxima-
+tion employed by different groups match each other. This
+indicates that there is no issue with the implementa-
+tion of these theories in our code.
+To support results
+from our methods further, we also compare Core and
+Valence contributions to the 6S − 7S transition in Table
+VI from the initial and final perturbed states through the
+
+22
+TABLE VII. Contributions to Core and Valence parts from different terms to E1P V of the 6s 2S1/2 − 7s 2S1/2 and 6s 2S1/2 −
+5d 2D3/2 transitions in 133Cs, in units 10−11i(−Qw/Nn)|e|a0 from Eqs. (89) and (90) of the CPDF-RPA method, which are
+quoted under ‘Expression a’ and ‘Expression b’ respectively. Results are given using the calculated ω value, ωex and ωex with
+the experimental energies of the initial and final states (denoted by Eexpt
+i,f ).
+Contribution
+Expression a
+Expression b
+6s 2S1/2 − 7s 2S1/2
+⟨7s|hw|6s+⟩
+⟨7s|uP V
+6s |6s+⟩
+Total
+⟨7sP V |d|6s⟩
+⟨7sP V |u+
+6s|6s⟩
+Total
+Core (ω)
+−0.0357
+−0.02257
+−0.05794
+−0.04299
+−0.01495
+−0.05794
+Valence (ω)
+1.06094
+0.27610
+1.33704
+1.30962
+0.02742
+1.33704
+Core (ωex)
+−0.03464
+−0.02211
+−0.05675
+−0.04299
+−0.01458
+−0.05757
+Valence (ωex)
+−0.19464
+−0.04598
+−0.24062
+1.30962
+0.02743
+1.33705
+Core (Eexpt
+i,f )
+−0.03546
+−0.02283
+−0.05829
+−0.04331
+−0.01498
+−0.05829
+Valence (Eexpt
+i,f )
+1.21721
+0.31956
+1.53677
+1.53384
+0.00293
+1.53677
+⟨7s|d|6sP V ⟩
+⟨7s|u+
+6s|6sP V ⟩
+Total
+⟨7s−|hw|6s⟩
+⟨7s−|uP V
+6s |6s⟩
+Total
+Core (ω)
+0.04099
+0.01864
+0.05963
+0.03564
+0.023399
+0.05963
+Valence (ω)
+−0.38508
+−0.06605
+−0.45113
+−0.35181
+−0.09932
+−0.45113
+Core (ωex)
+0.04099
+0.01915
+0.06014
+0.03651
+0.02458
+0.06109
+Valence (ωex)
+−0.38508
+−0.06644
+−0.45152
+−0.17081
+−0.05022
+−0.22103
+Core (Eexpt
+i,f )
+0.04210
+0.01999
+0.06209
+0.03686
+0.02523
+0.06209
+Valence (Eexpt
+i,f )
+−0.12128
+−0.05800
+−0.17928
+−0.13743
+−0.04185
+−0.17928
+6s 2S1/2 − 5d 2D3/2
+⟨5d3/2|hw|6s+⟩ ⟨5d3/2|uP V
+6s |6s+⟩
+Total
+⟨5dP V
+3/2|d|6s⟩
+⟨5dP V
+3/2|u+
+6s|6s⟩
+Total
+Core (ω)
+0.0
+−0.00616
+−0.00616
+−0.00451
+−0.00165
+−0.00616
+Valence (ω)
+0.0
+−0.27386
+−0.27386
+−0.27878
+0.00492
+−0.27386
+Core (ωex)
+0.0
+−0.00612
+−0.0612
+−0.00451
+−0.00164
+−0.00615
+Valence (ωex)
+0.0
+−0.23287
+−0.23287
+−0.27878
+0.00493
+−0.27385
+Core (Eexpt
+i,f )
+0.0
+−0.00628
+−0.00628
+−0.00459
+−0.001087
+−0.00628
+Valence (Eexpt
+i,f )
+0.0
+−0.83936
+−0.83936
+−0.87962
+0.04028
+−0.83936
+⟨5d3/2|d|6sP V ⟩ ⟨5d3/2|u+
+6s|6sP V ⟩
+Total
+⟨5d−
+3/2|hw|6s⟩ ⟨5d−
+3/2|uP V
+6s |6s⟩
+Total
+Core (ω)
+−0.19574
+−0.00596
+−0.20170
+−0.12037
+−0.08133
+−0.20170
+Valence (ω)
+−2.88646
+0.20172
+−2.68474
+−2.11585
+−0.56889
+−2.68474
+Core (ωex)
+−0.19574
+−0.00636
+−0.20210
+−0.12109
+−0.08182
+−0.20291
+Valence (ωex)
+−2.88646
+0.20201
+−2.68445
+−1.94829
+−0.52407
+−2.47236
+Core (Eexpt
+i,f )
+−0.20110
+−0.00744
+−0.20854
+−0.12363
+−0.08491
+−0.20854
+Valence(Eexpt
+i,f )
+−2.02306
+0.16022
+−1.86284
+−1.46451
+−0.39833
+−1.86284
+DHF, CPDF, RPA, CPDF-RPA* and CPDF-RPA meth-
+ods with the values reported in a Comment by Roberts
+and Ginges [30], and Martensson [11]. We find reasonably
+good agreement between our results with the earlier esti-
+mations. As it has been explained in the previous section,
+definitions of Core correlation effects arising through the
+CPDF, RPA and CPDF-RPA methods all differ. Thus,
+the exact reason for which sign of Core contribution to
+E1P V of the 6S − 7S transition in 133Cs differ between
+Ref. [24] and Ref. [25] is not clear to us as the exact
+method(s) employed in Ref. [24] for its estimation is not
+mentioned explicitly. Only from the sign of the Core con-
+
+23
+tribution quoted in Ref. [24] and by comparing it with
+the signs of Core contributions from the RMBPT(3)w,
+CPDF and RCCSD methods of the present work and
+from the RCCSD and RCCSDT methods of Ref. [29],
+we can assume that Ref. [24] estimates Core contribu-
+tion by considering HW as perturbation. In such a case,
+the Tail contributions to E1P V for the 6S − 7S transi-
+tion in 133Cs from Refs. [24] and [29] as well as from the
+RCCSD result of the present work should almost agree
+each other on the basis of the argument that the net
+X-factor value should agree irrespective of the fact that
+whether HW or D is treated as perturbation. Large dif-
+ferences between the X-factors from Refs. [24], [25] and
+this work, which report as 0.0175, 0.0256 and 0.0254 in
+units of 10−11i(−Qw/Nn)|e|a0, suggests that the former
+work underestimates the Tail contribution. It should be
+noted that the Tail contributions are estimated without
+using sum-over-states approach in all the works, so the
+difference in these values are mainly due to different lev-
+els of approximation made in the many-body methods
+employed for their estimations.
+Now, we wish to address the reason why Roberts and
+Ginges were able to get same sign for the Core contribu-
+tion to E1P V of the 7S − 6D3/2 transition in Ra+ using
+their RPA+BO method with that are reported using the
+RCC method in Ref. [40]. Since correlation trends to
+E1P V of the nS − (n − 1)D3/2 transitions are almost
+similar in Cs and Ra+, with the ground state princi-
+pal quantum number n of the respective system, we can
+understand the above point by analysing the Core con-
+tributions to E1P V of the 6S − 5D3/2 transition from
+different methods and comparing their trends with the
+6S − 7S transition of 133Cs. By looking at these contri-
+butions from Table V, it can be easily followed that there
+is one-order magnitude difference between the Core con-
+tribution in the 6S−7S transition from the RMBPT(3)w
+and RMBPT(3)d methods while there is a sign change
+between these results from the CPDF method and the
+RPA. However, the difference between the Core contri-
+butions from the RMBPT(3)w and RMBPT(3)d meth-
+ods in the 6S − 5D3/2 transition are small, and there
+is no sign difference between the CPDF and RPA re-
+sults. These trends can be explained as follows. In the
+6S − 7S transition, wave functions of both the associ-
+ated states have large overlap over the nucleus while in
+the 6S − 5D3/2 transition only the wave function of the
+ground state has large overlap with the nucleus. As a re-
+sult, strong core-polarization effects contribute through
+both the states in the former case. Also, contribution
+from individual diagram of the CPDF-RPA method is al-
+most comparable in the 6S −7S transition, while a selec-
+tive diagrams contribute predominantly in the 6S−5D3/2
+transition. Since core-polarization effects arising through
+the D operator are stronger and have opposite signs than
+that arise through HW , the net Core contributions in the
+S − S and S − D transitions behave very differently in
+the CDHF method and RPA, and the same propagates to
+the CPDF-RPA*/CPDF-RPA method. Since Core and
+Valence contributions are basically redistributed in the
+CPDF-RPA* and RCCSD methods, difference between
+the final values between Refs. [25] and [29] as well as
+from the present work are coming out to be very small in
+the 6S−7S transition, while it is slightly noticeable in the
+6S−5D3/2 transition (refer to Table I for the comparison
+of results from the Mixed-states and RCCSD methods) .
+The DCP contributions from our calculations can be
+estimated by taking the differences in the results from
+the CPDF-RPA* and CPDF-RPA methods. This differ-
+ence for the 6S − 7S transition from our work is com-
+pared with the corresponding values from Refs. [11] and
+[26]. From this comparison, we find that our result agrees
+better with Roberts than Martensson. However, our fi-
+nal CPDF-RPA result agrees well with Ref.
+[11] than
+Ref. [26]. We also intend to mention that the CPDF-
+RPA* results in Refs. [25] and [30] are also scaled by
+using ωex = 0.0844 a.u.. In the previous section, we have
+justified theoretically why such an approach would lead
+to errors in the determination of the E1P V values. To
+demonstrate it numerically, we have given results for both
+the 6S − 7S and 6S − 5D3/2 transitions from the CPDF-
+RPA method using Eqs. (89) and (90) in Table VII. We
+have given these values using ω, ωex and then also using
+ωex and experimental energies (denoted by Eexpt
+i,f ) of the
+6S, 7S and 5D3/2 states. From the comparison of the re-
+sults, we observe a very a interesting trend. When both
+ω and energies of the atomic states are considered either
+from theory or experiment, results from both Eqs. (89)
+and (90) match each other, otherwise large discrepancies
+are seen. In the RMBPT, RPA or CPDF-RPA method,
+it is possible to use ωex and experimental energies of the
+initial and final states simultaneously in the E1P V eval-
+uating expressions. However, one can either use ωex or
+ωex with experimental energy of only the valence state
+(whose perturbed state wave function is evaluated) in
+the complicated methods like the RCC method. Ener-
+gies of the double, triple excited configurations appear in
+the denominator of the RCC theory, their experimental
+energies cannot be used in the wave function determin-
+ing equations. By corroborating this fact with the above
+finding, it can be said that scaling the wave function
+by using experimental energy of the valence state alone
+may not always give accurate result, rather it may intro-
+duce additional error to the calculation. As explained in
+the previous section, this fact can be theoretically under-
+stood using Eq. (41). Nevertheless, it can be found from
+Table VII that our result with ωex value from the CPDF-
+RPA* method does not match with the corresponding
+results from Refs. [25, 30] for the 6S −7S transition. We
+are unable to understand the reason for this though re-
+sults with theoretical ω value from both the works agree
+quite well.
+As mentioned in Sec. IV, three different approaches
+can be adopted in any many-body theory framework for
+evaluation of the E1P V amplitudes. The same applies
+to the RCC theory as well. However, we adopt the ap-
+proach of evaluating the matrix element of the D op-
+
+24
+TABLE VIII. First-principle calculated E1P V values (in −i(QW /N)ea0×10−11) of the 6s 2S1/2−7s 2S1/2 and 6s 2S1/2−5d 2D3/2
+transitions in 133Cs from different terms of the RCCSD method. Both ab initio and scaled values are given for comparison.
+We have used two different types of scaling: (a) only scaling amplitudes of the the unperturbed S(0)
+v
+operators and (b) scaling
+amplitudes of both the S(0)
+v
+and S(1)
+v
+operators.
+Here, contributions under ‘Norm’ represents the difference between the
+contributions after and before normalizing the RCCSD wave functions. ‘Others’ denote contributions from those RCCSD terms
+that are not shown explicitly in this table.
+RCC term
+6s2S1/2 − 7s 2S1/2
+6s 2S1/2 − 5d 2D3/2
+Ab initio
+Scaled-a
+Scaled-b
+Ab initio
+Scaled-a
+Scaled-b
+Core contribution
+DT (1)
+1
+−0.04161
+−0.04161
+−0.04161
+−0.00062
+−0.00062
+−0.00062
+T (1)†
+1
+D
+0.03964
+0.03964
+0.03964
+−0.17132
+−0.17132
+−0.17132
+Others
+−0.00005
+−0.00005
+−0.00005
+0.01757
+0.01757
+0.01757
+Norm
+0.00005
+0.00005
+0.00005
+0.00692
+0.00670
+0.00670
+Valence contribution
+DS(1)
+1i
+−0.19363
+−0.19363
+−0.19688
+−2.96310
+−2.96310
+−2.97589
+S(1)†
+1f D
+1.80382
+1.80382
+1.80263
+−0.89993
+−0.89993
+−1.30760
+S(0)†
+1f DS(1)
+1i
+−0.23184
+−0.23187
+−0.23297
+−0.06863
+−0.06548
+−0.06487
+S(1)†
+1f DS(0)
+1i
+−0.41826
+−0.41895
+−0.41942
+0.10487
+0.10502
+0.14626
+DS(1)
+2i
+−0.00039
+−0.00039
+−0.00039
+0.00107
+0.00107
+0.00108
+S(1)†
+2f D
+0.00033
+0.00033
+0.00033
+−0.00023
+−0.00023
+−0.00023
+Others
+−0.04040
+−0.04222
+−0.04025
+0.24888
+0.24806
+0.27704
+Norm
+−0.02122
+−0.01942
+−0.02110
+0.16040
+0.15505
+0.15309
+Total
+0.89643
+0.89570
+0.88998
+−3.56412
+−3.56721
+−3.91879
+erator after considering HW as the external perturba-
+tion. Though this approach is in the line with the CPDF
+method, it is effectively takes care of electron correla-
+tion effects through both the HW and D operators as in
+the CPDF-RPA method. In fact, it goes much beyond
+the CPDF-RPA method to include the electronic corre-
+lation effects which will be evident from the follow-up
+discussions. It means that it is possible to deduce all the
+CPDF-RPA contributions from the RCC theory, which is
+even true at the level of the RCCSD method approxima-
+tion. In this sense that the RCCSDT method employed
+by Sahoo et al. [29] to estimate the E1P V amplitude of
+the 6S − 7S transition in 133Cs includes the RPA contri-
+butions that are mentioned in Refs. [25, 30]. However,
+some of these contributions are not a part of the Core
+contribution rather they come through the Valence con-
+tribution in our RCCSD method owing to the fact that
+the D operator is not treated as an external perturba-
+tion here to determine the perturbed atomic wave func-
+tions. This point can be comprehend from the compar-
+ison between the RMBPT(3)w and RMBPT(3)d results,
+which are propagated to all-orders in the RCC theory.
+To define Core contributions in the line of CPDF-RPA
+method, the RCC theory of E1P V can be derived ei-
+ther treating the HW and D operators simultaneously
+as external perturbation or perturbing wave functions by
+considering one of these operators as external perturba-
+tion and evaluating matrix element of the other operator
+in the normal-order RCC theory framework similar to
+that is discussed in Ref. [46]. Among the choices of con-
+sidering one of them as the external perturbation, it is
+advisable to use HW as perturbation in which evalua-
+tion of perturbed wave function in an iterative scheme
+can converge faster. In fact, this should also be the nat-
+ural choice from the APV theory point of view and is
+being adopted here.
+In order to understand the Core
+and Valence contributions to the E1P V amplitudes from
+our RCCSD method, we can take the help of the dia-
+grams from the RMBPT(3)w method. Since wave opera-
+tor amplitude determining equations for both the meth-
+ods follow the same Bloch’s prescription, all physical
+effects appear in the RMBPT(3)w method are present
+to all-orders in the RCCSD method.
+It means that
+the core-polarization effects and additional lower-order
+DCP contributions that arise in the RMBPT(3)w method
+are present to all-orders in the RCCSD method.
+In
+the CPDF-RPA method, core-polarization effects are ap-
+pearing through the single excitations while the DCP ef-
+fects are implicitly present through the double excitations
+in the RCCSD method. Similarly, the pair-correlation ef-
+fects of the RMBPT(3)w method are present to all-orders
+through the single excitations in the RCCSD method.
+Since both single and double excitation amplitude solv-
+ing equations are coupled in the RCCSD method, cor-
+relations among all these physical effects are taken into
+account in this method. Again through the non-linear
+
+25
+terms from the exponential form of the RCCSD method,
+higher-order correlation effects, neither a part of core-
+polarization or pair-correlation types, are also included
+in the RCCSD method. It is not possible to include these
+effects systematically using a blend of many-body meth-
+ods.
+In Table VIII, we present the E1P V values of the
+6S − 7S and 6S − 5D3/2 transitions in 133Cs from in-
+dividual RCCSD terms to fathom the discussions of the
+previous paragraph quantitatively. By using definitions
+of the T and Sv RCC excitation operators, we catego-
+rized the results into Core and Valence correlation con-
+tributions. By subtracting the Core contributions of the
+DHF method from the contributions of the ¯DT (1)
+1
+and its
+complex conjugate (c.c.) term, the net Core correlation
+contributions to E1P V in the RCCSD method can be
+inferred. Similarly by subtracting the Valence contribu-
+tions of the DHF method from the ¯DS(1)
+1i +S(1)†
+1f
+¯D terms
+and adding contributions from other Valence correlation
+contributing terms, we can get the net Valence correla-
+tion contributions to E1P V in the RCCSD method. The
+Core correlations arising through ¯DT (1)
+1
+and c.c. terms
+contain correlation contributions from both the singly
+and doubly excited configurations.
+By analysing the
+RMBPT(3)w diagrams contributing to the T (1)
+1
+ampli-
+tude determining equation shown in Fig. 14, it can be
+understood that the ¯DT (1)
+1
+and c.c. terms contain the
+Core contributions of the CPDF method, pair-correlation
+contributions of the RMBPT(3) method to all-orders and
+many more.
+By analyzing diagrams from ¯DT (1)
+1
+and
+their breakdown in terms of the RMBPT(3)w method
+carefully, it can be evident that this term does not in-
+clude Core contributions arising through the RPA and
+some of the contributions that arise through the CPDF-
+RPA method. Similarly, all the Valence correlation con-
+tributions from the CPDF method, RPA and CPDF-
+RPA* method are included through the ¯DS(1)
+1i + S(1)†
+1f
+¯D
+terms in the RCCSD method. In addition, they also in-
+clude many contributions that can appear through the
+BO-correlation technique and beyond.
+However, a lot
+more correlation contributions to E1P V arise through
+other RCCSD terms among which correlation contribu-
+tions arising through ¯DS(1)
+2i ,
+¯DT (1)
+1/2S(0)
+1/2i, T (1)†
+1/2 ¯DS(0)
+1/2i,
+such terms but replacing S(0/1)
+1/2i
+operators with S(0/1)†
+1/2f ,
+S(0)†
+1/2f ¯DS(1)
+1/2i, S(1)†
+1/2f ¯DS(1)
+1/2i etc.
+terms in the RCCSD
+method. Obviously, these contributions are not present
+in the CPDF-RPA* method and many of them cannot
+be considered as a part of the BO-correlation method.
+Moreover, corrections to the entire correlation contri-
+butions including that appear through the CPDF-RPA
+method due to normalization of the wave functions (given
+as ‘Norm’) are quoted separately in the above table
+and they are found to be non-negligible.
+The most
+prominent DCP contributions are absorbed through the
+¯DS(1)
+2i +S(1)†
+2f
+¯D terms in the RCCSD method. Along with
+some of Core contributions from the CPDF-RPA method
+(like the ones appears in the RPA) are also included
+through these terms in the RCCSD method. In addition,
+non-linear terms ¯DT (1)
+1/2S(0)
+2i , T (1)†
+2
+¯DS(0)
+1i , S(0)†
+2f
+¯DS(1)
+2i etc.
+including their c.c. terms posses a lot more Valence corre-
+lation contributions that are beyond the scope of consid-
+ering by the combined CPDF-RPA and BO-correlation
+methods.
+In Table VIII, contributions to E1P V of the 6S − 7S
+and 6S −5D3/2 transitions from the RCCSD terms using
+the scaled S(0)
+v
+and S(1)
+v
+amplitudes are also given. To
+show how the results vary with scaling both the unper-
+turbed and perturbed wave functions independently, we
+present results after (a) scaling only the amplitudes of
+the S(0)
+v
+operators and (b) then by scaling amplitudes of
+both the S(0)
+v
+and Sv(1) operators. Significant differences
+from both the scaled results are noticed. As mentioned
+before, it would not be correct to scale the T (0/1) ampli-
+tudes as orbitals used for their determination see V N po-
+tential against the amplitude determining equations for
+the S(0/1)
+v
+operators, where orbitals see V N−1 potential.
+Thus, substituting ωex value in the estimation of Core
+contribution may not be theoretically proper. Again, we
+can only substitute energy of the valence state from out-
+side in the wave function solving equations, whereas en-
+ergies of the intermediate states have to be generated
+implicitly in the RCC theory.
+It means that evaluat-
+ing the E1P V amplitudes through the scaling procedure
+using the RCC method may introduce numerical errors
+to the calculations. Nonetheless, comparison of the semi-
+empirical results obtained by using experimental energies
+with the ab initio values of E1P V for both the 6S−7S and
+6S − 5D3/2 transitions shows that there are significant
+differences among them. These differences can be min-
+imised by including higher-level excitations in the RCC
+theory. However, these higher-level excitations will not
+only improve the energy values but they will also change
+the matrix elements of the HW and D operators.
+As
+shown by Sahoo et al. [29, 47] inclusion of the triple ex-
+citations in the RCC theory, modify the energies and the
+matrix elements of the HW and D operators in such a way
+that the E1P V values from the RCCSD and RCCSDT
+methods almost remain to be same. From this argument,
+we cannot argue that the scaled E1P V values are more
+accurate than the ab initio values in the RCCSD method
+approximation.
+VI.
+SUMMARY
+By employing a number of relativistic many-body
+methods
+at
+different
+levels
+of
+approximation
+such
+as finite-order perturbation theory, coupled-perturbed
+Dirac-Fock method, random phase approximation, com-
+bined coupled-perturbed Dirac-Fock and random phase
+approximation method, and relativistic coupled-cluster
+theory, we investigated various roles of core and valence
+correlation effects in the calculations of the parity vio-
+
+26
+lating electric dipole amplitudes of the 6S → 7S and
+6S → 5D3/2 transitions in 133Cs.
+From this analysis,
+we were able to address a long standing issue of get-
+ting opposite signs to the core correlation contribution
+to the parity violating electric dipole amplitude of the
+aforementioned 6S → 7S transition using the combined
+coupled-perturbed Dirac-Fock and random phase ap-
+proximation methods. We also analysed results from the
+sum-over-states approach and first-principle calculations
+using the relativistic coupled-cluster method with sin-
+gles and doubles approximation to figure out the missing
+contributions in the former approach. Inclusion of these
+missing contributions through the combined coupled-
+perturbed Dirac-Fock, random phase approximation and
+Br¨uckener-orbital correlation methods is compared with
+the first-principle calculations using the coupled-cluster
+method. This comparison shows that the first-principle
+approach using the relativistic coupled-cluster theory in-
+corporates electron correlation effects due to the Dirac-
+Coulomb Hamiltonian more rigorously than the other
+methods mentioned above in the evaluation of parity vi-
+olating electric dipole amplitudes in the 133Cs atom.
+ACKNOWLEDGEMENT
+The computations reported in the present work were
+carried out using the Vikram-100 HPC cluster of the
+Physical Research Laboratory (PRL), Ahmedabad, Gu-
+jarat, India.
+[1] E. D. Commins, Quantum Mechanics An Experimental-
+ists Approach, Cambridge University Press, New York
+(2003).
+[2] J. Erler and P. Langacker, Phys. Rev. Lett. 105, 031801
+(2010).
+[3] W. J. Marciano and J. L. Rosner, Phys. Rev. Lett. 65,
+2963 (1990); 68, 898 (1992).
+[4] M. -A. Bouchiat and C. Bouchiat, Phys. Lett. B 48, 111
+(1974); J. Phys. (France) 35, 899 (1974).
+[5] P. Fayet, Phys. Lett. B 96, 83 (1980).
+[6] C. S. Wood, S. C. Bennet, D. Cho, B. P. Masterson, J. L.
+Roberts, C. E. Tanner and C. E. Wieman, Science 275,
+1759 (1997).
+[7] J. Choi and D. S. Elliott, Phys. Rev. A 93, 023432 (2016).
+[8] A. Kastberg, T. Aoki, B. K. Sahoo, Y. Sakemi and B. P.
+Das, Phys. Rev. A 100, 050101(R) (2019).
+[9] V. A. Dzuba, V. V. Flambaum, P. G. Silvestrov and O.
+P. Sushkov, Phys. Lett. A 103, 265 (1984).
+[10] V. A. Dzuba, V. V. Flambaum, P. G. Silvestrov and O.
+P. Sushkov, J. Phys. B: Atom. Mol. Phys. 18, 597 (1985).
+[11] A. -M. M˚artensson-Pendrill, J. Phys. 46, 11 (1985).
+[12] S. A. Blundell, J. Sapirstein and W. R. Johnson, Phys.
+Rev. D 45, 5 (1992).
+[13] V. A. Dzuba, V. V. Flambaum and J. S. M. Ginges, Phys.
+Rev. D 63, 062101 (2001).
+[14] V. A. Dzuba, V. V. Flambaum and O. P. Sushkov, Phys.
+Lett. A 141, 147 (1989).
+[15] A. I. Milstein, O. P. Sushkov and I. S. Terekhov, Phys.
+Rev. Lett. 89, 283003 (2002).
+[16] B. A. Brown, A. Derevianko and V. V. Flambaum, Phys.
+Rev. C 79, 035501 (2009).
+[17] A. Derevianko, Phys. Rev. Lett. 85, 1618 (2000); Phys.
+Rev. A 65, 012106 (2001).
+[18] M. G. Kozlov, S. G. Porsev and I. I. Tupitsyn, Phys. Rev.
+Lett. 86, 3260 (2001).
+[19] V. M. Shabaev, K. Pachucki, I. I. Tupitsyn and V. A.
+Yerokhin, Phys. Rev. Lett. 94, 213002 (2005).
+[20] V. A. Dzuba, V. V. Flambaum and J. S. M. Ginges, Phys.
+Rev. D 66, 076013 (2002).
+[21] B. P. Das, B. K. Sahoo, G. Gopakumar, and R. K.
+Chaudhuri, J. Mol. Str. 768, 141 (2006).
+[22] B.
+K.
+Sahoo,
+Ph.D.
+thesis,
+Manga-
+lore
+University,
+Karnataka,
+India,
+2006,
+http://prints.iiap.res.in/handle/2248/680.
+[23] B. K. Sahoo, J. Phys. B 43, 085005 (2010).
+[24] S. G. Porsev, K. Beloy and A. Derevianko, Phys. Rev. D
+82, 036008 (2010).
+[25] V. A. Dzuba, J. C. Berengut, V. V. Flambaum and B.
+Roberts, Phys. Rev. Lett. 109, 203003 (2012).
+[26] B. M. Roberts, V. A. Dzuba and V. V. Flambaum, Phys.
+Rev. A 88, 042507 (2013).
+[27] M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball,
+A. Derevianko and C. W. Clark, Rev. Mod. Phys. 90,
+025008 (2018).
+[28] C. Wieman and A. Derevianko, arXiv:1904.00281 (Un-
+published).
+[29] B. K. Sahoo, B. P. Das and H. Spiesberger, Phys. Rev.
+D 103, L111303 (2021).
+[30] B. M. Roberts and J. S. M. Ginges, Phys. Rev. D 105,
+018301 (2022).
+[31] H. B. Tran Tan, D. Xiao and A. Derevianko, Phys. Rev.
+A 105, 022803 (2022).
+[32] B. K. Sahoo, B. P. Das and H. Spiesberger, Phys. Rev.
+D 105, 018302 (2022).
+[33] R. N. Cahn and G. L. Kane, Phys. Lett. B 71, 348 (1977).
+[34] W. J. Marciano and A. I. Sanda, Phys. Rev. D 17, 3055
+(1978).
+[35] E. D. Commins, Phys. Scr. T46, 92 (1993).
+[36] I. Lindgren and J. Morrison, Atomic Many-Body Theory,
+Springer-Verlag Berlin Heidelberg (1986).
+[37] B. M. Roberts, V. A. Dzuba and V. V. Flambaum, Phys.
+Rev. A 88, 012510 (2013).
+[38] A. Kastberg, B. K. Sahoo, T. Aoki, Y. Sakemi and B. P.
+Das, Symmetry 12, 974 (2020).
+[39] B. K. Sahoo, B. P. Das, Phys. Rev. A 84, 010502(R)
+(2011).
+[40] L. W. Wansbeek, B. K. Sahoo, R. G. E. Timmermans,
+K. Jungmann, B. P. Das and D. Mukherjee, Phys. Rev.
+A 78, 050501(R)(2008).
+[41] L. Young, W. T. Hill III, S. J. Sibener, S. D. Price, C.
+E. Tanner, C. E. Wieman and S. R. Leone, Phys. Rev. A
+50, 2174 (1994).
+
+27
+[42] A. A. Vasilyev, I. M. Savukov, M. S. Safronova and H.
+G. Berry, Phys. Rev. A 66, 020101 (2002).
+[43] M.-A. Bouchiat, J. Guena, and L. Pottier, J. Phys.
+(Paris), Lett. 45, 523 (1984).
+[44] S. C. Bennett, J. L. Roberts and C. E. Wieman, Phys.
+Rev. A 59, R16 (1999).
+[45] A. Kramida, Yu. Ralchenko, J. Reader and NIST ASD
+Team, NIST Atomic Spectra Database (ver. 5.6.1) (Na-
+tional Institute of Standards and Technology, Gaithers-
+burg, MD, 2018).
+[46] B. K. Sahoo and B. P. Das, Phys. Rev. Letts. 120, 203001
+(2018).
+[47] B. K. Sahoo and B. P. Das, arXiv:2008.08941 [hep-ph]
+(2020) (Unpublished).
+
diff --git a/l9E1T4oBgHgl3EQfhAQw/content/tmp_files/load_file.txt b/l9E1T4oBgHgl3EQfhAQw/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..76e9823846f120e443b635160cbb35000e6314a8
--- /dev/null
+++ b/l9E1T4oBgHgl3EQfhAQw/content/tmp_files/load_file.txt
@@ -0,0 +1,2022 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf,len=2021
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='03235v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='atom-ph]  9 Jan 2023  Deciphering Core, Valence and Double-Core-Polarization Contributions to Parity  Violating Amplitudes in 133Cs using Different Methods  a,bA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Chakraborty and aB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Sahoo∗  aAtomic, Molecular and Optical Physics Division,  Physical Research Laboratory, Navrangpura, Ahmedabad 380009, India and  bIndian Institute of Technology Gandhinagar, Palaj, Gandhinagar 382355, India  (Dated: January 10, 2023)  As a prerequisite of probing physics beyond the Standard Model (BSM) of particle physics, it  is imperative to perform calculation of parity violating electric dipole (E1P V ) amplitudes within  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='5% accuracy in atomic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Latest high precision calculations of E1P V of the 6s 2S → 7s 2S  transition in 133Cs by three different groups claim achieving its accuracy below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='5%, but such  claims contradict on the view that their final values differ by 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  One of the major issues in  these calculations is the opposite signs among the core correlation contribution from different works  leading to 200% difference in its value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  In a review [Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  90, 025008 (2018)], a  Letter [Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' D 103, L111303 (2021)] and a Comment [Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' D 105, 018301 (2022)],  reliability of all these calculations are strongly contended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' We unearthed here the underlying reason  for getting sign discrepancies in various works by investigating how different electron correlation  effects are encapsulated through the undertaken methods in the above works to determine E1P V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Detailed discussions presented in this work would help in guiding theoretical studies to improve  accuracy of E1P V in atomic systems to probe BSM physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  INTRODUCTION  Atomic parity violation (APV) has implications for ex-  ploring physics beyond the Standard Model (SM) of par-  ticle physics [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' The neutral current weak interactions  due to exchange of the Z0 bosons between electrons and  nucleus in atomic systems lead to APV [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' APV stud-  ies can offer a fundamental quantity known as nuclear  weak charge QW , from which model independent val-  ues of electron-up and electron-down quarks coupling co-  efficients can be inferred [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Hence, any deviations  of these values from the SM can be used to probe new  physics or to complement some of the findings of the  Large Hadron Collider facility when it is upgraded at the  higher TeV energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' The nuclear spin-independent  (NSI) component of APV has been measured to an ac-  curacy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='35% in the 6s 2S1/2 − 7s 2S1/2 transition in  133Cs [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Advanced experimental techniques have been  proposed recently to improve accuracy of the above mea-  surement, as well as carrying out similar measurement  in the 6s 2S1/2 − 5d 2D3/2 transition of 133Cs [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In  order to extract QW value from these measurements, it is  imperative to perform calculation of the parity violating  electric dipole (E1P V ) amplitudes of the corresponding  transitions very precisely (arguably less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='5%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  There have been a long history of performing calcula-  tions of the E1P V amplitude for the 6s 2S1/2 − 7s 2S1/2  transition in 133Cs by employing different state-of-the-  art relativistic atomic many-body theories at different  levels of approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Among the early calculations,  Dzuba et al had employed the time-dependent Hartree-  Fock (TDHF) method [9, 10], while Martensson had  ∗ bijaya@prl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='in  applied the combined coupled-perturbed Dirac-Hartree-  Fock (CPDF) method and random-phase approximation  (RPA), together referred as CPDF-RPA method [11],  to investigate the roles of core-polarization effects to  E1P V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Both these methods are technically equivalent,  but Martensson had also provided results at the interme-  diate levels using approximations at the Dirac-Hartree-  Fock (DHF), CPDF and RPA methods as well as list-  ing contributions from double-core-polarization (DCP)  effects explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Later, Blundell et al had employed  a linearized version relativistic coupled-cluster method  in the singles and doubles excitation approximation (SD  method) to estimate the E1P V amplitude of the above  transition [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' However, they had adopted a sum-over-  states approach in which matrix elements of the electric  dipole (E1) operator and APV interaction Hamiltonian  were evaluated for the transitions involving np 2P1/2 in-  termediate states (called as “Main” contribution) with  the principal quantum number n = 6 − 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' This method  also utilized the calculated E1 matrix elements and mag-  netic dipole hyperfine structure constants to estimate  the uncertainty of E1P V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Uncertainties from the ener-  gies were removed by considering the experimental en-  ergies, while contributions from core orbitals (referred  as “Core” contribution henceforth) and higher np 2P1/2  intermediate states (hereafter called as “Tail” contribu-  tion) were estimated using lower-order methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Follow-  ing this work, Dzuba et al improved their calculation  of TDHF method by incorporating correlation contribu-  tions through the Br¨uckner orbitals (BO) and referred  the approach as RPA+BO method [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Higher-order  contributions from the Breit, lower-order QED and neu-  tron skin effects were added subsequently through dif-  ferent works to claim for more precise E1P V value for  extracting the BSM physics [14–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  It is worth men-  tioning here that all these higher-order effects were es-  2  timated through different types of many-body methods  and without accounting correlations among themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Soon after these theoretical results, relativistic coupled-  cluster (RCC) theory with singles and doubles approx-  imation (RCCSD method) was employed to treat both  the electromagnetic and weak interactions on an equal  footing [21–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Moreover, it also treated correlations  among the Main, Core and Tail contributions to E1P V  but its accuracy was an concern due to its ab initio nature  and use of a relatively smaller basis size in the available  computational resources at that time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  A decade ago, Porsev et al made further improve-  ment in the sum-over-states result of Blundell et al  by considering contributions from the non-linear terms  from RCCSD method to their SD method as well as  adding valence triple excitations (CCSDvT method) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Their claimed accuracy to the E1P V amplitude of the  6s 2S1/2 − 7s 2S1/2 transition in 133Cs was about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='27%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  However, the Core and Tail contributions were still es-  timated using a blend of many-body methods without  explicitly stating which physical effects were taken into  account for their evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' We refer these two contri-  butions together as X-factor in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In an attempt  to improve the calculated E1P V value further, Dzuba et  al estimated the X-factor contributions using their TDHF  approach but omitting the DCP contributions [25] in the  similar line to their earlier works [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' This calculation  showed an opposite sign of Core contribution than that  was reported by Porsev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In 2013, Roberts reported  the DCP contribution separately [26] and the result was  slightly different than the value of Martenssion [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' The  opposite sign of the Core contribution of Dzuba et al  with Porcev et al was criticized in two papers [27, 28],  which prompted for carrying out further investigation  on different correlation contributions to E1P V from the  first-principle approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In 2021, Sahoo et al improved  their calculation of the above E1P V amplitude by imple-  menting the singles, doubles and triples approximation  to both the unperturbed and perturbed wave functions  (RCCSDT method) and using a much bigger set of basis  functions [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' They also used the same V N−1 poten-  tial as in the previous cases and presented the Core and  Valence (Main and Tail together) contributions explic-  itly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' As per the convention adopted in this approach, the  Core contribution agreed with the earlier RCCSD result  [21–23], and was close to the reported value of Blundell  et al [12] and Porsev et al [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In a recent Comment,  Roberts and Ginges have argued in favour of opposite  sign of Core contribution than other findings by giving  intermediate results of their RPA+BO method [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In  another work, Tan el al estimated combined Core and  Tail contributions to the E1P V amplitude of the above  transition using mixed-parity orbitals through RPA [31]  and support the value reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' It is, there-  fore, abundantly clear that it is necessary to find out the  issue of sign problem with the Core contribution to the  above E1P V amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' More importantly, the basis of  dividing net E1P V result into Core, Main, Tail, DCP,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  contributions in an approach should be properly  defined and missing physical effects in a method com-  pared to others need to be well understood when a mix-  ture of methods are used to estimate these contributions  piece-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Any misinterpretation or misrepresentation  of these contributions can have repercussion effect when  they are used to infer beyond the SM (BSM) physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  The present work is devoted to addressing the afore-  mentioned sign issue with the Core contribution among  various works, bringing to notice the shortcomings of the  sum-over-states approach, and explaining the reason for  the coincidental agreement between the core contribu-  tions of Porsev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [24] and Sahoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' We start  with various procedures that can be adopted through a  general many-body method to evaluate E1P V amplitudes  in atomic systems and demonstrate how the definition  of Core contribution can vary from one procedure to an-  other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' With the help of lower-order many-body methods,  we find out the missing contributions in a typical sum-  over-states approach to estimate E1P V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' We then analyze  results from different methods to learn about how and  to what extent these missing effects are incorporated in  the previous calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' We also make a similar anal-  ysis for the E1P V amplitude of the 6s 2S1/2 − 5d 2D3/2  transition in 133Cs and compare it with the result of the  6s 2S1/2 − 7s 2S1/2 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' There are two important  reasons for quoting result of the S−D3/2 transition of Cs  here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' First, our analysis would be helpful to estimate its  E1P V amplitude more precisely which is required for the  ongoing experiments [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Second, we intend to address  a comment [30, 32] on why sign for Core contribution to  the S − D3/2 transitions from different methods agree in  contrast to the S − S transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  THEORY  The short range effective Lagrangian corresponding to  the vector- -axial-vector neutral weak current interaction  of an electron with up- and down-quarks in an atomic  system can be given by [4, 33–35]  Leq = GF  √  2  �  u,d  �  C1u ¯ψuγµψu + C1d ¯ψdγµψd  � ¯ψeγµγ5ψe  = GF  √  2  �  n  C1n ¯ψnγµψn ¯ψeγµγ5ψe,  (1)  where GF = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='16632×10−5 GeV−2 is the Fermi constant,  sums u, d and n stand for up-quark, down-quark and nu-  cleons respectively, and C1i with i = u, d and n repre-  sent coupling coefficients of the interaction of an electron  with up-quark, down-quark and nucleons (protons (Pn)  and neutrons (Nn)) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Adding them coher-  ently and taking the non-relativistic approximation for  nucleons, the temporal component can give the nuclear-  spin-independent APV interaction Hamiltonian as  HNSI  PV = − GF  2  √  2QW γ5ρ(r),  (2)  3  (a)  (b)  (c)  (d)  f  f  f  f  i  i  i  i  p  p  a  a  D  HW  D  HW  HW  HW  D  D  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Goldstone diagrams representing Core (a and b) and  Valence (c and d) contributions to E1P V in the V N−1 DHF  potential of an one valence atomic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Here, double ar-  rows represent initial (i) and final (f) valence orbitals, single  arrows going down (a) means occupied orbitals and single ar-  rows going up (p) means virtual orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' The operators D  and HW are shown in curly line and a line with bullet point  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  where ρ(r) is the averaged nuclear density and QW =  2[ZC1P n + (A − Z)C1Nn] is known as the nuclear weak  charge with Z and A representing for atomic and mass  numbers respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  It is obvious that QW  is a  model dependent quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Thus, the difference of its  actual value from the SM can provide signatures of  physics supporting other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  In the SM, C1u =  1  2  �  1 − 8  3 sin2 θSM  W  �  and C1d = − 1  2  �  1 − 4  3 sin2 θSM  W  �  [4, 33–  35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  This follows C1Nn = 2C1d + C1u = −1/2 and  C1P n = 2C1u + C1d = (1 − 4 sin2 θSM  W )/2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='04, which  are accurate up to 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Hence, it is necessary to evaluate  model independent QW in any atomic system within this  accuracy in order to infer physics beyond the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  EVALUATION PROCEDURES OF E1P V  In the presence of APV, the net atomic Hamiltonian  is given by  Hat = H + HNSI  PV  = H + GF HW ,  (3)  where H contains contributions from electromagnetic in-  teractions and HW is defined in order to treat GF as  a small parameter to include contributions from HNSI  PV  perturbatively through a many-body method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Using the  wave functions of Hat, we determine E1P V of a transition  between states |Ψi⟩ and |Ψf⟩ as  E1P V =  ⟨Ψf|D|Ψi⟩  �  ⟨Ψf|Ψf⟩⟨Ψi|Ψi⟩  ,  (4)  where D is the E1 operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In the earlier calculations  of E1P V amplitude in 133Cs, atomic wave functions were  determined by using the V N−1 potential with N is the  number of electrons of the atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Choice of this poten-  tial is convenient to produce both the ground and excited  states of Cs atom using the Fock-space formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' We  adopt the same formalism here, so that description of  different correlation effects and comparison of results are  consistent to each other in all the considered works in-  cluding in the definition of the Core contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Following a typical approach in the many-body prob-  lems, we define a suitable mean-field Hamiltonian H0 to  replace the exact Hamiltonian H to obtain a set of ap-  proximated solutions  H0|Φk⟩ = Ek|Φk⟩,  (5)  where subscript k is used to identify different states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Since [5p6] is the common closed-shell configuration in  the states of our interest in the present work, we ob-  tain the solution of this closed-core first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' We consider  the DHF method to obtain its mean-field wave function  |Φ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Then, the |Φk⟩ wave functions are obtained as  |Φv⟩ = a†  v|Φ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (6)  Starting with |Φv⟩, we can express the exact wave func-  tion of the state using Bloch’s prescription as [36]  |Ψv⟩ = Ωv|Φv⟩,  (7)  where Ωv is known as the wave operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In the V N−1  potential approximation, we first solve electron correla-  tion effects among electrons from the core orbitals of |Φ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Then, correlation effects involving electron from the va-  lence orbital are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Accordingly Ωv is divided into  two parts  Ωv = Ω0 + Ωv,  (8)  where Ω0 represents wave operator accounting correla-  tions of electrons only from the core orbitals while Ωv  takes care of correlations of electrons from all orbitals  including the valence orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  In a given many-body  method, we can solve amplitudes of the above wave op-  erators using the equations  [Ω0, H0]P0 = UresΩ0P0  (9)  and  [Ωv, H0]Pv = Ures(Ω0 + Ωv)Pv  −ΩvPvUres(Ω0 + Ωv)Pv,  (10)  where Pn = |Φn⟩⟨Φn| and Qn = 1 − Pn with n ≡ 0, v,  and Ures = H − H0 is known as the residual interaction  that contributes to the amplitudes of Ω0 and Ωv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  In  fact, the energy of the |Ψv⟩ state can be evaluated as the  expectation value of the effective Hamiltonian  Heff = PvUres(Ω0 + Ωv)Pv  (11)  with respect to the reference state |Φv⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  It should be  noted that energy of the state is given by  Ev = ⟨Φv|Heff|Φv⟩  (12)  For the choice of V N potential in the generation of single  particle orbitals, amplitudes of the Ωv operator can be  estimated as  [Ωv, H0]Pv = Ures(Ω0 + Ωv)Pv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (13)  4  It means that Ev that gives the energy of the state does  not appear in the wave function determining equation for  the case of V N potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Thus, the core-valence interac-  tion effects in the construction of DHF potential in case  of V N−1 is partly compensated through the wave oper-  ator amplitude determining equation through the extra  term with energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' If any method utilizes the V N−1 po-  tential without taking into account the above-mentioned  extra term, it can be termed as an improper theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Both |Ψi⟩ and |Ψf⟩ can be determined by solving the  equation-of-motion for Hat in the above formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' How-  ever, parity cannot be treated as a good quantum number  for Hat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' As a consequence, it will relax one degree of free-  dom in describing atomic states for which computations  of amplitudes for the Ω0 and Ωv operators will increase  by many folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Compared to H, the strength of HNSI  PV  in Hat is smaller by 1012 order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Thus, it is important  that contributions from H is accounted as much as pos-  sible in the determination of the above wave functions  with the available computational resources and only the  first-order effect due to HNSI  PV can be accounted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Anyway,  inclusion of higher-order effects from HNSI  PV will not serve  for any purpose in our study as they will be much smaller  then our interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Thus, we express atomic wave function  |Ψv⟩ of a general state with valence orbital v as  |Ψv⟩ = |Ψ(0)  v ⟩ + GF |Ψ(1)  v ⟩,  (14)  where |Ψ(0)  v ⟩ is the zeroth-order wave function containing  contributions only from H while |Ψ(1)  v ⟩ includes one-order  contribution from HW with respect to |Ψ(0)  v ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Substitut-  ing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (14) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (4) and keeping finite terms up to  first-order in GF , we get  E1P V ≃ GF  �  ⟨Ψ(0)  f |D|Ψ(1)  i ⟩  Nif  +  ⟨Ψ(1)  f |D|Ψ(0)  i ⟩  Nif  �  , (15)  where normalization factor Nif = �NfNi with Nv =  ⟨Ψ(0)  v |Ψ(0)  v ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Presenting results in this paper GF is ab-  sorbed with the E1P V value, so it does not appear ex-  plicitly in our calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' It can be noted that contri-  bution from the first term is referred as the initial per-  turbed state contribution whereas contribution from the  second term is referred as the final perturbed state con-  tribution in the above expression during the discussion of  results later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In order to treat both the electromagnetic  and weak interaction Hamiltonians on an equal footing  in a many-body method and consider correlations among  them, solutions of the unperturbed and first-order per-  turbed wave functions should satisfy  H|Ψ(0)  v ⟩ = E(0)  v |Ψ(0)  v ⟩  (16)  and  (H − E(0)  v )|Ψ(1)  v ⟩ = (E(1)  v  − HW )|Ψ(0)  v ⟩,  (17)  respectively, where E(1)  v  = 0 for odd-parity interaction  operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Using the wave operator formalism, we can express  |Ψ(0)  v ⟩ = Ωv(0)|Φv⟩  = (Ω(0)  0  + Ω(0)  v )|Φv⟩  (18)  and  |Ψ(1)  v ⟩ = Ωv(1)|Φv⟩  = (Ω(1)  0  + Ω(1)  v )|Φv⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (19)  Substituting the wave operators, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
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+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (20)  In the above expression,contribution from the first term is  referred as “Core” correlation contribution while the rest  is termed as “Valence” correlation contribution in this  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Analogously, contributions to Nv = ⟨Φ0|av(Ω(0)  0 +  Ω(0)  v )†(Ω(0)  0  + Ω(0)  v )a†  v|Φ0⟩ can also be divided into two  parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Sum-over-states approach  In the sum-over-states approach, the first-order wave  function of a general state can be expressed as  |Ψ(1)  v ⟩ =  �  I̸=v  |Ψ(0)  I ⟩ ⟨Ψ(0)  I |HW |Ψ(0)  v ⟩  Nv(E(0)  v  − E(0)  I )  ,  (21)  where |Ψ(0)  I ⟩ are the zeroth-order intermediate states and  E(0)  n  is the unperturbed energy of the nth level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Thus, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  5  (i)  (ii)  (iii)  (iv)  (v)  (vi)  (vii)  (viii)  (ix)  (x)  (xi)  (xii)  (xiii)  (xiv)  (xiv)  (xvi)  (xvii)  (xviii)  (xix)  (xx)  (xxi)  (xxii)  (xxiii)  (xxiv)  (xxv)  (xxvi)  (xxvii)  (xxviii)  (xxix)  (xxx)  (xxxi)  (xxxii)  (xxxiii)  (xxxiv)  (xxxv)  (xxxvi)  (xxxvii)  (xxxviii)  (xxxix)  (xL)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A few important electron correlation contributing di-  agrams to E1P V in the RMBPT(3) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' The dotted lines  represent the atomic Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Diagrams (i)-(viii) give  Core contributions in the RMBPT(3)w method, but they cor-  respond to Valence contributions in the RMBPT(3)d method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Diagrams (ix)-(xvi) follow other way around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (15) can be written as  E1P V =  �  I̸=i  ⟨Ψ(0)  f |D|Ψ(0)  I ⟩⟨Ψ(0)  I |HW |Ψ(0)  i ⟩  Nif(E(0)  i  − E(0)  I )  +  �  I̸=f  ⟨Ψ(0)  f |HW |Ψ(0)  I ⟩⟨Ψ(0)  I |D|Ψ(0)  i ⟩  Nif(E(0)  f  − E(0)  I )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (22)  In the above expression of E1P V , correlations among  the H and HW that appear through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (17) are omit-  ted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Secondly, there could be conflict with the definitions  of using Core, Main and Tail contributions to E1P V with  the definitions used in various first-principle based calcu-  lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  This is explicitly demonstrated later how for-  mula given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (20) can be altered to redefine Core  and Valence contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' To understand definitions of  Core, Main and Tail contributions used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [24], we  follow the work of Blundell et al [12] where these terms  were used for the first time in the context of estimating  E1P V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Division of the total E1P V value in this calcula-  tion was made as “Main”, “Core” and “Tail” contribu-  tions based on the mere assumption that |Ψ(0)  i ⟩, |Ψ(0)  f ⟩  and |Ψ(0)  I ⟩ are represented by only single Slater determi-  nants like in the DHF method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Thus, the intermediate  states |Ψ(0)  I ⟩ are considered to have only the np 2P1/2  configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In such assumption,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' the Core (C),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Main  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='(V ) and Tail (T ) contributions to the E1P V amplitude of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='the 6s 2S1/2−7s 2S1/2 transition in 133Cs were estimated  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='as  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='E1P V (C) =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='n≤5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='⟨7S1/2|D|nP1/2⟩⟨nP1/2|HW |6S1/2⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='(E6S1/2 − EnP1/2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='n≤5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='⟨7S1/2|HW |nP1/2⟩⟨nP1/2|D|6S1/2⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='(E7S1/2 − EnP1/2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='(23)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='E1P V (V ) =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='n=6−9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='⟨7S1/2|D|nP1/2⟩⟨nP1/2|HW |6S1/2⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='(E6S1/2 − EnP1/2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='n=6−9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='⟨7S1/2|HW |nP1/2⟩⟨nP1/2|D|6S1/2⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='(E7S1/2 − EnP1/2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='(24)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='E1P V (T ) =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='n≥10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='⟨7S1/2|D|nP1/2⟩⟨nP1/2|HW |6S1/2⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='(E6S1/2 − EnP1/2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='n≥10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='⟨7S1/2|HW |nP1/2⟩⟨nP1/2|D|6S1/2⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='(E7S1/2 − EnP1/2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='(25)  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' However, wave functions of multi-electron  atomic systems are determined through a many-body  method by expressing as a linear combination of many  Slater determinants which can differ by either single or  multiple entries of rows or columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' As a result, con-  tributions from cross-terms involving other Slater de-  terminants, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  excited configurations 5p56s7s of the  intermediate states with respect to both the 6s 2S1/2  and 7s 2S1/2 states, cannot appear through the above  6  (i)  (ii)  (iii)  (iv)  (v)  (vi)  (vii)  (viii)  (ix)  (x)  (xi)  (xii)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  A few representative DCP diagrams from the  RMBPT(3) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  All-order forms of diagrams from (i)  to (viii) appear in the CPDF-RPA method, but the rest are  not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' These missing contributions appear in the RCC theory  to all-orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  breakup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' One of such contributions is referred to as DCP  effects which arise through the CPDF-RPA (or TDHF)  method as described by Martensson [11] and Roberts  [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  There are other contributions that could come  through effects that are neither part of BO contributions  nor CPDF-RPA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' However those effects appear  through the first-principle approach of the RCC method  employed by Sahoo et al [32] as shown later part of this  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' These contributions are not small and demands  for an appropriate many-body method to account their  contributions at the par with the np 2P1/2 intermediate  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' This argument can be understood better with the  following explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  In the 133Cs atom, the low-lying excited states have a  common core [5p6] and differ by only a valence orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Thus, the DHF wave functions of these states can be  expressed as |ΦI⟩ = a†  I|Φ0⟩ and the exact wave functions  can be defined as  |Ψ(0)  I ⟩ = ΩI(0)|ΦI⟩  = (Ω(0)  0  + Ω(0)  I )|ΦI⟩,  (26)  where ΩI(0) and Ω(0)  I  are the total and valence corre-  lation contributing wave operators respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Using  these wave operators, we can express Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (22) as  E1P V =  �  I̸=i  ⟨Φ0|af(Ω(0)  0  + Ω(0)  f )†D(Ω(0)  0  + Ω(0)  I )a†  I|Φ0⟩  Nif  × ⟨Φ0|aI(Ω(0)  0  + Ω(0)  I )†HW (Ω(0)  0  + Ω(0)  i )a†  i|Φ0⟩  (E(0)  i  − E(0)  I )  +  �  I̸=f  ⟨Φ0|af(Ω(0)  0  + Ω(0)  f )†HW (Ω(0)  0  + Ω(0)  I )a†  I|Φ0⟩  Nif  × ⟨Φ0|aI(Ω(0)  0  + Ω(0)  I )†D(Ω(0)  0  + Ω(0)  i )a†  i|Φ0⟩  (E(0)  f  − E(0)  I )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (27)  Since wave operators from the initial, final and inter-  mediate states include linear combinations of configu-  rations describing one-hole–one-particle, two-hole–two-  particle etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' excitations, it is obvious that a sum-over-  states approach cannot include contributions from the  higher-level excited configurations contributing to the in-  termediate states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  First-principle approach  It is evident that it is imperative to determine the  E1P V amplitudes in atomic systems using the first prin-  ciple approaches that account contributions from all pos-  sible intermediate configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' This can be done using  either Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (15) or Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' It is desirable to solve both  Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (16) and (17) in the former case, while Bloch’s  equations for the unperturbed and perturbed wave op-  erators need to be solved for the later approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' The  amplitude solving Bloch’s equations for the unperturbed  operators Ω(0)  0  and Ω(0)  v  are similar to that are given by  Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (9) and (10), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' The Bloch’s equations  for the first-order perturbed wave operators can be given  by  [Ω(1)  0 , H0]P0 = (HW Ω(0)  0  + UresΩ(1)  0 )P0  (28)  and  [Ω(1)  v , H0]Pv = [HW (Ω(0)  0  + Ω(0)  v ) + Ures(Ω(1)  0  + Ω(1)  v )Pv  −Ω(1)  v E(0)  v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (29)  As mentioned in Introduction, several all-order meth-  ods in the CPDF, RPA, CPDF-RPA/TDHF and RCC  theory frameworks are employed to determine the E1P V  amplitudes in 133Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Here, we attempt to formulate all  these methods using wave operators so that in the end it  would be easier for us to make one-to-one relations among  various contributions arising through these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Es-  pecially, such exercise is going to be useful in explaining  the reason for which there is a sign difference between  the Core contributions to E1P V from the TDHF method  of Dzuba et al [25] and RCC method employed by Sahoo  et al [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  7  =  +  +  +  +  a  (a)  (b)  (c)  (d)  (e)  p  +  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (f)  ΩCPDF  0  +  (g)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Goldstone diagrams contributing to amplitude deter-  mining equation of ΩCP DF  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Through iterative scheme these  effects are included to all-orders in the CPDF method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  We can rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (15) as  E1P V =  ⟨Ψ(0)  f |HW |˜Ψ(1)  i ⟩  Nif  +  ⟨˜Ψ(1)  f |HW |Ψ(0)  i ⟩  Nif  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (30)  This can be equivalently expressed by either  E1P V =  ⟨Ψ(0)  f |D|Ψ(1)  i ⟩  Nif  +  ⟨Ψ(0)  f |HW |˜Ψ(1)  i ⟩  Nif  (31)  or  E1P V =  ⟨˜Ψ(1)  f |HW |Ψ(0)  i ⟩  Nif  +  ⟨Ψ(1)  f |D|Ψ(0)  i ⟩  Nif  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (32)  In the above expressions, we define  |˜Ψ(1)  i ⟩ =  �  I̸=f  |Ψ(0)  I ⟩  ⟨Ψ(0)  I |D|Ψ(0)  i ⟩  (E(0)  i  − E(0)  I  + ω)  (33)  and  |˜Ψ(1)  f ⟩ =  �  I̸=i  |Ψ(0)  I ⟩  ⟨Ψ(0)  I |D|Ψ(0)  f ⟩  (E(0)  f  − E(0)  I  − ω)  (34)  with ω = E(0)  f  − E(0)  i  is the excitation energy between  the initial and final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' It implies that mathemati-  cally Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (15), (30), (31) and (32) are equal in an ex-  act many-body method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Thus, any of these expressions  can be used in the determination of the E1P V ampli-  tude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' We shall demonstrate later that the CPDF, RPA,  CPDF-RPA and RCC methods are using different formu-  las as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' So it is important to understand  their relations and classifications of individual contribu-  tions through the above methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Since approximations  made to the unperturbed and perturbed wave functions  are not at the same level in these methods, it is obvious to  guess that results from these methods can be very differ-  ent unless electron correlation effects in an atomic system  are negligibly small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' It is also not clear whether classifi-  cations of Core and Tail contributions in these methods  are uniquely defined or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  To understand the above points, lets find out the Core  contributions from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (15) and (30) by expressing the  perturbed wave function due to the D operator in terms  of wave operators as  |˜Ψ(1)  v ⟩ = (˜Ω(1)  0  + ˜Ω(1)  v )|Φv⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (35)  With this, the Core contributing terms in both HW and  D perturbing approaches are given by  E1P V (C) = ⟨Φ0|af[Ω(0)†  0  DΩ(1)  0  + Ω(1)†  0  DΩ(0)  0 ]a†  i|Φ0⟩  Nif  (36)  and  E1P V (C) = ⟨Φ0|af[˜Ω(1)†  0  HW Ω(0)  0  + Ω(0)†  0  HW ˜Ω(1)  0 ]a†  i|Φ0⟩  Nif  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (37)  After a careful analysis it can be shown that the Core con-  tributions arising through the wave operators Ω(1)  0  and  ˜Ω(1)  0  can be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Similar arguments also hold for the  Tail contributions arising through the perturbed valence  operators Ω(1)  v  and ˜Ω(1)  v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' To get better inside of this argu-  ment, we can rewrite the sum-over-states formula given  by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (27) as  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
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+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='I̸=i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='⟨Φ0|afΩ(0)†  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='DΩ(0)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0 Ω(0)†  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='I  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='HW Ω(0)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='i a†  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='i|Φ0⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='Nif(E(0)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='f  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='− E(0)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='I  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='− ω)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='I̸=i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='⟨Φ0|afΩ(0)†  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='f  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='DΩ(0)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0 Ω(0)†  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='I  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='HW Ω(0)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0 a†  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='i|Φ0⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='Nif(E(0)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='f  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='− E(0)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='I  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='− ω)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='+ {HW ⇔ D}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='(38)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='Both Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (27) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (38) are equal but they are  given differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' These equations are nothing but the  expanded forms of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (20) and (27) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' How-  ever, different terms are rearranged to place them under  the categories of Core and Valence contributing terms  in the respective formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Thus, we may now outline  findings from the above discussions as follows  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' It is important to note that in the evaluation of  E1P V both the HW and D operators can be treated  symmetrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Thus, in an approximated method  where correlation effects through both these opera-  tors are not incorporated equivalently, distinctions  of “Core” and “Valence” contributions to E1P V  cannot be defined uniquely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' As a consequence of the above point, estimating  both the “Core” and “Valence” contributions using  a blend of many-body methods could mislead the  final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Numerical stability to the calculation of E1P V can  be verified by evaluating expressions given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (15), (30) and (31) simultaneously though it can be  a strenuous procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Scaling wave functions for estimating a part of con-  tribution or using experimental value of ω in a  method in which both the HW and D operators  are not treated symmetrically may not always im-  ply that the result is improved rather it could in-  troduce further uncertainty to the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  The last point mentioned above can be understood as  discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Lets use the experimental value for ω  (shown as ωex) to define the first-order perturbed wave  functions due to the D operator  |˜Ψ(1)  i ⟩ =  �  I̸=f  |Ψ(0)  I ⟩  ⟨Ψ(0)  I |D|Ψ(0)  i ⟩  (E(0)  i  − E(0)  I  + ωex)  (39)  =  +  +  +  +  p  (a)  (b)  (c)  (d)  (e)  v  +  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (f)  ΩCPDF  v  (g)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Diagrams denoting amplitude solving equation for  ΩCP DF  v  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' These core-polarization effects are included to all-  orders in the CPDF method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  and  |˜Ψ(1)  f ⟩ =  �  I̸=i  |Ψ(0)  I ⟩  ⟨Ψ(0)  I |D|Ψ(0)  f ⟩  (E(0)  f  − E(0)  I  − ωex)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (40)  Substituting these wave functions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (22), the sum-  over-states expression for E1P V can be given by  E1P V =  �  I̸=i  ⟨Ψ(0)  f |D|Ψ(0)  I ⟩⟨Ψ(0)  I |HW |Ψ(0)  i ⟩  Nif(E(0)  i  − E(0)  I  − δω)  +  �  I̸=f  ⟨Ψ(0)  f |HW |Ψ(0)  I ⟩⟨Ψ(0)  I |D|Ψ(0)  i ⟩  Nif(E(0)  f  − E(0)  I  + δω)  , (41)  where δω = ωex − ω, with ω being the theoretical value,  cannot be zero when ω is obtained using a particular  many-body method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' As can be seen, introduction of ωex  value affects contributions from the initial and final per-  turbed terms differently leading to inconsistency in the  evaluation of E1P V than the ab initio calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' This  can be better evident from the following inequalities in  an approximated many-body method  E1P V =  ⟨Ψ(0)  f |D|Ψ(1)  i ⟩  Nif  +  ⟨Ψ(1)  f |D|Ψ(0)  i ⟩  Nif  ̸=  ⟨Ψ(0)  f |HW |˜Ψ(1)  i ⟩  Nif  +  ⟨˜Ψ(1)  f |HW |Ψ(0)  i ⟩  Nif  ̸=  ⟨Ψ(0)  f |D|Ψ(1)  i ⟩  Nif  +  ⟨Ψ(0)  f |HW |˜Ψ(1)  i ⟩  Nif  ̸=  ⟨˜Ψ(1)  f |HW |Ψ(0)  i ⟩  Nif  +  ⟨Ψ(1)  f |D|Ψ(0)  i ⟩  Nif  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (42)  The above inequalities are the result of introduction of  δω in the first-order perturbed wave function due to D  after the substitution of ωex in place of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  MANY-BODY METHODS OF E1P V  The main objective in the APV study is to obtain the  E1P V amplitude within sub-one percent accuracy from  9  the atomic many-body theory perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  In view of  this, it is imperative to evaluate both the zeroth-order  and first-order atomic wave functions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (15) very  accurately by employing a powerful relativistic atomic  many-body method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Owing to complications in account-  ing for various contributions, the entire calculation is  usually performed in several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' The majority of the  contribution from H arises from electron correlation ef-  fects due to Coulomb interactions in the presence of APV  interactions, while corrections from the Breit and QED  interactions are added separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  It would be neces-  sary to include all these interactions through a com-  mon many-body theory in order to consider correlations  among themselves as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Since corrections from the  Breit and QED interactions to E1P V are very small, their  reported estimations are found to be almost consistent  to each other by various works [14–20, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Thus, we fo-  cus here mainly on the discussions considering the Dirac-  Coulomb (DC) interaction Hamiltonian in the determi-  nation of unperturbed wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Again, our aim in  the present paper is to demonstrate how to achieve high-  accuracy calculations of the E1P V amplitudes in 133Cs  by understanding about the roles of Core correlations to  these quantities and identifying contributions that can  arise through a particular many-body method but can be  missed by another method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' To explain these points ex-  plicitly, we discuss calculations of the E1P V amplitudes  in 133Cs using the following methods  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Relativistic  Many-body  Perturbation  Theory  (RMBPT):  This  method  is  employed  in  the  Rayleigh-Schr¨odinger perturbation theory frame-  work to fathom about significance of various  physical effects to E1P V arising at the lower-order  level and to learn how they propagate in the  all-order perturbative methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' CPDF method:  This method is employed in or-  der to reproduce previous results reported by other  groups using the same method using our basis func-  tions that are later considered in the RCC calcula-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' RPA: This method is employed for the same pur-  pose as the above and show importance of coore-  lation contributions arising through this method  which are missing in the CPDF method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' CPDF-RPA method:  This method is employed  again for the above purpose as well as understand-  ing the reason for which Core contribution from  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [25] has a different sign than other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' RCC method: This method is employed both in  the sum-over-states and first-principle approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Differences in both the results are further com-  pared with the values that are included through  the CPDF-RPA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (a)  (b)  (c)  (d)  f  f  f  f  i  i  i  i  p  p  a  a  ΩCPDF  0  ΩCPDF  v  ΩCPDF†  0  D  D  D  D  ΩCPDF†  v  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Property diagrams of the CPDF method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  These  diagrams are similar to the DHF method, but the HW op-  erators of the DHF diagrams are replaced by ΩCP DF  0  and  ΩCP DF  v  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Expansion of ΩCP DF  0  and ΩCP DF  v  in terms of am-  plitude diagrams shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 4 and 5 can reveal how the  DHF contributions and lower-order core-polarization effects  of the RMBPT(3) method are incorporated within the CPDF  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  The atomic Hamiltonian H with DC approximation  can be expressed as sum of one-body and two-body op-  erators  H =  �  i  �  c⃗αD · ⃗pi + (β − 1)c2 + Vnuc(ri)  �  +  �  i,j>i  1  rij  =  �  i  h(ri) +  �  i,j>i  g(rij),  (43)  where c is the speed of light, ⃗αD and β are the Dirac  matrices, ⃗p is the single particle momentum operator,  and �  i,j  1  rij represents the Coulomb potential between  the electrons located at the ith and jth positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' The  entire Hamiltonian can be divided into one-body part  (�  i h(ri)) and two-body part (�  i,j>i g(rij)) for conve-  nience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Using the DHF method, we can obtain the single  particle orbitals using the modified DHF Hamiltonian as  F = �  i f(ri) = �  i(hi + u(0)  i ) and residual interaction  as Vres = H − F in the DC approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Hence  f|i⟩ = ǫi|i⟩  (44)  with the single particle energies ǫi giving unperturbed  DHF energy E0  =  �Nc  b  ǫb and the DHF potential  UDHF = �  i u0(ri) is given by  u0(ri)|i⟩ =  Nc  �  b  [⟨b|g|b⟩|i⟩ − ⟨b|g|i⟩|b⟩]  (45)  for b summing over all occupied-orbitals Nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In the DHF  method, both Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (44) and (45) are solved iteratively to  obtain self-consistent solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' It can be followed from  the above expression that for the determinant expressed  by |Φk⟩ = a†  k|Φ0⟩, the DHF energy is given by Ek = E0 +  ǫk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Similarly, DHF energies of the excited configurations  are given by Ep,q,···  a,b,··· = E0 + ǫp + ǫp + · · · − ǫa − ǫb − · · ·  and {p, q, · · · } denoting for the occupied and unoccupied  orbitals respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' We discuss below E1P V expressions  10  =  +  +  +  +  a  (a)  (b)  (c)  (d)  (e)  p  +  (f)  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  ΩRPA  0  +  (g)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Graphical representation of ΩRP A  0  and its expansion  in terms of lower-order RMBPT(3) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  using the DHF method and different many-body methods  that are employed to account for the electron correlation  effects due to Vres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  DHF method  Using wave functions from the DHF method, we can  evaluate the E1P V amplitude in the mean-field approach  as  E1P V = ⟨Φf|D|Φ(1)  i ⟩ + ⟨Φ(1)  f |D|Φi⟩,  (46)  where |Φ(1)  n=i,f⟩ is the first-order perturbed wave func-  tion with respect to |Φn=i,f⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' We can express these wave  function as  |Φ(1)  n ⟩ =  �  I  |ΦI⟩⟨ΦI|HW |Φn⟩  En − EI  ,  (47)  where |ΦI⟩ are the intermediate states with mean-field  energies EI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Substituting this expression above, it yields  E1P V =  �  I̸=i  ⟨Φf|D|ΦI⟩⟨ΦI|HW |Φi⟩  Ei − EI  +  �  I̸=f  ⟨Φf|HW |ΦI⟩⟨ΦI|D|Φi⟩  Ef − EI  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (48)  Using HW = �  i hw(ri) and D = �  i d(ri), and following  the Slater-Condon rules, it gives  E1P V =  �  a  ⟨f|d|a⟩⟨a|hw|i⟩  ǫi − ǫa  +  �  a  ⟨f|hw|a⟩⟨a|d|i⟩  ǫf − ǫa  +  �  p̸=i  ⟨f|d|p⟩⟨p|hw|i⟩  ǫi − ǫp  +  �  p̸=f  ⟨f|hw|p⟩⟨p|d|i⟩  ǫf − ǫp  , (49)  where |k = a, p⟩ denotes kth single particle DHF orbital  with energy ǫk for the occupied orbitals denoted by a and  virtual orbitals denoted by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Contributions arising from  the first two terms of the above expression are referred  to as the lowest-order Core contributions while contri-  butions from the later two terms are said to be Valence  =  +  +  +  +  p  (a)  (b)  (c)  (d)  (e)  v  +  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (f)  (ΩRPA  v  )  +  (g)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Graphical representation of ΩRP A  v  and its expansion  in terms of lower-order RMBPT(3) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  contributions that include the lowest-orders to both Main  and Tail parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  In terms of wave operators, the DHF expression for  E1P V can be given by  E1P V = ⟨Φf|DΩi(0,1)|Φi⟩ + ⟨Φf|Ωf(0,1)†D|Φi⟩, (50)  where Ωv(0,1) = Ω(0,1)  0  + Ω(0,1)  v  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Further, Ω(0,1)  0  =  �  a,p  ⟨Φp  a|HW |Φ0⟩  E0−Ep  a  a†  paa = �  a,p  ⟨p|hw|a⟩  ǫa−ǫp a†  paa ≡ �  a,p Ωp  a  and Ω(0,1)  v  = �  p  ⟨Φp  v|HW |Φv⟩  Ev−Ep  v  a†  pav  =  ⟨p|hw|v⟩  ǫv−ǫp a†  paa  ≡  �  p Ωp  v for the intermediate states |Φp  a⟩ with energies  Ep  a = E0+ǫp−ǫa and |Φp  v⟩ with energies Ep  v = E0+ǫp with  respect to the |Φ0⟩ and |Φv⟩ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Representing  the wave operators in terms of the Goldstone diagrams,  we show the Core and Valence contributions to E1P V in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 1(a) and (b) correspond to Core contribut-  ing terms, while Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 1(c) and (d) correspond to Valence  contributing terms here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  RMBPT method  We employ the RMBPT method in the Rayleigh-  Schr¨odinger approach and estimate contributions only up  to third-order of perturbation (RMBPT(3) method) by  considering two orders of Vres and one-order of HW ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  the net Hamiltonian is expressed as  Hat = F + λ1Vres + λ2HW ,  (51)  where λ1 and λ2 are arbitrary parameters introduced to  count orders of Vres and HW in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Here, we  can calculate either matrix element of D after perturb-  ing wave functions by HW or matrix element of HW after  perturbing wave functions by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' We adopt here both ap-  proaches for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' First, it can help us to identify  the lower-order contributions terms to the CPDF and  RPA methods so that their inclusion through the RCC  method can be easily understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Second, it would be  interesting to see classification of the Core and Valence  contributions in both the approaches of the RMBPT  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In both the approaches, the unperturbed wave  11  operators in the RMBPT method can be given by  Ω(0)  n  =  �  k  Ω(k,0)  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (52)  Similarly, the first-order perturbed wave operators can  be denoted by  Ω(1)  n  =  �  k  Ω(k,1)  n  ,  (53)  where subscript n = 0, v stands for core or valence oper-  ators and superscript k denotes for order of Vres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Ampli-  tudes of the wave operators in the RMBPT method are  determined by  [Ω(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0)  0  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' F]P0 = Q0VresΩ(k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0)  0  P0  (54)  and  [Ω(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0)  v  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' F]Pv = QvVres(Ω(k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0)  0  + Ω(k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0)  v  )Pv  −  k−1  �  m=1  Ω(k−m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0)  v  E(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0)  v  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (55)  where E(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0)  v  is the mth-order perturbed energy of  the total energy E(0)  v  = �∞  m=1 E(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0)  v  and is evalu-  ated by using kth-order effective Hamiltonian H(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0)  eff  =  PvVres(Ω(m−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0)  0  +Ω(m−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0)  v  )Pv as part of the net effective  Hamiltonian H(0)  eff = �∞  m=1 H(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0)  eff  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Amplitudes of the perturbed wave operators due to  HW can be evaluated by  [Ω(k,1)  0  , F]P0 = Q0HwΩ(k,0)  0  P0 + Q0VresΩ(k−1,1)  0  P0 (56)  and  [Ω(k,1)  v  , F]Pv = QvVres(Ω(k−1,1)  0  + Ω(k−1,1)  v  )Pv  +QvHw(Ω(k,0)  0  + Ω(k,0)  v  )Pv  −  k−1  �  m=1  Ω(k−m,1)  v  E(m,0)  v  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (57)  In the above expression, the lower-order unperturbed  wave operators are given by Ω(0,0)  n  = 1 and Ω(1,0)  n  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  For the case of considering D as perturbation, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (56)  and (57) can be again used to solve amplitudes of the  ˜Ω(1)  0 , ˜Ω(1)  i  and ˜Ω(1)  f  operators in the RMBPT methods  by replacing Ωp  a/v by Ωp+  a/v =  ⟨p|d|a⟩  ǫa/v−ǫp−ωa†  paa/v and Ωp†  a/v  by complex conjugate of Ωp−  a/v =  ⟨p|d|v⟩  ǫa/v−ǫp+ωa†  paa/v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' This  follows, the nth-order E1P V expression as  E1P V =  1  �n−1  l=0 N l  if  �  n  �  k=0  �  ⟨Φf|(Ω(n−k,0)  0  + Ω(n−k,0)  f  )†D  (Ω(k,1)  0  + Ω(k,1)  i  )  �  |Φi⟩ +  n  �  k=0  �  ⟨Φf|(Ω(k,1)  0  +Ω(k,1)  f  )†D(Ω(n−k,0)  0  + Ω(n−k,0)  i  �  |Φi⟩)  �  ,  (58)  (a)  (b)  (c)  (d)  f  f  f  f  i  i  i  i  p  p  a  a  ΩRPA  0  ΩRPA  v  ΩRPA†  0  ΩRPA†  v  HW  HW  HW  HW  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Property contributing diagrams of the RPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' These  diagrams are similar to the CPDF diagrams, but the core-  polarization effects are included through the D operator in-  stead of HW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Expansion of the ΩRP A  0  and ΩRP A  v  in terms  of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 7 and 8 can show that the RPA property diagrams  include the DHF contributions and core-polarization effects  that are distinctly different than the CPDF method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  where HW  is considered in the perturbation with  N k  if  =  [(�  l⟨Φf|(Ω(k−l,0)  0  +  Ω(k−l,0)  f  )†(Ω(l,0)  0  +  Ω(l,0)  f  )|Φf⟩)(�  m⟨Φi|(Ω(k−m,0)  0  + Ω(k−m,0)  i  )†(Ω(m,0)  0  +  Ω(m,0)  i  )|Φi⟩)]1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  In the case of D as perturbation, it yields  E1P V =  1  �n−1  l=0 N l  if  �  n  �  k=0  �  ⟨Φf|(Ω(n−k,0)  0  + Ω(n−k,0)  f  )†HW  (˜Ω(k,1)  0  + ˜Ω(k,1)  i  )  �  |Φi⟩ +  n  �  k=0  �  ⟨Φf|(˜Ω(k,1)  0  +˜Ω(k,1)  f  )†HW (Ω(n−k,0)  0  + Ω(n−k,0)  i  �  |Φi⟩)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (59)  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 2, we show the important correlation contribut-  ing Goldstone diagrams to E1P V arising through the  RMBPT(3) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' It should be noted that the lowest-  order diagrams of the RMBPT(3) method are same as  the diagrams corresponding to the DHF method and  they are not shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Since both Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (58) and (59)  are equivalent at given level of approximation, the Gold-  stone diagrams are identical in both the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Thus,  the Core and Valence contributions arising through both  the expressions can be distinguished and quoted sepa-  rately by adopting the definitions of the respective wave  operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  This would help us identifying lower-order  Core and Valence correlation contributions to the CPDF,  RPA, CPDF-RPA and RCC methods that are going to be  discussed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In order to distinguish results while pre-  senting from both the approaches, we use the notations  RMBPT(3)w and RMBPT(3)d in place of RMBPT(3) for  the cases with HW as the perturbation and with D as the  perturbation respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  CPDF Method  In the CPDF method, it is possible to extend the E1P V  calculation to all-orders in a very simple manner by ex-  12  tending the DHF expression and with much less com-  putational effort compared to the RMBPT(3) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  This can be derived by starting with the DHF expres-  sion, given by  E1P V = ⟨f|d|i(1)⟩ + ⟨f (1)|d|i⟩,  (60)  where  |k(1)⟩ =  �  I̸=k  |I⟩⟨I|hw|k⟩  ǫk − ǫI  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (61)  In the CPDF method, the first-order perturbed sin-  gle particle orbital |k(1)⟩ is obtained by including core-  polarization effects due to Vres to all-orders (denoted by  |kP V ⟩) by defining an effective potential, similar to u(0)  i ,  in the presence of hw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' To arrive at this expression, we  consider the net Hamiltonian Hint to define the modified  single particle DHF Hamiltonian f P V  i  = fi + λ2hw and  potential as  f P V  i  |˜i⟩ = ˜ǫi|˜i⟩  (62)  and  ˜ui =  Nc  �  b  �  ⟨˜b|gbi|˜b⟩|˜i⟩ − ⟨˜b|gbi|˜i⟩|˜b⟩  �  ,  (63)  where the tilde symbol denotes solution for Hint in place  of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Now expanding |˜i⟩ = |i⟩ + λ2|iP V ⟩ + O(λ2  2) from  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (62) and retaining terms that are linear in λ2, we  can get  (fi − ǫi)|iP V ⟩ = −hw|i⟩ − uP V  i  |i⟩,  (64)  where  uP V  i  |i⟩ =  Nc  �  b  �  ⟨b|g|b⟩|iP V ⟩ − ⟨b|g|iP V ⟩|b⟩  +⟨bP V |g|b⟩|i⟩ − ⟨bP V |g|i⟩|b⟩  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (65)  Like Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (44) and (45), both Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (64) and (65)  are also solved iteratively to obtain the self-consistent  solutions to account for core-polarization effects to all-  orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Using the above APV modified orbitals, E1P V  can be evaluated as  E1P V = ⟨f|d|iP V ⟩ + ⟨f P V |d|i⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (66)  To make one-to-one comparison between contributions  arising through the CPDF method and other many-body  methods including lower-order terms of the RMBPT(3)  method,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' we can present the CPDF expression for E1P V  using the wave operators as  E1P V = ⟨Φf|DΩi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='CP DF |Φi⟩ + ⟨Φf|Ωf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='CP DF †D|Φi⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='(67)  where  Ωv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='CP DF  =  ΩCP DF  0  +  ΩCP DF  v  =  �∞  k=1  ��  a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='p Ω(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='1)  a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='p  + �  p Ω(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='1)  v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='p  �  with  subscripts  0  and v stand for operators responsible for including core  =  +  +  +  +  +  +  +  +  +  +  +  +  +  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (i)  (ii)  (iii)  (iv)  (v)  (vi)  (vii)  (viii)  (ix)  (x)  (xi)  (xii)  (xiii)  (xiv)  ΩCPDF+  (xv)  (xvi)  +  +  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Goldstone diagrams to the amplitude solving equa-  tions of the ΩCP DF ± operator that gives rise to DCP con-  tributions in the CPDF-RPA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Diagrams from (ix)  to (xvi) along with their exchanges are coming due to the  implementation of the orthogonalization condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  and valence correlations, and superscript k denotes order  of Vres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Amplitudes of these operators are given by  Ω(k,1)  a,p  = Ωp  a +  �  b,q  �[⟨pb|g|aq⟩ − ⟨pb|g|qa⟩]  ǫa − ǫp  Ω(k−1,1)  b,q  +Ω(k−1,1)†  b,q  [⟨pq|g|ab⟩ − ⟨pq|g|ba⟩]  ǫa − ǫp  �  (68)  and  Ω(k,1)  v,p  = Ωp  v +  �  b,q  �[⟨pb|g|vq⟩ − ⟨pb|g|qv⟩]  ǫv − ǫp  Ω(k−1,1)  b,q  +Ω(k−1,1)†  b,q  [⟨pq|g|vb⟩ − ⟨pq|g|bv⟩]  ǫv − ǫp  �  ,  (69)  where the sums over a, b and p, q represent occupied and  virtual operators respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' To compute amplitude of  the above operators, we set Ω(0,1)  a,p  ≈ Ωp  a and Ω(0,1)  v,p  ≈ Ωp  v  in the beginning to initiate the iteration procedure from  k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' As can be followed here that only the effective  singly excited configurations are contributing through  the Ωv,CP DF operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Thus, it completely misses out  pair-correlation contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  The Goldstone diagrams that contribute to the ampli-  tudes of ΩCP DF  0  are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Similarly, the Gold-  stone diagrams contributing to the amplitudes of ΩCP DF  v  are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Using these operators, we show  the final Goldstone diagrams that contribute to E1P V  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' By analyzing these diagrams in terms of the  Goldstone diagrams shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 4 and 5, it is easy to  follow how the core-polarization effects are included to  all-orders through the CPDF method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Again, compar-  ing those diagrams with the diagrams from the DHF and  13  ΩCPDF†  0  ΩRPA†  0  ΩCPDF  v  ΩRPA  v  ΩCPDF+  (i)  (ii)  (iii)  (iv)  (v)  (vi)  (vii)  (viii)  (ix)  (x)  (xi)  (xii)  (xiii)  (xiv)  (xv)  (xvi)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  D  HW  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Goldstone diagrams contributing to the E1P V am-  plitude in the CPDF-RPA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Comparing these dia-  grams with the diagrams of the RMBPT(3) method, can be  understood by expanding the ΩCP DF  0/v  , ΩRP A  0/v  and ΩCP DF ±  operators, it can be followed that the CPDF-RPA method  does not include contributions from pair-correlation effects  and some of the DCP effects representing diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  RMBPT(3) methods the relations among them can be  easily understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' One can also realize here that number  of Goldstone diagrams appear in the CPDF method are  very less than the RMBPT(3) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Thus, the com-  putational efforts in the CPDF method are much smaller  compared to the RMBPT(3) method though it is an all-  order theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  RPA  From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (66), it can be followed that the CPDF  method includes correlation effects in the first-order wave  functions through the HW operator only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Thus, it com-  pletely misses out correlation effects in the unperturbed  state that can arise through the D operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' It can also  be noted that the CPDF method is formulated based  on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Therefore proceeding with a similar man-  ner based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (30), it can lead to capturing core-  polarization effects through the D operator and the RPA  is formulated exactly on the same line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  To derive the RPA expression, we consider the net  Hamiltonian H±  int = H + λ3D ∓ ω to define the modi-  fied single particle DHF Hamiltonian f ±  i = fi + λ3d ∓ ω  and potential as  f ±  i |˜i±⟩ = ˜ǫ±  i |˜i±⟩  (70)  =  +  +  +  +  T(0)  1  T(0)  2  =  +  +  +  +  (ii)  (iii)  (iv)  (i)  (ii)  (iii)  (v)  (vi)  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (i)  (v)  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  +  (iv)  T(0)  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Goldstone diagrams demonstrating breakdown of  the T (0) operators in terms of lower-order perturbative exci-  tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  and  ˜u±  i =  Nc  �  b  �  ⟨˜b±|g|˜b±⟩|˜i±⟩ − ⟨˜b±|g|˜i±⟩|˜b±⟩  �  ,  (71)  respectively, where the superscript ± denotes solution for  H±  int in place of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Now expanding |i±⟩ = |i⟩ + λ3|i±⟩ +  O(λ2  3) and retaining terms that are linear in λ3, we can  get  (fi − ǫi ∓ ω)|i±⟩ = −d|i⟩ − u±  i |i⟩,  (72)  where  u±  i |i⟩ =  Nc  �  b  �  ⟨b|g|b⟩|i±⟩ − ⟨b|g|i±⟩|b⟩  +⟨b∓|g|b⟩|i⟩ − ⟨b∓|g|i⟩|b⟩  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (73)  Here also both Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (72) and (73) are solved iteratively  to obtain the self-consistent solutions to account for core-  polarization effects to all-orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Using the above D op-  erator modified orbitals, E1P V can be evaluated as  E1P V = ⟨Φf|HW Ωi,+|Φi⟩ + ⟨Φf|Ωf,−†HW |Φi⟩, (74)  where  Ωv,±  =  Ω±  0  +  Ω±  v  =  �∞  k=1  ��  a,p Ω±(k,1)  a,p  + �  p Ω±(k,1)  v,p  �  with  subscripts  0 and v stand for operators responsible for including  Core and Valence correlations, and superscript k denotes  order of Vres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Amplitudes of these operators are given  by  Ω±(k,1)  a,p  = Ωp±  a  +  �  b,q  �[⟨pb|g|aq⟩ − ⟨pb|g|qa⟩]  ǫa − ǫp ± ω  Ω±(k−1,1)  b,q  +Ω∓(k−1,1)†  b,q  [⟨pq|g|ab⟩ − ⟨pq|g|ba⟩]  ǫa − ǫp ± ω  �  (75)  14  =  +  +  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  =  +  +  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (i)  S(0)  1v  (ii)  (iii)  S(0)  2v  (i)  (ii)  (iii)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Goldstone diagrams demonstrating breakdown of  the S(0)  v  operators in terms of lower-order perturbative exci-  tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  and  Ω±(k,1)  v,p  = Ωp±  v  +  �  b,q  �[⟨pb|g|vq⟩ − ⟨pb|g|qv⟩]  ǫv − ǫp ± ω  Ω±(k−1,1)  b,q  +Ω∓(k−1,1)†  b,q  [⟨pq|g|vb⟩ − ⟨pq|g|bv⟩]  ǫv − ǫp ± ω  �  ,  (76)  where we assume Ω±(0,1)  a,p  ≈ Ωp±  a  and Ω±(0,1)  v,p  ≈ Ωp±  v  initially for the iteration procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' We also intend to  mention here is that in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (75) and (76) , ω value  can be used from the experiment while in the ab ini-  tio framework it is taken from the DHF method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Fol-  lowing the explanation in the previous sub-section, it is  obvious that the RPA wave operators will pick up core-  polarization correlations through the D operator to all-  orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Again based on the classification adopted in this  work, the Goldstone diagrams contributing to the ampli-  tude determining equation for Core operator are shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 7, while the diagrams contributing to the ampli-  tudes of the Valence operator are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Using the above operators, we show the final Goldstone  diagrams that contribute to E1P V of RPA in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' By  analyzing these diagrams in terms of the Goldstone di-  agrams shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 7 and 8, it can be followed how  the core-polarization effects are included through D to  all-orders through the RPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Again, comparing these dia-  grams with the diagrams from the DHF and RMBPT(3)  methods the relations among both the methods can be  understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Though the number of Goldstone diagrams  appear in the RPA and the CPDF method are same, but  amplitudes in the CPDF method are expected to con-  verge faster than the RPA owing to strong correlations  arising through D than HW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' It can be noticed here that  the Core correlations (without DHF contributions) aris-  ing in the RPA are distinctly different than that appear  via the CPDF method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  =  +  +  +  +  (i)  (iii)  (iv)  (v)  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (ii)  T(1)  1  =  T(1)  2  T (1)  1  (i)  +  (iii)  T (0)  2  +  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  T (0)  2  (iv)  +  (v)  T (0)  2  (ii)  T (1)  1  (vi)  +  T (1)  1  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Goldstone diagrams demonstrating breakdown of  the T (1) operators in terms of lower-order perturbative exci-  tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  CPDF-RPA/TDHF method  As realized above, the CPDF method and the RPA  include core-polarization effects only through the first-  order perturbed wave functions but the unperturbed  wave functions and energies in both the cases are used  from the DHF method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  In order to achieve core-  polarization effects through both the states it is necessary  to include the HW and D operators in the perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  The simplest approach for doing so would be to add both  the CPDF and RPA results and remove repeated appear-  ance of the DHF value from one of the approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' How-  ever, such an approach would omit correlations among  both the HW and D operators which may not be negli-  gible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Keeping in view of the above, we define the total  Hamiltonian as  Ht = H + λ2HW + λ3D  ≡ Hint + λ3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (77)  Treating both the HW and D operators perturbatively,  the exact atomic wave function (|Ψv⟩) of Ht can be ex-  pressed as  |Ψv⟩ = |Ψ(0,0)  v  ⟩ + λ2|Ψ(1,0)  v  ⟩ + λ3|Ψ(0,1)  v  ⟩  +λ2λ3|Ψ(1,1)  v  ⟩ + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (78)  As denoted earlier, |Ψ(m,n)  v  ⟩ represents consideration of  m orders of Hw and n orders of D in the atomic wave  function |Ψv⟩ of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In the wave operator formalism, it  is given by  Ωv|Φv⟩ = Ω(0,0)  v  |Φv⟩ + λ2Ω(1,0)  v  |Φv⟩ + λ3Ω(0,1)  v  |Φv⟩  +λ2λ3Ω(1,1)  v  |Φv⟩ + · · · ,  (79)  15  =  +  +  +  +  S(1)  1v  (i)  (ii)  (iii)  (iv)  =  +  +  +  S(1)  2v  (i)  (ii)  (iii)  (iv)  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  +  (v)  S(0)  1v  (vi)  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (v)  +  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Goldstone diagrams demonstrating breakdown of  the S(1)  v  operators in terms of lower-order perturbative exci-  tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  where superscripts denote the same meaning as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' It  can be noted that each Ω(m,n)  v  component will have parts  carrying Core and Valence correlations separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  In this case, we can determine the E1P V amplitude as  the transition amplitude of O ≡ λ2Hw + λ3D between  the initial perturbed state to the final unperturbed state  or between the initial unperturbed state to the final per-  turbed state (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (31) and (32)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  E1P V = ⟨Ψ(0,0)  f  |Ψ(1,1)  i  ⟩ + ⟨Ψ(0,0)  f  |D|Ψ(1,0)  i  ⟩  +⟨Ψ(0,0)  f  |HW |Ψ(0,1)  i  ⟩  = ⟨Φf|Ω(0,0)†  f  Ω(1,1)  i  |Φi⟩ + ⟨Φf|Ω(0,0)†  f  DΩ(1,0)  i  |Φi⟩  + ⟨Φf|Ω(0,0)†  f  HW Ω(0,1)  i  |Φi⟩  (80)  or  E1P V = ⟨Ψ(1,1)  f  |Ψ(0,0)  i  ⟩ + ⟨Ψ(1,0)  f  |D|Ψ(0,0)  i  ⟩  +⟨Ψ(0,1)  f  |HW |Ψ(0,0)  i  ⟩  = ⟨Φf|Ω(1,1)†  f  Ω(0,0)  i  |Φi⟩ + ⟨Φf|Ω(1,0)†  f  DΩ(0,0)  i  |Φi⟩  + ⟨Φf|Ω(0,1)†  f  HW Ω(0,0)  i  |Φi⟩,  (81)  keeping terms that are of the order of λ2λ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Note that  both HW and D operators are treated on an equal foot-  ing in this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Thus, definitions of both Core and  Valence contributions to E1P V will be identical for both  Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (80) and (81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Also, it would be prudent to use  both the expressions in an approximated method to ver-  ify numerical uncertainty to the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' However, if  ωex (some earlier studies have done it through the scaling  D  T(1)  1  T(1)†  1  S(1)  1i  S(1)†  1f  S(1)  2i  S(1)†  2f  S(0)  2i  S(1)†  1f  S(0)†  1f  S(1)  1i  S(0)†  2f  T(1)  1  S(0)  1i  S(1)†  2f  S(0)  2i  D  (ii)  (i)  (iii)  (iv)  (v)  (vi)  (vii)  (viii)  (ix)  (x)  S(0)†  2f  S(1)  1i  S(0)  2i  S(0)†  1f  T(1)†  1  (xi)  (xii)  (xiii)  S(1)†  2f  S(0)  2i  S(0)†  2f  S(1)  2i  (xiv)  T(1)†  1  S(1)  2i  (xv)  T(1)  1  S(1)†  2f  (xvi)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A few important E1P V evaluating diagrams in the  RCCSD method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Diagrams shown as DT (1) and its complex  conjugate c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=') term are the dominant Core contributing ef-  fects that include all the core-polarization effects of the CPDF  method, pair-correlation effects of the RMBPT(3) method  and their correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Similarly, diagrams shown as DS(1)  1i  and S(1)†  1f D contain Valence contributing core-polarization ef-  fects of the CPDF method, pair-correlation effects from the  RMBPT(3) method and their correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' DS(1)  2i and S(1)†  2f D  representing diagrams contain Core contributions from the  RPA and DCP contributions from the CPDF-RPA method  as well as that are absent in this method but appears in the  RMBPT(3) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Non-linear RCCSD terms incorporate  higher-order correlation effects of the above terms and ac-  count correlations among core-polarization, pair-correlation  and their intercombinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  procedure) is used then the results from both these equa-  tions may not agree to each other due to inconsistencies  in the treatment of the intermediate states through these  equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  The modified single particle Hamiltonian for the cor-  responding Hamiltonian Ht = Hint + λ3D in the CPDF-  RPA method can be written as f P V ±  i  = f P V  i  + λ3d ∓ ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  It follows  f P V ±  i  |i⟩ = ǫi|i⟩  (82)  and  ui =  Nc  �  b  �  ⟨b|g|b⟩|i⟩ − ⟨b|g|i⟩|b⟩  �  ,  (83)  where the bar symbol denotes solution for Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' By ex-  panding, we get |i⟩ = |iP V ⟩ + λ3|iP V ±⟩ + O(λ2  3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' It gives  (f P V  i  − ǫP V  i  ∓ ω)|iP V ±⟩ = −d|iP V ⟩ − uP V (1)  i  |iP V ⟩,(84)  16  where  uP V (1)  i  |iP V ⟩ =  Nc  �  b  �  ⟨bP V |g|bP V ⟩|iP V ±⟩  −⟨bP V |g|iP V ±⟩|bP V ⟩  +⟨bP V |g|bP V ⟩|iP V ±⟩  −⟨bP V |g|iP V ±⟩|bP V ⟩  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (85)  Further expanding Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (84) and (85), and retaining  terms of the order of λ2λ3 we get  (fi − ǫi ∓ ω)|iP V ±⟩ = −d|iP V ⟩ − u±  i |iP V ⟩ − hw|i±⟩  −uP V  i  |i±⟩ − uP V ±  i  |i⟩,  (86)  where  uP V ±  i  |i⟩ =  Nc  �  b  �  ⟨b∓|g|bP V ⟩|i⟩  −⟨b∓|g|i⟩|bP V ⟩  +⟨bP V |g|b±⟩|i⟩  −⟨bP V |g|i⟩|b±⟩  +⟨b|g|bP V ±⟩|i⟩  −⟨b|g|i⟩|bP V ±⟩  +⟨bP V ∓|g|b⟩|i⟩  −⟨bP V ∓|g|i⟩|b⟩  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (87)  It can be further noted that in the CPDF method the  perturbed core DHF orbital (|aP V ⟩) is orthogonal to the  unperturbed core orbital (|a⟩) and the same is also true  in the RPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' ⟨a|aP V ⟩ = 0 and ⟨a|a±⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' However,  ⟨a|aP V ±⟩ ̸= 0 in the CPDF-RPA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' This necessi-  tates to use the orthogonalized core orbitals (|ao±⟩) by  imposing the condition  |ao±⟩ = |aP V ±⟩ −  �  b  |b⟩⟨b|aP V ±⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (88)  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 10, we show the Goldstone diagrams contribut-  ing to the determination of the |aP V ±⟩ and also the extra  diagrams that are subtracted to obtain |ao±⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' It can be  understood below that it is not required to obtain the  modified orbitals for the virtual orbitals in the CPDF-  RPA method to estimate the E1P V amplitude, so we do  not show Goldstone diagrams contributing to the ampli-  tudes of valence operator here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Using the following expressions in the formula given by  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (80), we can write  E1P V = ⟨f|d + u+  i |iP V ⟩ + ⟨f|hw + uP V  i  |i+⟩  +⟨f|uP V +  i  |i⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (89)  Similarly, using the formula given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (81) we can get  E1P V = ⟨f P V |d + u+  i |i⟩ + ⟨f −|hw + uP V  i  |i⟩  +⟨f|uP V −  f  |i⟩  = ⟨f P V |d + u+  i |i⟩ + ⟨f −|hw + uP V  i  |i⟩  +⟨f|uP V +  i  |i⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (90)  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' E1P V values, in units 10−11i(−Qw/Nn)|e|a0, of  the 6s 2S1/2 − 7s 2S1/2 and 6s 2S1/2 − 5d 2D3/2 transitions  in 133Cs from DC Hamiltonian reported by various works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Methods shown as ‘Sum-over’ and ‘Mixed’ are obtained us-  ing sum-over-states approach and mixed many-body methods  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Results shown in bold fonts are claimed to be  within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='5% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Method  This work Others  This work Others  6s 2S1/2 − 7s 2S1/2  6s 2S1/2 − 5d 2D3/2  DHF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='7375  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='736 [11]  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='3933  RMBPT(3)w 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0902  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='4639  CPDF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='9226  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='924 [11]  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='7989  RPA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='7094  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='707 [11]  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='2362  CPDF-RPA∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='8876  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='8914 [25]  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='1665  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='70 [37]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='8923† [25]  CPDF-RPA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='8844  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='886 [11]  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0281  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='80 [26]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='907 [26]  RCCSD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='8964  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='8961 [29]  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='5641  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='210[38]  RCCSDT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='8967 [29]  Sum-over  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='9053 [24]  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='76 [13]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='8998† [24]  Mixed-states  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='8967 [25]  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='62 [13]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='8938† [25]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='9083‡ [25]  †Note: Scaled value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  ‡Scaled value + borrowed contribution from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  In the wave operator form, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (89) can be given by  E1P V = ⟨Φf|DΩi,CP DF |Φi⟩ + ⟨Φf|HwΩi,+|Φi⟩  +⟨Φf|ΩCP DF +|Φi⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (91)  From (90), we can write  E1P V = ⟨Φf|Ωf,CP DF †D|Φi⟩ + ⟨Φf|Ω−†Hw|Φi⟩  +⟨Φf|Ωf,CP DF −|Φi⟩  = ⟨Φf|Ωf,CP DF †D|Φi⟩ + ⟨Φf|Ωf,−†Hw|Φi⟩  +⟨Φf|ΩCP DF +|Φi⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (92)  In the above expressions, we define  ΩCP DF + =  �  i,j  (⟨f|u+  i |iP V ⟩ + ⟨f|uP V  i  |i+⟩  +⟨f|uP V +  i  |i⟩)a†  jai  (93)  and  ΩCP DF − =  �  i,j  (⟨f P V |u−  f |i⟩ + ⟨f −|uP V  f  |i⟩  +⟨f|uP V −  f  |i⟩)a†  jai  (94)  It is also worth noting that some of the works in the lit-  erature do not consider contribution from ⟨f|uP V +  i  |i⟩ in  the CPDF-RPA method, and their contributions are sep-  arately quoted as ‘DCP’ effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 11, we show the  17  TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Reduced E1 (in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=') and HW (in units of 10−11i(−Qw/Nn)|e|a0) matrix elements from the RCCSD method, and  the excitation energies (in cm−1) of the low-lying states of 133Cs that are used to estimate the ‘Main’ contribution of E1P V for  the 6S − 7S transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' E1 matrix elements and energies are compared with the available most precise experimental values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Transition  E1 amplitude  Excitation Energy  HW amplitude  This work  Experiment  This work  Experiment [45]  6P1/2-6S  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='5487  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='5097(74) [41]  −11243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='93  −11178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='27  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='2541  7P1/2-6S  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='3006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='2825(20) [42]  −21838.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='93  −21765.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='35  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='7135  8P1/2-6S  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0914  −25787.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='48  −25708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='83  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='4808  9P1/2-6S  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0388  −27735.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='96  −27637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='3471  6P1/2-7S  −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='2500  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='233(22)[43]  7352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='53  7357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='6067  7P1/2-7S  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='2967  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='308(15)[44]  −3242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='47  −3229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='3445  8P1/2-7S  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='9492  −7191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='02  −7173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='2320  9P1/2-7S  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='3867  −9139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='50  −9101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='47  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='1674  diagrams that are contributing to E1P V in the CPDF-  RPA method icluding the DCP effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Now compar-  ing diagrams from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 11 and the diagrams from the  RMBPT(3) method shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 2, it can be shown that  some of the Core and Valence correlation diagrams of the  RMBPT method are appearing as the Core diagrams of  the CPDF-RPA method (so also for Valence contribu-  tions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  This, therefore, clearly demonstrates that the  definitions of Core and Valence correlation contributions  to E1P V are not unique and their classifications differ  based on the approach adopted in a many-body method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Among the approximated methods where various physi-  cal effects or correlation effects through both the HW and  D operators are not included on an equal footing, it may  not be possible to make an one-to-one comparative anal-  ysis among contributions arising through the Core and  Valence correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In such a scenario, it is suggestive  to compare only the final results from different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  The advantages of using the CPDF-RPA method are  that this method includes core-polarization effects to all-  orders, treats both the HW and D operators on an equal  footing (which means the results would remain invari-  ant in what-so-ever order either HW or D is included  in Ht along with H), and gives DCP effects that are  not present in the CPDF method or RPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' However, it  still misses out many non-core-polarization effects includ-  ing pair-correlation contributions and correlations among  core-polarization, and pair-correlation effects in the de-  termination of E1P V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Again, orthogonalization of per-  turbed occupied orbitals is incorporated by hand while  the approach does not take care of them in a natural man-  ner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' We would like to mention that some of the earlier  calculations using the CPDF-RPA method neglected con-  tributions from the DCP effects (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Results  from such as an approximation are denoted as CPDF-  RPA* method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  In fact, omission of some of the DCP  contributions in this method is just due to a similar rea-  son.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Moreover, the CPDF, RPA and CPDF-RPA meth-  ods cannot be derived using Bloch’s prescription though  we have expressed them using wave operators just to  make one-to-one connection of these methods with the  TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Estimated ‘Main’ contributions to the E1P V val-  ues, in units 10−11i(−Qw/Nn)|e|a0, of the 6s 2S1/2 −7s 2S1/2  transition in 133Cs using matrix elements involving np 2P1/2  intermediate states in the sum-over-states approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Four  cases are being considered: (a) ab initio result in which cal-  culated values from Table II are used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (b) replacing calculated  E1 matrix elements by their experimental values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (c) retaining  calculated E1 matrix elements and using experimental ener-  gies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' and (d) using experimental values for both E1 matrix  elements and energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Approach  ⟨7S|D|6SP NC⟩  ⟨7SP NC|D|6S⟩  Total  (a)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='4461  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='3171  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='8710  (b)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='4373  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='3121  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='8748  (c)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='4522  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='3156  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='8634  (d)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='4434  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='3106  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='8672  RMBPT method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Thus, these methods are simply the  extension of the DHF method and cannot take into ac-  count the effects that may have been neglected in gen-  erating the single particle orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' For example, the ef-  fects of V N, V N−1, V N−2 etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  potentials used in the  determination of wave functions through the Bloch equa-  tion are taken care through an effective Hamiltonian like  Heff = PvVresΩ(0)  v Pv that appears in solving amplitude  of Ω(0)  v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' However, the wave operator amplitude solving  equations in the CPDF, RPA and CPDF-RPA methods  remain to be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Thus, the valence electron inter-  action neglected in the construction of DHF potential (in-  active valence orbital) is not amended through the above  methods like in the RMBPT method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' All these short-  comings of the CPDF-RPA method will be adequately  addressed by the RCC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  RCC method  The RCC method both in the non-relativistic and its  counter relativistic forms have been extensively consid-  ered these days to include electron correlation effects  18  in the determination of properties of different types of  many-body systems accurately such as nuclear, atomic,  molecular and solid state systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  This is the reason  for which these days it is commonly referred to as the  gold standard of many-body theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Compared to the  CPDF, RPA and CPDF-RPA methods, implementation  and computational efforts in the RCC method are ex-  tensively complex and expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' It can account correla-  tions through both the HW and D operators to all-orders,  as well take care other shortcomings of the CPDF-RPA  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  All CPDF-RPA effects along with other ef-  fects like pair-correlations, inter-correlations among core-  polarization and pair-correlation effects, corrections due  to choice of V N−1 DHF potential approximation etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' are  sub-summed within our RCC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' This theory was  already implemented and results using the method were  already reported for Cs [21, 23], Ba+ [21], Yb+ [39] and  Ra+ [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Here, we consider this theory in the singles  and doubles approximation (RCCSD method) only to  demonstrate how it captures correlations of previously  mentioned methods including appearance of orthogonal-  ization of perturbed core orbitals in a natural fashion, all-  order pair-correlations, additional DCP effects and nor-  malization of wave functions etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' compared to a mixed  many-body method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Since all these effects are present  within the RCC theory and the wave functions are ob-  tained through iterative scheme, all of these effects are  inter-correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Incorporation of additional effects from  the Breit and QED interactions can also be treated in the  similar fashion, if their corresponding interaction poten-  tial terms are added in the atomic Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Going  beyond the RCCSD method approximation in the RCC  theory such as the RCCSDT method, means it can cap-  ture even higher-order non core-polarization effects and  further inter-correlations among core-polarization and  pair-correlation effects that are beyond the reach of the  mixed many-body methods employed earlier to estimate  the E1P V amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' As shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [29] though the  energies, E1 matrix elements and magnetic dipole hy-  perfine structure constants from both the RCCSD and  RCCSDT methods were quite significant, the difference  in the E1P V values from both these two methods was  rather small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  This suggests that consideration of the  RCCSD method would be sufficient enough to address  the earlier mentioned concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  In the RCC theory framework, the exact wave function  of an atomic state can be given by  |Ψv⟩ = eS|Φv⟩,  (95)  where S is an excitation operator carrying out excitations  of electrons out of core orbitals from the DHF wave func-  tion |Φv⟩ to generate the excited state configurations due  to Vres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In other words, these excitation configurations  can be thought of as contributions taken from each or-  der of corrections to the wave function from the RMBPT  method to construct an all-order form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' We can further  define  S = T + Sv,  (96)  TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Core and Valence correlation contributions to  the E1P V values, in units 10−11i(−Qw/Nn)|e|a0, for the  6s 2S1/2−7s 2S1/2 and 6s 2S1/2−5d 2D3/2 transitions in 133Cs  from the RMBPT(3)w and RMBPT(3)d approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Results  (a) considering Heff effect due to V N−1 potential and (b)  using DHF orbital energies are shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Approach  RMBPT(3)w  RMBPT(3)d  Core  Valence  Core  Valence  6s 2S1/2 − 7s 2S1/2  (a)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00205  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='09220  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00003  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='24290  (b)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00206  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='43938  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00031  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='43763  6s 2S1/2 − 5d 2D3/2  (a)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='16391  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='30000  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='12208  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='55427  (b)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='18267  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='97004  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='12513  −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='02758  in order to distinguish excitations of electrons among the  core orbitals, denoted by T , and excitations of electrons  from valence orbital or valence orbital along with core  orbitals, denoted by Sv in the V N−1 framework of con-  structing the DHF wave function |Φv⟩ = a†  v|Φ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Accord-  ingly we can write  |Ψv⟩ = eT +Sv|Φv⟩  = eT {1 + Sv} |Φv⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (97)  Here eSv = 1 + Sv is the exact form for the atomic states  having one valence orbital v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In the RCCSD method, the  excitation operators are denoted as  T = T1 + T2  (98)  and  Sv = S1v + S2v,  (99)  where subscripts 1 and 2 stand for the singles and doubles  excitations respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  In the wave operator form given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (7) and (8),  it corresponds to  Ω0 = eT  (100)  and  Ωv = eT Sv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (101)  Following the Bloch’s equations given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (9) and  (10) in general form, amplitudes of the T and Sv excita-  tion operators are obtained by  ⟨Φ∗  0|(HeT )l|Φ0⟩ = 0  (102)  and  ⟨Φ∗  v|[(HeT )l − Ev]Sv + (HeT )l|Φv⟩ = −⟨Φ∗  v|(HeT )l|Φv⟩,  (103)  where subscript l denotes for the linked terms, bra states  with superscript ∗ means excited states with respect to  the respective DHF ket states appear in the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  19  TABLE V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Ab initio contributions to Core and Valence parts  of the E1P V values, in units 10−11i(−Qw/Nn)|e|a0, for the  6s 2S1/2−7s 2S1/2 and 6s 2S1/2−5d 2D3/2 transitions in 133Cs  from different methods considered in this work using the DC  Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Available results from previous calculations are  also given for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Method  This work  Others  Core  Valence  Core  Valence  6s 2S1/2 − 7s 2S1/2  DHF  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00173  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='73923  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00174[30]  RMBPT(3)w −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00205  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='09220  RMBPT(3)d −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00003  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='24290  CPDF  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00199  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='92454  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00201 [30]  RPA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00028  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='70912  CPDF-RPA∗  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00169  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='88591  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00170 [30]  CPDF-RPA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00169  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='88267  RCCSD  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00197  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='89840  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0019 [29] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='8980 [29]  6s 2S1/2 − 5d 2D3/2  DHF  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='11684 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='27646  RMBPT(3)w −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='16391 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='3000  RMBPT(3)d −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='12208 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='55427  CPDF  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='19122 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='60768  RPA  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='12037 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='11585  CPDF-RPA∗ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='20786 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='95860  CPDF-RPA  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='20786 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='82021  RCCSD  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='14745 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='41667  In the ab initio procedure, the energy of the |Ψv⟩ is de-  termined by calculating expectation value of the effective  Hamiltonian i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Ev = ⟨Φv|Heff|Φv⟩  = ⟨Φv|Pv(HeT )l {1 + Sv} Pv|Φv⟩,  (104)  with respect to |Φv⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' As can be noticed Ev is a function  of Sv and Sv itself depends on Ev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Thus, the non-linear  Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (103) and (104) are solved iteratively to obtain am-  plitudes of Ωv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' As pointed out earlier, appearance of Ev  in the determination of Sv amplitudes is a consequence  of using orbitals from V N−1 potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' It is also possible  to obtain amplitudes of the RCC operators by substitut-  ing experimental energy in the semi-empirical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Also, one can scale the Sv amplitudes by multiplying  by a suitable parameter λ to obtain the calculated Ev  value matching with the experimental energy in the sim-  ilar spirit of the scaling procedures adopted in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [24]  and [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  To evaluate E1P V , we need to express the RCC opera-  tors in terms of both the unperturbed and first-order per-  turbed operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' As explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' III, we have three  different options to obtain the E1P V amplitude in the  RCC theory framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' First, by adopting the approach  similar to the CPDF method, in which HW is considered  as external perturbation and matrix element of D can be  determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Second, considering D as the external per-  turbative operator as in the RPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Third and the most  effective approach would be along the line of the CPDF-  RPA method, in which, both the HW and D operators  can be treated as external perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' The implemen-  tation of the third approach would be more challenging  and computationally very expensive as it will demand to  store amplitudes of four different types of perturbed RCC  operators instead of storing only one type of perturbed  amplitudes in the first case and two types in the second  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Among the first two approaches, computational  efforts are almost similar but implementation-wise con-  sidering HW as perturbation will be more natural and  is easier to deal with its angular momentum couplings  owing to its scalar form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Moreover, amplitudes of the  perturbed operators due to HW will converge faster than  when D is treated as perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Again, it is possible  to use experimental energies in the first approach to ob-  tain semi-empirical results in case it is required while it  is a problem in the second case owing to the fact that is  already discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' From this view, we adopt the  first approach to estimate the E1P V value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  We expand the T and Sv operators by treating HW  as the perturbation to separate out the solutions for the  unperturbed and the first-order wave functions by ex-  pressing  T = T (0) + λ2T (1)  (105)  and  Sv = S(0)  v  + λ2S(1)  v ,  (106)  where the superscript meanings are same as specified ear-  lier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' This yields  |Ψ(0)  v ⟩ = (Ω(0)  0  + Ω(0)  v )|Φv⟩  (107)  and  |Ψ(0)  v ⟩ = (Ω(1)  0  + Ω(1)  v )|Φv⟩  (108)  with the definitions Ω(0)  0  = eT (0), Ω(1)  0  = eT (0)T (1),  Ω(0)  v  = eT (0)S(0)  v  and Ω(1)  v  = eT (0) �  T (1)S(0)  v  + S(1)  v  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' The  unperturbed operator amplitudes are obtained by solv-  ing the usual RCC theory equations as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  The first-order perturbed RCC operator amplitudes are  determined as  ⟨Φ∗  0|(HeT (0))lT (1)|Φ0⟩ = −⟨Φ∗  0|(HW eT (0))l|Φ0⟩(109)  and  ⟨Φ∗  v|[(HeT (0))l − E(0)  v ]S(1)  v |Φv⟩ = −⟨Φ∗  v|[(HW eT (0))l  +(HeT (0))lT (1)]{1 + S(0)  v }|Φv⟩ (110)  As can be seen, the exact calculated energy also enters  into the amplitude determining equation of S(1)  v  because  of the V N−1 potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  This is one of the advantages  20  of the RCC method over the CPDF-RPA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  In  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  12,13, 14, 15 we show some of the important  Goldstone diagrams contributing to the T (0)  2  , T (0)  1  , S(0)  1v ,  T (1)  2  ,T (1)  1  ,S(0)  2v , S(1)  1v and S(1)  2v amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' These diagrams  can be compared with the CPDF-RPA wave operator am-  plitude determining diagrams in order to understand how  they are embedded within the RCC operators irrespec-  tive of the fact that denominators in the RCC method  will contain the exact energy of the state instead of the  DHF energy in the CPDF-RPA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  The E1P V expression between the states |Ψi⟩ and |Ψf⟩  in the RCC theory is given by  E1P V =  ⟨Φf|{S(1)  f  + (S(0)†  f  + 1)T (1)†} ¯D{1 + S(0)  i  }|Φi⟩  ⟨Φf|{S(0)†  f  + 1} ¯N{1 + S(0)  i  }|Φi⟩  +  ⟨Φf|{S(0)†  f  + 1} ¯D{T (1)(1 + S(0)  i  ) + S(1)  i  }|Φi⟩  ⟨Φf|{S(0)†  f  + 1} ¯N{1 + S(0)  i  }|Φi⟩  ,  (111)  where ¯D = eT (0)+DeT (0) and ¯N = eT (0)+eT (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Unlike the  CPDF, RPA and CPDF-RPA methods, normalization  factors appear explicitly in the RCC expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Using  the wave operator notations, one can easily identify which  RCC terms contribute to the Core and Valence correla-  tions in the evaluation of E1P V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' It means basically, any  term is connected either with the Sn=i,f (0/1) operators or  with their conjugate operators will be a part of the Va-  lence correlation otherwise they will be a part of the Core  correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' It can be further clarified that the definitions  of Core and Valence correlation contributions to E1P V in  our RCC theory are in the line of the RMBPT(3)w and  CPDF methods, and different than the RPA and CPDF-  RPA methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  16 we show a few important  contributing Goldstone diagrams from the RCC method  to Core and Valence correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Also, for better under-  standing, the Goldstone diagrams of the RCCSD opera-  tors are further demonstrated as the sum of lower-order  Goldstone diagrams of the RMBPT(3) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  From  these relations, it can be followed that the RCC method  includes correlation effects from core-polarization, pair-  correlation, and DCP to all orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  It is also obvious  from the above diagrams that orthogonalization to core  orbitals and extra DCP contributions are also appearing  in a natural manner in our RCC theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Moreover, corre-  lations among all these effects are implicitly present due  to the fact that singles and doubles excitation amplitude  equations are coupled through many non-linear terms in  the RCC theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  RESULTS & DISCUSSIONS  We present the calculated values of E1P V  of the  6S − 7S and 6S − 5D3/2 transitions in  133Cs from  the DHF, RMBPT(3), CPDF, RPA, CPDF-RPA and  RCCSD methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  As mentioned in Introduction, the  main intention of carrying out this study is to demon-  TABLE VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Comparison of contributions from the initial and  final perturbed states to E1P V of the 6s 2S1/2 − 7s 2S1/2  transition of 133Cs, in units 10−11i(−Qw/Nn)|e|a0, at differ-  ent levels of approximation between the present work and that  are reported in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [11, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Method  ⟨7SP NC|D|6S⟩  ⟨7S|D|6SP NC⟩  Ours  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [11]  Ours  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [11]  Total contribution  DHF  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='01168  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='010  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='27418  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='274  CPDF  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='26664  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='267  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='34409  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='344  RPA  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='02557  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='023  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='31617  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='316  CPDF-RPA*  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='27910  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='279  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='39150  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='391  Ours  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [30]  Ours  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [30]  Core contribution  DHF  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='02638 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='02645  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='02465  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='02472  CPDF  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='04298 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='04319  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='04099  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='04119  RPA  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='03536  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='03564  CPDF-RPA* −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='05794 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='05822  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='05963  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='05992  Valence contribution  DHF  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='03806  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='29883  CPDF  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='30962  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='38508  RPA  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='06094  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='70912  CPDF-RPA*  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='33704  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='45113  strate similarities and differences among various con-  tributions to E1P V through the above methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  This  would be useful in addressing the issue of the sign for the  Core correlation contributions to the E1P V value of the  6S − 7S transition in 133Cs that are reported differently  by various groups [24, 25, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Moreover, this exercise  would be useful in understanding the missing contribu-  tions in a method compared to others that are under con-  sideration here so that accuracy of the earlier reported  results in an atomic system, including Cs, obtained us-  ing a particular method can be further improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' We  have allowed correlations from all the occupied orbitals  and allowed excitations of electrons from a given set of  virtual orbitals in all the considered many-body methods  to make comparative analysis of results from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In  order to show that we have taken a sufficiently large set  basis functions, we validate our calculations by compar-  ing our E1P V values from the DHF, CPDF, RPA and  CPDF-RPA methods with the earlier reported values by  Martensson [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  The reason for presenting the E1P V  value of the 6S − 5D3/2 transition in 133Cs is to answer  to a comment by Roberts and Ginges in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [30], where  they argue about the agreement of the sign of Core contri-  bution to the E1P V value of a S − D transition reported  earlier using the RCCSD method [40] while observing a  sign difference for the 6S − 7S transition in 133Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  In Table I, we present the E1P V values of the 6S − 7S  and 6S −5D3/2 transitions in 133Cs using the DC Hamil-  21  tonian from a number of methods including the DHF  method in order to understand the importance of corre-  lation effects in their determination and to demonstrate  that choice of a method matters a lot for their rigourous  inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' The values shown in bold fonts in this table  are claimed to be accurate within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='5% by the earlier  works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A careful look into these results reveal that some  of them differ by 1% from each other, which suggests  there could be issues with the estimation of accuracy in  these calculations that needs to be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Results  from sum-over-states approach, given as ‘Sum-over’ in  the table, uses scaled E1 matrix elements and energies  from the CCSDvT method to estimate the Main con-  tribution of E1P V for the 6S − 7S transition while the  X-factor is obtained using a blend of many-body meth-  ods [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [25], the same ‘Main’ contribution is  utilized but Core and Tail contributions to the X-factor  are estimated using the CPDF-RPA* method (denoted  as only RPA in the original paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Pair-correlation ef-  fects to these estimations were estimated using the BO-  correlation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Result from these RPA+BO meth-  ods is given under ‘Mixed-states’ in the above table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Thus large discrepancy seen in both the results from the  above table come from the X-factors estimated in Refs  [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' If the total X-factors had agreed between two  works but individual contributions would have differed,  then the difference in the results could have been at-  tributed to distribution of contributions under the Core  and Valence correlations in the considered approaches in  both the works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' From the significant differences seen be-  tween the X-factors from the Mixed-states approach and  our RCCSD method in both the 6S −7S and 6S −5D3/2  transitions, it does not support such distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  To understand the reasons for significant discrepancies  seen in the X-factors from various works, we first analyze  the Main contribution to the 6S − 7S transition by us-  ing the calculated properties from our RCCSD method  in the sum-over-states approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' We used the E1 ma-  trix elements and energies from our calculations as well  as from experiments to show the differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In Table II,  we present the HW matrix elements, E1 matrix elements  and energies obtained using the DC Hamiltonian in the  RCCSD method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' The calculated E1 matrix elements and  energies are also compared with the experimental values  [41–45] in the same table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Using these values we esti-  mate the Main contributions to E1P V of the 6S − 7S  transition and they are given in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Results (a)  from ab initio calculations, (b) using experimental E1  values with calculated energies, (c) calculated E1 values  with experimental energies and (d) using experimental  values for both the E1 matrix elements and energies are  given separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' This analysis shows that result from  (b) is larger than (a), but results from (c) and (d) are  lower than (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  It means that accuracy of energies in  a given method affect the results more than E1 matrix  elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Later we demonstrate explicitly that it intro-  duces error to E1P V estimation when we use experimen-  tal energy only of the initial or final state through the  first-principle calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Differences between the ab  initio and semi-empirical calculations can be minimised  by including contributions from the triples, quadruples  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' higher-level excitations of the RCC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [29], Sahoo et al have demonstrated difference between  the ab initio calculations of E1P V of the 6S − 7S tran-  sition using the RCCSD and RCCSDT methods is very  small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' They further showed that Core contributions are  almost same in both the methods, and the agreement of  the E1P V values from both the methods was the result  of opposite trends of correlation effects in the evaluation  of the HW matrix elements than the E1 matrix elements  and energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A similar trend was anticipated from the  Tail contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Subtracting the ab initio value of Main  from the final RCCSD result, we find the X-factor to  E1P V of the 6S −7S transition as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0254, against 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0175  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0256 of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [24] and [25] respectively in units of  10−11i(−Qw/Nn)|e|a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' It means that there is a large dif-  ference between the X-factor of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [24] and our work,  whereas these values almost agree between Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [25] and  the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Since there is a sign difference between  the Core contribution from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [25] and the RCCSD  value of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [29], the above analysis suggests that the  sign difference is solely due to different definitions used  for the Core contribution in both the works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  In order to explain how definition of Core contribution  changes depending upon the choice of an approach to  estimate E1P V , we present the Core and Valence contri-  butions separately to the 6S − 7S and 6S − 5D3/2 tran-  sitions from both the RMBPT(3)w and RMBPT(3)d ap-  proaches in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Just for the sake of demonstrating  how appearance of Heff in the wave function determining  equation to choice of V N−1 modify the result, we present  RMBPT(3) results considering effect of Heff (given re-  sults as (a) in the above table) and replacing it with  DHF energy as the case of the CPDF-RPA method (cor-  responding results are given under (b) in the above ta-  ble).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' As can be seen, the Core and Valence contributions  from both the RMBPT(3)w and RMBPT(3)d approaches  are coming out differently whereas the final results from  both the methods are almost close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' It can  also be realized that changes in the results for both the  transitions are enormous when Heff is considered in the  wave function solving equation than otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  To further figure out about the mismatch in the X-  factors from various works, we present the Core and Va-  lence contributions to the E1P V values separately for  both the 6S − 7S and 6S − 5D3/2 transitions arising  through the first-principle calculations in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' As  can be seen from the table, signs of Core contributions  to E1P V of both the transitions from the DHF method  and many-body methods at a given level of approxima-  tion employed by different groups match each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' This  indicates that there is no issue with the implementa-  tion of these theories in our code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  To support results  from our methods further, we also compare Core and  Valence contributions to the 6S − 7S transition in Table  VI from the initial and final perturbed states through the  22  TABLE VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Contributions to Core and Valence parts from different terms to E1P V of the 6s 2S1/2 − 7s 2S1/2 and 6s 2S1/2 −  5d 2D3/2 transitions in 133Cs, in units 10−11i(−Qw/Nn)|e|a0 from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (89) and (90) of the CPDF-RPA method, which are  quoted under ‘Expression a’ and ‘Expression b’ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Results are given using the calculated ω value, ωex and ωex with  the experimental energies of the initial and final states (denoted by Eexpt  i,f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Contribution  Expression a  Expression b  6s 2S1/2 − 7s 2S1/2  ⟨7s|hw|6s+⟩  ⟨7s|uP V  6s |6s+⟩  Total  ⟨7sP V |d|6s⟩  ⟨7sP V |u+  6s|6s⟩  Total  Core (ω)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0357  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='02257  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='05794  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='04299  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='01495  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='05794  Valence (ω)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='06094  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='27610  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='33704  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='30962  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='02742  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='33704  Core (ωex)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='03464  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='02211  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='05675  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='04299  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='01458  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='05757  Valence (ωex)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='19464  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='04598  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='24062  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='30962  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='02743  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='33705  Core (Eexpt  i,f )  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='03546  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='02283  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='05829  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='04331  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='01498  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='05829  Valence (Eexpt  i,f )  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='21721  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='31956  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='53677  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='53384  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00293  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='53677  ⟨7s|d|6sP V ⟩  ⟨7s|u+  6s|6sP V ⟩  Total  ⟨7s−|hw|6s⟩  ⟨7s−|uP V  6s |6s⟩  Total  Core (ω)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='04099  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='01864  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='05963  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='03564  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='023399  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='05963  Valence (ω)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='38508  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='06605  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='45113  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='35181  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='09932  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='45113  Core (ωex)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='04099  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='01915  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='06014  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='03651  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='02458  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='06109  Valence (ωex)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='38508  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='06644  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='45152  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='17081  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='05022  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='22103  Core (Eexpt  i,f )  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='04210  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='01999  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='06209  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='03686  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='02523  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='06209  Valence (Eexpt  i,f )  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='12128  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='05800  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='17928  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='13743  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='04185  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='17928  6s 2S1/2 − 5d 2D3/2  ⟨5d3/2|hw|6s+⟩ ⟨5d3/2|uP V  6s |6s+⟩  Total  ⟨5dP V  3/2|d|6s⟩  ⟨5dP V  3/2|u+  6s|6s⟩  Total  Core (ω)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00616  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00616  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00451  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00165  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00616  Valence (ω)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='27386  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='27386  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='27878  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00492  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='27386  Core (ωex)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00612  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0612  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00451  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00164  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00615  Valence (ωex)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='23287  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='23287  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='27878  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
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+page_content='27385  Core (Eexpt  i,f )  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
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+page_content='83936  ⟨5d3/2|d|6sP V ⟩ ⟨5d3/2|u+  6s|6sP V ⟩  Total  ⟨5d−  3/2|hw|6s⟩ ⟨5d−  3/2|uP V  6s |6s⟩  Total  Core (ω)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
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+page_content='02306  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='16022  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='86284  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='46451  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='39833  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='86284  DHF, CPDF, RPA, CPDF-RPA* and CPDF-RPA meth-  ods with the values reported in a Comment by Roberts  and Ginges [30], and Martensson [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' We find reasonably  good agreement between our results with the earlier esti-  mations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' As it has been explained in the previous section,  definitions of Core correlation effects arising through the  CPDF, RPA and CPDF-RPA methods all differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Thus,  the exact reason for which sign of Core contribution to  E1P V of the 6S − 7S transition in 133Cs differ between  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [24] and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [25] is not clear to us as the exact  method(s) employed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [24] for its estimation is not  mentioned explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Only from the sign of the Core con-  23  tribution quoted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [24] and by comparing it with  the signs of Core contributions from the RMBPT(3)w,  CPDF and RCCSD methods of the present work and  from the RCCSD and RCCSDT methods of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [29],  we can assume that Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [24] estimates Core contribu-  tion by considering HW as perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In such a case,  the Tail contributions to E1P V for the 6S − 7S transi-  tion in 133Cs from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [24] and [29] as well as from the  RCCSD result of the present work should almost agree  each other on the basis of the argument that the net  X-factor value should agree irrespective of the fact that  whether HW or D is treated as perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Large dif-  ferences between the X-factors from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [24], [25] and  this work, which report as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0175, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0256 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0254 in  units of 10−11i(−Qw/Nn)|e|a0, suggests that the former  work underestimates the Tail contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' It should be  noted that the Tail contributions are estimated without  using sum-over-states approach in all the works, so the  difference in these values are mainly due to different lev-  els of approximation made in the many-body methods  employed for their estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Now, we wish to address the reason why Roberts and  Ginges were able to get same sign for the Core contribu-  tion to E1P V of the 7S − 6D3/2 transition in Ra+ using  their RPA+BO method with that are reported using the  RCC method in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Since correlation trends to  E1P V of the nS − (n − 1)D3/2 transitions are almost  similar in Cs and Ra+, with the ground state princi-  pal quantum number n of the respective system, we can  understand the above point by analysing the Core con-  tributions to E1P V of the 6S − 5D3/2 transition from  different methods and comparing their trends with the  6S − 7S transition of 133Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' By looking at these contri-  butions from Table V, it can be easily followed that there  is one-order magnitude difference between the Core con-  tribution in the 6S−7S transition from the RMBPT(3)w  and RMBPT(3)d methods while there is a sign change  between these results from the CPDF method and the  RPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' However, the difference between the Core contri-  butions from the RMBPT(3)w and RMBPT(3)d meth-  ods in the 6S − 5D3/2 transition are small, and there  is no sign difference between the CPDF and RPA re-  sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' These trends can be explained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In the  6S − 7S transition, wave functions of both the associ-  ated states have large overlap over the nucleus while in  the 6S − 5D3/2 transition only the wave function of the  ground state has large overlap with the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' As a re-  sult, strong core-polarization effects contribute through  both the states in the former case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Also, contribution  from individual diagram of the CPDF-RPA method is al-  most comparable in the 6S −7S transition, while a selec-  tive diagrams contribute predominantly in the 6S−5D3/2  transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Since core-polarization effects arising through  the D operator are stronger and have opposite signs than  that arise through HW , the net Core contributions in the  S − S and S − D transitions behave very differently in  the CDHF method and RPA, and the same propagates to  the CPDF-RPA*/CPDF-RPA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Since Core and  Valence contributions are basically redistributed in the  CPDF-RPA* and RCCSD methods, difference between  the final values between Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [25] and [29] as well as  from the present work are coming out to be very small in  the 6S−7S transition, while it is slightly noticeable in the  6S−5D3/2 transition (refer to Table I for the comparison  of results from the Mixed-states and RCCSD methods) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  The DCP contributions from our calculations can be  estimated by taking the differences in the results from  the CPDF-RPA* and CPDF-RPA methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' This differ-  ence for the 6S − 7S transition from our work is com-  pared with the corresponding values from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [11] and  [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' From this comparison, we find that our result agrees  better with Roberts than Martensson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' However, our fi-  nal CPDF-RPA result agrees well with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [11] than  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' We also intend to mention that the CPDF-  RPA* results in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [25] and [30] are also scaled by  using ωex = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='0844 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='. In the previous section, we have  justified theoretically why such an approach would lead  to errors in the determination of the E1P V values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' To  demonstrate it numerically, we have given results for both  the 6S − 7S and 6S − 5D3/2 transitions from the CPDF-  RPA method using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (89) and (90) in Table VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' We  have given these values using ω, ωex and then also using  ωex and experimental energies (denoted by Eexpt  i,f ) of the  6S, 7S and 5D3/2 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' From the comparison of the re-  sults, we observe a very a interesting trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' When both  ω and energies of the atomic states are considered either  from theory or experiment, results from both Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (89)  and (90) match each other, otherwise large discrepancies  are seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In the RMBPT, RPA or CPDF-RPA method,  it is possible to use ωex and experimental energies of the  initial and final states simultaneously in the E1P V eval-  uating expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' However, one can either use ωex or  ωex with experimental energy of only the valence state  (whose perturbed state wave function is evaluated) in  the complicated methods like the RCC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Ener-  gies of the double, triple excited configurations appear in  the denominator of the RCC theory, their experimental  energies cannot be used in the wave function determin-  ing equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' By corroborating this fact with the above  finding, it can be said that scaling the wave function  by using experimental energy of the valence state alone  may not always give accurate result, rather it may intro-  duce additional error to the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' As explained in  the previous section, this fact can be theoretically under-  stood using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Nevertheless, it can be found from  Table VII that our result with ωex value from the CPDF-  RPA* method does not match with the corresponding  results from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [25, 30] for the 6S −7S transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' We  are unable to understand the reason for this though re-  sults with theoretical ω value from both the works agree  quite well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  As mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' IV, three different approaches  can be adopted in any many-body theory framework for  evaluation of the E1P V amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' The same applies  to the RCC theory as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' However, we adopt the ap-  proach of evaluating the matrix element of the D op-  24  TABLE VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' First-principle calculated E1P V values (in −i(QW /N)ea0×10−11) of the 6s 2S1/2−7s 2S1/2 and 6s 2S1/2−5d 2D3/2  transitions in 133Cs from different terms of the RCCSD method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Both ab initio and scaled values are given for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  We have used two different types of scaling: (a) only scaling amplitudes of the the unperturbed S(0)  v  operators and (b) scaling  amplitudes of both the S(0)  v  and S(1)  v  operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Here, contributions under ‘Norm’ represents the difference between the  contributions after and before normalizing the RCCSD wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' ‘Others’ denote contributions from those RCCSD terms  that are not shown explicitly in this table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  RCC term  6s2S1/2 − 7s 2S1/2  6s 2S1/2 − 5d 2D3/2  Ab initio  Scaled-a  Scaled-b  Ab initio  Scaled-a  Scaled-b  Core contribution  DT (1)  1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
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+page_content='00062  T (1)†  1  D  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
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+page_content='01757  Norm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
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+page_content='00670  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00670  Valence contribution  DS(1)  1i  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='19363  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='19363  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='19688  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='96310  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='96310  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='97589  S(1)†  1f D  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='80382  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
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+page_content='80263  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='89993  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='89993  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='30760  S(0)†  1f DS(1)  1i  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='23184  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
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+page_content='06548  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='06487  S(1)†  1f DS(0)  1i  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='41826  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='41895  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
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+page_content='14626  DS(1)  2i  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00039  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
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+page_content='00108  S(1)†  2f D  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
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+page_content='15309  Total  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='89643  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
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+page_content='88998  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='56412  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
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+page_content='91879  erator after considering HW as the external perturba-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Though this approach is in the line with the CPDF  method, it is effectively takes care of electron correla-  tion effects through both the HW and D operators as in  the CPDF-RPA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In fact, it goes much beyond  the CPDF-RPA method to include the electronic corre-  lation effects which will be evident from the follow-up  discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' It means that it is possible to deduce all the  CPDF-RPA contributions from the RCC theory, which is  even true at the level of the RCCSD method approxima-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In this sense that the RCCSDT method employed  by Sahoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [29] to estimate the E1P V amplitude of  the 6S − 7S transition in 133Cs includes the RPA contri-  butions that are mentioned in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [25, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' However,  some of these contributions are not a part of the Core  contribution rather they come through the Valence con-  tribution in our RCCSD method owing to the fact that  the D operator is not treated as an external perturba-  tion here to determine the perturbed atomic wave func-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' This point can be comprehend from the compar-  ison between the RMBPT(3)w and RMBPT(3)d results,  which are propagated to all-orders in the RCC theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  To define Core contributions in the line of CPDF-RPA  method, the RCC theory of E1P V can be derived ei-  ther treating the HW and D operators simultaneously  as external perturbation or perturbing wave functions by  considering one of these operators as external perturba-  tion and evaluating matrix element of the other operator  in the normal-order RCC theory framework similar to  that is discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Among the choices of con-  sidering one of them as the external perturbation, it is  advisable to use HW as perturbation in which evalua-  tion of perturbed wave function in an iterative scheme  can converge faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In fact, this should also be the nat-  ural choice from the APV theory point of view and is  being adopted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  In order to understand the Core  and Valence contributions to the E1P V amplitudes from  our RCCSD method, we can take the help of the dia-  grams from the RMBPT(3)w method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Since wave opera-  tor amplitude determining equations for both the meth-  ods follow the same Bloch’s prescription, all physical  effects appear in the RMBPT(3)w method are present  to all-orders in the RCCSD method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  It means that  the core-polarization effects and additional lower-order  DCP contributions that arise in the RMBPT(3)w method  are present to all-orders in the RCCSD method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  In  the CPDF-RPA method, core-polarization effects are ap-  pearing through the single excitations while the DCP ef-  fects are implicitly present through the double excitations  in the RCCSD method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Similarly, the pair-correlation ef-  fects of the RMBPT(3)w method are present to all-orders  through the single excitations in the RCCSD method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Since both single and double excitation amplitude solv-  ing equations are coupled in the RCCSD method, cor-  relations among all these physical effects are taken into  account in this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Again through the non-linear  25  terms from the exponential form of the RCCSD method,  higher-order correlation effects, neither a part of core-  polarization or pair-correlation types, are also included  in the RCCSD method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' It is not possible to include these  effects systematically using a blend of many-body meth-  ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  In Table VIII, we present the E1P V values of the  6S − 7S and 6S − 5D3/2 transitions in 133Cs from in-  dividual RCCSD terms to fathom the discussions of the  previous paragraph quantitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' By using definitions  of the T and Sv RCC excitation operators, we catego-  rized the results into Core and Valence correlation con-  tributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' By subtracting the Core contributions of the  DHF method from the contributions of the ¯DT (1)  1  and its  complex conjugate (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=') term, the net Core correlation  contributions to E1P V in the RCCSD method can be  inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Similarly by subtracting the Valence contribu-  tions of the DHF method from the ¯DS(1)  1i +S(1)†  1f  ¯D terms  and adding contributions from other Valence correlation  contributing terms, we can get the net Valence correla-  tion contributions to E1P V in the RCCSD method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' The  Core correlations arising through ¯DT (1)  1  and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' terms  contain correlation contributions from both the singly  and doubly excited configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  By analysing the  RMBPT(3)w diagrams contributing to the T (1)  1  ampli-  tude determining equation shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 14, it can be  understood that the ¯DT (1)  1  and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' terms contain the  Core contributions of the CPDF method, pair-correlation  contributions of the RMBPT(3) method to all-orders and  many more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  By analyzing diagrams from ¯DT (1)  1  and  their breakdown in terms of the RMBPT(3)w method  carefully, it can be evident that this term does not in-  clude Core contributions arising through the RPA and  some of the contributions that arise through the CPDF-  RPA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Similarly, all the Valence correlation con-  tributions from the CPDF method, RPA and CPDF-  RPA* method are included through the ¯DS(1)  1i + S(1)†  1f  ¯D  terms in the RCCSD method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In addition, they also in-  clude many contributions that can appear through the  BO-correlation technique and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  However, a lot  more correlation contributions to E1P V arise through  other RCCSD terms among which correlation contribu-  tions arising through ¯DS(1)  2i ,  ¯DT (1)  1/2S(0)  1/2i, T (1)†  1/2 ¯DS(0)  1/2i,  such terms but replacing S(0/1)  1/2i  operators with S(0/1)†  1/2f ,  S(0)†  1/2f ¯DS(1)  1/2i, S(1)†  1/2f ¯DS(1)  1/2i etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  terms in the RCCSD  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Obviously, these contributions are not present  in the CPDF-RPA* method and many of them cannot  be considered as a part of the BO-correlation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Moreover, corrections to the entire correlation contri-  butions including that appear through the CPDF-RPA  method due to normalization of the wave functions (given  as ‘Norm’) are quoted separately in the above table  and they are found to be non-negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  The most  prominent DCP contributions are absorbed through the  ¯DS(1)  2i +S(1)†  2f  ¯D terms in the RCCSD method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Along with  some of Core contributions from the CPDF-RPA method  (like the ones appears in the RPA) are also included  through these terms in the RCCSD method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' In addition,  non-linear terms ¯DT (1)  1/2S(0)  2i , T (1)†  2  ¯DS(0)  1i , S(0)†  2f  ¯DS(1)  2i etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  including their c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' terms posses a lot more Valence corre-  lation contributions that are beyond the scope of consid-  ering by the combined CPDF-RPA and BO-correlation  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  In Table VIII, contributions to E1P V of the 6S − 7S  and 6S −5D3/2 transitions from the RCCSD terms using  the scaled S(0)  v  and S(1)  v  amplitudes are also given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' To  show how the results vary with scaling both the unper-  turbed and perturbed wave functions independently, we  present results after (a) scaling only the amplitudes of  the S(0)  v  operators and (b) then by scaling amplitudes of  both the S(0)  v  and Sv(1) operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Significant differences  from both the scaled results are noticed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' As mentioned  before, it would not be correct to scale the T (0/1) ampli-  tudes as orbitals used for their determination see V N po-  tential against the amplitude determining equations for  the S(0/1)  v  operators, where orbitals see V N−1 potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Thus, substituting ωex value in the estimation of Core  contribution may not be theoretically proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Again, we  can only substitute energy of the valence state from out-  side in the wave function solving equations, whereas en-  ergies of the intermediate states have to be generated  implicitly in the RCC theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  It means that evaluat-  ing the E1P V amplitudes through the scaling procedure  using the RCC method may introduce numerical errors  to the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Nonetheless, comparison of the semi-  empirical results obtained by using experimental energies  with the ab initio values of E1P V for both the 6S−7S and  6S − 5D3/2 transitions shows that there are significant  differences among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' These differences can be min-  imised by including higher-level excitations in the RCC  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' However, these higher-level excitations will not  only improve the energy values but they will also change  the matrix elements of the HW and D operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  As  shown by Sahoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' [29, 47] inclusion of the triple ex-  citations in the RCC theory, modify the energies and the  matrix elements of the HW and D operators in such a way  that the E1P V values from the RCCSD and RCCSDT  methods almost remain to be same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' From this argument,  we cannot argue that the scaled E1P V values are more  accurate than the ab initio values in the RCCSD method  approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  SUMMARY  By employing a number of relativistic many-body  methods  at  different  levels  of  approximation  such  as finite-order perturbation theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' coupled-perturbed  Dirac-Fock method,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' random phase approximation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' com-  bined coupled-perturbed Dirac-Fock and random phase  approximation method,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' and relativistic coupled-cluster  theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' we investigated various roles of core and valence  correlation effects in the calculations of the parity vio-  26  lating electric dipole amplitudes of the 6S → 7S and  6S → 5D3/2 transitions in 133Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  From this analysis,  we were able to address a long standing issue of get-  ting opposite signs to the core correlation contribution  to the parity violating electric dipole amplitude of the  aforementioned 6S → 7S transition using the combined  coupled-perturbed Dirac-Fock and random phase ap-  proximation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' We also analysed results from the  sum-over-states approach and first-principle calculations  using the relativistic coupled-cluster method with sin-  gles and doubles approximation to figure out the missing  contributions in the former approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Inclusion of these  missing contributions through the combined coupled-  perturbed Dirac-Fock, random phase approximation and  Br¨uckener-orbital correlation methods is compared with  the first-principle calculations using the coupled-cluster  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' This comparison shows that the first-principle  approach using the relativistic coupled-cluster theory in-  corporates electron correlation effects due to the Dirac-  Coulomb Hamiltonian more rigorously than the other  methods mentioned above in the evaluation of parity vi-  olating electric dipole amplitudes in the 133Cs atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  ACKNOWLEDGEMENT  The computations reported in the present work were  carried out using the Vikram-100 HPC cluster of the  Physical Research Laboratory (PRL), Ahmedabad, Gu-  jarat, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Commins, Quantum Mechanics An Experimental-  ists Approach, Cambridge University Press, New York  (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Erler and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Langacker, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 105, 031801  (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [3] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Marciano and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Rosner, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 65,  2963 (1990);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 68, 898 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' -A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Bouchiat and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Bouchiat, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' B 48, 111  (1974);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' (France) 35, 899 (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [5] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Fayet, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' B 96, 83 (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [6] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Wood, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Bennet, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Cho, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Masterson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Roberts, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Tanner and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Wieman, Science 275,  1759 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Choi and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Elliott, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A 93, 023432 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Kastberg, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Aoki, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Sahoo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Sakemi and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Das, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A 100, 050101(R) (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [9] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Dzuba, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Flambaum, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Silvestrov and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Sushkov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A 103, 265 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [10] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Dzuba, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Flambaum, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Silvestrov and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Sushkov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' B: Atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 18, 597 (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' -M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' M˚artensson-Pendrill, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 46, 11 (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [12] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Blundell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Sapirstein and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Johnson, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' D 45, 5 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [13] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Dzuba, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Flambaum and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Ginges, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' D 63, 062101 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [14] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Dzuba, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Flambaum and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Sushkov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A 141, 147 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Milstein, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Sushkov and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Terekhov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 89, 283003 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [16] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Brown, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Derevianko and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Flambaum, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' C 79, 035501 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [17] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Derevianko, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 85, 1618 (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A 65, 012106 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [18] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Kozlov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Porsev and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Tupitsyn, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 86, 3260 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [19] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Shabaev, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Pachucki, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Tupitsyn and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Yerokhin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 94, 213002 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [20] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Dzuba, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Flambaum and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Ginges, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' D 66, 076013 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [21] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Das, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Sahoo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Gopakumar, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Chaudhuri, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 768, 141 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [22] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Sahoo,  Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  thesis,  Manga-  lore  University,  Karnataka,  India,  2006,  http://prints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='iiap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='in/handle/2248/680.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [23] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Sahoo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' B 43, 085005 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [24] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Porsev, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Beloy and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Derevianko, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' D  82, 036008 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [25] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Dzuba, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Berengut, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Flambaum and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Roberts, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 109, 203003 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [26] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Roberts, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Dzuba and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Flambaum, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A 88, 042507 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Safronova, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Budker, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' DeMille, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Kimball,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Derevianko and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Clark, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 90,  025008 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [28] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Wieman and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Derevianko, arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='00281 (Un-  published).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [29] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Sahoo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Das and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Spiesberger, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  D 103, L111303 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [30] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Roberts and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Ginges, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' D 105,  018301 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [31] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Tran Tan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Xiao and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Derevianko, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  A 105, 022803 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [32] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Sahoo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Das and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Spiesberger, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  D 105, 018302 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [33] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Cahn and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Kane, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' B 71, 348 (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [34] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Marciano and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Sanda, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' D 17, 3055  (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [35] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Commins, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' T46, 92 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [36] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Lindgren and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Morrison, Atomic Many-Body Theory,  Springer-Verlag Berlin Heidelberg (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [37] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Roberts, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Dzuba and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Flambaum, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A 88, 012510 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [38] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Kastberg, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Sahoo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Aoki, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Sakemi and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Das, Symmetry 12, 974 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [39] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Sahoo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Das, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A 84, 010502(R)  (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [40] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Wansbeek, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Sahoo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Timmermans,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Jungmann, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Das and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Mukherjee, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  A 78, 050501(R)(2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [41] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Young, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Hill III, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Sibener, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Price, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Tanner, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Wieman and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Leone, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A  50, 2174 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  27  [42] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Vasilyev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Savukov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Safronova and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Berry, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A 66, 020101 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [43] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Bouchiat, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Guena, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Pottier, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  (Paris), Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 45, 523 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [44] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Bennett, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Roberts and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Wieman, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' A 59, R16 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [45] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Kramida, Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Ralchenko, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Reader and NIST ASD  Team, NIST Atomic Spectra Database (ver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='1) (Na-  tional Institute of Standards and Technology, Gaithers-  burg, MD, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [46] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Sahoo and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Das, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Letts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' 120, 203001  (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='  [47] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Sahoo and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content=' Das, arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
+page_content='08941 [hep-ph]  (2020) (Unpublished).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E1T4oBgHgl3EQfhAQw/content/2301.03235v1.pdf'}
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+arXiv:2301.00206v1  [math.DS]  31 Dec 2022
+Melnikov’s persistence in degeneracy of high co-dimension.
+Jiayin Dua 1 , Yong Lia,b 2
+aCollege of Mathematics, Jilin University, Changchun 130012, P. R. China.
+bSchool of Mathematics and Statistics, and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal
+University, Changchun 130024, P. R. China.
+Abstract
+In this paper, we study the Melnikov’s persistence about lower-dimensional invariant tori for degenerate Hamiltonian
+systems with the following Hamiltonian
+H(x, y, u, v) = h(y) + g(u, v) + εP(x, y, u, v),
+(x, y, u, v) ∈ Tn × G × Rd × Rd,
+where n ≥ 2 and d ≥ 1 are positive integers, g = o(|u|2 + |v|2) admits certain transversality, and G ⊂ Rn is a
+bounded closed region. This is a try in studying lower-dimensional invariant tori in the normal degeneracy of high
+co-dimension, i.e., d > 1. Under R¨ussmann-like non-degenerate condition and transversality condition, we apply the
+homotopy invariance of topological degree to remove the first order terms about u and v and employ the quasi-linear
+KAM iterative procedure.
+Keywords: Degenerate Hamiltonian system, lower-dimensional tori, Melnikov’s persistence, high co-dimension.
+Contents
+1
+Introduction
+2
+2
+Main Results
+3
+3
+KAM Step
+6
+3.1
+Description of the 0-th KAM step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+3.2
+Induction from ν-th KAM step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+3.2.1
+Description of the ν-th KAM step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+3.2.2
+Construct a symplectic transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+3.2.3
+Truncation
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+3.2.4
+Homological Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+3.2.5
+Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+13
+3.2.6
+Eliminate the first order terms about z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+13
+3.2.7
+Estimate on N+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+15
+3.2.8
+Estimate on Φ+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+16
+3.2.9
+Frequency Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+19
+3.2.10 Estimate on P+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+20
+4
+Proof of Theorem 1
+21
+4.1
+Iteration Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+21
+4.2
+Convergence
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+25
+4.3
+Measure estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+27
+1E-mail address : dujy20@mails.jlu.edu.cn
+2Corresponding author’s e-mail address : liyong@jlu.edu.cn
+Preprint submitted to a
+January 3, 2023
+
+5
+Appendix A.
+28
+6
+Appendix B.
+29
+1. Introduction
+It is well known that by the classical KAM theory [1, 12, 21, 29, 32], most non-resonant Lagrangian tori of a
+non-degenerate integrable Hamiltonian system survive small perturbations with only slight deformation. In contrast
+to this, the destruction of those resonant tori appears inevitable under perturbations in general. Nevertheless, their
+disintegration does not imply the nonexistence of regular motions such as periodic or quasi-periodic motions, as
+pointed out in [5]. Some studies indicate that the resonant torus foliated by non-resonant lower-dimensional tori is not
+destroyed completely and that there are some lower-dimensional tori which survive perturbation if the Hamiltonian
+satisfies a certain non-degenerate condition, see [8, 24, 27, 36, 39]. So, in the perturbation theory of Hamiltonian
+systems, the persistence of lower-dimensional invariant tori plays an important role in understanding the destruction
+mechanism of resonant Lagrangian tori.
+Let’s consider the unperturbed system with the following normal form:
+N = ⟨ω, y⟩ + 1
+2⟨z, Ωz⟩,
+where (y, z) ∈ Rn × R2d, n ≥ 2, d ≥ 1. In non-degenerate case, i.e., Ω is non-singular, if all the eigenvalues of JΩ
+belong to iR1\{0} (J is the standard symplectic matrix), we say that the lower-dimensional invariant torus is elliptic,
+which corresponds to the following normal form:
+N = ⟨ω, y⟩ + 1
+2
+d
+�
+j≥1
+Ω j(u2
+j + v2
+j),
+where (y, z) ∈ Rn × R2d, z = (u, v) ∈ Rd × Rd. In 1965, Melnikov [28] announced that the elliptic lower dimensional
+invariant tori can persist for any small perturbations under certain coupling nonresonance conditions, called Melnikov
+conditions:
+⟨k, ω⟩ + ⟨l, Λ⟩ � 0,
+∀(k, l) ∈ (Zn × Z2d)\{0}, |l| ≤ 2,
+where Λ is the column vector formed by the eigenvalues of Ω. A detailed and rigorous proof of the above Melnikov’s
+theorem was due to Eliasson [13] and then an infinite dimensional analogue was obtained by Kuksin [23], P¨oschel
+[33], Bourgain [2] and Wayne [41]. On the other hand, if all the eigenvalues of JΩ are not on the imaginary axis, the
+lower-dimensional invariant torus is called hyperbolic, which is associated with the unperturbed system
+N = ⟨ω, y⟩ + 1
+2
+d
+�
+j≥1
+Ω j(u2
+j − v2
+j).
+The persistence problem for hyperbolic tori was first considered in the work of Moser [30] and later studied by Graff
+[17] and Zehnder [46] for a fixed toral frequency ω satisfying the following Diophantine condition:
+|⟨k, ω⟩| > γ
+|k|τ ,
+k ∈ Zn \ {0},
+(1.1)
+where k = (k1, · · · , kn), |k| = |k1| + · · · + |kn|, γ > 0 and τ > n − 1. Moreover, the persistence of lower-dimensional
+invariant tori has been extensively studied in various (normally) non-degenerate cases, see [3, 4, 5, 6, 8, 9, 10, 11,
+13, 14, 15, 20, 26, 28, 33, 35, 39] and references therein. However, in degenerate case, i.e., Ω is singular, generally,
+the lower dimensional invariant tori might be destroyed under small perturbations. Then a natural question is what
+degenerate systems admit Melnikov’s persistence, which is an essential problem, for example, see [22]. A normal
+form in the simplest degenerate case is, see Takens [38],
+N = ⟨ω, y⟩ + 1
+2v2 + f(u, v),
+2
+
+where y ∈ Rn, f(u, v) = o(|u|2+|v|2) � 0, u, v ∈ R1, which is called co-dimension 1. You in [45] proved the persistence
+of a hyperbolic-type degenerate lower-dimensional torus for co-dimension 1 Hamiltonian systems with the following
+normal form
+N = ⟨ω, y⟩ + 1
+2v2 − u2l,
+l ≥ 2,
+where (y, u, v) ∈ Rn × R1 × R1. For more details see [42, 43]. Recently, Li and Yi showed the persistence of lower-
+dimensional tori of general types, under a Melnikov-type of non-resonance condition, for the Hamiltonian system
+with the following general normal form
+N = e + ⟨ω, y⟩ + 1
+2⟨
+� y
+z
+�
+, M
+� y
+z
+�
+⟩ + O(|(y, z)|3),
+where (y, z) ∈ Rn × R2d, d ≥ 1, and M is nonsingular. For some other related studies, see [16, 18, 19, 44]. In this
+paper, we will consider the Melnikov’s persistence when the unperturbed systems possess both high-degeneracy and
+high co-dimension.
+More precisely, we consider the following more general degenerate Hamiltonian system
+H(x, y, u, v) = h(y) + g(u, v) + εP(x, y, u, v),
+(1.2)
+where n ≥ 2 and d ≥ 1 are positive integers, (x, y, u, v) ∈ Tn × G × Rd × Rd, Tn is the standard n-torus, G ⊂ Rn is a
+bounded closed region, g = o(|u|2 + |v|2), h, g and P are real analytic functions in (x, y, u, v), ε > 0 is a small parameter
+and εP is the small perturbation. We verify the persistence of lower-dimensional invariant tori for the degenerate
+Hamiltonian system with high co-dimension under certain high degeneracy conditions. We further construct some
+sufficient conditions in term of the topological degree condition as well as the weak convexity condition for g(u, v),
+see (A0) in Section 2 for details of these two conditions. We also give an example to state that our condition (A0) is
+sharp to Melnikov’s persistence, see Proposition 2.
+The paper is organized as follows. In Section 2, we state our main results (Theorem 1, Proposition 1 and Propo-
+sition 2). Section 3 contains the detailed construction and estimates for one cycle of KAM steps. In Section 4, we
+complete the proof of Theorem 1 by deriving an iteration lemma and showing the convergence of KAM iterations.
+Finally, the proof of Proposition 1 and Proposition 2 can be found in Appendix A and Appendix B, respectively.
+2. Main Results
+To state our main results we need first to introduce a few definitions and notations.
+• Given a domain D, we let ¯D, ∂D denote the closure of D and the boundary of D, respectively. Do := ¯D \ ∂D
+refers to the interior.
+• We shall use the same symbol | · | to denote the Euclidean norm of vectors, and use | · |D to denote the supremum
+norm of a function on a domain D.
+• For any two complex column vectors ξ, η of the same dimensional, ⟨ξ, η⟩ always stands for ξ⊤η, i.e., the
+transpose of ξ times η.
+• id is the identical mapping, and Id is the d-order unit matrix.
+• For a vector value function f, Df denotes the Jacobian matrix of f, and J f = detDf its Jacobian determinant.
+• All Hamiltonians in the sequel are endowed with the standard symplectic structure.
+• Bδ(ζ) stands for a sphere with ζ as center and δ as radius.
+• For simplicity we let z = (u, v) ∈ R2d, ∇g(z) = ( ∂g
+∂u, ∂g
+∂v).
+3
+
+• Denote the complex neighborhoods of Tn × {0} × {0}
+D(s, r) = {(x, y, z) : |Imx| < r, |y| < s2, |z| < s}
+(2.1)
+with 0 < s, r < 1.
+Let Ω ⊂ Rn be a bounded and open domain. We give the definition of the degree for f ∈ C2( ¯Ω, Rn), see [31].
+Definition 2.1. If f ∈ C2( ¯Ω, Rn) and p ∈ Rn \ f(∂Ω).
+(1) Denote Nf = {x ∈ Ω|J f(x) = 0}. If p � f(Nf ), then
+deg(f, Ω, p) :=
+�
+x∈ f −1(p)
+sign(J f(x)),
+setting deg(f, Ω, p) = 0 if f −1(p) = Ø.
+(2) If p ∈ f(Nf ), by Sard theorem (see [31]), deg(f, Ω, p) is defined as deg(f, Ω, p) := deg(f, Ω, p1), for some ( and
+all) p1 � f(Nf ) such that |p1 − p| ≪ 1.
+Let ξ ∈ G be arbitrarily given. The Taylor expansion of the Hamiltonian (1.2) about ξ in a small neighborhood of
+ξ in G reads
+H(x, y, z, ξ) = e(ξ) + ⟨ω(ξ), y − ξ⟩ + ˜h(y − ξ) + g(z) + εP(x, y, z),
+where e(ξ) = h(ξ), ω(ξ) = ∂h(ξ)
+∂y , and ˜h(y − ξ) = O(|y − ξ|2). Using the transformation (y − ξ) → y in the above, we have
+H(x, y, z, ξ) = e(ξ) + ⟨ω(ξ), y⟩ + ˜h(y, ξ) + g(z) + εP(x, y, z, ξ),
+where (x, y, z) ∈ D(s, r), ξ ∈ G.
+We are finally led to consider the following Hamiltonian
+� H : Tn × G × R2d × G → R1,
+H(x, y, z, ξ) = e(ξ) + ⟨ω(ξ), y⟩ + ˜h(y, ξ) + g(z) + εP(x, y, z, ξ).
+(2.2)
+First, we make the following assumptions:
+(A0) For fix ζ0 = 0 ∈ Oo ⊂ R2d, where O is a bounded closed domain, there are σ > 0, L ≥ 2 such that
+deg(∇g(z) − ∇g(ζ0), Oo, 0) � 0,
+|∇g(z) − ∇g(ζ0)| ≥ σ|z − ζ0|L.
+(A1) There is a positive integer M such that
+rank{∂i
+ξA0(ξ) : 0 ≤ |i| ≤ M} = 1
+for all ξ ∈ G,
+where
+A0(ξ) = ⟨ k
+|k|, ω(ξ)⟩, k ∈ Zn\{0}, |k| = |k1| + · · · + |kn|.
+We can now state our main results:
+Theorem 1. Consider the Hamiltonian (2.2). Assume that (A0)-(A1) hold and ζ0 = 0 is a equilibrium point of
+g(z), i.e.,
+g(z) = o(|z|2),
+∇g(ζ0) = 0.
+(2.3)
+4
+
+Let ε0 > 0 be sufficiently small. Then as 0 < ε < ε0, there exist Cantor sets G∗ ⊂ G with |G\G∗| → 0 as ε → 0 as well
+as a Cm−1 Whitney smooth family of symplectic transformations
+Ψ∗ = Φ1 ◦ Φ2 ◦ · · · ◦ Φ∞ : D( s
+2, r
+2) → D(s, r),
+ξ ∈ G∗,
+which is analytic in x, Cm−1 Whitney smooth in y, z and ξ, and close to the identity, such that
+H∗ = H ◦ Ψ∗ = e∗(ξ) + ⟨ω∗(ξ), y⟩ + ˜h∗(y, ξ) + g∗(z, ξ) + ¯g∗(y, z, ξ) + P∗(x, y, z, ξ)
+(2.4)
+with
+˜h∗(y, ξ) = O(|y|2),
+g∗(z, ξ) = g(z) + ε
+(m−2)(5m+4)
+8m(m+1) O(|z|2),
+¯g∗(y, z, ξ) =
+�
+2|ı|+| |≤m,1≤|ı|,| |
+¯gı (ξ)yız,
+∂i
+y∂ j
+zP∗|(y,z)=(0,0) = 0,
+where e∗, ω∗, ˜h∗, g∗, ¯g∗, P∗ are real analytic in their associated variables, 2|i| + |j| ≤ m, and
+m ≥ L +
+√
+L2 + 16L + 16
+4
+.
+(2.5)
+Thus for each 0 < ε < ε0 the Hamiltonian system (2.2) admits a Whitney smooth family of real analytic, quasi-periodic
+n-tori Tε
+ξ, ξ ∈ G∗.
+Remark 2.1. In Theorem 1, one can allow d > 1. Hence it seems to be the first result about lower-dimensional
+invariant tori in the normal degeneracy of high co-dimension.
+Remark 2.2. Recall z = (u, v) ∈ Rd × Rd. For partially degenerate Hamiltonian (1.2) with
+g(u, v) = 1
+2
+d
+�
+i=1
+λiv2
+i + f(u), λi � 0, i = 1, · · · , d, d ≥ 1, f(u) = O(|u|4),
+we will study it in a forthcoming paper.
+Remark 2.3. (2.4) in Theorem 1 indicates that the Hamiltonian system (2.2) is conjugated to a nonlinear system,
+not a linear one. Moreover, in order to ensure the persistence of low-dimensional invariant tori, the conjugated system
+cannot have first order terms about z, i.e., g∗(z) = O(|z|2), otherwise there is no invariant torus because one cannot
+get invariant set in the z direction as ∇g∗(0) � 0.
+Indeed, as an application of Theorem 1, we have the following proposition:
+Proposition 1. Consider the Hamiltonian (2.2) with
+g(u, v) = 1
+2l0
+|u|2l0 + 1
+2k0
+|v|2k0,
+where l0, k0 > 1 are integers, the dimension of u might be any positive integer, hence high co-dimensional. Assume
+ω(ξ) satisfies the condition (A1). Then the perturbed Hamiltonian system (1.2) admits a family of lower-dimensional
+tori.
+See Appendix A for the complete proof.
+Next, we will give an example to state that our assumption (A0) is sharp to Melnikov’s persistence. See below for
+a counter example:
+5
+
+Proposition 2. Consider the Hamiltonian (1.2) with
+H = ω · y + 1
+3u3 + 1
+3v3 + ε2u,
+(2.6)
+where y ∈ R1, (u, v) ∈ R1 × R1, and ε2u is a perturbation term. Assume ω satisfies Diophantine condition (1.1). Then
+the Hamiltonian system does not admit any lower-dimensional torus.
+The proof can be found in Appendix B.
+3. KAM Step
+In this section, we will show the detailed construction and estimates for one cycle of KAM steps, see [7, 18, 25,
+26, 33].
+3.1. Description of the 0-th KAM step.
+Recall the integer m satisfying
+m ≥ L +
+√
+L2 + 16L + 16
+4
+,
+(3.7)
+where L ≥ 2 is defined as in (A0). Denote ρ =
+1
+2(m+1), and let η > 0 be an integer such that (1 + ρ)η > 2. We put
+γ = ε
+1
+2m(m+1)(2d)m .
+(3.8)
+Consider the perturbed Hamiltonian (2.2). First we define the following 0-th KAM step parameters:
+r0 = r,
+γ0 = γ,
+µ0 = ε
+1
+8(m+1) ,
+s0 = sε
+1
+8(m+1) γ(m+1)(2d)m
+0
+, σ0 = σ, β0 = s,
+e0 = e(ξ),
+ω0 = ω(ξ),
+˜h0(y) = ˜h(y),
+g0(z) = g(z),
+¯g0 = 0,
+G0 = ˆG,
+(3.9)
+D(s0, r0) := {(x, y, z, ξ) : |y| < s2
+0, |z| < s0, |Imx| < r0},
+where 0 < r0, s0, s, γ0 ≤ 1, σ is defined in (A0). Therefore, we have that
+H0 =: H(x, y, z, ξ) = N0 + P0,
+N0 =: e0 + ⟨ω0, y⟩ + ˜h0(y) + g0(z) + ¯g0,
+P0 =: εP(x, y, z, ξ).
+We first prove an estimate.
+Lemma 3.1. Assume that
+(H0) : ε
+m
+8(m+1)
+0
+|∂ℓ
+ξP|D(s0,r0)
+sm
+≤ 1, |ℓ| ≤ m.
+For |ℓ| ≤ m,
+|∂ℓ
+ξP0|D(s0,r0) ≤ γ(m+1)(2d)m
+0
+sm
+0 µ0.
+(3.10)
+Proof. Using the fact γ0 = ε
+1
+2m(m+1)(2d)m , s0 = sε
+1
+8(m+1) γ(m+1)(2d)m
+0
+and µ0 = ε
+1
+8(m+1) , we have
+sm
+0 = smε
+m
+8(m+1) γm(m+1)(2d)m
+0
+= smε
+m
+8(m+1) + 1
+2 ,
+and
+γ(m+1)(2d)m
+0
+sm
+0 µ0 = ε
+1
+2m smε
+m
+8(m+1) + 1
+2 ε
+1
+8(m+1) ≥ smε
+1
+4 + 5
+8 +
+1
+8(m+1) .
+(3.11)
+6
+
+Then, by (H0), for any 0 < ε < ε0, we get
+ε
+m
+8(m+1) |∂ℓ
+ξP|D(s0,r0)
+sm
+≤ 1, |ℓ| ≤ m,
+i.e.,
+ε
+m
+8(m+1) |∂ℓ
+ξP|D(s0,r0) ≤ sm, |ℓ| ≤ m.
+(3.12)
+Thus, by (3.11) and (3.12), for |ℓ| ≤ m,
+|∂ℓ
+ξP0|D(s0,r0) = ε|∂ℓ
+ξP|D(s0,r0) ≤ ε
+1
+4 + 5
+8 +
+1
+8(m+1) ε
+m
+8(m+1) |∂ℓ
+ξP|D(s0,r0)
+≤ smε
+1
+4 + 5
+8 +
+1
+8(m+1) ≤ γ(m+1)(2d)m
+0
+sm
+0 µ0,
+which implies (3.10).
+The proof is complete.
+3.2. Induction from ν-th KAM step
+3.2.1. Description of the ν-th KAM step
+We first define the ν-th KAM step parameters:
+rν = rν−1
+2
++ r0
+4 ,
+sν = 1
+8 s2ρ+1
+ν−1 ,
+µν = 8msρ
+ν−1µν−1.
+Now, suppose that at ν-th step, we have arrived at the following real analytic Hamiltonian:
+Hν = Nν + Pν,
+(3.13)
+with
+Nν = eν(ξ) + ⟨ων(ξ), y⟩ + ˜hν(y, ξ) + gν(z, ξ) + ¯gν(y, z, ξ),
+(3.14)
+˜hν(y, ξ) =
+�
+2≤|i|
+˜hi0(ξ)yi,
+gν(z, ξ) = g(z) +
+ν−1
+�
+i=0
+γ(m+1)(2d)m
+i
+sm−2
+i
+µiO(|z|2),
+¯gν(y, z, ξ) =
+�
+2|i|+| j|≤m,1≤|i|,| j|
+¯gi j(ξ)yizj,
+(3.15)
+defined on D(sν, rν),
+∇gν(0) = ∇gν−1(ζν − ζν−1) + ∇z[Rν−1](0, ζν − ζν−1) = 0,
+ζν ∈ B(sm−2
+ν−1 µν−2)
+1
+L (ζν−1),
+and
+| ∂ℓ
+ξPν |D(sν,rν)≤ γ(m+1)(2d)m
+ν
+sm
+ν µν.
+(3.16)
+For simplicity, we will omit the index for all quantities of the present KAM step (at ν-th step), use + to index
+all quantities (Hamiltonian, domains, normal form, perturbation, transformation, etc.) in the next KAM step
+(at (ν+1)-th step), and use − to index all quantities in the previous KAM step (at (ν-1)-th step). To simplify
+notations, we will not specify the dependence of P, P+ etc. All the constants c1 − c6 (they will be defined in different
+lemmas-Lemma 3.2, 3.3, 3.5, 3.6, 3.8, 3.9) are positive and independent of the iteration process, and we will also use
+c to denote any intermediate positive constant which is independent of the iteration process. Define
+r+ = r
+2 + r0
+4 ,
+7
+
+β+ = β
+2 + β0
+4 ,
+σ+ = σ
+2 + σ0
+4 ,
+γ+ = γ
+2 + γ0
+4 ,
+s+ = 1
+8αs,
+α = s2ρ = s
+1
+(m+1) ,
+µ+ = 8mc0µsρ,
+c0 = max{1, c1, c2, · · · , c6},
+K+ = ([log 1
+s ] + 1)3η,
+(1 + ρ)η > 2,
+D(s) = {(y, z) ∈ Cn × R2d : |y| < s2, |z| < s},
+ˆD(s) = D(s, r+ + 7
+8(r − r+)),
+˜D = D(β+, r+ + 5
+8(r − r+)),
+D i
+8 α = D( i
+8αs, r+ + i − 1
+8
+(r − r+)), i = 1, 2, · · · , 8,
+D+ = D 1
+8 α = D(s+, r+),
+Γ(r − r+) =
+�
+0<|k|≤K+
+|k|(m+1)(2d)mτ+me−|k| r−r+
+8 ,
+G+ = {ξ ∈ G : |⟨k, ω⟩| > γ
+|k|τ , A⊤ı Aı  > γ2
+|k|2τ I(2d)ı+, for k ∈ Zn,
+0 < |k| < K+, 2|ı| + || ≤ m, 1 ≤ ||}
+3.2.2. Construct a symplectic transformation
+We will construct a symplectic transformation Φ+:
+Φ+ : D(s+, r+) × G+ → D(s, r) × G
+such that it transforms Hamiltonian (3.13) into the Hamiltonian of the next KAM cycle (at (ν+1)-th step), i.e.,
+H+ = H ◦ Φ+ = N+ + P+,
+where N+ and P+ have similar properties as N and P respectively on D(sν+1, rν+1).
+Next, we show the detailed construction of Φ+ and the estimates of P+.
+3.2.3. Truncation
+Consider the Taylor-Fourier series of P:
+P =
+�
+k∈Zn, ı, ∈Zn
++
+pkı yıze
+√
+−1⟨k,x⟩,
+and let R be the truncation of P of the form
+R =
+�
+|k|≤K+, 2|ı|+| |≤m
+pkı yıze
+√
+−1⟨k,x⟩
+=
+�
+|k|≤K+, 2|ı|+| |≤m
+pkı yı1
+1 · · · yınn z1
+1 · · · z2d
+2d e
+√
+−1⟨k,x⟩,
+where |ı| = |ı1| + · · · + |ın|, || = |1| + · · · + |2d|.
+Next, we will prove that |P − R| is much smaller than |P| by truncating appropriately, see the below lemma.
+8
+
+Lemma 3.2. Assume that
+(H1) :
+� ∞
+K+
+tne−t r−r+
+16 dt ≤ s.
+Then there is a constant c1 such that for all ξ ∈ G, |ℓ| ≤ m,
+|∂ℓ
+ξ(P − R)|Dα ≤ c1γ(m+1)(2d)m sm+1µ,
+(3.17)
+|∂ℓ
+ξR|Dα ≤ c1γ(m+1)(2d)m smµ.
+(3.18)
+Proof. Denote
+I =
+�
+|k|>K+, ı, ∈Zn
++
+pkı yıze
+√
+−1⟨k,x⟩,
+II =
+�
+|k|≤K+, 2|ı|+| |>m
+pkı yıze
+√
+−1⟨k,x⟩.
+Then
+P − R = I + II.
+To estimate I, we notice by (3.16) that
+|
+�
+ı∈Zn
++
+∂ℓ
+ξpkıyı| ≤ |∂ℓ
+ξP|D(s,r)e−|k|r ≤ γ(m+1)(2d)m smµe−|k|r,
+where the first inequality has been frequently used in [7, 8, 18, 25, 33, 34, 37] and for the detailed proof, see [37].
+This together with (H1) and (3.16) yields
+|∂ℓ
+ξI| ˆD(s) ≤
+�
+|k|>K+
+|∂ℓ
+ξP|D(s,r)e−|k| r−r+
+8
+≤ γ(m+1)(2d)m smµ
+∞
+�
+κ=K+
+κne−κ r−r+
+8
+≤ γ(m+1)(2d)m smµ
+� ∞
+K+
+tne−t r−r+
+16 dt ≤ γ(m+1)(2d)m sm+1µ.
+(3.19)
+It follows from (3.16) and (3.19) that
+|∂ℓ
+ξ(P − I)| ˆD(s) ≤ |∂ℓ
+ξP|D(s,r) + |∂ℓ
+ξI| ˆD(s) ≤ 2γ(m+1)(2d)m smµ.
+For 2|p| + |q| = m + 1, let
+�
+be the obvious antiderivative of ∂(p,q)
+∂ypzq . Then the Cauchy estimate of P − I on D∗ yields
+|∂ℓ
+ξII|Dα = |∂ℓ
+ξ
+�
+∂(p,q)
+∂ypzq
+�
+|k|≤K+, 2|ı|+| |>m
+pkı yıze
+√
+−1⟨k,x⟩dydz|Dα
+≤ |
+�
+| ∂(p,q)
+∂ypzq ∂ℓ
+ξ(P − I)|dydz|Dα
+≤ | c
+sm+1
+�
+|∂ℓ
+ξ(P − I)|D∗dydz|Dα
+≤ 2 c
+sm+1 γ(m+1)(2d)m smµ(αs)m+1
+≤ cγ(m+1)(2d)m sm+1µ.
+Thus,
+|∂ℓ
+ξ(P − R)|Dα ≤ cγ(m+1)(2d)m sm+1µ,
+and therefore,
+|∂ℓ
+ξR|Dα ≤ |∂ℓ
+ξ(P − R)|Dα + |∂ℓ
+ξP|D(s,r) ≤ cγ(m+1)(2d)m smµ.
+The proof is complete.
+9
+
+3.2.4. Homological Equation
+As usual, we shall construct a symplectic transformation as the time 1-map φ1
+F of the flow generated by a Hamil-
+tonian F of the form
+F =
+�
+0<|k|≤K+, 2|ı|+| |≤m
+Fkı yıze
+√
+−1⟨k,x⟩
+=
+�
+0<|k|≤K+, 2|ı|+| |≤m
+Fkı yı1
+1 · · · yınn z1
+1 · · · z2d
+2d e
+√
+−1⟨k,x⟩,
+(3.20)
+where Fkı  are (ξ, y, z)-dependent vectors or matrices of obvious dimensions.
+The Hamiltonian F can be determined by the following quasi-linear homological equation
+{N, F} + R − [R] − Q = 0,
+(3.21)
+where [R] =
+1
+(2π)n
+�
+Tn R(x, y, z)dx is the average of the truncation R, and the correction term
+Q = (∂zg + ∂z¯g)J∂zF|2|ı|+| |>m,
+(3.22)
+corresponds to the (m + 1)-order and higher-order terms in ⟨∇z¯g, ˜J∇zF⟩.
+Recall (3.14), i.e.,
+N = e(ξ) + ⟨ω(ξ), y⟩ + ˜h(y, ξ) + g(z, ξ) + ¯g(y, z, ξ),
+with
+˜h(y, ξ) =
+�
+2≤|i|
+˜h0i0(ξ)yi,
+g(z, ξ) =
+�
+2≤| j|
+g00 j(ξ)zj,
+¯g(y, z, ξ) =
+�
+2|i|+| j|≤m,1≤|i|,| j|
+¯g0i j(ξ)yizj.
+Notice that
+{N, F} = −∂yN∂xF + ∂xN∂yF + ∂zNJ∂zF
+= −
+√
+−1⟨k, ω + ∇˜h(y) + ∂y¯g⟩Fkı yıze
+√
+−1⟨k,x⟩ + (∂zg + ∂z¯g)J∂zF.
+(3.23)
+In order to simplify the notations, we sometimes omit the subscript of � and only use � to represent the sum to the
+index over the corresponding range of variation.
+Now we calculate the last term of (3.23):
+(∂zg + ∂z¯g)J∂zF = (∂zg + ∂z¯g)J∂zF|2|ı|+| |≤m + Q.
+(3.24)
+Specially,
+∂zgJ∂zF|2|ı|+| |≤m = ∂zgJ
+�
+0<|k|≤K+, 2|ı|+| |≤m
+Fkı yı∂z(z)e
+√
+−1⟨k,x⟩
++ ∂zgJ
+�
+0<|k|≤K+, 2|ı|+| |≤m
+∂z(Fkı )yıze
+√
+−1⟨k,x⟩
+=:
+�
+S ı Fkı yıze
+√
+−1⟨k,x⟩ +
+�
+2≤| j′|
+S j′∂zFkı(− j′+1)yıze
+√
+−1⟨k,x⟩
+=:
+�
+˜S ı Fkı yıze
+√
+−1⟨k,x⟩ +
+�
+(S ı  − ˜S ı )Fkı yıze
+√
+−1⟨k,x⟩
+(3.25)
++
+�
+2≤| j′|
+S j′∂zFkı(− j′+1)yıze
+√
+−1⟨k,x⟩,
+10
+
+where 0 < |k| ≤ K+, 2|ı| + || ≤ m, 2 ≤ |j′|, S ı  is a (|ı| + ||) order tensor and concerned with ∂2
+zg(z) and J, S j′ is
+concerned with ∂ j′
+z g(0) and J, ˜S ı  is a (|ı| + ||) order tensor and concerned with ∂2
+zg(0) and J. And
+∂z¯gJ∂zF|2|ı|+| |≤m =∂z¯gJ
+�
+0<|k|≤K+, 2|ı|+| |≤m
+Fkı yı∂z(z)e
+√
+−1⟨k,x⟩
++ ∂z¯gJ
+�
+0<|k|≤K+, 2|ı|+| |≤m
+∂z(Fkı )yıze
+√
+−1⟨k,x⟩
+=:
+�
+S i j(Fk(ı−i)(+2− j) + ∂zFk(ı−i)(+1− j))yıze
+√
+−1⟨k,x⟩
+(3.26)
+where S i j is concerned with ∂i
+y∂ j
+z ¯g(0, 0) and J, and 2|i| + |j| ≤ m, 1 ≤ |i|, 1 ≤ |j|, 0 < |k| ≤ K+, 2|ı| + || ≤ m.
+Substituting (3.24), (3.25), and (3.26) into (3.23), putting (3.22) and (3.23) into (3.21), comparing the coefficients,
+we thus obtain the following quasi-linear equations for all 0 < |k| ≤ K+, 2|ı| + || ≤ m:
+(
+√
+−1⟨k, ω + ∇˜h(y) + ∇y¯g⟩I(2d)ı+ − S ı )Fkı  = pkı  + S j′∂zFkı(− j′+1)
+(3.27)
++ S i jFk(ı−i)(+2− j) + S i j∂zFk(ı−i)(+1− j),
+where i, j, j′ are defined as above, S i jFk(ı−i)(+2− j) stands for �
+2|i|+| j|≤m,1≤|i|,| j| S i jFk(ı−i)(+2− j), and the rest terms are
+analogously defined.
+The above equations (3.27) are solvable if the coefficient matrices are nonsingular. We denote
+G+ ={ξ ∈ G : |⟨k, ω⟩| > γ
+|k|τ , A⊤ı Aı  > γ2
+|k|2τ I(2d)ı+, for k ∈ Zn,
+(3.28)
+0 < |k| < K+, 2|ı| + || ≤ m, 1 ≤ ||},
+with
+Aı  =
+√
+−1⟨ k
+|k|, ω⟩I(2d)ı+ +
+˜S ı 
+|k| ,
+(3.29)
+where ˜S ı  is concerned with ∂2g(0)
+∂z2
+and J. Then, we can solve equations (3.27) on G+. The details can be seen in the
+following lemma:
+Lemma 3.3. Assume that
+(H2) :
+max
+|ℓ|≤m,|i|≤m|∂ℓ
+ξ∂i
+y˜h − ∂ℓ
+ξ∂i
+y˜h0|D(s)×G+ ≤ s
+1
+2
+0 ,
+(H3) : 4s <
+γ − γ+
+(M∗ + 2)Kτ+1
++
+,
+where
+M∗ =
+max
+|ℓ|≤m,|i|≤m|∂ℓ
+ξ∂i
+y˜h0(y)|D(s)×G+.
+The quasi-linear equations (3.27) can be uniquely solved on D(s) × G+ to obtain the coefficient Fkı  which satisfy the
+following equality:
+|∂ℓ
+ξ∂i
+y∂ j
+zFkı |D(s)×G+ ≤ c2|k||ℓ|+|i|+| j|+(|ℓ|+|i|+| j|+1)(2d)τsm−2|ı|−| |µe−|k|r,
+for all 0 < |k| ≤ K+, 2|ı| + || ≤ m, |ℓ| + |i| + |j| ≤ m, where c2 is a constant.
+Proof. For ∀(y, ξ) ∈ D(s) × G+, by (H2), (H3),
+|∇˜h| = |(∇˜h − ∇˜h0) + ∇˜h0| ≤ (1 + M∗)|y| < (1 + M∗)s <
+γ
+4|k|τ+1 .
+(3.30)
+11
+
+Recalling that ¯g(y, z) comes from the perturbation and it is formulized in (3.15), we have
+|∇y¯g| ≤ cs <
+γ
+4|k|τ+1 ,
+(3.31)
+with c ≤ �ν−1
+i=0 γ(m+1)(2d)m
+i
+sm−3
+i
+µi ≪ 1, where the last equality follows from (H3). Notice by (3.30) and (3.31) that
+| det⟨k, ∇˜h + ∇y¯g⟩I(2d)ı+| ≤ ( γ
+4|k|τ)(2d)ı+, 2|ı| + || ≤ m.
+It follows from the definition of S ı , ˜S ı  and g that
+| det(S ı  − ˜S ı )| ≤ | det(|s|I(2d)ı+)| < ( γ
+4|k|τ )(2d)ı+, 2|ı| + || ≤ m.
+This together with (3.27) and (3.28) yields
+| det Bı | >
+γ(2d)ı+
+2(|k|τ)(2d)ı+ , 2|ı| + || ≤ m
+(3.32)
+where Bı  is the coefficient matrix of (3.27). Then, by (3.32),
+|Bı |−1 = | adjBı 
+det Bı 
+| ≤ c|k|τ(2d)+(2d)−1
+γ(2d)
+, 2|ı| + || ≤ m
+where adjBı  is the algebraic minor of Bı . Applying the identity
+∂ j
+zB−1
+ı  = −
+| j|
+�
+| j′|=1
+� j
+j′
+�
+(∂ j− j′
+z
+B−1
+ı  ∂ j′
+z Bı )B−1
+ı 
+inductively, we have for |ℓ| + |i| + |j| ≤ m
+|∂ℓ
+ξ∂i
+y∂ j
+zB−1
+ı  |D(s)×G+ ≤ c|k||ℓ|+|i|+| j||B−1
+ı  ||ℓ|+|i|+| j|+1
+≤ c|k||ℓ|+|i|+| j|+(|ℓ|+|i|+| j|+1)(2d)τ
+γ(|ℓ|+|i|+| j|+1)(2d)
+, 2|ı| + || ≤ m,
+which was precisely proved in [24, 25]. We note by the Cauchy estimate that for 2|ı| + || ≤ m,
+|∂ℓ
+ξpkı |G+ ≤ |∂ℓ
+ξP|D(s)×G+ s−2|ı|−| |e−|k|r ≤ γ(m+1)(2d)m sm−2|ı|−| |µe−|k|r.
+So, for |ℓ| + |i| + |j| ≤ m, 2|ı| + || ≤ m,
+|∂ℓ
+ξ∂i
+y∂ j
+zFkı |D(s)×G+ ≤ c|k||ℓ|+|i|+| j|+(|ℓ|+|i|+| j|+1)(2d)τ
+γ(|ℓ|+|i|+| j|+1)(2d)
+γ(m+1)(2d)m sm−2|ı|−| |µe−|k|r
+≤ c|k||ℓ|+|i|+| j|+(|ℓ|+|i|+| j|+1)(2d)τsm−2|ı|−| |µe−|k|r.
+The proof is complete.
+Next, we apply the above transformation φ1
+F to Hamiltonian H, i.e.,
+H ◦ φ1
+F = (N + R) ◦ φ1
+F + (P − R) ◦ φ1
+F
+= (N + R) + {N, F} +
+� 1
+0
+{(1 − t){N, F} + R, F} ◦ φt
+Fdt + (P − R) ◦ φ1
+F
+= N + [R] +
+� 1
+0
+{Rt, F} ◦ φt
+Fdt + (P − R) ◦ φ1
+F + Q
+12
+
+=: ¯N+ + ¯P+,
+where
+¯N+ = N + [R],
+¯P+ =
+� 1
+0
+{Rt, F} ◦ φt
+Fdt + (P − R) ◦ φ1
+F + Q,
+(3.33)
+Rt = (1 − t)Q + (1 − t)[R] + tR.
+3.2.5. Translation
+In this subsection, we will construct a translation so as to eliminate the first order terms about z.
+Consider the translation
+φ : x → x,
+y → y,
+z → z + ζ+ − ζ,
+where z = (u, v), ζ+ is to be determined. Let
+Φ+ = φ1
+F ◦ φ.
+Then
+H ◦ Φ+ = N+ + P+,
+N+ = ¯N+ ◦ φ,
+P+ = ¯P+ ◦ φ,
+(3.34)
+with
+N+ = ¯N+ ◦ φ = (N + [R]) ◦ φ = (e + ⟨ω, y⟩ + ˜h(y) + g(z) + ¯g(y, z) + [R](y, z)) ◦ φ
+= e + ⟨ω, y⟩ + ˜h(y) + g(z + ζ+ − ζ) + ¯g(y, z + ζ+ − ζ) + [R](y, z + ζ+ − ζ)
+=: e+ + ⟨ω+, y⟩ + ˜h+ + g+ + ¯g+,
+where
+e+ = e + g(ζ+ − ζ) + [R](0, ζ+ − ζ),
+(3.35)
+ω+ = ω + ∇y[R](0, ζ+ − ζ) + ∇y¯g(0, ζ+ − ζ),
+(3.36)
+˜h+ = ˜h(y) + [R](y, ζ+ − ζ) − [R](0, ζ+ − ζ) − ⟨∇y[R](0, ζ+ − ζ), y⟩
+(3.37)
++ ¯g(y, ζ+ − ζ) − ⟨∇y¯g(0, ζ+ − ζ), y⟩,
+g+ = g(z + ζ+ − ζ) − g(ζ+ − ζ) + [R](0, z + ζ+ − ζ) − [R](0, ζ+ − ζ),
+(3.38)
+¯g+ = ¯g(y, z + ζ+ − ζ) − ¯g(y, ζ+ − ζ) + [R](y, z + ζ+ − ζ) − [R](y, ζ+ − ζ)
+(3.39)
+− [R](0, z + ζ+ − ζ) + [R](0, ζ+ − ζ).
+3.2.6. Eliminate the first order terms about z
+In this subsection, we will appropriately choose ζ+ to remove the first order terms about z. The concrete details
+are expressed as the following lemma, which is crucial to our proof.
+Lemma 3.4. Notice that
+∇g+(0) = ∇g(ζ+ − ζ) + ∇z[R](0, ζ+ − ζ).
+(3.40)
+There exists ζ+ ∈ B(sm−1
+−
+µ−)
+1
+L (ζ) such that
+∇g+(0) = ∇g(0) = · · · = ∇g0(0) = 0.
+13
+
+Proof. The proof will be completed by induction on ν. We start with the case ν = 0. It follows from (2.3) and (A0)
+that
+∇g0(ζ0) = ∇g0(0) = 0,
+deg(∇g0(·) − ∇g0(0), Oo, 0) � 0,
+|∇g0(z) − ∇g0(z∗)| ≥ σ0|z − z∗|L.
+Now assume that for some ν ≥ 1 we have got
+∇gi(0) = ∇gi−1(ζi − ζi−1) + ∇z[Ri−1](0, ζi − ζi−1) = 0,
+deg(∇gi(·) − ∇gi(0), Oo, 0) � 0,
+(3.41)
+|∇gi(z) − ∇gi(z∗)| ≥ σi|z − z∗|L,
+(3.42)
+where i = 1, 2, · · · , ν, ζi ∈ B(sm−1
+i−2 µi−2)
+1
+L (ζi−1), s−1 = s0, µ−1 = µ0, z∗ ∈ O, z ∈ O\B(sm−1
+i−1 µi−1)
+1
+L (z∗). Then, we need to find ζ+
+near ζ such that ∇g+(0) = ∇g(0).
+Consider homotopy Ht(z) : [0, 1] × O → R2d,
+Ht(z) =: ∇g(z − ζ) − ∇g(0) + t∇z[R](0, z − ζ).
+Notice that
+|∇z[R](y, z)| ≤ γ(m+1)(2d)m sm−1µ.
+(3.43)
+For any z ∈ ∂O, t ∈ [0, 1], by (3.42) and (3.43), we have
+|Ht(z)| ≥ |∇g(z − ζ) − ∇g(0)| − |∇z[R](0, z − ζ)|
+≥ σ|z − ζ|L − γ(m+1)(2d)m sm−1µ
+> σδL
+2 ,
+where δ := min{|z − ζ|, ∀z ∈ ∂O}. So, it follows from the homotopy invariance and (3.41) that
+deg(H1(·), Oo, 0) = deg(H0(·), Oo, 0) � 0.
+(3.44)
+We note by (3.42) and (3.43) that for any z ∈ O\B(sm−1
+−
+µ−)
+1
+L (ζ),
+|H1(z)| = |∇g(z − ζ) − ∇g(0) + ∇z[R](0, z − ζ)|
+≥ |∇g(z − ζ) − ∇g(0)| − |∇z[R](0, z − ζ)|
+≥ σ|z − ζ|L − γ(m+1)(2d)m sm−1µ
+≥ σsm−1
+−
+µ− − γ(m+1)(2d)m sm−1µ
+≥ σ
+2 sm−1
+−
+µ−.
+Hence, by excision and (3.44),
+deg(H1(·), B(sm−1
+−
+µ−)
+1
+L (ζ), 0) = deg(H1(·), Oo, 0) � 0,
+then there exist at least a ζ+ ∈ B(sm−1
+−
+µ−)
+1
+L (ζ), such that
+H1(ζ+) = 0,
+i.e.,
+∇g(ζ+ − ζ) + ∇z[R](0, ζ+ − ζ) = ∇g(0),
+14
+
+thus, by (3.40),
+∇g+(0) = ∇g(0) = · · · = ∇g0(0) = 0.
+(3.45)
+Next, we need to prove
+deg(∇g+(·) − ∇g+(0), Oo, 0) � 0,
+(3.46)
+|∇g+(z) − ∇g+(z∗)| ≥ σ+|z − z∗|L.
+(3.47)
+By (3.38),
+∇g+(z) = ∇g(z + ζ+ − ζ) + ∇z[R](0, z + ζ+ − ζ).
+Then
+∇g+(z) − ∇g(z) = ∇g(z + ζ+ − ζ) − ∇g(z) + ∇z[R](0, z + ζ+ − ζ),
+(3.48)
+and
+∇g+(z) − ∇g+(z∗) = ∇g(z + ζ+ − ζ) − ∇g(z∗ + ζ+ − ζ)
+(3.49)
++ ∇z[R](0, z + ζ+ − ζ) − ∇z[R](0, z∗ + ζ+ − ζ).
+In view of (3.43), (3.48), and ζ+ ∈ B(sm−1
+−
+µ−)
+1
+L (ζ), we get
+|∇g+(z) − ∇g(z)| ≤ c(sm−1
+−
+µ−)
+1
+L ,
+so, this together with the property of degree, (3.41) and (3.45) that (3.46) holds, i.e.,
+deg(∇g+(·) − ∇g+(0), Oo, 0) = deg(∇g+(·) − ∇g(·) + ∇g(·) − ∇g(0), Oo, 0)
+= deg(∇g(·) − ∇g(0), Oo, 0) � 0.
+It follows from (3.42), (3.43) and (3.49) that for any z ∈ O\B(sm−1µ)
+1
+L (z∗)
+|∇g+(z) − ∇g+(z∗)| ≥ σ|z − z∗|L − 2γ(m+1)(2d)m sm−1µ ≥ σ+|z − z∗|L,
+which implies (3.47).
+The proof is complete.
+3.2.7. Estimate on N+
+Now, we give the estimate of N+.
+Lemma 3.5. There is a constant c3 such that for all |ℓ| ≤ m:
+|∂ℓ
+ξ(ζ+ − ζ)|G+ ≤ c3(sm−1
+−
+µ−)
+1
+L ,
+(3.50)
+|∂ℓ
+ξ(e+ − e)|G+ ≤ c3(sm−1
+−
+µ−)
+1
+L ,
+(3.51)
+|∂ℓ
+ξ(ω+ − ω)|G+ ≤ c3(sm−1
+−
+µ−)
+1
+L ,
+(3.52)
+|∂ℓ
+ξ(˜h+ − ˜h)|D(s+)×G+ ≤ c3(sm−1
+−
+µ−)
+1
+L ,
+(3.53)
+|∂ℓ
+ξ(g+ − g)|D(s+)×G+ ≤ c3(sm−1
+−
+µ−)
+1
+L ,
+(3.54)
+|∂ℓ
+ξ(¯g+ − ¯g)|D(s+)×G+ ≤ c3(sm−1
+−
+µ−)
+1
+L .
+(3.55)
+15
+
+Proof. Differentiating (3.40) with respect to ξ yields
+(∇2g(ζ+ − ζ) + ∇2
+z[R](0, ζ+ − ζ))∂ξ(ζ+ − ζ) = ∂ξ∇g(ζ+ − ζ) + ∂ξ∇z[R](0, ζ+ − ζ).
+Applying this identity inductively and using ζ+ ∈ B(sm−1
+−
+µ−)
+1
+L (ζ) in Lemma 3.4, we can immediately get (3.50). Recall
+e+ and ω+ in (3.35) and (3.36), respectively, we see that
+|∂ℓ
+ξ(e+ − e)|G+ ≤ c|∂ℓ
+ξ(ζ+ − ζ)|2 + γ(m+1)(2d)m smµ ≤ c(sm−1
+−
+µ−)
+1
+L ,
+|∂ℓ
+ξ(ω+ − ω)|G+ ≤ c|∂ℓ
+ξ(ζ+ − ζ)| + γ(m+1)(2d)m sm−2µ ≤ c(sm−1
+−
+µ−)
+1
+L ,
+which implies (3.51) and (3.52).
+In view of (3.37), we have that
+|∂ℓ
+ξ(˜h+(y) − ˜h(y))|D(s+) = |∂ℓ
+ξ([R](y, ζ+ − ζ) − [R](0, ζ+ − ζ) − ⟨∇y[R](0, ζ+ − ζ), y⟩)|
++ |∂ℓ
+ξ(¯g(yı(ζ+ − ζ) ) − ⟨∇y¯g((ζ+ − ζ) ), y⟩)|
+≤ γ(m+1)(2d)m smµ + (sm−1
+−
+µ−)
+1
+L s4
++
+≤ c(sm−1
+−
+µ−)
+1
+L .
+It follows from (3.38) that
+|∂ℓ
+ξ(g+(z) − g(z))|D(s+) = |∂ℓ
+ξ(g(z + ζ+ − ζ) − g(ζ+ − ζ) − g(z))|
++ |∂ℓ
+ξ([R](0, z + ζ+ − ζ) − [R](0, ζ+ − ζ))|
+≤ c(sm−1
+−
+µ−)
+1
+L s2
++ + γ(m+1)(2d)m smµ
+≤ c(sm−1
+−
+µ−)
+1
+L .
+According to (3.39), we obtain that
+|∂ℓ
+ξ(¯g+(yız) − ¯g(yız))|D(s+) ≤ |∂ℓ
+ξ[R](yız)| ≤ c4γ(m+1)(2d)m smµ ≤ c(sm−1
+−
+µ−)
+1
+L .
+3.2.8. Estimate on Φ+
+Recall that F is as in (3.20) with the coefficients and its estimate is given by Lemma 3.3. Then, we have the
+following estimate on F.
+Lemma 3.6. There is a constant c4 such that for all |ℓ| + |l| + |i| + |j| ≤ m,
+|∂ℓ
+ξ∂l
+x∂i
+y∂ j
+zF| ˆD(s)×G+ ≤
+� c4sm−2|i|−| j|µΓ(r − r+),
+2|i| + |j| ≤ m,
+c4µΓ(r − r+),
+m ≤ 2|i| + |j|.
+(3.56)
+Proof. Let
+a(i, j) =
+� sm−2|i|−| j|,
+if 2|i| + |j| ≤ m,
+0,
+otherwise.
+By (3.20) and Lemma 3.3, we have
+|∂ℓ
+ξ∂l
+x∂i
+y∂ j
+zF| ˆD(s)×G+ ≤
+�
+0<|k|≤K+,2|ı|+| |≤m
+|k|l|∂ℓ
+ξ∂i
+y∂ j
+z(Fkı yız)|e|k|(r++ 7
+8 (r−r+))
+≤
+�
+0<|k|≤K+
+c|k||l|+|ℓ|+|i|+| j|+(|ℓ|+|i|+| j|+1)(2d)τsa(i, j)µe−|k| r−r+
+8
+≤ csa(i, j)µΓ(r − r+).
+This proves (3.56). The proof is complete.
+16
+
+To obtain the symplectic transformation stated in Theorem 1, we need to extend the function F smoothly to the
+domain ˆD(β0) × G0.
+Lemma 3.7. Assume (H2)-(H3). Then F and ζ+ − ζ can be smoothly extended to functions of H¨older class
+Cm−1+̺,m−1+̺( ˆD(β0) × G0) and Cm−1+̺ respectively, where 0 < ̺ < 1 is fixed. Moreover, there is a constant c such that
+∥F∥Cm−1+̺,m−1+̺( ˆD(β0)×G0) ≤ cµΓ(r − r+),
+(3.57)
+∥ζ+ − ζ∥Cm−1+̺(G0) ≤ c(sm−1
+−
+µ−)
+1
+L ,
+(3.58)
+where ∥F∥Cm−1+̺,m−1+̺( ˆD(β0)×G0) =: |∂ℓ
+ξ∂l
+x∂i
+y∂ j
+zF| ˆD(β0)×G0, |ℓ| ≤ m − 1 + ̺, |l| + |i| + |j| ≤ m − 1 + ̺.
+Proof. It follows from the standard Whitney extension theorem that F and ζ+−ζ can be extended smoothly to functions
+˜F and
+˜
+(ζ+ − ζ) of H¨older class Cm−1+̺,m−1+̺( ˆD(β0 × G0)), Cm−1+̺(G0), respectively, such that
+∥ ˜F∥m−1+̺,m−1+̺
+C
+( ˆD(β0 × G0)) ≤ c∥F∥Cm,m( ˆD(s)×G+),
+∥
+˜
+ζ+ − ζ∥Cm−1+̺(G0) ≤ c∥ζ+ − ζ∥Cm(G+),
+where c is a constant depending only on m, ̺ and the dimensions n, d. The lemma now follows from (3.56) in Lemma
+3.6 and (3.50) in Lemma 3.5.
+Lemma 3.8. In addition to (H2)-(H3), assume that
+(H4) : c3(sm−1
+−
+µ−)
+1
+L < 1
+8αs,
+(H5) : c4µΓ(r − r+) < 1
+4(r − r+),
+(H6) : c4sm−1µΓ(r − r+) < 1
+8αs,
+(H7) : c3µΓ(r − r+) + c3(sm−1µ)
+1
+L < β − β+.
+Then the following hold:
+(1) For all 0 ≤ t ≤ 1,
+φt
+F : D 1
+4 α → D 1
+2 α,
+(3.59)
+φ : D 1
+8 α → D 1
+4 α,
+(3.60)
+are well defined.
+(2) Let Φ+ = φ1
+F ◦ φ. Then for all ξ ∈ G+,
+Φ+ : D+ → D,
+˜D+ → D(β, r).
+(3.61)
+(3) There is a constant c5 such that
+|∂ℓ
+ξDi
+(x,y,z)(φt
+F − id)|D α
+4 ×G+ ≤ c5µΓ(r − r+),
+|ℓ| + |i| ≤ m,
+where Di
+(x,y,z)F stands for the |i|-order partial derivative with respect to (x, y, z).
+(4)
+|∂ℓ
+ξDi
+(x,y,z)(Φ+ − id)| ˜D×G+ ≤ c5µΓ(r − r+),
+|ℓ| + |i| ≤ m.
+17
+
+Proof. (1) For ∀(x, y, z) ∈ D 1
+8 α, we note by ζ+ ∈ B(sm−1
+−
+µ−)
+1
+L (ζ) and (H4) that
+|z + ζ+ − ζ| < |z| + |ζ+ − ζ| < 1
+8αs + c(sm−1
+−
+µ−)
+1
+L < 1
+4αs,
+which implies (3.60).
+To verify (3.59), we denote φt
+F1, φt
+F2, φt
+F3 as the components of φt
+F in the x, y, z coordinates, respectively. Let
+XF = (Fy, −Fx, ˜JFz)⊤ be the vector field generated by F. Then
+φt
+F = id +
+� t
+0
+XF ◦ φλ
+Fdλ,
+0 ≤ t ≤ 1.
+(3.62)
+For any (x, y, z) ∈ D 1
+4 α, we let t∗ = sup{t ∈ [0, 1] : φt
+F(x, y, z) ∈ Dα}. Then for any 0 ≤ t ≤ t∗, in view of (x, y, z) ∈ D 1
+4 α,
+(3.56) in Lemma 3.6, (H5) and (H6)
+|φt
+F1(x, y, z)|D 1
+4 α ≤ |x| +
+� t
+0
+|Fy ◦ φλ
+F|D 1
+4 αdλ
+≤ r+ + 1
+8(r − r+) + c4sm−2µΓ(r − r+)
+< r+ + 3
+8(r − r+),
+|φt
+F2(x, y, z)|D 1
+4 α ≤ |y| +
+� t
+0
+| − Fx ◦ φλ
+F|D 1
+4 αdλ
+≤ 1
+4αs + c4smµΓ(r − r+)
+< 3
+8αs,
+|φt
+F3(x, y, z)|D 1
+4 α ≤ |z| +
+� t
+0
+| ˜JFz ◦ φλ
+F|D 1
+4 αdλ
+≤ 1
+4αs + c4sm−1µΓ(r − r+)
+< 3
+8αs.
+Thus, φt
+F ∈ D 1
+2 α ⊂ Dα, i.e. t∗ = 1 and (1) holds.
+(2) Using (H5), (H7), (3.57) and a similar argument as above, one sees that φt
+F : ¯D+ → D(β, r) is well defined for
+all 0 ≤ t ≤ 1, where ¯D+ = D(r+ + 5
+8(r − r+), β+ + c(sm−1
+−
+µ−)
+1
+L ). Thus, Φ+ : ˜D+ → D(β, r) is also well defined.
+(3) Note that
+|∂ℓ
+ξXF| ˜D×G+ ≤ c|∂ℓ
+ξD(x,y,x)F| ˜D×G+.
+(3.63)
+Using (3.56) in Lemma 3.6 and (3.62), we immediately have
+|φt
+F − id|D α
+4 ×G+ ≤ cµΓ(r − r+).
+Differentiating (3.62) yields
+D(x,y,z)φt
+F = I2n+2d +
+� t
+0
+(D(x,y,z)XF)D(x,y,z)φλ
+Fdλ
+= I2n+2d +
+� t
+0
+J(D2
+(x,y,z)F)D(x,y,z)φλ
+Fdλ,
+0 ≤ t ≤ 1.
+18
+
+It follows from Lemma 3.6, (3.62) and Gronwall Inequality that
+|D(x,y,z)φt
+F − I2n+2d|D α
+4 ×G+ ≤ |
+� t
+0
+D(x,y,z)XF ◦ φλ
+FD(x,y,z)φλ
+Fdλ|D α
+4 ×G+
+≤
+� t
+0
+|D(x,y,z)XF ◦ φλ
+F||D(x,y,z)φλ
+F − I2n+2d| ˆD(s)×G+dλ
++
+� t
+0
+|D(x,y,z)XF ◦ φλ
+F| ˆD(s)×G+dλ
+≤ cµΓ(r − r+).
+Similarly, by using (3.56) in Lemma 3.6, Gronwall’s inequality and (3.63) inductively, we have
+|∂ℓ
+ξDi
+(x,y,z)φt
+F|D α
+4 ×G+ ≤ cµΓ(r − r+),
+|ℓ| + |i| ≤ m.
+(4) Observe that
+Φ+ − id = (φ1
+F − id) ◦ φ +
+
+0
+0
+ζ+ − ζ
+ .
+Then,
+|∂ℓ
+ξDi
+(x,y,z)(Φ+ − id)| ˜D×G+ ≤ |∂ℓ
+ξDi
+(x,y,z)(φ1
+F − id)| ˜D×G+|∂ℓ
+ξDi
+(x,y,z)φ| ˜D×G+
++ |∂ℓ
+ξ(ζ+ − ζ)| ˜D×G+
+≤ cµΓ(r − r+)(sm−1
+−
+µ−)
+1
+L + (sm−1
+−
+µ−)
+1
+L
+≤ cµΓ(r − r+)αs + αs
+≤ cµΓ(r − r+),
+where the second equality follows from (3.57) and (3.58) in Lemma 3.7, the third equality follows from (H4), and the
+last equality follows from the definition of µ and s.
+The proof is complete.
+3.2.9. Frequency Property
+To estimate the new frequency, we first assume that
+(H8) : 3sK2τ+1
++
+≤ {γ − γ+
+γ0
+, γ2 − γ2
++
+γ2
+0
+}.
+It is obvious that for all 0 < |k| ≤ K+, ξ ∈ G+,
+|⟨k, ∇y[R](0, ζ+ − ζ)⟩| ≤ cK+γ(m+1)(2d)m sm−2µ,
+|⟨k, ∇y¯g((ζ+ − ζ) )⟩| ≤ cK+γ(m+1)(2d)m(sm−1
+−
+µ−)
+1
+L ≤ cK+γ(m+1)(2d)mαs.
+Then this together with (3.36) and (H8) yields
+|⟨k, ω+⟩| ≥ |⟨k, ω⟩| − |⟨k, ∇y[R](0, ζ+ − ζ) + ∇y¯g((ζ+ − ζ) )⟩|
+≥ γ
+|k|τ − γ − γ+
+|k|τ
+= γ+
+|k|τ .
+Recall that
+A+
+ı  =
+√
+−1⟨ k
+|k|, ω+⟩I(2d)ı+ +
+˜S +
+ı 
+|k| = Aı  +
+√
+−1⟨ k
+|k|, ω+ − ω⟩I(2d)ı+ +
+˜S +
+ı  − ˜S ı 
+|k|
+=: Aı  + ˜Aı ,
+19
+
+where
+Aı  =
+√
+−1⟨ k
+|k|, ω⟩I(2d)ı+ +
+˜S ı 
+|k| ,
+˜Aı  =
+√
+−1⟨ k
+|k|, ω+ − ω⟩I(2d)ı+ +
+˜S +
+ı  − ˜S ı 
+|k|
+.
+Then,
+A+⊤
+ı  A+
+ı  = A⊤ı Aı  + A⊤ı  ˜Aı  + ˜A⊤ı Aı  + ˜A⊤ı  ˜Aı .
+(3.64)
+By (3.50) in Lemma (3.5) , we have
+|
+√
+−1⟨ k
+|k|, ω+ − ω⟩| ≤ (sm−1
+−
+µ−)
+1
+L ≤ cαs.
+(3.65)
+Note that ˜S +
+ı  − ˜S ı  is concerned with ∂2
+zg+(0) − ∂2
+zg(0), then, by (3.38), (3.18) in Lemma 3.2,
+| ˜S +
+ı  − ˜S ı |
+|k|
+≤ cγ(m+1)(2d)ı+ sm−2µ.
+(3.66)
+Thus, by (3.64), (3.65), (3.66) and (H8),
+A+⊤
+ı  A+
+ı  ≥ γ2
+|k|2τ I(2d)ı+ − γ2 − γ2
++
+3|k|2τ I(2d)ı+ − γ2 − γ2
++
+3|k|2τ I(2d)ı+ − γ2 − γ2
++
+3|k|2τ I(2d)ı+ ≥ γ2
++
+|k|2τ I(2d)ı+.
+3.2.10. Estimate on P+
+In the following, we estimate the next step P+.
+Lemma 3.9. Denote
+∆ = αm+1sm+1µ(sm−2µΓ2(r − r+) + Γ(r − r+)) + γ(m+1)(2d)m s2m−2µ2Γ(r − r+)
++ γ(m+1)(2d)m sm+1µc.
+Assume (H0)-(H8). Then there is a constant c6 such that for all |ℓ| ≤ m
+|∂ℓ
+ξ ¯P+|D+×G+ ≤ c6∆.
+(3.67)
+Moreover, if
+(H9) : c6∆ ≤ γ(m+1)(2d)m
++
+sm
++µ+,
+then
+|∂ℓ
+ξP+|D+×G+ ≤ γ(m+1)(2d)m
++
+sm
++µ+.
+(3.68)
+Proof. Recall the definition of Q as in (3.22). Observe by (3.39) and (3.56) that
+|∂ℓ
+ξQ|D 1
+4 α×G+ ≤ c(αs)m+1µΓ(r − r+)
+(3.69)
+and
+|∂ℓ
+ξ{Q, F}|D 1
+4 α×G+ ≤ cαm+1sm+1sm−2µ2Γ2(r − r+) + cαm+1sm−1smµ2Γ2(r − r+)
++ cαm+1smsm−1µ2Γ2(r − r+)
+≤ cαm+1s2m−1µ2Γ2(r − r+).
+(3.70)
+20
+
+Using (3.18) in Lemma 3.2 and (3.56) in Lemma 3.6, we also have
+|∂ℓ
+ξ[R]| ˆD×G+ + |∂ℓ
+ξR| ˆD×G+ ≤ γ(m+1)(2d)m smµ,
+and
+|∂ℓ
+ξ{[R], F}|D 1
+4 α×G+ + |∂ℓ
+ξ{R, F}|D 1
+4 α×G+ ≤ cγ(m+1)(2d)m s2m−2µ2Γ(r − r+).
+(3.71)
+Let us recall the definition of ¯P+ in (3.33), that is
+¯P+ =
+� 1
+0
+{Rt, F} ◦ φt
+Fdt + (P − R) ◦ φ1
+F + Q,
+Rt = (1 − t)Q + (1 − t)[R] + tR.
+Then by (3.17), (3.69), (3.70) and (3.71),
+|∂ℓ
+ξ ¯P+|D 1
+4 α×G+ ≤ cαm+1s2m−1µ2Γ2(r − r+) + cγ(m+1)(2d)m s2m−2µ2Γ(r − r+)
++ cγ(m+1)(2d)m sm+1µ + c(αs)m+1µΓ(r − r+)
+≤ αm+1sm+1µ(sm−2µΓ2(r − r+) + Γ(r − r+))
++ γ(m+1)(2d)m sm+1µ(sm−3µΓ(r − r+) + c),
+which implies (3.67).
+Moreover, by (H9) and the definition of P+ as in (3.34), we see
+|∂ℓ
+ξP+|D+×G+ ≤ γ(m+1)(2d)m
++
+sm
++µ+.
+The proof is complete.
+This completes one cycle of KAM steps.
+4. Proof of Theorem 1
+4.1. Iteration Lemma
+In this section, we will prove an iteration lemma which guarantees the inductive construction of the transformations
+in all KAM steps.
+Let r0, s0, γ0, µ0, H0, N0, P0 be given at the beginning of Section 3 and let D0 = D(s0, r0), ˜D0 = D(β0, r0), K0 = 0,
+Φ0 = id. We define the following sequence inductively for all ν = 1, 2, · · ·.
+rν = r0(1
+2 +
+1
+2ν+1 ),
+βν = β0(1
+2 +
+1
+2ν+1 ),
+γν = γ0(1
+2 +
+1
+2ν+1 ),
+sν = 1
+8αν−1sν−1,
+αν = s2ρ
+ν = s
+1
+m+1
+ν
+,
+µν = 8mc0µν−1sρ
+ν−1,
+Kν = ([log( 1
+sν−1
+)] + 1)3η,
+Dν = D(sν, rν),
+˜Dν = D(βν, rν + 5
+8(rν−1 − rν)),
+21
+
+D(sν) = {(y, z) : |y| < s2
+ν, |z| < sν},
+Gν = {ξ ∈ Gν−1 : |⟨k, ων−1⟩| > γ
+|k|τ , Aν−1⊤
+ı 
+Aν−1
+ı 
+> γ2
+|k|2τ I(2d)ı+, for k ∈ Zn, 0 < |k| < Kν,
+2|ı| + || ≤ m, 1 ≤ ||}, where Aν−1
+ı 
+is defined as in (3.29).
+Lemma 4.1. Denote
+µ∗ = s2ε
+1
+4(m+1) γ2(m+1)(2d)m
+0
+.
+If ε is small enough, then the KAM step described in Section 3 is valid for all ν = 0, 1, · · ·, resulting in the sequences
+eν, ων, ˜hν, gν, ¯gν, Hν, Pν, Φν,
+ν = 1, 2, · · · , with the following properties:
+(1) For all |ℓ| ≤ m,
+|∂ℓ
+ξ(eν+1 − eν)|D(sν+1)×Gν+1 ≤ µ
+1
+2∗
+2ν ,
+(4.72)
+|∂ℓ
+ξ(eν − e0)|D(sν)×Gν ≤ 2µ
+1
+2∗ ,
+(4.73)
+|∂ℓ
+ξ(ων+1 − ων)|D(sν+1)×Gν+1 ≤ µ
+1
+2∗
+2ν ,
+(4.74)
+|∂ℓ
+ξ(ων − ω0)|D(sν)×Gν ≤ 2µ
+1
+2∗ ,
+(4.75)
+|∂ℓ
+ξ(˜hν+1 − ˜hν)|D(sν+1)×Gν+1 ≤ µ
+1
+2∗
+2ν ,
+(4.76)
+|∂ℓ
+ξ(˜hν − ˜h0)|D(sν)×Gν ≤ 2µ
+1
+2∗ ,
+(4.77)
+|∂ℓ
+ξ(gν+1 − gν)|D(sν+1)×Gν+1 ≤ µ
+1
+2∗
+2ν ,
+(4.78)
+|∂ℓ
+ξ(gν − g0)|D(sν)×Gν ≤ 2µ
+1
+2∗ ,
+(4.79)
+|∂ℓ
+ξ(¯gν+1 − ¯gν)|D(sν+1)×Gν+1 ≤ µ
+1
+2∗
+2ν ,
+(4.80)
+|∂ℓ
+ξ(¯gν − ¯g0)|D(sν)×Gν ≤ 2µ
+1
+2∗ ,
+(4.81)
+|∂ℓ
+ξ(ζν+1 − ζν)|Gν+1 ≤ (sm−1
+ν−1 µν−1)
+1
+L ,
+(4.82)
+|∂ℓ
+ξPν|Dν×Gν ≤ γ(m+1)(2d)m
+ν
+sm
+ν µν.
+(4.83)
+(2) Gν+1 = {ξ ∈ Gν : |⟨k, ων⟩| > γν
+|k|τ , Aν⊤
+ı  Aν
+ı  >
+γ2
+ν
+|k|2τ I(2d)ı+, for k ∈ Zn, Kν < |k| ≤ Kν+1, 2|ı| + || ≤ m, 1 ≤ ||}.
+(3) There exists a family of symplectic coordinate transformations Φν+1 : ˜Dν+1×Gν+1 → ˜Dν and a subset Gν+1 ⊂ Gν,
+Gν+1 = Gν \
+�
+Kν<|k|≤Kν+1
+Rν+1
+k
+(γν),
+(4.84)
+where
+Rν+1
+k
+(γν) ={ξ ∈ Gν : |⟨k, ων⟩| ≤ γν
+|k|τ , Aν⊤
+ı  Aν
+ı  ≤ γ2
+ν
+|k|2τ I(2d)ı+,
+for k ∈ Zn, Kν < |k| ≤ Kν+1, 2|ı| + || ≤ m, 1 ≤ ||},
+22
+
+such that on ˜Dν+1 × Gν+1
+Hν+1 = Hν ◦ Φν+1 = Nν+1 + Pν+1,
+and
+|∂ℓ
+ξ∂i
+(x,y,z)(Φν+1 − id)| ˜Dν+1×Gν+1 ≤ µ
+1
+2∗
+2ν .
+(4.85)
+Proof. The proof amounts to the verification of (H0)-(H9) for all ν. For simplicity, we let r0 = 1. We can make s0
+small by picking ε0 sufficiently small. So, we see that (H0) holds and (H1)-(H9) hold for ν = 0.
+Note that
+µν = (8mc0)ν(1
+8)
+(m+2)((1+
+1
+m+1 )ν−1)−(ν−1)
+2
+s
+1
+2 ((1+
+1
+m+1 )ν−1)
+0
+µ0,
+(4.86)
+sν = (1
+8)(m+1)((1+
+1
+m+1 )ν−1)s
+(1+
+1
+m+1 )ν
+0
+.
+(4.87)
+Let θ ≫ 1 be fixed and s0 be small enough so that
+s0 < (8
+θ )
+1
+2ρ < 1.
+(4.88)
+Then
+s1 = 1
+8 s1+2ρ
+0
+< 1
+θ s0 < 1,
+s2 = 1
+8 s1+2ρ
+1
+< 1
+θ s1 < 1
+θ2 s0,
+...
+sν = 1
+8 s1+2ρ
+ν−1 < · · · < 1
+θν s0.
+(4.89)
+Denote
+Γν = Γ(rν − rν+1).
+We notice that
+rν − rν+1
+r0
+=
+1
+2ν+2 .
+(4.90)
+Since
+Γν ≤
+� ∞
+1
+t(m+1)(2d)mτ+me−
+t
+2ν+5 dt
+≤ ((m + 1)(2d)mτ + m)!2(ν+5)((m+1)(2d)mτ+m),
+it is obvious that for θ large enough,
+s
+ρ
+2ν Γi
+ν < s
+ρ
+2ν (Γi
+ν + Γν) ≤ 1,
+i = 1, 2,
+(4.91)
+and
+sνΓν ≤ s1−ρ
+ν
+≤
+s1−ρ
+0
+θ(1−ρ)ν .
+Recall αν = s
+1
+m+1
+ν
+, then αm+1
+ν
+= sν. In view of s0 = sε
+1
+8(m+1) γ(m+1)(2d)m
+0
+, we have γ(m+1)(2d)m
+0
+> s0. It is obvious by the
+definition of γν and (4.87) that γ(m+1)(2d)m
+ν
+> sν if ε0 is small enough. Moreover,
+γ(m+1)(2d)m
+ν
+s
+1
+4(m+1)
+ν
+= (1
+2 +
+1
+2ν+1 )(m+1)(2d)m(1
+8)
+1
+4 ((1+
+1
+m+1 )ν−1)s
+1
+4(m+1) (1+
+1
+m+1 )ν
+0
+23
+
+≤ (1
+2 +
+1
+2ν+2 )(m+1)(2d)m = γ(m+1)(2d)m
+ν+1
+as s0 small enough. This together with the definition of ∆ν as in Lemma 3.9 yields
+∆ν = αm+1
+ν
+sm+1
+ν
+µν(sm−2
+ν
+µνΓ2(rν − rν+1) + Γ(rν − rν+1))
++ γ(m+1)(2d)m
+ν
+s2m−2
+ν
+µ2
+νΓ(rν − rν+1) + γ(m+1)(2d)m
+ν
+sm+1
+ν
+µνc
+≤ γ(m+1)(2d)m
+ν
+s
+1
+4(m+1)
+ν
+s
+(1+
+1
+m+1 )m
+ν
+s
+1
+2(m+1)
+ν
+µν(s
+1
+4(m+1)
+ν
+sm−2
+ν
+µνΓ2(rν − rν+1)
++ s
+1
+4(m+1)
+ν
+Γ(rν − rν+1) + s
+1
+4(m+1)
+ν
+sm−3
+ν
+µΓ(rν − rν+1) + s
+1
+4(m+1)
+ν
+c)
+≤ γ(m+1)(2d)m
+ν+1
+sm
+ν+1µν+1,
+which implies that (H9) holds for all ν ≥ 1.
+We note that for any constant a > 0, b > 1, sa(log 1
+s + 1)b → 0 as s → 0, then
+3γ2
+0sνKτ+1
+ν+1 ≤ 3γ2
+0sν([log 1
+sν
+] + 1)3η(2τ+1)
+≤ (1
+2 +
+1
+2ν+1 )2 − (1
+2 +
+1
+2ν+2 )2
+= γ2
+ν − γ2
+ν+1,
+if ε0 is small enough. Hence, (H8) holds.
+By (4.90) and (4.91), it is easy to verify that (H5)-(H7) hold for all ν ≥ 1 if θ is large enough and ε0 is small
+enough.
+It is obvious by m ≥ L+
+√
+L2+16L+16
+4
+, L ≥ 2 as in (3.7) that
+m − 1
+L
+> 2m + 2
+m + 1 > (m + 2
+m + 1)2,
+then
+s
+m−1
+L
+−
+< s
+( m+2
+m+1 )2
+−
+,
+i.e.,
+(sm−1
+−
+µ−)
+1
+L < (1
+8)1+ m+2
+m+1 s
+( m+2
+m+1 )2
+−
+= 1
+8 s
+m+2
+m+1 = 1
+8αs,
+which implies (H4).
+To verify (H3), we observe by (4.87) and (4.89) that
+4(M∗ + 2)sνKτ+1
+ν+1 <
+s0
+2ν+2 < γ0
+2ν+2 ,
+as θ is large enough, which verifies (H3) for all ν ≥ 1.
+Let θ1−ρ ≥ 2 in (4.88), (4.89). We have that for all ν ≥ 1
+sν ≤ s0
+2ν ≤ µ
+1
+2∗
+2ν .
+(4.92)
+The verification of (H2) follows from (4.92) and an inductive application of (3.53) in Lemma 3.5 for all ν = 0, 1, · · · .
+Since (1 + ρ)η > 2, we have
+r0
+2ν+6 ([log 1
+s ] + 1)η ≥
+r0
+2ν+6 (−(1 + ρ)ν log s0)η
+≥ − r0
+2ν+6 (1 + ρ)ην(log µ0)η
+≥ 1.
+24
+
+It follows from the above that
+log(n + 1)! + (ν + 6)n log 2 + 3nη log([log 1
+s ] + 1) −
+r0
+2(ν+6) ([log 1
+s ] + 1)3η
+≤ log(n + 1)! + (ν + 6)n log 2 + 3nη log(log 1
+s + 2) − (log 1
+s )2η
+≤ − log 1
+s,
+as µ is small, which is ensured by making ε small. Thus,
+� ∞
+Kν+1
+tne−
+tr0
+2ν+6 dt ≤ (n + 1)!2(ν+6)nKn
+ν+1e−
+Kν+1
+2ν+6 ≤ s,
+i.e. (H1) holds.
+Above all, the KAM steps described in Section 3 are valid for all ν, which gives the desired sequences stated in
+the lemma.
+Now, (4.72), (4.74), (4.76), (4.78) and (4.80) follow from (3.53), (3.54) and (3.55) in Lemma 3.5 and (4.92);
+by adding up (4.72), (3.52), (4.76), (4.78) and (4.80) for all ν = 0, 1, · · ·, we can get (4.73), (4.75) , (4.77), (4.79)
+and (4.81) respectively; (4.83) follows from (3.68) in Lemma 3.9; (4.82) follows from Lemma 3.5; (2) follows from
+Lemma 3.8.
+Note that (2) automatically holds for ν = 0. We now let ν > 0. By Subsubsection 3.2.9, it is obvious that
+Gν = {ξ ∈ Gν : |⟨k, ων⟩| > γν
+|k|τ , Aν⊤
+ı  Aν
+ı  > γ2
+ν
+|k|2τ I(2d)ı+, for k ∈ Zn, 0 < |k| ≤ Kν, 2|ı| + || ≤ m, 1 ≤ ||}.
+Denote
+ˆGν+1 = {ξ ∈ Gν : |⟨k, ων⟩| > γν
+|k|τ , Aν⊤
+ı  Aν
+ı  > γ2
+ν
+|k|2τ I(2d)ı+, for k ∈ Zn, Kν < |k| ≤ Kν+1, 2|ı| + || ≤ m, 1 ≤ ||}.
+Then
+Gν+1 ={ξ ∈ Gν : |⟨k, ων⟩| > γν
+|k|τ , Aν⊤
+ı  Aν
+ı  > γ2
+ν
+|k|2τ I(2d)ı+, for k ∈ Zn, 0 < |k| ≤ Kν+1, 2|ı| + || ≤ m, 1 ≤ ||}
+=Gν ∩ ˆGν+1 = ˆGν+1,
+which implies (4.84).
+The proof is complete.
+4.2. Convergence
+The convergence is standard. For the sake of completeness, we briefly give the outline of the proof. Let
+Ψν = Φ0 ◦ Φ2 ◦ · · · ◦ Φν.
+Recalling that Φν+1 : ˜Dν+1 → ˜Dν in Lemma 4.1, we have
+Ψν : ˜Dν → ˜D0,
+H0 ◦ Ψν = Hν = Nν + Pν,
+ν = 0, 1, · · · , where Ψ0 = id. Let G∗ = �∞
+ν=0 Gν. First, we show the uniform convergence of Ψν on D( β0
+2 , r0
+2 ) × G∗.
+Note that
+Ψν = id +
+ν
+�
+i=1
+(Ψi − Ψi−1),
+25
+
+where, for each i = 1, 2, · · ·,
+Ψi − Ψi−1 = Φ0 ◦ · · · ◦ Φi − Φ0 ◦ · · · ◦ Φi−1
+≤
+� 1
+0
+D(Φ0 ◦ · · · ◦ Φi−1)(id + θ(Φi − id))dθ(Φi − id).
+Since, by (4.85), on D( β0
+2 , r0
+2 ) × G∗,
+|D(Φ0 ◦ · · · ◦ Φi−1)(id + θ(Φi − id))|
+≤ |DΦ0(Φ1 ◦ · · · ◦ Φi−1)(id + θ(Φi − id))| · · ·|DΦi−1(id + θ(Φi − id))|
+≤ (1 + µ
+1
+2∗ )(1 + µ
+1
+2∗
+2 ) · · ·(1 + µ
+1
+2∗
+2i−1 ) ≤ e1+ 1
+2 +···+
+1
+2i−1 ≤ e2,
+we have
+|Ψi − Ψi−1|D( β0
+2 , r0
+2 )×G∗ ≤ e2|Φi − id|D( β0
+2 , r0
+2 )×G∗ ≤ e2 µ
+1
+2∗
+2i ,
+for all i = 1, 2, · · · . Thus, Ψν converges uniformly on D( β0
+2 , r0
+2 ) × G∗. We denote its limit by Ψ∗. Then
+Ψ∗ = Ψ0 +
+∞
+�
+i=1
+(Ψi − Ψi−1),
+(4.93)
+which is uniformly close to the identity. By a standard argument using the Whitney Extension Theorem, one can
+further show that Ψ∗ is Whitney smooth with respect to ξ ∈ G∗ (see [18], [24] ,[25], [32] for details). By Lemma
+4.1,we see that eν, ων, ˜hν, gν, ¯gν and ζν are uniformly convergent and denote the limits by e∗, ω∗, ˜h∗, g∗, ¯g∗ and ζ∗,
+respectively. It follows from Lemma 3.4 that
+∇g∗(0) = · · · = ∇gν(0) = · · · = ∇g0(ζ0) = 0.
+Then, Nν converge uniformly to
+N∗ = e∗ + ⟨ω∗, y⟩ + ˜h∗(y) + g∗(z) + ¯g∗(yız),
+with
+˜h∗(y) = O(|y|2),
+g∗(z) = g(z) +
+∞
+�
+i=0
+γ(m+1)(2d)m
+i
+sm−2
+i
+µiO(|z|2)
+= g(z) + ε
+(m−2)(5m+4)
+8m(m+1) O(|z|2),
+(4.94)
+¯g∗(y, z, ξ) =
+�
+2|ı|+| |≤m,1≤|ı|,| |
+¯gı (ξ)yız,
+where (4.94) follows from (4.92) and the definition of µ∗ in Lemma 4.1.
+Hence
+Pν = H0 ◦ Ψν − Nν
+converges uniformly to
+P∗ = H0 ◦ Ψ∗ − N∗.
+Since
+|Pν|Dν ≤ γ(m+1)(2d)m
+ν
+sm
+ν µν,
+26
+
+the Cauchy estimate implies that
+|∂i
+y∂ j
+zPν|D( sν
+2 ,rν) ≤ γ(m+1)(2d)m
+ν
+s(m−2i− j)
+ν
+µν,
+for all 2|i| + |j| ≤ m. By (4.86) and (4.87), it is easy to see that the right hand side of the above converges to 0 as
+ν → ∞. Hence, on D(0, r0
+2 ) × G∗,
+∂ j
+y∂ j
+zP∗|(y,z)=(0,0) = 0,
+for all x ∈ T n, ω ∈ G∗, 2|i| + |j| ≤ m. It follows that for each ξ ∈ G∗, T n × {0} × {0} is an analytic, quasi-periodic,
+invariant n-torus associated to the Hamiltonian H∞ = H ◦ Ψ∞ with frequency ω∗.
+4.3. Measure estimate
+Lemma 4.2.
+|G0\G∗| → 0, as ε → 0.
+Proof. Since all the eigenvalues of normal direction are 0 in 0-th step, the measure estimate of G1 in the first step
+can be reduced to the estimate of ⟨k, ω⟩. Moreover, if ˜S ν
+ı  ≡ 0, the measure estimate of Gν in every step also can be
+reduced to the estimate of ⟨k, ω⟩. For details, see [8, 24]. Otherwise, if ˜S ν
+ı  � 0, the measure estimate of Gν is referred
+to [35]. For the sake of completeness, we give the outline of the proof.
+Recall
+Rν+1
+k
+(γν) ={ξ ∈ Gν : |⟨k, ων⟩| ≤ γν
+|k|τ , Aν⊤
+ı  Aν
+ı  ≤ γ2
+ν
+|k|2τ I(2d)ı+,
+for k ∈ Zn, Kν < |k| ≤ Kν+1, 2|ı| + || ≤ m, 1 ≤ ||}
+⊂S1 ∪ S2,
+where
+S1 = {ξ ∈ Gν : |⟨k, ων⟩| ≤ γν
+|k|τ },
+S2 = {ξ ∈ Gν : Aν⊤
+ı  Aν
+ı  ≤ γ2
+ν
+|k|2τ I(2d)ı+}.
+With Taylor series,
+Aı (ξ) = L(ξ0)Θ(ξ),
+where
+L(ξ0) = (Aı (ξ0), ∂ξAı (ξ0), · · · , ∂i
+ξAı (ξ0),
+� 1
+0
+(1 − t)|i+1|∂i+1
+ξ Aı (ξ0 + tˆξ)dt),
+Θ(ξ) = (I, ˆξI, · · · , ˆξiI, ˆξi+1I)⊤,
+ˆξ = ξ − ξ0.
+Noting that ˜S ı  comes from the perturbation and (A1), we have
+rank{C j : 1 ≤ j ≤ (2d)ı+ } = (2d)ı+ ,
+(4.95)
+where A j
+ı  is the j-th column of Aı , C j is a column of {∂i
+ξA j
+ı  : 0 ≤ |i| ≤ M} and Aı  is defined as in (3.29). By (4.95)
+and the continuity of determinant, there is an orthogonal matrix Qξ0 such that
+L(ξ)Qξ0 = (E(ξ), F(ξ)),
+for ξ ∈ ¯Gξ0,
+where E(ξ) is a (2d)ı+  × (2d)ı+  nonsingular matrix on ¯Gξ0, and ¯Gξ0 is the closure of a neighborhood Gξ0 of ξ0.
+Moreover,
+Aı  = L(ξ)Qξ0Q−1
+ξ0 Θ = (E(ξ), F(ξ)) ˜Θ,
+27
+
+A⊤ı  = ˜Θ⊤(E(ξ), F(ξ))⊤,
+˜Θ = Q−1
+ξ0 Θ =
+� ˜Θ1
+˜Θ2
+�
+,
+where ˜Θ1 is a (2d)ı+  × (2d)ı+ -order matrix. Obviously,
+rank
+� E⊤E
+E⊤F
+F⊤E
+F⊤F
+�
+= (2d)ı+ .
+Then there is an unitary matrix Uξ such that
+U−1
+ξ
+� E⊤E
+E⊤F
+F⊤E
+F⊤F
+�
+Uξ =
+� diag(λ1, · · · , λ(2d)ı+)
+0
+0
+0
+�
+,
+(4.96)
+where λi, 1 ≤ i ≤ (2d)ı+ , is nonvanishing eigenvalue of
+� E⊤E
+E⊤F
+F⊤E
+F⊤F
+�
+. By Poincar´e Separation Theorem (for
+example, p. 96, [40]),
+A⊤ı Aı  ≥ ˜Θ⊤
+� E⊤E
+E⊤F
+F⊤E
+F⊤F
+�
+˜Θ
+≥ ˜Θ⊤
+� diag(λ1, · · · , λ(2d)ı+)
+0
+0
+0
+�
+˜Θ
+≥
+�
+˜Θ⊤
+1
+˜Θ⊤
+2
+� � diag(λ1, · · · , λ(2d)ı+)
+0
+0
+0
+� � ˜Θ1
+˜Θ2
+�
+≥ min
+i
+min
+¯Oξ0
+λi ˜Θ⊤
+1 ˜Θ1
+≥ min
+i
+min
+¯Oξ0
+λi(min
+j |ˆξ j|)2M+2I(2d)ı+,
+where ”≥” is L¨ower order (for specific definition, see [40]) and the last equality has been explicitly explained on page
+12 of [35]. Hence, with finite cover theorem, we have
+|S2| = |{ξ ∈ G, 0 ≤ A⊤ı Aı  ≤ γ2
+|k|2τ I(2d)ı+}| ≤ c γ
+1
+M+1
+|k|
+τ
+M+1 .
+Similarly,
+|S1| = |{ξ ∈ G, |⟨k, ω(ξ)⟩ ≤ γ
+|k|τ | ≤ c γ
+1
+M+1
+|k|
+τ
+M+1 .
+Then |Rν+1| <
+γ
+1
+M+1
+0
+|k|
+τ
+M+1 . Consequently,
+|G0\G∗| = |
+∞
+�
+ν=0
+�
+Kν<|k|≤Kν+1
+Rν+1
+k
+| ≤
+∞
+�
+ν=0
+�
+Kν<|k|≤Kν+1
+γ
+1
+M+1
+0
+|k|
+τ
+M+1 → 0, as ε → 0.
+5. Appendix A.
+Proof of Proposition 1.
+28
+
+Proof. Notice that
+∇g(u, v) = (u|u|2l0−2, v|v|2k0−2),
+∇g(0, 0) = 0.
+For 0 < δ < 1, Bδ(0) denotes the open ball centered at the origin with radius δ. We have that ∇g(u, v) − ∇g(0, 0) is odd
+and unequal to zero on ∂Bδ(0), i.e.,
+∇g(−u, −v) − ∇g(0, 0) = (−u|u|2l0−2, −v|v|2k0−2) = −(∇g(u, v) − ∇g(0, 0)),
+and
+∇g(u, v) − ∇g(0, 0) � 0, ∀y ∈ ∂Bδ(0).
+It follows from Borsuk’s theorem in [31] that
+deg(∇g(u, v) − ∇g(0, 0), Bδ(0), 0) � 0.
+There exist σ = 1 and L = 2 min{l0, k0} − 1 such that
+|∇g(u, v) − ∇g(0, 0)| > σ|(u, v)|L.
+So, by Theorem 1, the results in Proposition 1 hold.
+6. Appendix B.
+Proof of Proposition 2
+Proof. Note that the motion equation of (2.6) is
+
+˙x = ω,
+˙y = 0,
+˙u = v2,
+˙v = −u2 − ε2.
+Since for any ε � 0, the equation u2 + ε2 = 0 has no real solution, we see that the Hamiltonian system does not admit
+any lower-dimensional torus. Let g(u, v) = 1
+3u3 + 1
+3v3. By simple calculation, we have
+deg(∇g(u, v), Bδ(0), 0) = 0,
+which implies that (A0) fails. Hence our assumption (A0) is sharp to Melnikov’s persistence.
+Acknowledgments
+The second author (Li Yong) is supported by National Basic Research Program of China (grant No. 2013CB834100),
+National Natural Science Foundation of China (grant No. 11571065, 11171132, 12071175), and Natural Science
+Foundation of Jilin Province (grant No. 20200201253JC).
+29
+
+References
+[1] V. I. Arnold, Proof of A.N. Kolmogorov’s theorem on the conservation of conditionally periodic motions with a small variation in the Hamil-
+tonian, Uspehi Mat. Nauk. 18(1963), no. 5, 9-36
+[2] J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE,
+Internat. Math. Res. Notices. 11(1994), 475-497
+[3] J. Bourgain, On Melnikov’s persistency problem, Math. Res. Lett. 4(1997), 445-458
+[4] H. W. Broer, H. Hanßmann, and J. G. You, On the destruction of resonant lagrangean tori in Hamiltonian systems, Springer Proc. Math. Stat.
+35(2013)
+[5] C. Q. Cheng, Lower dimensional invariant tori in the regions of instability of nearly integrable Hamiltonian systems, Comm. Math. Phys.
+203(1999), no. 2, 385-419
+[6] L. Chierchia and D. B. Qian, Moser’s theorem for lower dimensional tori, J. Differential Equations 206(2004), no. 1, 55-93
+[7] S. N. Chow, Y. Li, and Y. F. Yi, Persistence of invariant tori on submanifolds in Hamiltonian systems, J. Nonlinear Sci. 12(2002), 585-617
+[8] F. Z. Cong, T. K¨upper, Y. Li, and J. G. You, KAM-type theorem on resonant surfaces for nearly integrable Hamiltonian systems, J. Nonlinear
+Sci. 10(2000), no. 1, 49-68
+[9] R. de la Llave and L. Xu, Whiskered tori for presymplectic dynamical systems, J. Dynam. Differential Equatioins 33(2021), no. 1, 1-34
+[10] A. Delshams and P. Guti´errez, Effective stability and KAM theory, J. Differential Equations 128(1996), no. 2, 415-490
+[11] A. Delshams and P. Guti´errez, Estimates on invariant tori near an elliptic equilibrium point of a Hamiltonian system, J. Differential Equations
+131(1996), no. 2, 277-303
+[12] A. Delshams and P. Guti´errez, Splitting potential and the Poincar´e-Melnikov method for whiskered tori in Hamiltonian systems, J. Nonlinear
+Sci. 10(2000), no. 4, 433-476
+[13] L. H. Eliasson, Perturbations of stable invariant tori for Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 15(1988), no. 1, 115-147
+[14] L. H. Eliasson, Biasymptotic solutions of perturbed integrable Hamiltonian systems, Bol. Soc. Brasil. Mat. 25(1994), no. 1, 57-76
+[15] G. Gallatti and G. Gentile, Hyperbolic low-dimensional invariant tori and summations of divergent series, Comm. Math. Phys. 227(2002),
+no. 3, 421-460
+[16] G. Gentile, Degenerate lower-dimensional tori under the Bryuno condition, Ergodic Theory Dynam. Systems 27(2007), no. 2, 427-457
+[17] S. M. Graff, On the continuation of hyperbolic invariant tori for Hamiltonian systems, J. Differential Equations 15(1974), 1-69
+[18] Y. C. Han, Y. Li, and Y. F. Yi, Degenerate lower-dimensional tori in Hamiltonian systems, J. Differential Equations 227(2006), no. 2, 670-691
+[19] S. Q. Hu and B. Liu, Completely degenerate lower-dimensional invariant tori for Hamiltonian system, J. Differential Equations 266(2019),
+no. 11, 7459-7480
+[20]
+`A. Jorba and J. Villanueva, On the persistence of lower dimensional invariant tori under quasi-periodic perturbations, J. Nonlinear Sci.
+7(1997), no. 5, 427-473
+[21] A. N. Kolmogorov, On preservation of conditionally periodic motions under a small change in the Hamiltonian function, In Dokl. Akad.
+Nauk SSSR 98(1954), 527-530
+[22] S. B. Kuksin, Nearly integrable infinite dimensional Hamiltonian systems, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1993
+[23] S. B. Kuksin, Hamiltonian perturbations of infinite dimensional linear systems with an imaginary spectrum, Funktsional. Anal. i Prilozhen.
+21(1987), no.3, 22-37
+[24] Y. Li and Y. F. Yi, A quasi-periodic Poincar´e’s theorem, Math. Ann. 326(2003), 649-690
+[25] Y. Li and Y. F. Yi, Persistence of lower dimensional tori of general types in Hamiltonian systems, Trans. Amer. Math. Soc. 357(2005), no. 4,
+1565-1600
+[26] Y. Li and Y. F. Yi, Persistence of hyperbolic tori in Hamiltonian systems, J. Differential Equations 208(2005), no. 2, 344-387
+[27] A. G. Medvedev, A. I. Neidhtadt, and D. V. Treschev, Lagrangian tori near resonances of near-integrable Hamiltonian systems, Nonlinearity.
+28(2015), no. 7, 2105-2130
+[28] V. K. Melnikov, On certain cases of conservation of almost periodic motions with a small change of the Hamiltonian function, Dokl. Akad.
+Nauk SSSR 165(1965), 1245-1248
+[29] J. Moser, On invariant curves of area-preserving mapping of an annulus, Nachr. Akad. Wiss. G¨ottingen.II 1962(1962), 1-20
+[30] J. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann. 169(1967), 136-176
+[31] D. Motreanu, V. V. Motreanu, and N. Papageorgiou, Topological and variational methods with applications to nonlinear boundary value
+problems, Springer Science+Business Media, LLC, 2014
+[32] J. P¨oschel, Integrability of Hamiltonian systems on Cantor-like sets, Comm. Pure Appl. Math. 35(1982), no. 5, 653-696
+[33] J. P¨oschel, On elliptic lower-dimensional tori in Hamiltonian systems, Math. Z. 202(1989), no. 4, 559-608
+[34] W. C. Qian, Y. Li, and X. Yang, Multiscale KAM theorem for Hamiltonian systems, J. Differential Equations 266(2019), 70-86
+[35] W. C. Qian, Y. Li, and X. Yang, Melnikov’s conditions in matrices, J. Dynam. Differential Equations (2020)32, no. 4, 1779-1795
+[36] M. Rudnev and S. Wiggins, KAM theory near multiplicity one resonant surfaces in perturbations of a priori stable Hamiltonian systems, J.
+Nonlinear Sci. 7(1997), no. 2, 177-209
+[37] D. A. Salamon, The Kolmogorov-Arnold-Moser theorem, Math. Phys. Electron. J. 10(2004), 1-37
+[38] F. Takens, Singularities of vector fields, Inst. Hautes ´Etudes Sci. Publ. Math. 43(1974), 47-100
+[39] D. V. Treshchev, The mechanism of destruction resonance tori in Hamiltonian systems, Math. USSR-Sb. 68(1991), no. 1, 181-203
+[40] S. G. Wang, M. X. Wu and Z. Z. Jia, Matrix Inequality, Science Press, Beijing (2006)
+[41] C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys. 127(1990), no. 3,
+479-528
+[42] J. X. Xu, Persistence of elliptic lower dimensional invariant tori for small perturbation of degenerate integrable Hamiltonian systems, J.
+Math. Anal. Appl. 208(1997), no. 2, 372-387
+[43] J. X. Xu and J. G. You, Persistence of hyperbolic-type degenerate lower-dimensional invariant tori with perscribed frequencies in Hamioto-
+nian systems, Regul. Chaotic Dyn. 25(2020), no. 6, 616-650
+30
+
+[44] L. Xu, Y. Li, and Y. F. Yi, Lower-dimensional tori in multi-scale nearly integrable Hamiltonian systems, Ann. Henri Poincar´e 18(2017), no.
+1, 53-83
+[45] J. G. You, A KAM theorem for hyperbolic-type degenerate lower dimensional tori in Hamiltonian systems, Comm. Math. Phys. 192(1998),
+no. 1, 145-168
+[46] E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems I and II, Comm. Pure Appl. Math.
+28(1975), 91-140; 29(1976), no. 1, 49-111
+31
+
diff --git a/mNAyT4oBgHgl3EQfYffL/content/tmp_files/load_file.txt b/mNAyT4oBgHgl3EQfYffL/content/tmp_files/load_file.txt
new file mode 100644
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+++ b/mNAyT4oBgHgl3EQfYffL/content/tmp_files/load_file.txt
@@ -0,0 +1,1662 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf,len=1661
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='00206v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='DS]  31 Dec 2022  Melnikov’s persistence in degeneracy of high co-dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Jiayin Dua 1 , Yong Lia,b 2  aCollege of Mathematics, Jilin University, Changchun 130012, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  bSchool of Mathematics and Statistics, and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal  University, Changchun 130024, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Abstract  In this paper, we study the Melnikov’s persistence about lower-dimensional invariant tori for degenerate Hamiltonian  systems with the following Hamiltonian  H(x, y, u, v) = h(y) + g(u, v) + εP(x, y, u, v),  (x, y, u, v) ∈ Tn × G × Rd × Rd,  where n ≥ 2 and d ≥ 1 are positive integers, g = o(|u|2 + |v|2) admits certain transversality, and G ⊂ Rn is a  bounded closed region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' This is a try in studying lower-dimensional invariant tori in the normal degeneracy of high  co-dimension, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=', d > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Under R¨ussmann-like non-degenerate condition and transversality condition, we apply the  homotopy invariance of topological degree to remove the first order terms about u and v and employ the quasi-linear  KAM iterative procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Keywords: Degenerate Hamiltonian system, lower-dimensional tori, Melnikov’s persistence, high co-dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Contents  1  Introduction  2  2  Main Results  3  3  KAM Step  6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='1  Description of the 0-th KAM step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
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+page_content='5  Translation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
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+page_content='  25  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='3  Measure estimate .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
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+page_content='  27  1E-mail address : dujy20@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='jlu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='cn  2Corresponding author’s e-mail address : liyong@jlu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='cn  Preprint submitted to a  January 3, 2023  5  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  28  6  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  29  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Introduction  It is well known that by the classical KAM theory [1, 12, 21, 29, 32], most non-resonant Lagrangian tori of a  non-degenerate integrable Hamiltonian system survive small perturbations with only slight deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' In contrast  to this, the destruction of those resonant tori appears inevitable under perturbations in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Nevertheless, their  disintegration does not imply the nonexistence of regular motions such as periodic or quasi-periodic motions, as  pointed out in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Some studies indicate that the resonant torus foliated by non-resonant lower-dimensional tori is not  destroyed completely and that there are some lower-dimensional tori which survive perturbation if the Hamiltonian  satisfies a certain non-degenerate condition, see [8, 24, 27, 36, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' So, in the perturbation theory of Hamiltonian  systems, the persistence of lower-dimensional invariant tori plays an important role in understanding the destruction  mechanism of resonant Lagrangian tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Let’s consider the unperturbed system with the following normal form:  N = ⟨ω, y⟩ + 1  2⟨z, Ωz⟩,  where (y, z) ∈ Rn × R2d, n ≥ 2, d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' In non-degenerate case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=', Ω is non-singular, if all the eigenvalues of JΩ  belong to iR1\\{0} (J is the standard symplectic matrix), we say that the lower-dimensional invariant torus is elliptic,  which corresponds to the following normal form:  N = ⟨ω, y⟩ + 1  2  d  �  j≥1  Ω j(u2  j + v2  j),  where (y, z) ∈ Rn × R2d, z = (u, v) ∈ Rd × Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' In 1965, Melnikov [28] announced that the elliptic lower dimensional  invariant tori can persist for any small perturbations under certain coupling nonresonance conditions, called Melnikov  conditions:  ⟨k, ω⟩ + ⟨l, Λ⟩ � 0,  ∀(k, l) ∈ (Zn × Z2d)\\{0}, |l| ≤ 2,  where Λ is the column vector formed by the eigenvalues of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' A detailed and rigorous proof of the above Melnikov’s  theorem was due to Eliasson [13] and then an infinite dimensional analogue was obtained by Kuksin [23], P¨oschel  [33], Bourgain [2] and Wayne [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' On the other hand, if all the eigenvalues of JΩ are not on the imaginary axis, the  lower-dimensional invariant torus is called hyperbolic, which is associated with the unperturbed system  N = ⟨ω, y⟩ + 1  2  d  �  j≥1  Ω j(u2  j − v2  j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  The persistence problem for hyperbolic tori was first considered in the work of Moser [30] and later studied by Graff  [17] and Zehnder [46] for a fixed toral frequency ω satisfying the following Diophantine condition:  |⟨k, ω⟩| > γ  |k|τ ,  k ∈ Zn \\ {0},  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='1)  where k = (k1, · · · , kn), |k| = |k1| + · · · + |kn|, γ > 0 and τ > n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Moreover, the persistence of lower-dimensional  invariant tori has been extensively studied in various (normally) non-degenerate cases, see [3, 4, 5, 6, 8, 9, 10, 11,  13, 14, 15, 20, 26, 28, 33, 35, 39] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' However, in degenerate case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=', Ω is singular, generally,  the lower dimensional invariant tori might be destroyed under small perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Then a natural question is what  degenerate systems admit Melnikov’s persistence, which is an essential problem, for example, see [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' A normal  form in the simplest degenerate case is, see Takens [38],  N = ⟨ω, y⟩ + 1  2v2 + f(u, v),  2  where y ∈ Rn, f(u, v) = o(|u|2+|v|2) � 0, u, v ∈ R1, which is called co-dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' You in [45] proved the persistence  of a hyperbolic-type degenerate lower-dimensional torus for co-dimension 1 Hamiltonian systems with the following  normal form  N = ⟨ω, y⟩ + 1  2v2 − u2l,  l ≥ 2,  where (y, u, v) ∈ Rn × R1 × R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' For more details see [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Recently, Li and Yi showed the persistence of lower-  dimensional tori of general types, under a Melnikov-type of non-resonance condition, for the Hamiltonian system  with the following general normal form  N = e + ⟨ω, y⟩ + 1  2⟨  � y  z  �  , M  � y  z  �  ⟩ + O(|(y, z)|3),  where (y, z) ∈ Rn × R2d, d ≥ 1, and M is nonsingular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' For some other related studies, see [16, 18, 19, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' In this  paper, we will consider the Melnikov’s persistence when the unperturbed systems possess both high-degeneracy and  high co-dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  More precisely, we consider the following more general degenerate Hamiltonian system  H(x, y, u, v) = h(y) + g(u, v) + εP(x, y, u, v),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2)  where n ≥ 2 and d ≥ 1 are positive integers, (x, y, u, v) ∈ Tn × G × Rd × Rd, Tn is the standard n-torus, G ⊂ Rn is a  bounded closed region, g = o(|u|2 + |v|2), h, g and P are real analytic functions in (x, y, u, v), ε > 0 is a small parameter  and εP is the small perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' We verify the persistence of lower-dimensional invariant tori for the degenerate  Hamiltonian system with high co-dimension under certain high degeneracy conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' We further construct some  sufficient conditions in term of the topological degree condition as well as the weak convexity condition for g(u, v),  see (A0) in Section 2 for details of these two conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' We also give an example to state that our condition (A0) is  sharp to Melnikov’s persistence, see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' In Section 2, we state our main results (Theorem 1, Proposition 1 and Propo-  sition 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Section 3 contains the detailed construction and estimates for one cycle of KAM steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' In Section 4, we  complete the proof of Theorem 1 by deriving an iteration lemma and showing the convergence of KAM iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Finally, the proof of Proposition 1 and Proposition 2 can be found in Appendix A and Appendix B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Main Results  To state our main results we need first to introduce a few definitions and notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Given a domain D, we let ¯D, ∂D denote the closure of D and the boundary of D, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Do := ¯D \\ ∂D  refers to the interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  We shall use the same symbol | · | to denote the Euclidean norm of vectors, and use | · |D to denote the supremum  norm of a function on a domain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  For any two complex column vectors ξ, η of the same dimensional, ⟨ξ, η⟩ always stands for ξ⊤η, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=', the  transpose of ξ times η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  id is the identical mapping, and Id is the d-order unit matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  For a vector value function f, Df denotes the Jacobian matrix of f, and J f = detDf its Jacobian determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  All Hamiltonians in the sequel are endowed with the standard symplectic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Bδ(ζ) stands for a sphere with ζ as center and δ as radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  For simplicity we let z = (u, v) ∈ R2d, ∇g(z) = ( ∂g  ∂u, ∂g  ∂v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  3  Denote the complex neighborhoods of Tn × {0} × {0}  D(s, r) = {(x, y, z) : |Imx| < r, |y| < s2, |z| < s}  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='1)  with 0 < s, r < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Let Ω ⊂ Rn be a bounded and open domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' We give the definition of the degree for f ∈ C2( ¯Ω, Rn), see [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' If f ∈ C2( ¯Ω, Rn) and p ∈ Rn \\ f(∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (1) Denote Nf = {x ∈ Ω|J f(x) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' If p � f(Nf ), then  deg(f, Ω, p) :=  �  x∈ f −1(p)  sign(J f(x)),  setting deg(f, Ω, p) = 0 if f −1(p) = Ø.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (2) If p ∈ f(Nf ), by Sard theorem (see [31]), deg(f, Ω, p) is defined as deg(f, Ω, p) := deg(f, Ω, p1), for some ( and  all) p1 � f(Nf ) such that |p1 − p| ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Let ξ ∈ G be arbitrarily given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' The Taylor expansion of the Hamiltonian (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2) about ξ in a small neighborhood of  ξ in G reads  H(x, y, z, ξ) = e(ξ) + ⟨ω(ξ), y − ξ⟩ + ˜h(y − ξ) + g(z) + εP(x, y, z),  where e(ξ) = h(ξ), ω(ξ) = ∂h(ξ)  ∂y , and ˜h(y − ξ) = O(|y − ξ|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Using the transformation (y − ξ) → y in the above, we have  H(x, y, z, ξ) = e(ξ) + ⟨ω(ξ), y⟩ + ˜h(y, ξ) + g(z) + εP(x, y, z, ξ),  where (x, y, z) ∈ D(s, r), ξ ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  We are finally led to consider the following Hamiltonian  � H : Tn × G × R2d × G → R1,  H(x, y, z, ξ) = e(ξ) + ⟨ω(ξ), y⟩ + ˜h(y, ξ) + g(z) + εP(x, y, z, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2)  First, we make the following assumptions:  (A0) For fix ζ0 = 0 ∈ Oo ⊂ R2d, where O is a bounded closed domain, there are σ > 0, L ≥ 2 such that  deg(∇g(z) − ∇g(ζ0), Oo, 0) � 0,  |∇g(z) − ∇g(ζ0)| ≥ σ|z − ζ0|L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (A1) There is a positive integer M such that  rank{∂i  ξA0(ξ) : 0 ≤ |i| ≤ M} = 1  for all ξ ∈ G,  where  A0(ξ) = ⟨ k  |k|, ω(ξ)⟩, k ∈ Zn\\{0}, |k| = |k1| + · · · + |kn|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  We can now state our main results:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Consider the Hamiltonian (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Assume that (A0)-(A1) hold and ζ0 = 0 is a equilibrium point of  g(z), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=',  g(z) = o(|z|2),  ∇g(ζ0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='3)  4  Let ε0 > 0 be sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Then as 0 < ε < ε0, there exist Cantor sets G∗ ⊂ G with |G\\G∗| → 0 as ε → 0 as well  as a Cm−1 Whitney smooth family of symplectic transformations  Ψ∗ = Φ1 ◦ Φ2 ◦ · · · ◦ Φ∞ : D( s  2, r  2) → D(s, r),  ξ ∈ G∗,  which is analytic in x, Cm−1 Whitney smooth in y, z and ξ, and close to the identity, such that  H∗ = H ◦ Ψ∗ = e∗(ξ) + ⟨ω∗(ξ), y⟩ + ˜h∗(y, ξ) + g∗(z, ξ) + ¯g∗(y, z, ξ) + P∗(x, y, z, ξ)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='4)  with  ˜h∗(y, ξ) = O(|y|2),  g∗(z, ξ) = g(z) + ε  (m−2)(5m+4)  8m(m+1) O(|z|2),  ¯g∗(y, z, ξ) =  �  2|ı|+| \uf6be|≤m,1≤|ı|,| \uf6be|  ¯gı \uf6be(ξ)yız\uf6be,  ∂i  y∂ j  zP∗|(y,z)=(0,0) = 0,  where e∗, ω∗, ˜h∗, g∗, ¯g∗, P∗ are real analytic in their associated variables, 2|i| + |j| ≤ m, and  m ≥ L +  √  L2 + 16L + 16  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='5)  Thus for each 0 < ε < ε0 the Hamiltonian system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2) admits a Whitney smooth family of real analytic, quasi-periodic  n-tori Tε  ξ, ξ ∈ G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' In Theorem 1, one can allow d > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Hence it seems to be the first result about lower-dimensional  invariant tori in the normal degeneracy of high co-dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Recall z = (u, v) ∈ Rd × Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' For partially degenerate Hamiltonian (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2) with  g(u, v) = 1  2  d  �  i=1  λiv2  i + f(u), λi � 0, i = 1, · · · , d, d ≥ 1, f(u) = O(|u|4),  we will study it in a forthcoming paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='4) in Theorem 1 indicates that the Hamiltonian system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2) is conjugated to a nonlinear system,  not a linear one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Moreover, in order to ensure the persistence of low-dimensional invariant tori, the conjugated system  cannot have first order terms about z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=', g∗(z) = O(|z|2), otherwise there is no invariant torus because one cannot  get invariant set in the z direction as ∇g∗(0) � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Indeed, as an application of Theorem 1, we have the following proposition:  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Consider the Hamiltonian (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2) with  g(u, v) = 1  2l0  |u|2l0 + 1  2k0  |v|2k0,  where l0, k0 > 1 are integers, the dimension of u might be any positive integer, hence high co-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Assume  ω(ξ) satisfies the condition (A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Then the perturbed Hamiltonian system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2) admits a family of lower-dimensional  tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  See Appendix A for the complete proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Next, we will give an example to state that our assumption (A0) is sharp to Melnikov’s persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' See below for  a counter example:  5  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Consider the Hamiltonian (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2) with  H = ω · y + 1  3u3 + 1  3v3 + ε2u,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='6)  where y ∈ R1, (u, v) ∈ R1 × R1, and ε2u is a perturbation term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Assume ω satisfies Diophantine condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Then  the Hamiltonian system does not admit any lower-dimensional torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  The proof can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' KAM Step  In this section, we will show the detailed construction and estimates for one cycle of KAM steps, see [7, 18, 25,  26, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Description of the 0-th KAM step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Recall the integer m satisfying  m ≥ L +  √  L2 + 16L + 16  4  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='7)  where L ≥ 2 is defined as in (A0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Denote ρ =  1  2(m+1), and let η > 0 be an integer such that (1 + ρ)η > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' We put  γ = ε  1  2m(m+1)(2d)m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='8)  Consider the perturbed Hamiltonian (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' First we define the following 0-th KAM step parameters:  r0 = r,  γ0 = γ,  µ0 = ε  1  8(m+1) ,  s0 = sε  1  8(m+1) γ(m+1)(2d)m  0  , σ0 = σ, β0 = s,  e0 = e(ξ),  ω0 = ω(ξ),  ˜h0(y) = ˜h(y),  g0(z) = g(z),  ¯g0 = 0,  G0 = ˆG,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='9)  D(s0, r0) := {(x, y, z, ξ) : |y| < s2  0, |z| < s0, |Imx| < r0},  where 0 < r0, s0, s, γ0 ≤ 1, σ is defined in (A0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Therefore, we have that  H0 =: H(x, y, z, ξ) = N0 + P0,  N0 =: e0 + ⟨ω0, y⟩ + ˜h0(y) + g0(z) + ¯g0,  P0 =: εP(x, y, z, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  We first prove an estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Assume that  (H0) : ε  m  8(m+1)  0  |∂ℓ  ξP|D(s0,r0)  sm  ≤ 1, |ℓ| ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  For |ℓ| ≤ m,  |∂ℓ  ξP0|D(s0,r0) ≤ γ(m+1)(2d)m  0  sm  0 µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='10)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Using the fact γ0 = ε  1  2m(m+1)(2d)m , s0 = sε  1  8(m+1) γ(m+1)(2d)m  0  and µ0 = ε  1  8(m+1) , we have  sm  0 = smε  m  8(m+1) γm(m+1)(2d)m  0  = smε  m  8(m+1) + 1  2 ,  and  γ(m+1)(2d)m  0  sm  0 µ0 = ε  1  2m smε  m  8(m+1) + 1  2 ε  1  8(m+1) ≥ smε  1  4 + 5  8 +  1  8(m+1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='11)  6  Then, by (H0), for any 0 < ε < ε0, we get  ε  m  8(m+1) |∂ℓ  ξP|D(s0,r0)  sm  ≤ 1, |ℓ| ≤ m,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=',  ε  m  8(m+1) |∂ℓ  ξP|D(s0,r0) ≤ sm, |ℓ| ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='12)  Thus, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='12), for |ℓ| ≤ m,  |∂ℓ  ξP0|D(s0,r0) = ε|∂ℓ  ξP|D(s0,r0) ≤ ε  1  4 + 5  8 +  1  8(m+1) ε  m  8(m+1) |∂ℓ  ξP|D(s0,r0)  ≤ smε  1  4 + 5  8 +  1  8(m+1) ≤ γ(m+1)(2d)m  0  sm  0 µ0,  which implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Induction from ν-th KAM step  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Description of the ν-th KAM step  We first define the ν-th KAM step parameters:  rν = rν−1  2  + r0  4 ,  sν = 1  8 s2ρ+1  ν−1 ,  µν = 8msρ  ν−1µν−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Now, suppose that at ν-th step, we have arrived at the following real analytic Hamiltonian:  Hν = Nν + Pν,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='13)  with  Nν = eν(ξ) + ⟨ων(ξ), y⟩ + ˜hν(y, ξ) + gν(z, ξ) + ¯gν(y, z, ξ),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='14)  ˜hν(y, ξ) =  �  2≤|i|  ˜hi0(ξ)yi,  gν(z, ξ) = g(z) +  ν−1  �  i=0  γ(m+1)(2d)m  i  sm−2  i  µiO(|z|2),  ¯gν(y, z, ξ) =  �  2|i|+| j|≤m,1≤|i|,| j|  ¯gi j(ξ)yizj,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='15)  defined on D(sν, rν),  ∇gν(0) = ∇gν−1(ζν − ζν−1) + ∇z[Rν−1](0, ζν − ζν−1) = 0,  ζν ∈ B(sm−2  ν−1 µν−2)  1  L (ζν−1),  and  | ∂ℓ  ξPν |D(sν,rν)≤ γ(m+1)(2d)m  ν  sm  ν µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='16)  For simplicity, we will omit the index for all quantities of the present KAM step (at ν-th step), use + to index  all quantities (Hamiltonian, domains, normal form, perturbation, transformation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=') in the next KAM step  (at (ν+1)-th step), and use − to index all quantities in the previous KAM step (at (ν-1)-th step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' To simplify  notations, we will not specify the dependence of P, P+ etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' All the constants c1 − c6 (they will be defined in different  lemmas-Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='3, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='8, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='9) are positive and independent of the iteration process, and we will also use  c to denote any intermediate positive constant which is independent of the iteration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Define  r+ = r  2 + r0  4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  7  β+ = β  2 + β0  4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  σ+ = σ  2 + σ0  4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  γ+ = γ  2 + γ0  4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  s+ = 1  8αs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  α = s2ρ = s  1  (m+1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  µ+ = 8mc0µsρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  c0 = max{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' c6},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  K+ = ([log 1  s ] + 1)3η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (1 + ρ)η > 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  D(s) = {(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' z) ∈ Cn × R2d : |y| < s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' |z| < s},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  ˆD(s) = D(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' r+ + 7  8(r − r+)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  ˜D = D(β+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' r+ + 5  8(r − r+)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  D i  8 α = D( i  8αs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' r+ + i − 1  8  (r − r+)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  D+ = D 1  8 α = D(s+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' r+),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Γ(r − r+) =  �  0<|k|≤K+  |k|(m+1)(2d)mτ+me−|k| r−r+  8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  G+ = {ξ ∈ G : |⟨k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' ω⟩| > γ  |k|τ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' A⊤ı \uf6beAı \uf6be > γ2  |k|2τ I(2d)ı+\uf6be,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' for k ∈ Zn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  0 < |k| < K+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 2|ı| + |\uf6be| ≤ m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 1 ≤ |\uf6be|}  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Construct a symplectic transformation  We will construct a symplectic transformation Φ+:  Φ+ : D(s+, r+) × G+ → D(s, r) × G  such that it transforms Hamiltonian (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='13) into the Hamiltonian of the next KAM cycle (at (ν+1)-th step), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=',  H+ = H ◦ Φ+ = N+ + P+,  where N+ and P+ have similar properties as N and P respectively on D(sν+1, rν+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Next, we show the detailed construction of Φ+ and the estimates of P+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Truncation  Consider the Taylor-Fourier series of P:  P =  �  k∈Zn, ı, \uf6be∈Zn  +  pkı \uf6beyız\uf6bee  √  −1⟨k,x⟩,  and let R be the truncation of P of the form  R =  �  |k|≤K+, 2|ı|+| \uf6be|≤m  pkı \uf6beyız\uf6bee  √  −1⟨k,x⟩  =  �  |k|≤K+, 2|ı|+| \uf6be|≤m  pkı \uf6beyı1  1 · · · yınn z\uf6be1  1 · · · z\uf6be2d  2d e  √  −1⟨k,x⟩,  where |ı| = |ı1| + · · · + |ın|, |\uf6be| = |\uf6be1| + · · · + |\uf6be2d|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Next, we will prove that |P − R| is much smaller than |P| by truncating appropriately, see the below lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  8  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Assume that  (H1) :  � ∞  K+  tne−t r−r+  16 dt ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Then there is a constant c1 such that for all ξ ∈ G, |ℓ| ≤ m,  |∂ℓ  ξ(P − R)|Dα ≤ c1γ(m+1)(2d)m sm+1µ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='17)  |∂ℓ  ξR|Dα ≤ c1γ(m+1)(2d)m smµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='18)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Denote  I =  �  |k|>K+, ı, \uf6be∈Zn  +  pkı \uf6beyız\uf6bee  √  −1⟨k,x⟩,  II =  �  |k|≤K+, 2|ı|+| \uf6be|>m  pkı \uf6beyız\uf6bee  √  −1⟨k,x⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Then  P − R = I + II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  To estimate I, we notice by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='16) that  |  �  ı∈Zn  +  ∂ℓ  ξpkıyı| ≤ |∂ℓ  ξP|D(s,r)e−|k|r ≤ γ(m+1)(2d)m smµe−|k|r,  where the first inequality has been frequently used in [7, 8, 18, 25, 33, 34, 37] and for the detailed proof, see [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  This together with (H1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='16) yields  |∂ℓ  ξI| ˆD(s) ≤  �  |k|>K+  |∂ℓ  ξP|D(s,r)e−|k| r−r+  8  ≤ γ(m+1)(2d)m smµ  ∞  �  κ=K+  κne−κ r−r+  8  ≤ γ(m+1)(2d)m smµ  � ∞  K+  tne−t r−r+  16 dt ≤ γ(m+1)(2d)m sm+1µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='19)  It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='16) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='19) that  |∂ℓ  ξ(P − I)| ˆD(s) ≤ |∂ℓ  ξP|D(s,r) + |∂ℓ  ξI| ˆD(s) ≤ 2γ(m+1)(2d)m smµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  For 2|p| + |q| = m + 1, let  �  be the obvious antiderivative of ∂(p,q)  ∂ypzq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Then the Cauchy estimate of P − I on D∗ yields  |∂ℓ  ξII|Dα = |∂ℓ  ξ  �  ∂(p,q)  ∂ypzq  �  |k|≤K+, 2|ı|+| \uf6be|>m  pkı \uf6beyız\uf6bee  √  −1⟨k,x⟩dydz|Dα  ≤ |  �  | ∂(p,q)  ∂ypzq ∂ℓ  ξ(P − I)|dydz|Dα  ≤ | c  sm+1  �  |∂ℓ  ξ(P − I)|D∗dydz|Dα  ≤ 2 c  sm+1 γ(m+1)(2d)m smµ(αs)m+1  ≤ cγ(m+1)(2d)m sm+1µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Thus,  |∂ℓ  ξ(P − R)|Dα ≤ cγ(m+1)(2d)m sm+1µ,  and therefore,  |∂ℓ  ξR|Dα ≤ |∂ℓ  ξ(P − R)|Dα + |∂ℓ  ξP|D(s,r) ≤ cγ(m+1)(2d)m smµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Homological Equation  As usual, we shall construct a symplectic transformation as the time 1-map φ1  F of the flow generated by a Hamil-  tonian F of the form  F =  �  0<|k|≤K+, 2|ı|+| \uf6be|≤m  Fkı \uf6beyız\uf6bee  √  −1⟨k,x⟩  =  �  0<|k|≤K+, 2|ı|+| \uf6be|≤m  Fkı \uf6beyı1  1 · · · yınn z\uf6be1  1 · · · z\uf6be2d  2d e  √  −1⟨k,x⟩,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='20)  where Fkı \uf6be are (ξ, y, z)-dependent vectors or matrices of obvious dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  The Hamiltonian F can be determined by the following quasi-linear homological equation  {N, F} + R − [R] − Q = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='21)  where [R] =  1  (2π)n  �  Tn R(x, y, z)dx is the average of the truncation R, and the correction term  Q = (∂zg + ∂z¯g)J∂zF|2|ı|+| \uf6be|>m,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='22)  corresponds to the (m + 1)-order and higher-order terms in ⟨∇z¯g, ˜J∇zF⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Recall (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='14), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=',  N = e(ξ) + ⟨ω(ξ), y⟩ + ˜h(y, ξ) + g(z, ξ) + ¯g(y, z, ξ),  with  ˜h(y, ξ) =  �  2≤|i|  ˜h0i0(ξ)yi,  g(z, ξ) =  �  2≤| j|  g00 j(ξ)zj,  ¯g(y, z, ξ) =  �  2|i|+| j|≤m,1≤|i|,| j|  ¯g0i j(ξ)yizj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Notice that  {N, F} = −∂yN∂xF + ∂xN∂yF + ∂zNJ∂zF  = −  √  −1⟨k, ω + ∇˜h(y) + ∂y¯g⟩Fkı \uf6beyız\uf6bee  √  −1⟨k,x⟩ + (∂zg + ∂z¯g)J∂zF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='23)  In order to simplify the notations, we sometimes omit the subscript of � and only use � to represent the sum to the  index over the corresponding range of variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Now we calculate the last term of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='23):  (∂zg + ∂z¯g)J∂zF = (∂zg + ∂z¯g)J∂zF|2|ı|+| \uf6be|≤m + Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='24)  Specially,  ∂zgJ∂zF|2|ı|+| \uf6be|≤m = ∂zgJ  �  0<|k|≤K+, 2|ı|+| \uf6be|≤m  Fkı \uf6beyı∂z(z\uf6be)e  √  −1⟨k,x⟩  + ∂zgJ  �  0<|k|≤K+, 2|ı|+| \uf6be|≤m  ∂z(Fkı \uf6be)yız\uf6bee  √  −1⟨k,x⟩  =:  �  S ı \uf6beFkı \uf6beyız\uf6bee  √  −1⟨k,x⟩ +  �  2≤| j′|  S j′∂zFkı(\uf6be− j′+1)yız\uf6bee  √  −1⟨k,x⟩  =:  �  ˜S ı \uf6beFkı \uf6beyız\uf6bee  √  −1⟨k,x⟩ +  �  (S ı \uf6be − ˜S ı \uf6be)Fkı \uf6beyız\uf6bee  √  −1⟨k,x⟩  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='25)  +  �  2≤| j′|  S j′∂zFkı(\uf6be− j′+1)yız\uf6bee  √  −1⟨k,x⟩,  10  where 0 < |k| ≤ K+, 2|ı| + |\uf6be| ≤ m, 2 ≤ |j′|, S ı \uf6be is a (|ı| + |\uf6be|) order tensor and concerned with ∂2  zg(z) and J, S j′ is  concerned with ∂ j′  z g(0) and J, ˜S ı \uf6be is a (|ı| + |\uf6be|) order tensor and concerned with ∂2  zg(0) and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' And  ∂z¯gJ∂zF|2|ı|+| \uf6be|≤m =∂z¯gJ  �  0<|k|≤K+, 2|ı|+| \uf6be|≤m  Fkı \uf6beyı∂z(z\uf6be)e  √  −1⟨k,x⟩  + ∂z¯gJ  �  0<|k|≤K+, 2|ı|+| \uf6be|≤m  ∂z(Fkı \uf6be)yız\uf6bee  √  −1⟨k,x⟩  =:  �  S i j(Fk(ı−i)(\uf6be+2− j) + ∂zFk(ı−i)(\uf6be+1− j))yız\uf6bee  √  −1⟨k,x⟩  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='26)  where S i j is concerned with ∂i  y∂ j  z ¯g(0, 0) and J, and 2|i| + |j| ≤ m, 1 ≤ |i|, 1 ≤ |j|, 0 < |k| ≤ K+, 2|ı| + |\uf6be| ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='24), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='25), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='26) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='23), putting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='22) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='23) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='21), comparing the coefficients,  we thus obtain the following quasi-linear equations for all 0 < |k| ≤ K+, 2|ı| + |\uf6be| ≤ m:  (  √  −1⟨k, ω + ∇˜h(y) + ∇y¯g⟩I(2d)ı+\uf6be − S ı \uf6be)Fkı \uf6be = pkı \uf6be + S j′∂zFkı(\uf6be− j′+1)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='27)  + S i jFk(ı−i)(\uf6be+2− j) + S i j∂zFk(ı−i)(\uf6be+1− j),  where i, j, j′ are defined as above, S i jFk(ı−i)(\uf6be+2− j) stands for �  2|i|+| j|≤m,1≤|i|,| j| S i jFk(ı−i)(\uf6be+2− j), and the rest terms are  analogously defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  The above equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='27) are solvable if the coefficient matrices are nonsingular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' We denote  G+ ={ξ ∈ G : |⟨k, ω⟩| > γ  |k|τ , A⊤ı \uf6beAı \uf6be > γ2  |k|2τ I(2d)ı+\uf6be, for k ∈ Zn,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='28)  0 < |k| < K+, 2|ı| + |\uf6be| ≤ m, 1 ≤ |\uf6be|},  with  Aı \uf6be =  √  −1⟨ k  |k|, ω⟩I(2d)ı+\uf6be +  ˜S ı \uf6be  |k| ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='29)  where ˜S ı \uf6be is concerned with ∂2g(0)  ∂z2  and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Then, we can solve equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='27) on G+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' The details can be seen in the  following lemma:  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Assume that  (H2) :  max  |ℓ|≤m,|i|≤m|∂ℓ  ξ∂i  y˜h − ∂ℓ  ξ∂i  y˜h0|D(s)×G+ ≤ s  1  2  0 ,  (H3) : 4s <  γ − γ+  (M∗ + 2)Kτ+1  +  ,  where  M∗ =  max  |ℓ|≤m,|i|≤m|∂ℓ  ξ∂i  y˜h0(y)|D(s)×G+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  The quasi-linear equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='27) can be uniquely solved on D(s) × G+ to obtain the coefficient Fkı \uf6be which satisfy the  following equality:  |∂ℓ  ξ∂i  y∂ j  zFkı \uf6be|D(s)×G+ ≤ c2|k||ℓ|+|i|+| j|+(|ℓ|+|i|+| j|+1)(2d)\uf6beτsm−2|ı|−| \uf6be|µe−|k|r,  for all 0 < |k| ≤ K+, 2|ı| + |\uf6be| ≤ m, |ℓ| + |i| + |j| ≤ m, where c2 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' For ∀(y, ξ) ∈ D(s) × G+, by (H2), (H3),  |∇˜h| = |(∇˜h − ∇˜h0) + ∇˜h0| ≤ (1 + M∗)|y| < (1 + M∗)s <  γ  4|k|τ+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='30)  11  Recalling that ¯g(y, z) comes from the perturbation and it is formulized in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='15), we have  |∇y¯g| ≤ cs <  γ  4|k|τ+1 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='31)  with c ≤ �ν−1  i=0 γ(m+1)(2d)m  i  sm−3  i  µi ≪ 1, where the last equality follows from (H3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Notice by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='30) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='31) that  | det⟨k, ∇˜h + ∇y¯g⟩I(2d)ı+\uf6be| ≤ ( γ  4|k|τ)(2d)ı+\uf6be, 2|ı| + |\uf6be| ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  It follows from the definition of S ı \uf6be, ˜S ı \uf6be and g that  | det(S ı \uf6be − ˜S ı \uf6be)| ≤ | det(|s|I(2d)ı+\uf6be)| < ( γ  4|k|τ )(2d)ı+\uf6be, 2|ı| + |\uf6be| ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  This together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='27) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='28) yields  | det Bı \uf6be| >  γ(2d)ı+\uf6be  2(|k|τ)(2d)ı+\uf6be , 2|ı| + |\uf6be| ≤ m  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='32)  where Bı \uf6be is the coefficient matrix of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Then, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='32),  |Bı \uf6be|−1 = | adjBı \uf6be  det Bı \uf6be  | ≤ c|k|τ(2d)\uf6be+(2d)\uf6be−1  γ(2d)\uf6be  , 2|ı| + |\uf6be| ≤ m  where adjBı \uf6be is the algebraic minor of Bı \uf6be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Applying the identity  ∂ j  zB−1  ı \uf6be = −  | j|  �  | j′|=1  � j  j′  �  (∂ j− j′  z  B−1  ı \uf6be ∂ j′  z Bı \uf6be)B−1  ı \uf6be  inductively, we have for |ℓ| + |i| + |j| ≤ m  |∂ℓ  ξ∂i  y∂ j  zB−1  ı \uf6be |D(s)×G+ ≤ c|k||ℓ|+|i|+| j||B−1  ı \uf6be ||ℓ|+|i|+| j|+1  ≤ c|k||ℓ|+|i|+| j|+(|ℓ|+|i|+| j|+1)(2d)\uf6beτ  γ(|ℓ|+|i|+| j|+1)(2d)\uf6be  , 2|ı| + |\uf6be| ≤ m,  which was precisely proved in [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' We note by the Cauchy estimate that for 2|ı| + |\uf6be| ≤ m,  |∂ℓ  ξpkı \uf6be|G+ ≤ |∂ℓ  ξP|D(s)×G+ s−2|ı|−| \uf6be|e−|k|r ≤ γ(m+1)(2d)m sm−2|ı|−| \uf6be|µe−|k|r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  So, for |ℓ| + |i| + |j| ≤ m, 2|ı| + |\uf6be| ≤ m,  |∂ℓ  ξ∂i  y∂ j  zFkı \uf6be|D(s)×G+ ≤ c|k||ℓ|+|i|+| j|+(|ℓ|+|i|+| j|+1)(2d)\uf6beτ  γ(|ℓ|+|i|+| j|+1)(2d)\uf6be  γ(m+1)(2d)m sm−2|ı|−| \uf6be|µe−|k|r  ≤ c|k||ℓ|+|i|+| j|+(|ℓ|+|i|+| j|+1)(2d)\uf6beτsm−2|ı|−| \uf6be|µe−|k|r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Next, we apply the above transformation φ1  F to Hamiltonian H, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=',  H ◦ φ1  F = (N + R) ◦ φ1  F + (P − R) ◦ φ1  F  = (N + R) + {N, F} +  � 1  0  {(1 − t){N, F} + R, F} ◦ φt  Fdt + (P − R) ◦ φ1  F  = N + [R] +  � 1  0  {Rt, F} ◦ φt  Fdt + (P − R) ◦ φ1  F + Q  12  =: ¯N+ + ¯P+,  where  ¯N+ = N + [R],  ¯P+ =  � 1  0  {Rt, F} ◦ φt  Fdt + (P − R) ◦ φ1  F + Q,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='33)  Rt = (1 − t)Q + (1 − t)[R] + tR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Translation  In this subsection, we will construct a translation so as to eliminate the first order terms about z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Consider the translation  φ : x → x,  y → y,  z → z + ζ+ − ζ,  where z = (u, v), ζ+ is to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Let  Φ+ = φ1  F ◦ φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Then  H ◦ Φ+ = N+ + P+,  N+ = ¯N+ ◦ φ,  P+ = ¯P+ ◦ φ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='34)  with  N+ = ¯N+ ◦ φ = (N + [R]) ◦ φ = (e + ⟨ω, y⟩ + ˜h(y) + g(z) + ¯g(y, z) + [R](y, z)) ◦ φ  = e + ⟨ω, y⟩ + ˜h(y) + g(z + ζ+ − ζ) + ¯g(y, z + ζ+ − ζ) + [R](y, z + ζ+ − ζ)  =: e+ + ⟨ω+, y⟩ + ˜h+ + g+ + ¯g+,  where  e+ = e + g(ζ+ − ζ) + [R](0, ζ+ − ζ),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='35)  ω+ = ω + ∇y[R](0, ζ+ − ζ) + ∇y¯g(0, ζ+ − ζ),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='36)  ˜h+ = ˜h(y) + [R](y, ζ+ − ζ) − [R](0, ζ+ − ζ) − ⟨∇y[R](0, ζ+ − ζ), y⟩  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='37)  + ¯g(y, ζ+ − ζ) − ⟨∇y¯g(0, ζ+ − ζ), y⟩,  g+ = g(z + ζ+ − ζ) − g(ζ+ − ζ) + [R](0, z + ζ+ − ζ) − [R](0, ζ+ − ζ),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='38)  ¯g+ = ¯g(y, z + ζ+ − ζ) − ¯g(y, ζ+ − ζ) + [R](y, z + ζ+ − ζ) − [R](y, ζ+ − ζ)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='39)  − [R](0, z + ζ+ − ζ) + [R](0, ζ+ − ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Eliminate the first order terms about z  In this subsection, we will appropriately choose ζ+ to remove the first order terms about z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' The concrete details  are expressed as the following lemma, which is crucial to our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Notice that  ∇g+(0) = ∇g(ζ+ − ζ) + ∇z[R](0, ζ+ − ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='40)  There exists ζ+ ∈ B(sm−1  −  µ−)  1  L (ζ) such that  ∇g+(0) = ∇g(0) = · · · = ∇g0(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  13  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' The proof will be completed by induction on ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' We start with the case ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='3) and (A0)  that  ∇g0(ζ0) = ∇g0(0) = 0,  deg(∇g0(·) − ∇g0(0), Oo, 0) � 0,  |∇g0(z) − ∇g0(z∗)| ≥ σ0|z − z∗|L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Now assume that for some ν ≥ 1 we have got  ∇gi(0) = ∇gi−1(ζi − ζi−1) + ∇z[Ri−1](0, ζi − ζi−1) = 0,  deg(∇gi(·) − ∇gi(0), Oo, 0) � 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='41)  |∇gi(z) − ∇gi(z∗)| ≥ σi|z − z∗|L,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='42)  where i = 1, 2, · · · , ν, ζi ∈ B(sm−1  i−2 µi−2)  1  L (ζi−1), s−1 = s0, µ−1 = µ0, z∗ ∈ O, z ∈ O\\B(sm−1  i−1 µi−1)  1  L (z∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Then, we need to find ζ+  near ζ such that ∇g+(0) = ∇g(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Consider homotopy Ht(z) : [0, 1] × O → R2d,  Ht(z) =: ∇g(z − ζ) − ∇g(0) + t∇z[R](0, z − ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Notice that  |∇z[R](y, z)| ≤ γ(m+1)(2d)m sm−1µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='43)  For any z ∈ ∂O, t ∈ [0, 1], by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='42) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='43), we have  |Ht(z)| ≥ |∇g(z − ζ) − ∇g(0)| − |∇z[R](0, z − ζ)|  ≥ σ|z − ζ|L − γ(m+1)(2d)m sm−1µ  > σδL  2 ,  where δ := min{|z − ζ|, ∀z ∈ ∂O}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' So, it follows from the homotopy invariance and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='41) that  deg(H1(·), Oo, 0) = deg(H0(·), Oo, 0) � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='44)  We note by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='42) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='43) that for any z ∈ O\\B(sm−1  −  µ−)  1  L (ζ),  |H1(z)| = |∇g(z − ζ) − ∇g(0) + ∇z[R](0, z − ζ)|  ≥ |∇g(z − ζ) − ∇g(0)| − |∇z[R](0, z − ζ)|  ≥ σ|z − ζ|L − γ(m+1)(2d)m sm−1µ  ≥ σsm−1  −  µ− − γ(m+1)(2d)m sm−1µ  ≥ σ  2 sm−1  −  µ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Hence, by excision and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='44),  deg(H1(·), B(sm−1  −  µ−)  1  L (ζ), 0) = deg(H1(·), Oo, 0) � 0,  then there exist at least a ζ+ ∈ B(sm−1  −  µ−)  1  L (ζ), such that  H1(ζ+) = 0,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=',  ∇g(ζ+ − ζ) + ∇z[R](0, ζ+ − ζ) = ∇g(0),  14  thus, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='40),  ∇g+(0) = ∇g(0) = · · · = ∇g0(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='45)  Next, we need to prove  deg(∇g+(·) − ∇g+(0), Oo, 0) � 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='46)  |∇g+(z) − ∇g+(z∗)| ≥ σ+|z − z∗|L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='47)  By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='38),  ∇g+(z) = ∇g(z + ζ+ − ζ) + ∇z[R](0, z + ζ+ − ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Then  ∇g+(z) − ∇g(z) = ∇g(z + ζ+ − ζ) − ∇g(z) + ∇z[R](0, z + ζ+ − ζ),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='48)  and  ∇g+(z) − ∇g+(z∗) = ∇g(z + ζ+ − ζ) − ∇g(z∗ + ζ+ − ζ)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='49)  + ∇z[R](0, z + ζ+ − ζ) − ∇z[R](0, z∗ + ζ+ − ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  In view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='43), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='48), and ζ+ ∈ B(sm−1  −  µ−)  1  L (ζ), we get  |∇g+(z) − ∇g(z)| ≤ c(sm−1  −  µ−)  1  L ,  so, this together with the property of degree, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='41) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='45) that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='46) holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=',  deg(∇g+(·) − ∇g+(0), Oo, 0) = deg(∇g+(·) − ∇g(·) + ∇g(·) − ∇g(0), Oo, 0)  = deg(∇g(·) − ∇g(0), Oo, 0) � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='42), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='43) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='49) that for any z ∈ O\\B(sm−1µ)  1  L (z∗)  |∇g+(z) − ∇g+(z∗)| ≥ σ|z − z∗|L − 2γ(m+1)(2d)m sm−1µ ≥ σ+|z − z∗|L,  which implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Estimate on N+  Now, we give the estimate of N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' There is a constant c3 such that for all |ℓ| ≤ m:  |∂ℓ  ξ(ζ+ − ζ)|G+ ≤ c3(sm−1  −  µ−)  1  L ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='50)  |∂ℓ  ξ(e+ − e)|G+ ≤ c3(sm−1  −  µ−)  1  L ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='51)  |∂ℓ  ξ(ω+ − ω)|G+ ≤ c3(sm−1  −  µ−)  1  L ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='52)  |∂ℓ  ξ(˜h+ − ˜h)|D(s+)×G+ ≤ c3(sm−1  −  µ−)  1  L ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='53)  |∂ℓ  ξ(g+ − g)|D(s+)×G+ ≤ c3(sm−1  −  µ−)  1  L ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='54)  |∂ℓ  ξ(¯g+ − ¯g)|D(s+)×G+ ≤ c3(sm−1  −  µ−)  1  L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='55)  15  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Differentiating (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='40) with respect to ξ yields  (∇2g(ζ+ − ζ) + ∇2  z[R](0, ζ+ − ζ))∂ξ(ζ+ − ζ) = ∂ξ∇g(ζ+ − ζ) + ∂ξ∇z[R](0, ζ+ − ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Applying this identity inductively and using ζ+ ∈ B(sm−1  −  µ−)  1  L (ζ) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='4, we can immediately get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Recall  e+ and ω+ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='35) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='36), respectively, we see that  |∂ℓ  ξ(e+ − e)|G+ ≤ c|∂ℓ  ξ(ζ+ − ζ)|2 + γ(m+1)(2d)m smµ ≤ c(sm−1  −  µ−)  1  L ,  |∂ℓ  ξ(ω+ − ω)|G+ ≤ c|∂ℓ  ξ(ζ+ − ζ)| + γ(m+1)(2d)m sm−2µ ≤ c(sm−1  −  µ−)  1  L ,  which implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='51) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  In view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='37), we have that  |∂ℓ  ξ(˜h+(y) − ˜h(y))|D(s+) = |∂ℓ  ξ([R](y, ζ+ − ζ) − [R](0, ζ+ − ζ) − ⟨∇y[R](0, ζ+ − ζ), y⟩)|  + |∂ℓ  ξ(¯g(yı(ζ+ − ζ) \uf6be) − ⟨∇y¯g((ζ+ − ζ) \uf6be), y⟩)|  ≤ γ(m+1)(2d)m smµ + (sm−1  −  µ−)  1  L s4  +  ≤ c(sm−1  −  µ−)  1  L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='38) that  |∂ℓ  ξ(g+(z) − g(z))|D(s+) = |∂ℓ  ξ(g(z + ζ+ − ζ) − g(ζ+ − ζ) − g(z))|  + |∂ℓ  ξ([R](0, z + ζ+ − ζ) − [R](0, ζ+ − ζ))|  ≤ c(sm−1  −  µ−)  1  L s2  + + γ(m+1)(2d)m smµ  ≤ c(sm−1  −  µ−)  1  L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  According to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='39), we obtain that  |∂ℓ  ξ(¯g+(yız\uf6be) − ¯g(yız\uf6be))|D(s+) ≤ |∂ℓ  ξ[R](yız\uf6be)| ≤ c4γ(m+1)(2d)m smµ ≤ c(sm−1  −  µ−)  1  L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Estimate on Φ+  Recall that F is as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='20) with the coefficients and its estimate is given by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Then, we have the  following estimate on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' There is a constant c4 such that for all |ℓ| + |l| + |i| + |j| ≤ m,  |∂ℓ  ξ∂l  x∂i  y∂ j  zF| ˆD(s)×G+ ≤  � c4sm−2|i|−| j|µΓ(r − r+),  2|i| + |j| ≤ m,  c4µΓ(r − r+),  m ≤ 2|i| + |j|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='56)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Let  a(i, j) =  � sm−2|i|−| j|,  if 2|i| + |j| ≤ m,  0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='20) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='3, we have  |∂ℓ  ξ∂l  x∂i  y∂ j  zF| ˆD(s)×G+ ≤  �  0<|k|≤K+,2|ı|+| \uf6be|≤m  |k|l|∂ℓ  ξ∂i  y∂ j  z(Fkı \uf6beyız\uf6be)|e|k|(r++ 7  8 (r−r+))  ≤  �  0<|k|≤K+  c|k||l|+|ℓ|+|i|+| j|+(|ℓ|+|i|+| j|+1)(2d)\uf6beτsa(i, j)µe−|k| r−r+  8  ≤ csa(i, j)µΓ(r − r+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  This proves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  16  To obtain the symplectic transformation stated in Theorem 1, we need to extend the function F smoothly to the  domain ˆD(β0) × G0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Assume (H2)-(H3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Then F and ζ+ − ζ can be smoothly extended to functions of H¨older class  Cm−1+̺,m−1+̺( ˆD(β0) × G0) and Cm−1+̺ respectively, where 0 < ̺ < 1 is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Moreover, there is a constant c such that  ∥F∥Cm−1+̺,m−1+̺( ˆD(β0)×G0) ≤ cµΓ(r − r+),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='57)  ∥ζ+ − ζ∥Cm−1+̺(G0) ≤ c(sm−1  −  µ−)  1  L ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='58)  where ∥F∥Cm−1+̺,m−1+̺( ˆD(β0)×G0) =: |∂ℓ  ξ∂l  x∂i  y∂ j  zF| ˆD(β0)×G0, |ℓ| ≤ m − 1 + ̺, |l| + |i| + |j| ≤ m − 1 + ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' It follows from the standard Whitney extension theorem that F and ζ+−ζ can be extended smoothly to functions  ˜F and  ˜  (ζ+ − ζ) of H¨older class Cm−1+̺,m−1+̺( ˆD(β0 × G0)), Cm−1+̺(G0), respectively, such that  ∥ ˜F∥m−1+̺,m−1+̺  C  ( ˆD(β0 × G0)) ≤ c∥F∥Cm,m( ˆD(s)×G+),  ∥  ˜  ζ+ − ζ∥Cm−1+̺(G0) ≤ c∥ζ+ − ζ∥Cm(G+),  where c is a constant depending only on m, ̺ and the dimensions n, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' The lemma now follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='56) in Lemma  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='6 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='50) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' In addition to (H2)-(H3), assume that  (H4) : c3(sm−1  −  µ−)  1  L < 1  8αs,  (H5) : c4µΓ(r − r+) < 1  4(r − r+),  (H6) : c4sm−1µΓ(r − r+) < 1  8αs,  (H7) : c3µΓ(r − r+) + c3(sm−1µ)  1  L < β − β+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Then the following hold:  (1) For all 0 ≤ t ≤ 1,  φt  F : D 1  4 α → D 1  2 α,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='59)  φ : D 1  8 α → D 1  4 α,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='60)  are well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (2) Let Φ+ = φ1  F ◦ φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Then for all ξ ∈ G+,  Φ+ : D+ → D,  ˜D+ → D(β, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='61)  (3) There is a constant c5 such that  |∂ℓ  ξDi  (x,y,z)(φt  F − id)|D α  4 ×G+ ≤ c5µΓ(r − r+),  |ℓ| + |i| ≤ m,  where Di  (x,y,z)F stands for the |i|-order partial derivative with respect to (x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (4)  |∂ℓ  ξDi  (x,y,z)(Φ+ − id)| ˜D×G+ ≤ c5µΓ(r − r+),  |ℓ| + |i| ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  17  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' (1) For ∀(x, y, z) ∈ D 1  8 α, we note by ζ+ ∈ B(sm−1  −  µ−)  1  L (ζ) and (H4) that  |z + ζ+ − ζ| < |z| + |ζ+ − ζ| < 1  8αs + c(sm−1  −  µ−)  1  L < 1  4αs,  which implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  To verify (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='59), we denote φt  F1, φt  F2, φt  F3 as the components of φt  F in the x, y, z coordinates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Let  XF = (Fy, −Fx, ˜JFz)⊤ be the vector field generated by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Then  φt  F = id +  � t  0  XF ◦ φλ  Fdλ,  0 ≤ t ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='62)  For any (x, y, z) ∈ D 1  4 α, we let t∗ = sup{t ∈ [0, 1] : φt  F(x, y, z) ∈ Dα}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Then for any 0 ≤ t ≤ t∗, in view of (x, y, z) ∈ D 1  4 α,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='56) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='6, (H5) and (H6)  |φt  F1(x, y, z)|D 1  4 α ≤ |x| +  � t  0  |Fy ◦ φλ  F|D 1  4 αdλ  ≤ r+ + 1  8(r − r+) + c4sm−2µΓ(r − r+)  < r+ + 3  8(r − r+),  |φt  F2(x, y, z)|D 1  4 α ≤ |y| +  � t  0  | − Fx ◦ φλ  F|D 1  4 αdλ  ≤ 1  4αs + c4smµΓ(r − r+)  < 3  8αs,  |φt  F3(x, y, z)|D 1  4 α ≤ |z| +  � t  0  | ˜JFz ◦ φλ  F|D 1  4 αdλ  ≤ 1  4αs + c4sm−1µΓ(r − r+)  < 3  8αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Thus, φt  F ∈ D 1  2 α ⊂ Dα, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' t∗ = 1 and (1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (2) Using (H5), (H7), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='57) and a similar argument as above, one sees that φt  F : ¯D+ → D(β, r) is well defined for  all 0 ≤ t ≤ 1, where ¯D+ = D(r+ + 5  8(r − r+), β+ + c(sm−1  −  µ−)  1  L ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Thus, Φ+ : ˜D+ → D(β, r) is also well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3) Note that  |∂ℓ  ξXF| ˜D×G+ ≤ c|∂ℓ  ξD(x,y,x)F| ˜D×G+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='63)  Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='56) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='6 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='62), we immediately have  |φt  F − id|D α  4 ×G+ ≤ cµΓ(r − r+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Differentiating (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='62) yields  D(x,y,z)φt  F = I2n+2d +  � t  0  (D(x,y,z)XF)D(x,y,z)φλ  Fdλ  = I2n+2d +  � t  0  J(D2  (x,y,z)F)D(x,y,z)φλ  Fdλ,  0 ≤ t ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  18  It follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='6, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='62) and Gronwall Inequality that  |D(x,y,z)φt  F − I2n+2d|D α  4 ×G+ ≤ |  � t  0  D(x,y,z)XF ◦ φλ  FD(x,y,z)φλ  Fdλ|D α  4 ×G+  ≤  � t  0  |D(x,y,z)XF ◦ φλ  F||D(x,y,z)φλ  F − I2n+2d| ˆD(s)×G+dλ  +  � t  0  |D(x,y,z)XF ◦ φλ  F| ˆD(s)×G+dλ  ≤ cµΓ(r − r+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Similarly, by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='56) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='6, Gronwall’s inequality and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='63) inductively, we have  |∂ℓ  ξDi  (x,y,z)φt  F|D α  4 ×G+ ≤ cµΓ(r − r+),  |ℓ| + |i| ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (4) Observe that  Φ+ − id = (φ1  F − id) ◦ φ +  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed  0  0  ζ+ − ζ  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Then,  |∂ℓ  ξDi  (x,y,z)(Φ+ − id)| ˜D×G+ ≤ |∂ℓ  ξDi  (x,y,z)(φ1  F − id)| ˜D×G+|∂ℓ  ξDi  (x,y,z)φ| ˜D×G+  + |∂ℓ  ξ(ζ+ − ζ)| ˜D×G+  ≤ cµΓ(r − r+)(sm−1  −  µ−)  1  L + (sm−1  −  µ−)  1  L  ≤ cµΓ(r − r+)αs + αs  ≤ cµΓ(r − r+),  where the second equality follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='57) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='58) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='7, the third equality follows from (H4), and the  last equality follows from the definition of µ and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Frequency Property  To estimate the new frequency, we first assume that  (H8) : 3sK2τ+1  +  ≤ {γ − γ+  γ0  , γ2 − γ2  +  γ2  0  }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  It is obvious that for all 0 < |k| ≤ K+, ξ ∈ G+,  |⟨k, ∇y[R](0, ζ+ − ζ)⟩| ≤ cK+γ(m+1)(2d)m sm−2µ,  |⟨k, ∇y¯g((ζ+ − ζ) \uf6be)⟩| ≤ cK+γ(m+1)(2d)m(sm−1  −  µ−)  1  L ≤ cK+γ(m+1)(2d)mαs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Then this together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='36) and (H8) yields  |⟨k, ω+⟩| ≥ |⟨k, ω⟩| − |⟨k, ∇y[R](0, ζ+ − ζ) + ∇y¯g((ζ+ − ζ) \uf6be)⟩|  ≥ γ  |k|τ − γ − γ+  |k|τ  = γ+  |k|τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Recall that  A+  ı \uf6be =  √  −1⟨ k  |k|, ω+⟩I(2d)ı+\uf6be +  ˜S +  ı \uf6be  |k| = Aı \uf6be +  √  −1⟨ k  |k|, ω+ − ω⟩I(2d)ı+\uf6be +  ˜S +  ı \uf6be − ˜S ı \uf6be  |k|  =: Aı \uf6be + ˜Aı \uf6be,  19  where  Aı \uf6be =  √  −1⟨ k  |k|, ω⟩I(2d)ı+\uf6be +  ˜S ı \uf6be  |k| ,  ˜Aı \uf6be =  √  −1⟨ k  |k|, ω+ − ω⟩I(2d)ı+\uf6be +  ˜S +  ı \uf6be − ˜S ı \uf6be  |k|  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Then,  A+⊤  ı \uf6be A+  ı \uf6be = A⊤ı \uf6beAı \uf6be + A⊤ı \uf6be ˜Aı \uf6be + ˜A⊤ı \uf6beAı \uf6be + ˜A⊤ı \uf6be ˜Aı \uf6be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='64)  By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='50) in Lemma (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='5) , we have  |  √  −1⟨ k  |k|, ω+ − ω⟩| ≤ (sm−1  −  µ−)  1  L ≤ cαs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='65)  Note that ˜S +  ı \uf6be − ˜S ı \uf6be is concerned with ∂2  zg+(0) − ∂2  zg(0), then, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='38), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='18) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2,  | ˜S +  ı \uf6be − ˜S ı \uf6be|  |k|  ≤ cγ(m+1)(2d)ı+\uf6be sm−2µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='66)  Thus, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='64), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='65), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='66) and (H8),  A+⊤  ı \uf6be A+  ı \uf6be ≥ γ2  |k|2τ I(2d)ı+\uf6be − γ2 − γ2  +  3|k|2τ I(2d)ı+\uf6be − γ2 − γ2  +  3|k|2τ I(2d)ı+\uf6be − γ2 − γ2  +  3|k|2τ I(2d)ı+\uf6be ≥ γ2  +  |k|2τ I(2d)ı+\uf6be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Estimate on P+  In the following, we estimate the next step P+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Denote  ∆ = αm+1sm+1µ(sm−2µΓ2(r − r+) + Γ(r − r+)) + γ(m+1)(2d)m s2m−2µ2Γ(r − r+)  + γ(m+1)(2d)m sm+1µc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Assume (H0)-(H8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Then there is a constant c6 such that for all |ℓ| ≤ m  |∂ℓ  ξ ¯P+|D+×G+ ≤ c6∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='67)  Moreover, if  (H9) : c6∆ ≤ γ(m+1)(2d)m  +  sm  +µ+,  then  |∂ℓ  ξP+|D+×G+ ≤ γ(m+1)(2d)m  +  sm  +µ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='68)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Recall the definition of Q as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Observe by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='39) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='56) that  |∂ℓ  ξQ|D 1  4 α×G+ ≤ c(αs)m+1µΓ(r − r+)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='69)  and  |∂ℓ  ξ{Q, F}|D 1  4 α×G+ ≤ cαm+1sm+1sm−2µ2Γ2(r − r+) + cαm+1sm−1smµ2Γ2(r − r+)  + cαm+1smsm−1µ2Γ2(r − r+)  ≤ cαm+1s2m−1µ2Γ2(r − r+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='70)  20  Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='18) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='56) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='6, we also have  |∂ℓ  ξ[R]| ˆD×G+ + |∂ℓ  ξR| ˆD×G+ ≤ γ(m+1)(2d)m smµ,  and  |∂ℓ  ξ{[R], F}|D 1  4 α×G+ + |∂ℓ  ξ{R, F}|D 1  4 α×G+ ≤ cγ(m+1)(2d)m s2m−2µ2Γ(r − r+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='71)  Let us recall the definition of ¯P+ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='33), that is  ¯P+ =  � 1  0  {Rt, F} ◦ φt  Fdt + (P − R) ◦ φ1  F + Q,  Rt = (1 − t)Q + (1 − t)[R] + tR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Then by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='17), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='69), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='70) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='71),  |∂ℓ  ξ ¯P+|D 1  4 α×G+ ≤ cαm+1s2m−1µ2Γ2(r − r+) + cγ(m+1)(2d)m s2m−2µ2Γ(r − r+)  + cγ(m+1)(2d)m sm+1µ + c(αs)m+1µΓ(r − r+)  ≤ αm+1sm+1µ(sm−2µΓ2(r − r+) + Γ(r − r+))  + γ(m+1)(2d)m sm+1µ(sm−3µΓ(r − r+) + c),  which implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Moreover, by (H9) and the definition of P+ as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='34), we see  |∂ℓ  ξP+|D+×G+ ≤ γ(m+1)(2d)m  +  sm  +µ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  This completes one cycle of KAM steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Proof of Theorem 1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Iteration Lemma  In this section, we will prove an iteration lemma which guarantees the inductive construction of the transformations  in all KAM steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Let r0, s0, γ0, µ0, H0, N0, P0 be given at the beginning of Section 3 and let D0 = D(s0, r0), ˜D0 = D(β0, r0), K0 = 0,  Φ0 = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' We define the following sequence inductively for all ν = 1, 2, · · ·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  rν = r0(1  2 +  1  2ν+1 ),  βν = β0(1  2 +  1  2ν+1 ),  γν = γ0(1  2 +  1  2ν+1 ),  sν = 1  8αν−1sν−1,  αν = s2ρ  ν = s  1  m+1  ν  ,  µν = 8mc0µν−1sρ  ν−1,  Kν = ([log( 1  sν−1  )] + 1)3η,  Dν = D(sν, rν),  ˜Dν = D(βν, rν + 5  8(rν−1 − rν)),  21  D(sν) = {(y, z) : |y| < s2  ν, |z| < sν},  Gν = {ξ ∈ Gν−1 : |⟨k, ων−1⟩| > γ  |k|τ , Aν−1⊤  ı \uf6be  Aν−1  ı \uf6be  > γ2  |k|2τ I(2d)ı+\uf6be, for k ∈ Zn, 0 < |k| < Kν,  2|ı| + |\uf6be| ≤ m, 1 ≤ |\uf6be|}, where Aν−1  ı \uf6be  is defined as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Denote  µ∗ = s2ε  1  4(m+1) γ2(m+1)(2d)m  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  If ε is small enough, then the KAM step described in Section 3 is valid for all ν = 0, 1, · · ·, resulting in the sequences  eν, ων, ˜hν, gν, ¯gν, Hν, Pν, Φν,  ν = 1, 2, · · · , with the following properties:  (1) For all |ℓ| ≤ m,  |∂ℓ  ξ(eν+1 − eν)|D(sν+1)×Gν+1 ≤ µ  1  2∗  2ν ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='72)  |∂ℓ  ξ(eν − e0)|D(sν)×Gν ≤ 2µ  1  2∗ ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='73)  |∂ℓ  ξ(ων+1 − ων)|D(sν+1)×Gν+1 ≤ µ  1  2∗  2ν ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='74)  |∂ℓ  ξ(ων − ω0)|D(sν)×Gν ≤ 2µ  1  2∗ ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='75)  |∂ℓ  ξ(˜hν+1 − ˜hν)|D(sν+1)×Gν+1 ≤ µ  1  2∗  2ν ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='76)  |∂ℓ  ξ(˜hν − ˜h0)|D(sν)×Gν ≤ 2µ  1  2∗ ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='77)  |∂ℓ  ξ(gν+1 − gν)|D(sν+1)×Gν+1 ≤ µ  1  2∗  2ν ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='78)  |∂ℓ  ξ(gν − g0)|D(sν)×Gν ≤ 2µ  1  2∗ ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='79)  |∂ℓ  ξ(¯gν+1 − ¯gν)|D(sν+1)×Gν+1 ≤ µ  1  2∗  2ν ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='80)  |∂ℓ  ξ(¯gν − ¯g0)|D(sν)×Gν ≤ 2µ  1  2∗ ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='81)  |∂ℓ  ξ(ζν+1 − ζν)|Gν+1 ≤ (sm−1  ν−1 µν−1)  1  L ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='82)  |∂ℓ  ξPν|Dν×Gν ≤ γ(m+1)(2d)m  ν  sm  ν µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='83)  (2) Gν+1 = {ξ ∈ Gν : |⟨k, ων⟩| > γν  |k|τ , Aν⊤  ı \uf6be Aν  ı \uf6be >  γ2  ν  |k|2τ I(2d)ı+\uf6be, for k ∈ Zn, Kν < |k| ≤ Kν+1, 2|ı| + |\uf6be| ≤ m, 1 ≤ |\uf6be|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (3) There exists a family of symplectic coordinate transformations Φν+1 : ˜Dν+1×Gν+1 → ˜Dν and a subset Gν+1 ⊂ Gν,  Gν+1 = Gν \\  �  Kν<|k|≤Kν+1  Rν+1  k  (γν),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='84)  where  Rν+1  k  (γν) ={ξ ∈ Gν : |⟨k, ων⟩| ≤ γν  |k|τ , Aν⊤  ı \uf6be Aν  ı \uf6be ≤ γ2  ν  |k|2τ I(2d)ı+\uf6be,  for k ∈ Zn, Kν < |k| ≤ Kν+1, 2|ı| + |\uf6be| ≤ m, 1 ≤ |\uf6be|},  22  such that on ˜Dν+1 × Gν+1  Hν+1 = Hν ◦ Φν+1 = Nν+1 + Pν+1,  and  |∂ℓ  ξ∂i  (x,y,z)(Φν+1 − id)| ˜Dν+1×Gν+1 ≤ µ  1  2∗  2ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='85)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' The proof amounts to the verification of (H0)-(H9) for all ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' For simplicity, we let r0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' We can make s0  small by picking ε0 sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' So, we see that (H0) holds and (H1)-(H9) hold for ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Note that  µν = (8mc0)ν(1  8)  (m+2)((1+  1  m+1 )ν−1)−(ν−1)  2  s  1  2 ((1+  1  m+1 )ν−1)  0  µ0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='86)  sν = (1  8)(m+1)((1+  1  m+1 )ν−1)s  (1+  1  m+1 )ν  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='87)  Let θ ≫ 1 be fixed and s0 be small enough so that  s0 < (8  θ )  1  2ρ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='88)  Then  s1 = 1  8 s1+2ρ  0  < 1  θ s0 < 1,  s2 = 1  8 s1+2ρ  1  < 1  θ s1 < 1  θ2 s0,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  sν = 1  8 s1+2ρ  ν−1 < · · · < 1  θν s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='89)  Denote  Γν = Γ(rν − rν+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  We notice that  rν − rν+1  r0  =  1  2ν+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='90)  Since  Γν ≤  � ∞  1  t(m+1)(2d)mτ+me−  t  2ν+5 dt  ≤ ((m + 1)(2d)mτ + m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2(ν+5)((m+1)(2d)mτ+m),  it is obvious that for θ large enough,  s  ρ  2ν Γi  ν < s  ρ  2ν (Γi  ν + Γν) ≤ 1,  i = 1, 2,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='91)  and  sνΓν ≤ s1−ρ  ν  ≤  s1−ρ  0  θ(1−ρ)ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Recall αν = s  1  m+1  ν  , then αm+1  ν  = sν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' In view of s0 = sε  1  8(m+1) γ(m+1)(2d)m  0  , we have γ(m+1)(2d)m  0  > s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' It is obvious by the  definition of γν and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='87) that γ(m+1)(2d)m  ν  > sν if ε0 is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Moreover,  γ(m+1)(2d)m  ν  s  1  4(m+1)  ν  = (1  2 +  1  2ν+1 )(m+1)(2d)m(1  8)  1  4 ((1+  1  m+1 )ν−1)s  1  4(m+1) (1+  1  m+1 )ν  0  23  ≤ (1  2 +  1  2ν+2 )(m+1)(2d)m = γ(m+1)(2d)m  ν+1  as s0 small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' This together with the definition of ∆ν as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='9 yields  ∆ν = αm+1  ν  sm+1  ν  µν(sm−2  ν  µνΓ2(rν − rν+1) + Γ(rν − rν+1))  + γ(m+1)(2d)m  ν  s2m−2  ν  µ2  νΓ(rν − rν+1) + γ(m+1)(2d)m  ν  sm+1  ν  µνc  ≤ γ(m+1)(2d)m  ν  s  1  4(m+1)  ν  s  (1+  1  m+1 )m  ν  s  1  2(m+1)  ν  µν(s  1  4(m+1)  ν  sm−2  ν  µνΓ2(rν − rν+1)  + s  1  4(m+1)  ν  Γ(rν − rν+1) + s  1  4(m+1)  ν  sm−3  ν  µΓ(rν − rν+1) + s  1  4(m+1)  ν  c)  ≤ γ(m+1)(2d)m  ν+1  sm  ν+1µν+1,  which implies that (H9) holds for all ν ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  We note that for any constant a > 0, b > 1, sa(log 1  s + 1)b → 0 as s → 0, then  3γ2  0sνKτ+1  ν+1 ≤ 3γ2  0sν([log 1  sν  ] + 1)3η(2τ+1)  ≤ (1  2 +  1  2ν+1 )2 − (1  2 +  1  2ν+2 )2  = γ2  ν − γ2  ν+1,  if ε0 is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Hence, (H8) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='90) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='91), it is easy to verify that (H5)-(H7) hold for all ν ≥ 1 if θ is large enough and ε0 is small  enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  It is obvious by m ≥ L+  √  L2+16L+16  4  , L ≥ 2 as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='7) that  m − 1  L  > 2m + 2  m + 1 > (m + 2  m + 1)2,  then  s  m−1  L  −  < s  ( m+2  m+1 )2  −  ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=',  (sm−1  −  µ−)  1  L < (1  8)1+ m+2  m+1 s  ( m+2  m+1 )2  −  = 1  8 s  m+2  m+1 = 1  8αs,  which implies (H4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  To verify (H3), we observe by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='87) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='89) that  4(M∗ + 2)sνKτ+1  ν+1 <  s0  2ν+2 < γ0  2ν+2 ,  as θ is large enough, which verifies (H3) for all ν ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Let θ1−ρ ≥ 2 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='88), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='89).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' We have that for all ν ≥ 1  sν ≤ s0  2ν ≤ µ  1  2∗  2ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='92)  The verification of (H2) follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='92) and an inductive application of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='53) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='5 for all ν = 0, 1, · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Since (1 + ρ)η > 2, we have  r0  2ν+6 ([log 1  s ] + 1)η ≥  r0  2ν+6 (−(1 + ρ)ν log s0)η  ≥ − r0  2ν+6 (1 + ρ)ην(log µ0)η  ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  24  It follows from the above that  log(n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' + (ν + 6)n log 2 + 3nη log([log 1  s ] + 1) −  r0  2(ν+6) ([log 1  s ] + 1)3η  ≤ log(n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' + (ν + 6)n log 2 + 3nη log(log 1  s + 2) − (log 1  s )2η  ≤ − log 1  s,  as µ is small, which is ensured by making ε small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Thus,  � ∞  Kν+1  tne−  tr0  2ν+6 dt ≤ (n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2(ν+6)nKn  ν+1e−  Kν+1  2ν+6 ≤ s,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' (H1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Above all, the KAM steps described in Section 3 are valid for all ν, which gives the desired sequences stated in  the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Now, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='72), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='74), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='76), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='78) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='80) follow from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='53), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='54) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='55) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='5 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='92);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  by adding up (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='72), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='52), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='76), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='78) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='80) for all ν = 0, 1, · · ·, we can get (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='73), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='75) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='77), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='79)  and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='81) respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='83) follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='68) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='82) follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' (2) follows from  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Note that (2) automatically holds for ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' We now let ν > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' By Subsubsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='9, it is obvious that  Gν = {ξ ∈ Gν : |⟨k, ων⟩| > γν  |k|τ , Aν⊤  ı \uf6be Aν  ı \uf6be > γ2  ν  |k|2τ I(2d)ı+\uf6be, for k ∈ Zn, 0 < |k| ≤ Kν, 2|ı| + |\uf6be| ≤ m, 1 ≤ |\uf6be|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Denote  ˆGν+1 = {ξ ∈ Gν : |⟨k, ων⟩| > γν  |k|τ , Aν⊤  ı \uf6be Aν  ı \uf6be > γ2  ν  |k|2τ I(2d)ı+\uf6be, for k ∈ Zn, Kν < |k| ≤ Kν+1, 2|ı| + |\uf6be| ≤ m, 1 ≤ |\uf6be|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Then  Gν+1 ={ξ ∈ Gν : |⟨k, ων⟩| > γν  |k|τ , Aν⊤  ı \uf6be Aν  ı \uf6be > γ2  ν  |k|2τ I(2d)ı+\uf6be, for k ∈ Zn, 0 < |k| ≤ Kν+1, 2|ı| + |\uf6be| ≤ m, 1 ≤ |\uf6be|}  =Gν ∩ ˆGν+1 = ˆGν+1,  which implies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='84).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Convergence  The convergence is standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' For the sake of completeness, we briefly give the outline of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Let  Ψν = Φ0 ◦ Φ2 ◦ · · · ◦ Φν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Recalling that Φν+1 : ˜Dν+1 → ˜Dν in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='1, we have  Ψν : ˜Dν → ˜D0,  H0 ◦ Ψν = Hν = Nν + Pν,  ν = 0, 1, · · · , where Ψ0 = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Let G∗ = �∞  ν=0 Gν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' First, we show the uniform convergence of Ψν on D( β0  2 , r0  2 ) × G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Note that  Ψν = id +  ν  �  i=1  (Ψi − Ψi−1),  25  where, for each i = 1, 2, · · ·,  Ψi − Ψi−1 = Φ0 ◦ · · · ◦ Φi − Φ0 ◦ · · · ◦ Φi−1  ≤  � 1  0  D(Φ0 ◦ · · · ◦ Φi−1)(id + θ(Φi − id))dθ(Φi − id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Since, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='85), on D( β0  2 , r0  2 ) × G∗,  |D(Φ0 ◦ · · · ◦ Φi−1)(id + θ(Φi − id))|  ≤ |DΦ0(Φ1 ◦ · · · ◦ Φi−1)(id + θ(Φi − id))| · · ·|DΦi−1(id + θ(Φi − id))|  ≤ (1 + µ  1  2∗ )(1 + µ  1  2∗  2 ) · · ·(1 + µ  1  2∗  2i−1 ) ≤ e1+ 1  2 +···+  1  2i−1 ≤ e2,  we have  |Ψi − Ψi−1|D( β0  2 , r0  2 )×G∗ ≤ e2|Φi − id|D( β0  2 , r0  2 )×G∗ ≤ e2 µ  1  2∗  2i ,  for all i = 1, 2, · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Thus, Ψν converges uniformly on D( β0  2 , r0  2 ) × G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' We denote its limit by Ψ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Then  Ψ∗ = Ψ0 +  ∞  �  i=1  (Ψi − Ψi−1),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='93)  which is uniformly close to the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' By a standard argument using the Whitney Extension Theorem, one can  further show that Ψ∗ is Whitney smooth with respect to ξ ∈ G∗ (see [18], [24] ,[25], [32] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' By Lemma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='1,we see that eν, ων, ˜hν, gν, ¯gν and ζν are uniformly convergent and denote the limits by e∗, ω∗, ˜h∗, g∗, ¯g∗ and ζ∗,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' It follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='4 that  ∇g∗(0) = · · · = ∇gν(0) = · · · = ∇g0(ζ0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Then, Nν converge uniformly to  N∗ = e∗ + ⟨ω∗, y⟩ + ˜h∗(y) + g∗(z) + ¯g∗(yız\uf6be),  with  ˜h∗(y) = O(|y|2),  g∗(z) = g(z) +  ∞  �  i=0  γ(m+1)(2d)m  i  sm−2  i  µiO(|z|2)  = g(z) + ε  (m−2)(5m+4)  8m(m+1) O(|z|2),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='94)  ¯g∗(y, z, ξ) =  �  2|ı|+| \uf6be|≤m,1≤|ı|,| \uf6be|  ¯gı \uf6be(ξ)yız\uf6be,  where (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='94) follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='92) and the definition of µ∗ in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Hence  Pν = H0 ◦ Ψν − Nν  converges uniformly to  P∗ = H0 ◦ Ψ∗ − N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Since  |Pν|Dν ≤ γ(m+1)(2d)m  ν  sm  ν µν,  26  the Cauchy estimate implies that  |∂i  y∂ j  zPν|D( sν  2 ,rν) ≤ γ(m+1)(2d)m  ν  s(m−2i− j)  ν  µν,  for all 2|i| + |j| ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='86) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='87), it is easy to see that the right hand side of the above converges to 0 as  ν → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Hence, on D(0, r0  2 ) × G∗,  ∂ j  y∂ j  zP∗|(y,z)=(0,0) = 0,  for all x ∈ T n, ω ∈ G∗, 2|i| + |j| ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' It follows that for each ξ ∈ G∗, T n × {0} × {0} is an analytic, quasi-periodic,  invariant n-torus associated to the Hamiltonian H∞ = H ◦ Ψ∞ with frequency ω∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Measure estimate  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  |G0\\G∗| → 0, as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Since all the eigenvalues of normal direction are 0 in 0-th step, the measure estimate of G1 in the first step  can be reduced to the estimate of ⟨k, ω⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Moreover, if ˜S ν  ı \uf6be ≡ 0, the measure estimate of Gν in every step also can be  reduced to the estimate of ⟨k, ω⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' For details, see [8, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Otherwise, if ˜S ν  ı \uf6be � 0, the measure estimate of Gν is referred  to [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' For the sake of completeness, we give the outline of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Recall  Rν+1  k  (γν) ={ξ ∈ Gν : |⟨k, ων⟩| ≤ γν  |k|τ , Aν⊤  ı \uf6be Aν  ı \uf6be ≤ γ2  ν  |k|2τ I(2d)ı+\uf6be,  for k ∈ Zn, Kν < |k| ≤ Kν+1, 2|ı| + |\uf6be| ≤ m, 1 ≤ |\uf6be|}  ⊂S1 ∪ S2,  where  S1 = {ξ ∈ Gν : |⟨k, ων⟩| ≤ γν  |k|τ },  S2 = {ξ ∈ Gν : Aν⊤  ı \uf6be Aν  ı \uf6be ≤ γ2  ν  |k|2τ I(2d)ı+\uf6be}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  With Taylor series,  Aı \uf6be(ξ) = L(ξ0)Θ(ξ),  where  L(ξ0) = (Aı \uf6be(ξ0), ∂ξAı \uf6be(ξ0), · · · , ∂i  ξAı \uf6be(ξ0),  � 1  0  (1 − t)|i+1|∂i+1  ξ Aı \uf6be(ξ0 + tˆξ)dt),  Θ(ξ) = (I, ˆξI, · · · , ˆξiI, ˆξi+1I)⊤,  ˆξ = ξ − ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Noting that ˜S ı \uf6be comes from the perturbation and (A1), we have  rank{C j : 1 ≤ j ≤ (2d)ı+ \uf6be} = (2d)ı+ \uf6be,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='95)  where A j  ı \uf6be is the j-th column of Aı \uf6be, C j is a column of {∂i  ξA j  ı \uf6be : 0 ≤ |i| ≤ M} and Aı \uf6be is defined as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='95)  and the continuity of determinant, there is an orthogonal matrix Qξ0 such that  L(ξ)Qξ0 = (E(ξ), F(ξ)),  for ξ ∈ ¯Gξ0,  where E(ξ) is a (2d)ı+ \uf6be × (2d)ı+ \uf6be nonsingular matrix on ¯Gξ0, and ¯Gξ0 is the closure of a neighborhood Gξ0 of ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Moreover,  Aı \uf6be = L(ξ)Qξ0Q−1  ξ0 Θ = (E(ξ), F(ξ)) ˜Θ,  27  A⊤ı \uf6be = ˜Θ⊤(E(ξ), F(ξ))⊤,  ˜Θ = Q−1  ξ0 Θ =  � ˜Θ1  ˜Θ2  �  ,  where ˜Θ1 is a (2d)ı+ \uf6be × (2d)ı+ \uf6be-order matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Obviously,  rank  � E⊤E  E⊤F  F⊤E  F⊤F  �  = (2d)ı+ \uf6be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Then there is an unitary matrix Uξ such that  U−1  ξ  � E⊤E  E⊤F  F⊤E  F⊤F  �  Uξ =  � diag(λ1, · · · , λ(2d)ı+\uf6be)  0  0  0  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='96)  where λi, 1 ≤ i ≤ (2d)ı+ \uf6be, is nonvanishing eigenvalue of  � E⊤E  E⊤F  F⊤E  F⊤F  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' By Poincar´e Separation Theorem (for  example, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 96, [40]),  A⊤ı \uf6beAı \uf6be ≥ ˜Θ⊤  � E⊤E  E⊤F  F⊤E  F⊤F  �  ˜Θ  ≥ ˜Θ⊤  � diag(λ1, · · · , λ(2d)ı+\uf6be)  0  0  0  �  ˜Θ  ≥  �  ˜Θ⊤  1  ˜Θ⊤  2  � � diag(λ1, · · · , λ(2d)ı+\uf6be)  0  0  0  � � ˜Θ1  ˜Θ2  �  ≥ min  i  min  ¯Oξ0  λi ˜Θ⊤  1 ˜Θ1  ≥ min  i  min  ¯Oξ0  λi(min  j |ˆξ j|)2M+2I(2d)ı+\uf6be,  where ”≥” is L¨ower order (for specific definition, see [40]) and the last equality has been explicitly explained on page  12 of [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Hence, with finite cover theorem, we have  |S2| = |{ξ ∈ G, 0 ≤ A⊤ı \uf6beAı \uf6be ≤ γ2  |k|2τ I(2d)ı+\uf6be}| ≤ c γ  1  M+1  |k|  τ  M+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Similarly,  |S1| = |{ξ ∈ G, |⟨k, ω(ξ)⟩ ≤ γ  |k|τ | ≤ c γ  1  M+1  |k|  τ  M+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Then |Rν+1| <  γ  1  M+1  0  |k|  τ  M+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Consequently,  |G0\\G∗| = |  ∞  �  ν=0  �  Kν<|k|≤Kν+1  Rν+1  k  | ≤  ∞  �  ν=0  �  Kν<|k|≤Kν+1  γ  1  M+1  0  |k|  τ  M+1 → 0, as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  28  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Notice that  ∇g(u, v) = (u|u|2l0−2, v|v|2k0−2),  ∇g(0, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  For 0 < δ < 1, Bδ(0) denotes the open ball centered at the origin with radius δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' We have that ∇g(u, v) − ∇g(0, 0) is odd  and unequal to zero on ∂Bδ(0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=',  ∇g(−u, −v) − ∇g(0, 0) = (−u|u|2l0−2, −v|v|2k0−2) = −(∇g(u, v) − ∇g(0, 0)),  and  ∇g(u, v) − ∇g(0, 0) � 0, ∀y ∈ ∂Bδ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  It follows from Borsuk’s theorem in [31] that  deg(∇g(u, v) − ∇g(0, 0), Bδ(0), 0) � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  There exist σ = 1 and L = 2 min{l0, k0} − 1 such that  |∇g(u, v) − ∇g(0, 0)| > σ|(u, v)|L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  So, by Theorem 1, the results in Proposition 1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Proof of Proposition 2  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Note that the motion equation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='6) is  \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3  ˙x = ω,  ˙y = 0,  ˙u = v2,  ˙v = −u2 − ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Since for any ε � 0, the equation u2 + ε2 = 0 has no real solution, we see that the Hamiltonian system does not admit  any lower-dimensional torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Let g(u, v) = 1  3u3 + 1  3v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' By simple calculation, we have  deg(∇g(u, v), Bδ(0), 0) = 0,  which implies that (A0) fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Hence our assumption (A0) is sharp to Melnikov’s persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Acknowledgments  The second author (Li Yong) is supported by National Basic Research Program of China (grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 2013CB834100),  National Natural Science Foundation of China (grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 11571065, 11171132, 12071175), and Natural Science  Foundation of Jilin Province (grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 20200201253JC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  29  References  [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Arnold, Proof of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Kolmogorov’s theorem on the conservation of conditionally periodic motions with a small variation in the Hamil-  tonian, Uspehi Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Nauk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 18(1963), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 5, 9-36  [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE,  Internat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Notices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 11(1994), 475-497  [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Bourgain, On Melnikov’s persistency problem, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 4(1997), 445-458  [4] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Broer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Hanßmann, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' You, On the destruction of resonant lagrangean tori in Hamiltonian systems, Springer Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  35(2013)  [5] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Cheng, Lower dimensional invariant tori in the regions of instability of nearly integrable Hamiltonian systems, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  203(1999), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 2, 385-419  [6] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Chierchia and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Qian, Moser’s theorem for lower dimensional tori, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Differential Equations 206(2004), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 1, 55-93  [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Chow, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Li, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Yi, Persistence of invariant tori on submanifolds in Hamiltonian systems, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Nonlinear Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 12(2002), 585-617  [8] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Cong, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' K¨upper, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Li, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' You, KAM-type theorem on resonant surfaces for nearly integrable Hamiltonian systems, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Nonlinear  Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 10(2000), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 1, 49-68  [9] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' de la Llave and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Xu, Whiskered tori for presymplectic dynamical systems, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Dynam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Differential Equatioins 33(2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 1, 1-34  [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Delshams and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Guti´errez, Effective stability and KAM theory, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Differential Equations 128(1996), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 2, 415-490  [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Delshams and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Guti´errez, Estimates on invariant tori near an elliptic equilibrium point of a Hamiltonian system, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Differential Equations  131(1996), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 2, 277-303  [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Delshams and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Guti´errez, Splitting potential and the Poincar´e-Melnikov method for whiskered tori in Hamiltonian systems, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Nonlinear  Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 10(2000), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 4, 433-476  [13] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Eliasson, Perturbations of stable invariant tori for Hamiltonian systems, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Scuola Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Sup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Pisa Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 15(1988), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 1, 115-147  [14] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Eliasson, Biasymptotic solutions of perturbed integrable Hamiltonian systems, Bol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Brasil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 25(1994), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 1, 57-76  [15] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Gallatti and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Gentile, Hyperbolic low-dimensional invariant tori and summations of divergent series, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 227(2002),  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 3, 421-460  [16] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Gentile, Degenerate lower-dimensional tori under the Bryuno condition, Ergodic Theory Dynam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Systems 27(2007), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 2, 427-457  [17] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Graff, On the continuation of hyperbolic invariant tori for Hamiltonian systems, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Differential Equations 15(1974), 1-69  [18] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Han, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Li, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Yi, Degenerate lower-dimensional tori in Hamiltonian systems, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Differential Equations 227(2006), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 2, 670-691  [19] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Hu and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Liu, Completely degenerate lower-dimensional invariant tori for Hamiltonian system, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Differential Equations 266(2019),  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 11, 7459-7480  [20]  `A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Jorba and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Villanueva, On the persistence of lower dimensional invariant tori under quasi-periodic perturbations, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Nonlinear Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  7(1997), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 5, 427-473  [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Kolmogorov, On preservation of conditionally periodic motions under a small change in the Hamiltonian function, In Dokl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Nauk SSSR 98(1954), 527-530  [22] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Kuksin, Nearly integrable infinite dimensional Hamiltonian systems, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1993  [23] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Kuksin, Hamiltonian perturbations of infinite dimensional linear systems with an imaginary spectrum, Funktsional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' i Prilozhen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  21(1987), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='3, 22-37  [24] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Li and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Yi, A quasi-periodic Poincar´e’s theorem, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 326(2003), 649-690  [25] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Li and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Yi, Persistence of lower dimensional tori of general types in Hamiltonian systems, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 357(2005), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 4,  1565-1600  [26] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Li and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Yi, Persistence of hyperbolic tori in Hamiltonian systems, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Differential Equations 208(2005), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 2, 344-387  [27] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Medvedev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Neidhtadt, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Treschev, Lagrangian tori near resonances of near-integrable Hamiltonian systems, Nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  28(2015), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 7, 2105-2130  [28] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Melnikov, On certain cases of conservation of almost periodic motions with a small change of the Hamiltonian function, Dokl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Nauk SSSR 165(1965), 1245-1248  [29] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Moser, On invariant curves of area-preserving mapping of an annulus, Nachr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Wiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' G¨ottingen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='II 1962(1962), 1-20  [30] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Moser, Convergent series expansions for quasi-periodic motions, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 169(1967), 136-176  [31] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Motreanu, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Motreanu, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Papageorgiou, Topological and variational methods with applications to nonlinear boundary value  problems, Springer Science+Business Media, LLC, 2014  [32] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' P¨oschel, Integrability of Hamiltonian systems on Cantor-like sets, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 35(1982), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 5, 653-696  [33] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' P¨oschel, On elliptic lower-dimensional tori in Hamiltonian systems, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 202(1989), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 4, 559-608  [34] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Qian, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Li, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Yang, Multiscale KAM theorem for Hamiltonian systems, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Differential Equations 266(2019), 70-86  [35] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Qian, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Li, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Yang, Melnikov’s conditions in matrices, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Dynam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Differential Equations (2020)32, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 4, 1779-1795  [36] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Rudnev and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Wiggins, KAM theory near multiplicity one resonant surfaces in perturbations of a priori stable Hamiltonian systems, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Nonlinear Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 7(1997), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 2, 177-209  [37] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Salamon, The Kolmogorov-Arnold-Moser theorem, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 10(2004), 1-37  [38] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Takens, Singularities of vector fields, Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Hautes ´Etudes Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 43(1974), 47-100  [39] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Treshchev, The mechanism of destruction resonance tori in Hamiltonian systems, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' USSR-Sb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 68(1991), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 1, 181-203  [40] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Wu and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Jia, Matrix Inequality, Science Press, Beijing (2006)  [41] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 127(1990), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 3,  479-528  [42] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Xu, Persistence of elliptic lower dimensional invariant tori for small perturbation of degenerate integrable Hamiltonian systems, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 208(1997), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 2, 372-387  [43] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Xu and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' You, Persistence of hyperbolic-type degenerate lower-dimensional invariant tori with perscribed frequencies in Hamioto-  nian systems, Regul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Chaotic Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 25(2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 6, 616-650  30  [44] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Li, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Yi, Lower-dimensional tori in multi-scale nearly integrable Hamiltonian systems, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Henri Poincar´e 18(2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  1, 53-83  [45] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' You, A KAM theorem for hyperbolic-type degenerate lower dimensional tori in Hamiltonian systems, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 192(1998),  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 1, 145-168  [46] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Zehnder, Generalized implicit function theorems with applications to some small divisor problems I and II, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content='  28(1975), 91-140;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 29(1976), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
+page_content=' 1, 49-111  31' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAyT4oBgHgl3EQfYffL/content/2301.00206v1.pdf'}
diff --git a/mtE0T4oBgHgl3EQf8QKN/content/tmp_files/2301.02786v1.pdf.txt b/mtE0T4oBgHgl3EQf8QKN/content/tmp_files/2301.02786v1.pdf.txt
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@@ -0,0 +1,2612 @@
+The JCMT SCUBA-2 Survey of the James Webb Space Telescope North Ecliptic Pole
+Time-Domain Field
+Minhee Hyun1,2
+, Myungshin Im2
+, Ian R. Smail3
+, William D. Cotton4
+, Jack E. Birkin3
+, Satoshi Kikuta5
+,
+Hyunjin Shim6
+, Christopher N. A. Willmer7
+, James J. Condon4
+, Rogier A. Windhorst8
+, Seth H. Cohen8
+,
+Rolf A. Jansen8
+, Chun Ly7
+, Yuichi Matsuda9
+, Giovanni G. Fazio10
+, A. M. Swinbank3
+, and Haojing Yan11
+1 Korea Astronomy and Space Science Institute, 776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea; minhee@kasi.re.kr
+2 SNU Astronomy Research Center, Astronomy Program, Department of Physics & Astronomy, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826,
+Republic of Korea; myungshin.im@gmail.com
+3 Centre for Extragalactic Astronomy, Department of Physics, Durham University, South Road, Durham, DH1 3LE, UK
+4 National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903, USA
+5 Center for Computational Sciences, University of Tsukuba, Ten-nodai, 1-1-1 Tsukuba, Ibaraki 305-8577, Japan
+6 Department of Earth Science Education, Kyungpook National University, 80 Daehak-ro, Buk-gu, Daegu 41566, Republic of Korea
+7 Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA
+8 School of Earth & Space Exploration, Arizona State University, Tempe, AZ 85287-1404, USA
+9 National Astronomical Observatory of Japan, Mitaka, Japan
+10 Center for Astrophysics, Harvard & Smithsonian, 60 Garden Street, MS-65, Cambridge, MA 02138-1516, USA
+11 Department of Physics and Astronomy, University of Missouri, Columbia, MO 65211, USA
+Received 2022 May 20; revised 2022 September 26; accepted 2022 September 26; published 2023 January 5
+Abstract
+The James Webb Space Telescope Time-Domain Field (JWST-TDF) is an ∼14′ diameter field near the North
+Ecliptic Pole that will be targeted by one of the JWST Guaranteed Time Observations programs. Here, we describe
+our James Clerk Maxwell Telescope SCUBA-2 850 μm imaging of the JWST-TDF and present the submillimeter
+source catalog and properties. We also present a catalog of radio sources from Karl J. Jansky Very Large Array
+3 GHz observations of the field. These observations were obtained to aid JWSTʼs study of dust-obscured galaxies
+that contribute significantly to cosmic star formation at high redshifts. Our deep 850 μm map covers the JWST-
+TDF at a noise level of σ850mm = 1.0 mJy beam−1, detecting 83/31 sources in the main/supplementary signal-to-
+noise ratio (S/N > 4 / S/N = 3.5–4) sample, respectively. The 3 GHz observations cover a 24′ diameter field with
+a 1σ noise of 1 μJy beam−1 at a 0 7 FWHM. We identified eighty-five 3 GHz counterparts to sixty-six 850 μm
+sources and then matched these with multiwavelength data from the optical to the mid-infrared wave bands. We
+performed spectral energy distribution fitting for 61 submillimeter galaxies (SMGs) matched with optical/near-
+infrared data, and found that SMGs at S/N > 4 have a median value of zphot = 2.22 ± 0.12, star formation rates of
+300 ± 40 Me yr−1 (Chabrier initial mass function), and typical cold dust masses of 5.9 ± 0.7 × 108 Me, in line
+with bright SMGs from other surveys. The large cold dust masses indicate correspondingly large cool gas masses,
+which we suggest are a key factor necessary to drive the high star formation rates seen in this population.
+Unified Astronomy Thesaurus concepts: Galaxy evolution (594); High-redshift galaxies (734); Galaxy formation
+(595); Submillimeter astronomy (1647); Galaxy counts (588); Ultraluminous infrared galaxies (1735)
+Supporting material: extended figure, machine-readable tables
+1. Introduction
+Tracing the formation of galaxies and the associated star
+formation across cosmic time is a key topic for understanding the
+evolution of our universe. Most of the star formation activity over
+cosmic history is hidden by surrounding dust, making it often
+difficult to observe this activity through UV and optical studies
+(e.g., Madau & Dickinson 2014; Driver et al. 2016; Koushan et al.
+2021). As such, it is important to study this hidden star formation
+in other wave bands to draw a complete picture of the evolution of
+galaxies, especially those in the early universe.
+Dust around young stars efficiently absorbs their UV/optical
+radiation and this is reradiated in the mid-infrared (MIR) and far-
+infrared (FIR). Consequently, MIR and FIR surveys of galaxies
+have revealed a population of luminous infrared galaxies, whose
+contribution to the cosmic star formation rate (SFR) has been
+found to rise toward z ∼ 1.5 (e.g., Goto et al. 2010, 2019;
+Magnelli et al. 2013; Kim et al. 2015). At higher redshifts,
+z ? 1.5, the FIR emission from galaxies is redshifted into the
+submillimeter (submm) window. As a result, surveys in the
+submm are well placed to find strongly star-forming, dust-
+obscured galaxies at high redshifts, which are commonly termed
+submm galaxies (SMGs)—defined as galaxies that are bright in
+the submm, S850mm > 1 mJy (Hodge & da Cunha 2020).
+From their first discovery (Smail et al. 1997; Barger et al. 1998;
+Hughes et al. 1998; Eales et al. 1999), SMGs have been an
+important population for understanding the most obscured phases
+of galaxy formation and evolution at high redshifts. These galaxies
+are bolometrically luminous (∼1012–13 L☉; Barger et al. 1998) and
+massive (;1011 M☉; Hainline et al. 2011; Michałowski et al. 2012;
+Dudzevičiūtė et al. 2020) and contain a large amount of gas
+(∼1011 M☉; Greve et al. 2005; Tacconi et al. 2008; Carilli et al.
+2010; Bothwell et al. 2013; Birkin et al. 2021). SMGs are
+considered the progenitors of elliptical or spheroidal galaxies in the
+local universe (Lilly et al. 1999; Miller et al. 2018) from the fact
+that they reside in massive dark matter halos with M ∼ 1012−13 M☉
+The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January
+https://doi.org/10.3847/1538-4365/ac9bf4
+© 2023. The Author(s). Published by the American Astronomical Society.
+Original content from this work may be used under the terms
+of the Creative Commons Attribution 4.0 licence. Any further
+distribution of this work must maintain attribution to the author(s) and the title
+of the work, journal citation and DOI.
+1
+
+(Blain et al. 2004; Farrah et al. 2006; Magliocchetti et al. 2007;
+Wilkinson et al. 2017; Stach et al. 2021) and from their rapid
+formation (Lilly et al. 1999; Swinbank et al. 2006; Simpson et al.
+2014). With the benefits of the strong negative K-correction in the
+submm, SMGs are relatively easy to identify even at very high
+redshifts (z ∼ 6, e.g., Riechers et al. 2017; Gruppioni et al. 2020).
+They are now widely studied with various facilities and in
+numerous fields (Scott et al. 2002; Coppin et al. 2006; Weiß et al.
+2009; Ikarashi et al. 2011; Geach et al. 2017; Simpson et al. 2019;
+Shim et al. 2022; see also Casey et al. 2014).
+One of the key questions about SMGs is what physical
+mechanism triggers their star formation activity. We might
+expect that most SMGs arise from galaxy mergers based on
+studies in the local universe that show that the majority of
+ultraluminous infrared galaxies (ULIRGs) are merging systems
+(Sanders & Mirabel 1996; Lonsdale et al. 2006). Merging
+provides an effective mechanism to drive intense star formation
+and
+active
+galactic
+nucleus
+(AGN)
+activity
+(Barnes
+&
+Hernquist 1991; Mihos & Hernquist 1996; Hopkins et al.
+2005; Springel et al. 2005; Hong et al. 2015; McAlpine et al.
+2019). Many observational studies have found distorted
+morphologies for SMGs (Swinbank et al. 2004; Menendez-
+Delmestre et al. 2007; Tacconi et al. 2008; Engel et al. 2010;
+Chen et al. 2015) suggestive of mergers. Some semianalytic
+models also support the scenario that the vigorous star
+formation in SMGs is driven by galaxy merging (e.g., Chen
+et al. 2015). However, the very strong dust obscuration, even in
+the rest-frame near-infrared (NIR), makes it difficult to
+distinguish
+truly
+interacting
+systems
+from
+those
+where
+structured dust obscuration creates apparent asymmetric or
+disturbed morphologies. The stellar-mass morphologies of
+SMGs are one area where the recently launched James Webb
+Space Telescope (JWST) is expected to make significant
+contributions, through high-resolution MIR imaging that is
+much less sensitive to dust obscuration.
+The JWST Time-Domain Field (JWST-TDF; Jansen &
+Windhorst 2018) is an ∼14′ diameter region in JWSTʼs
+northern continuous viewing zone near the North Ecliptic Pole
+(NEP). The JWST-TDF has a unique advantage as an
+extragalactic survey field in that it can be observed at any
+cadence with JWST and is also expected to be frequently
+visited due to spacecraft housekeeping activities. In addition,
+the field is free from sources brighter than mAB ∼ 16 and has
+low zodiacal foreground, as well as low Galactic extinction.
+The planned JWST observations (GTO 1176, PI: R. Windhorst)
+include eight-filter 0.8–5.0 μm imaging with the Near-Infrared
+Camera (NIRCam) and 1.75–2.23 μm grism spectroscopy and
+2 μm direct imaging with the Near Infrared Imager and Slitless
+Spectrograph (NIRISS). These observations will detect sources
+down to mAB ; 29 mag, and this sensitivity (and spatial
+resolution) makes the JWST-TDF ideal for exploring the
+morphologies of SMGs, especially those that are faint in the
+rest-frame optical/NIR.
+To take advantage of the JWST-TDF for SMG studies, we
+have undertaken a submm survey covering the full JWST-TDF
+using the Submillimetre Common-user Bolometer Array 2
+(SCUBA-2; Holland et al. 2013) of the James Clerk Maxwell
+Telescope (JCMT). We call this survey the SCUBA-2 JWST-
+TDF Survey (S2TDF) and to fully exploit its data we have also
+obtained sensitive, high-resolution 3 GHz observations of this
+region using the Karl J. Jansky Very Large Array (VLA).
+Together these two observational data sets support the wider
+multiwavelength studies by the JWST-TDF GTO team. In
+particular, we expect that our SCUBA-2 and VLA surveys of
+the JWST-TDF will provide critical information on the
+redshifted FIR and radio emission, and thus on the dust-
+obscured star formation rates and AGN activity, in galaxies to
+be detected by JWST and will be useful for the wider
+astronomical community for years to come. In this paper, we
+present the S2TDF, and the catalog of SCUBA-2 sources in
+this field, as well as details of the VLA 3 GHz survey of this
+region used to identify the counterparts to these SCUBA-2
+sources. Section 2 presents the SCUBA-2 850 μm observations
+and the data reduction process (the corresponding description
+of the 3 GHz VLA observations and their reduction and
+cataloging is given in Appendix A). In Section 3, we describe
+the data analysis methods used to construct the catalog of
+850 μm sources, such as jackknife simulations for deriving flux
+deboosting, completeness, the false detection rate, and posi-
+tional accuracy. In Section 4, we discuss the matching of the
+SCUBA-2 sources with the 3 GHz radio and optical/NIR data
+to robustly identify the SMG counterparts of the SCUBA-2
+sources, followed by the analysis of the spectral energy
+distributions (SEDs) of these counterparts using the available
+multiwavelength data. Section 5 presents the 850 μm number
+counts in the JWST-TDF and examines the properties of the
+radio- and optical/NIR-matched SMGs such as their redshifts,
+SFRs, and dust masses. In Section 6, we summarize our results.
+Throughout this paper, all magnitudes are given in the AB
+magnitude system (Oke & Gunn 1983) and we adopt
+cosmological parameters of H0 = 70 km s−1 Mpc−1, ΩΛ = 0.7,
+and ΩM = 0.3. For the derivation of SMG properties with
+SED fitting, we use the Chabrier initial mass function
+(Chabrier 2003).
+2. Observation and Data Reduction
+In the following section we describe in detail the SCUBA-2
+observations of the JWST-TDF. The corresponding description
+of the VLA 3 GHz observations is given in Appendix A.
+2.1. SCUBA-2 Observations
+The SCUBA-2 850 μm data were obtained from 2018
+December
+to
+2020
+December
+(programs:
+M18BP026,
+M19BP031, and R20XP003) with the SCUBA-2 (Holland
+et al. 2013), the submm continuum bolometer array mounted
+on the JCMT. We used the PONG900 mapping mode, which
+generates a 15′ diameter circular map with nearly uniform
+sensitivity (Chapin et al. 2013). As a consequence, our data
+cover the full area of the 14′ diameter JWST-TDF, centered on
+17 22 47.9,
++65 49 22
+(J2000).
+Figure
+1
+illustrates
+the
+SCUBA-2 area, overlaid on the field of view of other
+multiwavelength coverages of the JWST-TDF. While we
+observed simultaneously at 450 and 850 μm, the weather
+conditions were generally not favorable for deep 450 μm
+observations12 (Figure 2). Therefore, we focus only on the deep
+850 μm data in this paper.
+The observation was split into 62 MSBs, each of which
+corresponds to a map with an on-source exposure time of
+∼40 minutes. The total on-source exposure time is 41.3 hr
+(Figure 2). The sky opacity (τ225), representing the weather
+12 The rms noise was ∼16 mJy beam−1 even in the deepest region of the
+stacked 450 μm map using only Band 1 data. The only significant source
+detected is a radio-loud QSO (TDF.0005).
+2
+The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January
+Hyun et al.
+
+conditions of the observation, ranged from 0.02 to 0.11, with a
+median value of 0.09. While we were awarded Band 3
+SCUBA-2 time for the 850 μm imaging, 13% and 21% of the
+data were obtained in better weather conditions in Band 1
+(τ225 < 0.05) and Band 2 (0.05 < τ225 < 0.08). The observation
+log is summarized in Figure 2.
+2.2. Data Reduction
+The SCUBA-2 data were reduced using the Dynamical
+Iterative Map Maker (DIMM) within SMURF (Sub-millimeter
+User Reduction Facility) with tools from the STARLINK KAPPA
+software package (Warren-Smith & Wallace 1993; Jenness
+et al. 2009) to deal with NDF format images. We utilized the
+Figure 1. Illustration of the multiwavelength data coverage in the JWST-TDF survey region. The inner dashed green and dotted dark blue circles are the NIRISS and
+NIRCam survey areas, respectively (10′ and 14′ diameters, respectively). The data coverage from other facilities is shown in different colors as indicated (for more
+details, see Jansen & Windhorst 2018). The solid blue circle represents the JCMT SCUBA-2 850 μm survey area analyzed in this work and the VLA 3 GHz
+observations, with a 24′ diameter field of view, cover the full field.
+Figure 2. The distribution of the JCMT-TDF SCUBA-2 observations undertaken in the 2018B, 2019B, and 2020X semesters. Colors represent the weather conditions
+of the observations, defined in “bands” calculated from the sky opacity, τ225. The data for 2018B were obtained typically in better conditions than those in which the
+data for the 2019B and 2020X semesters were obtained. Each observation block (minimum schedulable block (MSB)) consists of an ∼40 minute exposure with the
+PONG900 scanning mode. The total exposure time is 41.3 hr (corresponding to 62 MSBs).
+3
+The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January
+Hyun et al.
+
+“REDUCE_SCAN_FAINT_POINT_SOURCES”
+recipe
+in
+the
+ORAC-DR data reduction pipeline for a nearly automatic
+reduction of SCUBA-2 data using SMURF and KAPPA. For
+more detailed information on the full data reduction process
+with SMURF, see Chapin et al. (2013).
+Flux calibration was applied to convert the picowatt units in the
+reduced map into units of janskys. The calibrated maps were
+produced
+with
+the
+standard
+flux
+conversion
+factor
+of
+537 Jy beam−1 pW−1 during the application of the PICARD
+recipes. For more information, see Dempsey et al. (2013). Finally,
+maps of each ∼40 minute observation were generated with 4 0
+pixel sampling.
+Using all the reduced and calibrated maps, we constructed a
+stacked final map by applying the PICARD recipe “MOSAIC_JCM-
+T_IMAGES.” Since our main targets are faint and compact, we
+used a matched filter recipe “SCUBA2_MATCHED_FILTER” to
+improve their detection. The matched filter process first smooths
+the map using a large Gaussian kernel and then subtracts the
+smoothed map from the original to eliminate large-scale residual
+noise. Then, the map was convolved with the SCUBA-2 beam.
+The flux calibration of the SCUBA-2 maps was affected by the
+matched filtering, resulting in a ;10% decline in flux (Simpson
+et al. 2019). To calibrate this factor, we inserted artificial bright
+sources into the map and recovered their flux after filtering. Then,
+we cropped the map within a radius of 600″ by using the PICARD
+recipe “CROP_SCUBA2_IMAGES” to isolate the region where the
+sensitivity is high and uniform (a mean of σ850mm = 1.0
+mJy beam−1) to detect sources, but noting that this region entirely
+covers the full planned JWST-TDF footprint.
+Figure 3 shows the final signal-to-noise ratio (S/N) map of the
+850 μm SCUBA-2 coverage of the JWST-TDF. The noise in the
+deepest regions of the map reaches σrms = 0.8 mJy beam−1, which
+is
+close
+to
+the
+SCUBA-2
+850 μm
+confusion
+limit,13
+σ850mm ∼ 0.7 mJy beam−1. Figure 4 presents the cumulative
+area of the SCUBA-2 field as a function of instrumental noise.
+We plot the corresponding values for the S2CLS EGS (Zavala
+et al. 2017), the S2CLS GOODS-N (Geach et al. 2017), the
+SCUBA-2 survey in the COSMOS field (Casey et al. 2013),
+SUPER GOODS-N (Cowie et al. 2017), SUPER GOODS-S
+(Cowie et al. 2018), the S2CLS NEP (Geach et al. 2017), the
+S2COSMOS (Simpson et al. 2019), and the NEPSC2 (Shim
+et al. 2020), which have a variable survey area and depth, on
+the figure for comparison.
+3. Source Catalog
+3.1. Source Detection
+The final SCUBA-2 flux map is shown in Figure 5. For source
+detection, we adopted a simple top-down peak-finding method,
+which is widely used in submm studies of cosmological survey
+fields as this approach is effective in deblending sources with a
+large difference in fluxes (Geach et al. 2017; Simpson et al. 2019;
+Shim et al. 2020). Specifically we followed the approach used by
+Simpson et al. (2019). In the filtered map, optimized to detect faint
+point sources, we searched for sources with prominent peaks in
+S/N in the region with σinst < 1.5 mJy beam−1 (Figure 3) and
+recorded their properties (such as their positions, flux densities,
+etc.) in a “first-pass” catalog. After the first detection pass, we
+subtracted these sources from the map by modeling their emission
+with an empirical point-spread function (PSF) made by using all
+Figure 3. The S/N map for the 41.3 hr of SCUBA-2 observations of the JWST-
+TDF. The cyan dashed circle is the area (r = 10′) where we performed source
+detection (σinst< 1.5 mJy beam−1) and the noise levels (1.0, 1.5, 2.0, and 3.0 mJy)
+are indicated by white contours. In the deepest region of the field, the instrumental
+noise reaches 0.8 mJy beam−1, and the mean of σinst is 1.0 mJy beam−1.
+Figure 4. Cumulative area as a function of sensitivity of the S2TDF (black solid
+line). The instrumental sensitivity of our map reaches σinst = 0.8 mJy beam−1 in
+the deepest region of the field covering 0.087 deg2 and this is quite close to the
+confusion limit of SCUBA-2 850 μm, 0.7 mJy beam−1 (red dashed line). For
+comparison, the S2CLS EGS (blue dot; Zavala et al. 2017) covering 0.019 deg2
+with a 1σ depth has σinst = 0.46 mJy beam−1; the S2CLS GOODS-N (green
+asterisk; Geach et al. 2017) covering 0.07 deg2 with a 1σ depth has σinst = 1.1 mJy
+beam−1; the SCUBA-2 survey in the COSMOS field (light green triangle; Casey
+et al. 2013) covering 0.109 deg2 with a 1σ depth has σinst = 0.8 mJy beam−1;
+SUPER GOODS (pink cross; Cowie et al. 2017, 2018) covering 0.233 deg2 with a
+1σ depth has σinst = 1.8 mJy beam−1 and reaches σinst = 0.4 mJy beam−1 in the
+deepest central field covering 0.018 deg2 (brown cross); the S2CLS NEP (purple
+diamond; Geach et al. 2017) covering 0.6 deg2 with a 1σ depth has σinst = 1.2 mJy
+beam−1; the S2COSMOS (orange square; Simpson et al. 2019) covering 1.6 deg2
+with a median noise level has σinst = 1.2 mJy beam−1; and the NEPSC2 (yellow
+star; Shim et al. 2020) covering 2 deg2 with a median noise level has
+σinst = 2.3 mJy beam−1. The horizontal cyan solid line shows the final area
+(r = 600″) of this work where we performed source detection (see Figure 3).
+13 There have been several values of the confusion limit derived from various
+studies; however we adopted the value 0.7 mJy beam−1, which is presented on the
+official JCMT website: https://www.eaobservatory.org/jcmt/instrumentation/
+continuum/scuba-2/time-and-sensitivity/.
+4
+The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January
+Hyun et al.
+
+>5σ sources in the map. If two sources lay within 40″ of each
+other, we used a double-PSF model. Using the map where the
+“first-pass” sources had been subtracted, we repeated the detection
+step for a second pass. If additional sources were detected within
+7 5 of the first-pass sources, we regarded the fluxes as the same
+sources as the first-pass sources and recorded the information from
+the prior detections only. This iteration went on until all sources
+above a minimum threshold of 3.5σ were found. More details are
+given in Simpson et al. (2019).
+In this manner, we detected 83 submm sources above a 4σ
+significance limit and a total of 114 above a 3.5σ limit in the
+850 μm map. Figure 5 shows the positions of all detected
+sources in the field. The thicker symbols represent the sources
+with S/N > 4. A histogram of the fluxes of the sources is
+shown in Figure 6.
+3.2. Jackknife Simulation
+Jackknife simulations are a widely used method to estimate
+the completeness, flux deboosting rate, false detection rate, and
+positional accuracy of submm surveys (e.g., Scott et al. 2008;
+Simpson et al. 2019; Shim et al. 2020). To construct a jackknife
+map, we randomly divided our SCUBA-2 MSBs into two
+groups, constructed a combined map from each group, and
+subtracted one from the other to remove any real astronomical
+signals. Next, the map where sources had been removed was
+scaled down by the square root of the total integration time to
+match the noise level in the actual final map.
+In this source-free “noise” map, we injected artificial sources
+based on the expected source number counts. We assumed the
+number density distribution followed a Schechter function with
+the parameters from the 850 μm number counts in Geach et al.
+(2017), N0 = 7180, S0 = 2.5, and γ = 1.5:
+⎜
+⎟⎜
+⎟
+⎜
+⎟
+⎛
+⎝
+⎞
+⎠
+⎛
+⎝
+⎞
+⎠
+⎛
+⎝
+⎞
+⎠
+=
+-
+g
+-
+dN
+dS
+N
+S
+S
+S
+S
+S
+exp
+.
+1
+0
+0
+0
+0
+( )
+Based on the expected number counts and our map area we
+inserted 500 sources into each noise map with 850 μm fluxes of
+1–20 mJy, using the above number distribution. We placed the
+Figure 5. SCUBA-2 flux map of JWST-TDF showing 83 detected sources at S/N > 4 and a further 31 with S/N = 3.5–4. The color of the sources represents their
+deboosted flux (see color bar) and thicker symbols represent S/N > 4 sources. By matching the 850 μm sources to the VLA 3 GHz catalog using a corrected
+Poissonian probability estimate, we found that 66 submm sources have radio counterparts (box symbols). Circle symbols are submm sources without radio
+identifications. The JWST NIRISS and NIRCam survey areas are indicated by dotted and dashed circles, respectively.
+Figure 6. The deboosted 850 μm flux density distribution of the 114 detected
+SCUBA-2 sources. SNR is the S/N from the observation. Eighty-two and
+thirty-two sources have S/N > 4.0 (red histogram) and S/N = 3.5–4.0 (pink
+histogram), respectively, and every source having a deboosted flux of >2.3
+mJy is in this regime.
+5
+The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January
+Hyun et al.
+
+sources at random positions in the jackknife map, neglecting
+any clustering. We produced 1000 jackknife-simulated maps
+resulting in a mock catalog consisting of 500,000 sources in
+total. The source detection processes were the same as the
+process applied to the science maps described in Section 3.1.
+Finally, we compared the output source properties to those in
+the original input catalog.
+3.3. Completeness
+From the jackknife simulation, we estimated the survey
+completeness using the recovery success rate of injected
+sources. Figure 7 shows the completeness as a function of the
+intrinsic flux density. The 50% and 80% completeness limits
+are roughly 3.0 mJy and 4.0 mJy, respectively.
+The completeness is not uniform across the survey area.
+Therefore, the completeness presented in Figure 7 is an average
+value over the survey area. To obtain a more accurate
+completeness, we derived a two-dimensional completeness
+map as a function of the local instrumental noise (σinst) and the
+deboosted source flux (see Section 3.5 for the deboosting
+process). Here, the instrumental noise was taken as the standard
+deviation of the jackknife noise map at the source position. The
+two-dimensional completeness map is presented in Appendix B.
+When we measured the number counts of 850 μm sources in
+Section 5.1, we applied the two-dimensional completeness
+correction to every source.
+3.4. False Detection Rate
+At low S/N, there is a possibility that noise fluctuations will
+appear as spurious sources. To assess the reliability of the
+source catalog, we performed source detection in the same way
+as the real source detection on both the jackknife “noise” map
+and the jackknife map with artificial sources added (termed the
+“source” map).
+By comparing the number of detections in the “noise” map and
+that in the “source” map, we evaluated the false detection rate as a
+function of S/N (Figure 7). Here, the false detection rate was
+defined as the ratio between the number of fake sources (false
+detections) and the number of detected sources. At >3.5σ, the false
+detection rate is ;8% and it drops to ;1% at >4.0σ.
+A detection threshold of >3.5σ gives a good balance
+between the sample size and the false detection rate for
+statistical studies of the SMG population (e.g., counts), while a
+threshold of >4σ gives a sample with high reliability for
+studying their detailed properties.
+3.5. Deboosting of Fluxes
+Due to the large PSF size of the SCUBA-2 map, submm
+fluxes of sources, especially those near the flux limit of the
+survey, were often artificially boosted due to blending with
+positive noise peaks or faint sources. The correction of the
+boosted fluxes is called “deboosting.”
+For flux deboosting, we derived the boosting factor (B) from
+the ratio of the output (observed) flux density to the input flux
+density of sources in the mock catalog from the jackknife
+simulation. The average boosting factor at a given flux is
+illustrated in Figure 7, which is well fitted by a power law in
+S/N (Geach et al. 2017; Shim et al. 2020) as given below for
+our sources:
+⎛
+⎝
+⎞
+⎠
+=
++
+´
+-
+B
+1
+0.3
+S N
+6.1
+.
+2
+1.6
+( )
+As noted earlier, in practice, the flux limits vary across the
+survey field. Therefore, we derived the boosting factor as a
+function of the map sensitivity as represented by the instrument
+noise, σinst, and the flux density (before the boosting correction)
+as described in Appendix B. All the flux values were deboosted
+with the relevant boosting factors. Furthermore, we derived the
+additional uncertainties in the flux values that were introduced
+during the deboosting process (see Appendix B), and note the
+uncertainties as σdeb.
+3.6. Positional Uncertainty of Sources
+We can estimate the positional uncertainties of the detected
+sources from the jackknife simulation by comparing the input
+position with the detected (recovered) position. We derived the
+rms dispersion of the positional differences (σ) in the R.A. and
+decl. coordinates from the jackknife simulation. The σ values
+were found to be inversely proportional to the S/N of the
+source as expected, and to have nearly identical values in both
+coordinates as given in the equation below:
+⎛
+⎝
+⎞
+⎠
+s =
+
+´
+-
+1. 8
+S N
+5
+.
+3
+1.0
+( )
+Additionally, we must consider an S/N-independent intrinsic
+positional error (σ0) associated with errors in the telescope
+pointing. This intrinsic positional error is independent of S/N and
+can usually be described by a Gaussian distribution with a
+constant rms σ0 of bright SMGs whose σ values are very small
+and negligible compared to σ0 (Condon 1997). We estimated the
+intrinsic positional error by matching high-S/N SCUBA-2
+sources (S/N > 10) with VLA source positions and excluding
+SCUBA-2 sources with multiple VLA source matches. This gave
+five SCUBA-2 sources, and since the number of sources is small,
+we used all of the R.A. and decl. offsets together to derive σ0
+assuming the intrinsic positional errors had an identical distribu-
+tion. We found that the S/N-independent positional error is
+σ0 ∼ 0 9. Considering the VLA positional errors (0 1) are much
+smaller than this value, we ignored their contribution to σ0.
+In sum, the positional uncertainty in both the R.A. and decl.
+directions can be taken as the equation below:
+s
+s
+s
+=
++
+.
+4
+tot
+2
+0
+2
+( )
+In Figure 7, we show σtot in bins of S/N, with the
+contributions from the S/N-dependent (Equation (3)) and
+intrinsic (σ0) terms indicated.
+To check the consistency between the positional uncertainty
+based on the data and the σtot explained above, we present the
+standard deviation of the R.A. and decl. positional offsets
+between the SCUBA-2 sources and matched VLA sources and
+their errors estimated from bootstrapping.
+For the Gaussian distributions of R.A. and decl. errors, the
+radial position offset (ρ) has a distribution that follows the
+Rayleigh distribution of r s
+r
+s
+-
+exp
+2
+tot
+2
+2
+tot
+2
+[
+(
+)]. For this
+distribution, 68% and 95.6% of the radial offsets would lie
+within 1.5σtot and 2.5σtot, respectively (Condon 1997; Ivison
+et al. 2007; Geach et al. 2017).
+6
+The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January
+Hyun et al.
+
+3.7. Astrometric Calibration Using VLA
+During JCMT observations, the calibration process checks and
+revises the telescopeʼs pointing, and usually, the values of the shift
+are of the order of ∼1″. However, it is important to determine
+precise absolute positions if we wish to identify the counterparts to
+the submm sources and derive their properties. Even a slight offset
+will make a significant difference in the calculation of the matching
+probability with other higher-resolution data. This astrometric
+calibration is typically achieved using radio maps, for example
+those from the VLA, with arcsecond resolution that is tied to the
+FK5 to a better than ∼0 01 absolute astrometric precision (e.g.,
+Ivison et al. 1998, 2002).
+We used the sources in the VLA 3 GHz catalog of this field
+described in Appendix A to improve the absolute astrometry of
+our JWST-TDF SCUBA-2 map. From the radio source catalog,
+detecting 756 sources at S/N > 5, we selected sources with a 3
+GHz flux density of …10.0 μJy, resulting in 557 VLA sources
+across our survey area. We used these to search for counterparts to
+our submm sources using a matching radius of 7 5.14 We
+Figure 7. (a) Completeness of the JWST-TDF SCUBA-2 survey calculated from the ratio of the number of recovered sources with S/N > 3.5 to the number of input
+sources. The 50% and 80% completeness limits are 3.0 and 4.0 mJy (deboosted flux), respectively. (b) The false detection rate, the ratio between the number of
+sources detected in the jackknife map and that of sources detected in the flux density map (inner panel), as a function of S/N. With an S/N > 3.5 cut the probability
+that a detected source is spurious is ;8%, and at S/N > 4 it is ;1%; this drops to 0% at S/N > 4.25. (c) Average flux boosting factor as a function of S/N, showing
+that the results are well fitted with a power law. However, we evaluated boosting rates for each source based on the observed flux density and the local instrumental
+noise by constructing a two-dimensional parameter space. (d) The total positional error in both R.A. and decl. directions (solid line), which is the quadratic sum of the
+intrinsic positional error (σ0, dashed line) and the positional difference with the bootstrapping error (σ, dotted line), as a function of S/N. Red filled circles are the
+standard deviation of the R.A. and decl. position offsets between the SCUBA-2 sources and their VLA counterparts.
+14 We set the maximum matching radius to 7 5, which is about half of the
+JCMT SCUBA-2 effective beam FWHM (θ1/2 = 14 6; Dempsey et al. 2013)
+at λ = 850 μm. This has been widely used as a matching radius for radio
+identification (e.g., Shim et al. 2020; Zhang et al. 2022), corresponding to ;3
+times the expected total positional error (2 4 for a 4.0σ detection and 2 7 for a
+3.5σ detection) with a 14 6 FWHM (see Equation (4)).
+7
+The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January
+Hyun et al.
+
+identified possible radio counterparts for 68 submm sources. To
+determine the astrometric offset of the SCUBA-2 positions
+with respect to the radio source positions, we then stacked
+thumbnails of the SCUBA-2 sources centered on the matched
+radio source positions and made a 74″ × 74″ stacked image
+(Figure 8). Through this procedure, we found a small mean
+coordinate shift for the SCUBA-2 map of 0
+83 ± 0 22 and
+−0 34 ± 0 03 in R.A. and decl., respectively. We corrected
+our cataloged SCUBA-2 source positions for this offset. We
+also tested the sensitivity of the derived offset to the adopted
+pixel scale of the SCUBA-2 map by repeating this process with
+data regridded to finer pixel scales of 0 5 and 2 0 sampling,
+finding no significant differences.
+3.8. JWST-TDF 850 μm Source Catalog
+We constructed two catalogs of the SCUBA-2 sources in the
+JWST-TDF (Table 4 in Appendix D) for the main (S/N > 4)
+and
+supplementary
+sources
+(S/N = 3.5–4.0),
+comprising
+eighty-three and thirty-one 850 μm sources, respectively. The
+coordinates of the sources include the astrometric corrections
+derived in Section 3.7.
+There are three different flux uncertainties given in the
+catalog: the instrumental noise (σinst), the deboosting uncertainty
+(σdeb), and the total uncertainty (σtotal). The total flux uncertainty
+is the square root of the squared sum of the instrumental
+noise, the deboosting uncertainty, and the confusion noise (σc)
+resulting from faint sources below the detection flux limit within
+the JCMT beam. We calculated the confusion noise using
+Equation (3) of Simpson et al. (2019):
+s
+s
+s
+=
+-
+5
+c
+total
+2
+inst
+2
+( )
+where σtotal and σinst are derived from the standard deviation of
+the 850 μm map where sources over the detection limit were
+removed, and from that of the jackknife map, respectively. The
+confusion noise in this study from Equation (5) is 0.14 mJy
+beam−1. The S/N of the sources were not deboosted.
+4. Band Merging and SED Analysis
+4.1. Multiwavelength Matching
+We matched the detected 850 μm sources in the JWST-TDF
+to radio counterparts from the 3 GHz catalog described in
+Appendix A. These radio counterparts are essential to provid-
+ing the precise positions necessary to determine the multi-
+wavelength properties of the likely SMGs from the optical to
+radio. For SMGs with radio counterparts and optical/NIR
+photometric detections, we derived their redshifts and other
+properties through SED fitting.
+4.1.1. Radio Counterparts
+To reliably identify the multiwavelength counterparts to the
+submm sources, we exploited the correlation between the radio
+and FIR luminosities of normal or star-forming galaxies (e.g.,
+Rickard & Harvey 1984; Helou et al. 1985; Condon 1992;
+Ivison et al. 2004, 2005, 2008), which has been widely used in
+previous submm studies (e.g., Barger et al. 2000; Smail et al.
+2000; Ivison et al. 2007; An et al. 2019). We first crossmatched
+the SCUBA-2 sources with our VLA 3 GHz radio catalog (see
+Appendix A). For matching, we used the 557 radio sources
+with 3 GHz flux densities above the uniform limit of …10 μJy
+(as used in Section 3.7).
+The corrected Poissonian probability, p, used in the
+matching analysis was defined as follows:
+=
+-
+-
+p
+E
+1
+exp
+6
+(
+)
+( )
+⎧
+⎨⎩
+=
++
+*
+*
+*
+*
+E
+P
+P
+P
+P
+P
+P
+P
+P
+1
+ln
+7
+c
+c
+c
+c
+{
+(
+)}
+( )
+
+
+p
+=
+P
+r N
+8
+c
+s
+T
+2
+( )
+p
+=
+*
+P
+r N.
+9
+i
+i
+2
+( )
+In line with previous work we adopted a maximum search
+radius (see Section 3.7), rs, of 7 5. The variable ri is the
+angular distance between the submm centroid and the matching
+radio source (with 3 GHz flux Si). NT and Ni are the radio
+source number densities, with S3GHz > 10 μJy and S3GHz > Si,
+respectively. For each SCUBA-2 source, we calculated the
+probability, p, for any radio sources within rs, and retained only
+sources having radio counterparts with p „ 0.065 based on the
+balance of recovery and false-positive rates in the analysis of an
+Atacama Large Millimeter/submillimeter Array (ALMA)-
+identified sample of SMGs by An et al. (2018).
+Figure 9 summarizes the results of the radio identifications.
+The SCUBA-2 sources with radio identification are listed in
+Table 4 along with the corresponding probabilities. Note that
+the number after the decimal point in the source ID indicates
+the radio counterpart identifications, ranked on the probability
+of the match; it can be larger than 0 if there are multiple
+matches. Using a maximum search radius of 7 5, there are 89
+radio identifications. Four radio sources have likelihoods of
+being false matches of p > 0.065, and removing these reduces
+the total number of robust matches to 85 radio sources. There
+are 52 submm sources with single radio counterparts, nine with
+two radio counterparts, and five with three counterparts. The
+SCUBA-2 sources with multiple radio counterparts have an
+850 μm flux density distribution that is indistinguishable from
+Figure 8. A 28″ × 28″ zoomed-in image of the stacked SCUBA-2 850 μm
+image at the positions of the radio counterparts matched to 67 SCUBA-2
+sources. The red cross is the center from the two-dimensional Gaussian fitting
+of each stacked image. This shows a small mean coordinate shift of
+0 83 ± 0 22 and −0 34 ± 0 03 in R.A. and decl., respectively, which we
+corrected for in our SCUBA-2 catalog positions.
+8
+The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January
+Hyun et al.
+
+that of the parent population of all SCUBA-2 sources with
+radio counterparts (S850μm = 3.3 ± 0.3 mJy versus S850μm =
+3.6 ± 0.7 mJy), while the fluxes of the SCUBA-2 sources
+lacking radio counterparts are somewhat fainter on average:
+S850μm = 2.8 ± 0.2 mJy. For the main sample, 54 out of 83
+SCUBA-2 sources matched with 70 radio sources including
+five/six triple/double-counterpart cases. For the supplementary
+sample, 12 out of 31 SCUBA-2 sources matched with 15 radio
+sources, among which there are three double-counterpart cases
+and no triple-counterpart cases.
+To summarize, 85 VLA radio sources are matched to 66 of the
+114 SCUBA-2 sources, and these matches are typically sources
+that are at the brighter end of the 850 μm flux distribution. The
+overall matched fraction corresponds to an identification rate of
+58% (65% and 39% for the main and the supplementary sample),
+which is broadly consistent with that seen in other radio-identified
+SMG samples at these depths (60%–80%, e.g., Ivison et al. 2007;
+Casey et al. 2013; Miettinen et al. 2015).
+4.1.2. Optical/NIR Counterparts
+For those SCUBA-2 sources with radio counterparts, we used
+the precise radio positions and our multiwavelength coverage to
+characterize the properties of these galaxies. We used the radio
+positions and matched these to the ground-based band-merged
+optical and NIR catalog of the JWST-TDF (C. N. A. Willmer
+et al. 2022, in preparation) with a search radius of 1 0. This radius
+is small enough that we expected no false matches.
+The band-merged catalog comprises optical photometry in
+the g, i, and z bands based on source detections in the combined
+i+z image (appropriate for detecting red galaxies such as
+SMGs). These data were obtained with Hyper Suprime-Cam
+(HSC; Bosch et al. 2018; Miyazaki et al. 2018), on the Subaru
+telescope, and have 3σ aperture magnitude limits of g = 26.0,
+i = 25.2, and z = 25.1 in 3 0 diameter apertures. The 3 0
+aperture magnitude is roughly identical to total magnitudes for
+compact/faint galaxies.
+This optical catalog has been merged with NIR catalogs in
+the Y, J, H, and K bands from the Magellan Infrared
+Spectrograph (MMIRS; McLeod et al. 2012) on the Multi-
+Mirror Telescope (MMT). The 3σ aperture magnitude limits of
+the MMIRS images are Y = 24.7, J = 23.7, H = 23.0, and
+K = 22.5, again in 3 0 diameter apertures, to provide total
+magnitudes. This NIR coverage has been supplemented by the
+WISE 3.6–22 μm counterparts, matched to the radio sources
+using a 1 0 search radius.
+There are 61 radio-identified SMGs matched to the merged
+optical/NIR catalog. Of these 61, 54 are detected above 3σ in
+the NIR (16 are detected with WISE), and the remaining seven
+have optical HSC detections and upper limits in the other
+bands.
+4.2. SED Fitting Analysis
+We performed an SED analysis of the 61 radio-identified,
+optical/NIR-matched SMGs to derive both photometric red-
+shifts and other physical properties (we similarly modeled a
+further 24 sources that had radio and submm constraints but no
+optical or NIR detections). We employed the MAGPHYS galaxy
+SED-fitting code (da Cunha et al. 2008; Cunha et al. 2015;
+Battisti et al. 2019).
+MAGPHYS uses an energy balance
+technique to combine observations from the optical/NIR,
+MIR/FIR, and submm wave bands to constrain the physical
+properties of sources, including the stellar population and dust
+content. The “high-redshift” version of the code, MAGPHYS
++PHOTOZ, also allows the redshift of the source to be estimated
+(Battisti et al. 2019). The ability of the code to include
+information from the dust-reprocessed emission seen in the
+MIR/FIR and submm is important when attempting to model
+SMGs, which are frequently faint or undetected in the optical
+and NIR wave bands used by other photometric modeling
+codes. MAGPHYS+PHOTOZ has been used to model samples of
+SMGs, for example by Dudzevičiūtė et al. (2020), who
+provided an extensive discussion of the calibration and testing
+Figure 9. VLA 3 GHz flux densities of 89 radio sources matched to the 68 SCUBA-2 source positions, as a function of corrected Poissonian probability, p (left), and
+as a function of the angular separation between the radio sources and the SCUBA-2 sources (right). SCUBA-2 sources with S/N > 4 are marked with blue filled
+circles. There are 37 multiple radio counterpart matches to 16 SCUBA-2 sources, and these are indicated with red diamond marks. After the probability cut p = 0.065
+is applied, 33 multiple matches to 14 SCUBA-2 sources remain. The red vertical line is the probability cut adopted from the analysis of ALMA identifications of
+SMGs by An et al. (2019). With that criterion, four radio sources are removed from the radio-identified sample. Hence, in total 85 radio sources are matched to 66
+SCUBA-2 sources.
+9
+The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January
+Hyun et al.
+
+of the code on dust-obscured galaxies, including its application
+to modeling SEDs derived from radiative transfer models of
+simulated galaxies from the
+EAGLE simulation (see also
+McAlpine et al. 2019).
+Our SED modeling used the optical/NIR/MIR photometry for
+the matched counterparts, and for the SMGs, also included our
+deboosted 850 μm observations and 3 GHz VLA data. In the
+analysis, we followed a similar approach to that used in
+Dudzevičiūtė et al. (2020), first applying MAGPHYS+PHOTOZ to
+the collated photometry of a sample of 264 sources, including four
+SMGs and 66 VLA sources, with robust spectroscopic redshifts
+(spanning zspec = 0−1.6) in our field (C. N. A. Willmer, private
+communication; see Appendix C), to check for systematic errors
+in either the total magnitudes or the absolute calibration of the
+photometry. For each filter, we determined the distribution of
+differences between the observed and model photometry for the
+best-fit MAGPHYS model SEDs of the sources at their known
+spectroscopic redshifts. This analysis confirmed that there were no
+significant, >0.10 mag, systematic offsets in any of the filters, and
+so we proceeded to model the galaxies allowing the code to fit for
+the redshift. This returned a median difference between the
+predicted
+photometric
+and
+spectroscopic
+redshifts
+of
+Δz/
+(1 + zspec) = −0.10 ± 0.02 for the 264 galaxies, indicating that
+the photometric redshifts are unbiased, with a scaled median
+absolute deviation of σz/(1 + zspec) = 0.19.
+We next applied MAGPHYS+PHOTOZ to the 61 SMGs in our
+field that had 3 GHz radio detections and suitable optical and/
+or NIR photometry (and also the 25 with just radio and submm
+detections). The best-fit model SEDs for the sources are shown
+in Figure 17 in Appendix D.
+MAGPHYS+PHOTOZ returns estimates of various physical
+quantities of the modeled galaxies, not all of which are well
+constrained by the observable properties (see the discussion in
+Dudzevičiūtė et al. 2020). Given the limited photometric
+coverage used in our analysis, we consider the redshifts and
+dust masses as the most robust output parameters, although we
+also discuss the estimates of the SFR and stellar mass in
+Sections 5.2 and 5.3.
+The estimated 16th–84th percentile errors on the various
+physical parameters from MAGPHYS+PHOTOZ are as follows:
+photometric
+redshift,
+δzphot/(1 + zphot) = 0.21;
+stellar
+mass,
+δM* = 0.37 dex; SFR, δSFR = 0.35 dex; dust mass, δMd =
+0.24 dex; and dust-to-stellar mass ratio (DSR), δMd/M* =
+0.43 dex. In the case of the main sample (S/N > 4), the errors
+are slightly smaller: photometric redshift, δzphot/(1 + zphot) =
+0.22; stellar mass, δM* = 0.36 dex; SFR, δSFR = 0.34 dex; dust
+mass, δMd = 0.23 dex; and DSR, δMd/M* = 0.43 dex. These
+uncertainties are typically 1.4 ± 0.2 times larger than the
+equivalent values derived for the 22-band coverage in AS2UDS
+by Dudzevičiūtė et al. (2020) reflecting the limitation of the
+current photometric coverage of the TDF region.
+5. Results and Discussion
+5.1. 850 μm Number Count
+The number density of sources as a function of flux density
+is one of the most basic observables available for a population
+and as such provides a fundamental test for galaxy formation
+models (e.g., Baugh et al. 2005). First, we present the number
+counts of the submm sources detected in the JWST-TDF and
+compare these with those of other surveys.
+Table 1 provides a summary of the number counts and
+Figure 10 shows the cumulative and differential counts of 850 μm
+sources in the ∼0.087 deg2 area of the JWST-TDF SCUBA-2
+survey. The counts use deboosted fluxes and are corrected using
+the completeness curve constructed in Section 3.3. The errors
+reflect the uncertainties from the deboosting process, which were
+derived from constructing 1000 mock catalogs of 850 μm sources
+by sampling the deboosted flux distributions for the sources and
+adopting the 16th and 84th percentiles of the resulting distribution
+as the uncertainties. The final number counts were derived from
+the mean values of these mock catalogs (Table 1). We also plot
+the best Schechter function fits to the number counts in the
+AS2COSMOS (Simpson et al. 2020), S2COSMOS (Simpson
+et al. 2019), and S2CLS (Geach et al. 2017) surveys for
+comparison.
+The JWST-TDF number counts in both panels trace the
+Schechter function–like trend well, and agree with the results
+from other surveys (e.g., Chen et al. 2013; Karim et al. 2013;
+Hsu et al. 2016; Geach et al. 2017; Stach et al. 2018; Simpson
+et al. 2019, 2020; Shim et al. 2020), especially at fainter flux
+densities (S850mm < 10 mJy). There is a slight excess at
+S850mm > 10 mJy. We believe that this is likely caused by
+cosmic variance and reflects four bright sources with fluxes
+exceeding S850mm = 10 mJy in our relatively small survey area
+(∼0.087 deg2). To quantify this, we randomly placed TDF-
+sized regions in the much larger S2COSMOS survey field
+(Simpson et al. 2019) and found that regions containing …5
+SCUBA-2
+sources
+brighter
+than
+S850mm = 10 mJy
+were
+detected ∼3% of the time—suggesting that the excess can be
+explained by cosmic variance.
+5.2. Redshifts
+We found that the best-fit photometric redshifts for the radio-
+identified and optical/NIR-detected SMGs with S/N > 4.0
+(>3.5) in our sample give a median redshift of z = 2.22 ± 0.12
+(z = 1.96 ± 0.08)
+and
+a
+16th–84th
+percentile
+range
+of
+z = 1.50–3.10 (z = 1.44–2.70). We note that there are high-
+quality spectroscopic redshifts for only four of the dust-selected
+SMGs (the low rate likely reflecting the relative faintness of the
+galaxies in the optical/NIR). Nevertheless, in three of these
+four cases, the spectroscopic redshifts and photometric redshift
+estimates appear to agree with each other (the exception is
+source 0056.0, with zspec = 0.03, a spectroscopic redshift
+that likely refers to a nearby foreground galaxy): 0023.0,
+zspec = 1.51, zphot =
+-
++
+1.55
+;
+0.27
+0.25
+0031.0, zspec = 1.36, zphot =
+-
++
+1.44
+;
+0.05
+0.05 and 0051.0, zspec = 1.51, zphot =
+-
++
+1.48 0.11
+0.08.
+Table 1
+The Number Counts of JWST-TDF SCUBA-2
+S850mm
+N(>S850mm)
+S850mm
+dN/dS850mm
+(mJy)
+(deg−2)
+(mJy)
+(deg−2 mJy−1)
+4.0
+-
++
+462.9 77.4
+90.5
+4.6
+-
++
+181.6 49.7
+65.9
+5.1
+-
++
+252.7 54.7
+71.2
+5.9
+-
++
+78.6 27.9
+40.0
+6.6
+-
++
+137.9 42.5
+54.1
+7.5
+-
++
+37.0 18.1
+24.2
+8.5
+-
++
+68.8 27.2
+40.3
+9.7
+-
++
+14.3 11.3
+13.4
+10.9
+-
++
+34.4 21.9
+35.3
+12.4
+-
++
+3.7 3.1
+9.3
+14.0
+-
++
+23.0 18.7
+30.1
+15.9
+-
++
+2.9 3.8
+7.2
+17.9
+-
++
+11.5 14.9
+26.2
+20.5
+-
++
+2.3 2.9
+5.2
+Note. Uncertainties were derived from the standard deviation in simulations for
+each bin using the source deboosting and completeness corrections.
+10
+The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January
+Hyun et al.
+
+The redshift distribution of our radio-identified 850 μm
+sources is presented in Figure 11. For comparison, we show the
+photometric redshift distribution for the purely submm-selected
+AS2UDS survey from Dudzevičiūtė et al. (2020), and also the
+equivalent distribution after applying an additional radio
+detection and K < 22.5 magnitude limit (to mimic the selection
+of our radio-identified, mostly NIR-detected sample). We also
+show the spectroscopic redshift distribution for the similarly
+radio-identified SMG sample from Chapman et al. (2005). In
+contrast to the flux-limited AS2UDS sample, which shows a
+tail of sources out to z ∼ 6, both of the radio/NIR-limited
+samples exhibit a sharp cutoff at z ∼ 4 and median redshifts at
+z ∼ 2, in reasonable agreement with those in the JWST-TDF.
+Figure 12 shows the S850μm and S850μm/S3GHz versus the
+photometric redshifts of the 51 (61) submm sources at
+S/N > 4.0 (S/N > 3.5). Also plotted are the K-band and
+radio limited sample of AS2UDS, mimicking the selection of
+the S2TDF sources, and the median redshifts of the submm
+flux-limited sample of AS2UDS. We conclude that the S2TDF
+sources follow the distribution of those in AS2UDS when a
+similar radio/NIR selection is applied to them.
+5.3. SFRs and Stellar Masses
+The distribution of SFRs with stellar mass for those S2TDF
+radio-identified SMGs with optical/NIR counterparts is plotted
+in Figure 13. The S2TDF sample shows a median SFR of
+300 ± 40 Me yr−1 and 240 ± 40 Me yr−1 for SCUBA-2
+sources with S/N > 4.0 and S/N > 3.5, respectively. The
+maximum SFR is ∼3000 Me yr−1 for both cases.
+These galaxies have stellar masses with a 16th–84th percentile
+range of 0.8–4 × 1011 Me, and a median of 1.7 ± 0.3 ×
+1011 Me, showing that most of them are already very massive.
+Adding the supplementary subset associated with SCUBA-2
+sources with S/N = 3.5–4.0, we found their stellar masses are
+very similar with a median M* = 1.6 ± 0.3 × 1011 Me and a
+16th–84th percentile range of 0.6–5 × 1011 Me. With their high
+SFRs, they are likely to become massive early-type galaxies
+today, as suggested by other SMG studies (Lilly et al. 1999;
+Birkin et al. 2021). Figure 13 also compares the SFR–M*
+distribution of the S2TDF SMGs to that of the K-band and radio
+limited subsample of AS2UDS, which shows a similar distribu-
+tion to the S2TDF SMGs. To compare the distribution to the trend
+in more typical “main-sequence” star-forming galaxies, we also
+show the SFR–M* relation for these galaxies in Figure 13. This
+comparison sample was constructed by matching the redshifts and
+stellar masses of the S2TDF SMGs to those of their nearest
+Figure 10. Cumulative and differential number counts of the 850 μm 114 sources (S850mm = 2–20 mJy, S/N > 3.5) in the JWST-TDF area. For comparison, we show
+results from previous submm surveys (Chen et al. 2013; Karim et al. 2013; Hsu et al. 2016; Geach et al. 2017; Stach et al. 2018; Simpson et al. 2019, 2020). The best-
+fitting results from Simpson et al. (2019, 2020), Shim et al. (2020), and Geach et al. (2017) are also presented. In general, our number counts are in reasonable
+agreement with those from the other surveys, although there is a slight excess at the bright end, S850mm > 10 mJy, arising from a small number of bright sources in
+the field.
+Figure 11. The photometric redshift distribution of radio-identified SMGs in
+the JWST-TDF. The red hatched histogram shows the distribution of samples
+at S/N > 4.0, yielding a median redshift of z = 2.22 ± 0.12 and a 16th–84th
+percentile range of z = 1.50−3.11. If we include the sample with 850 μm
+S/N = 3.5–4 (pink shaded histogram), then we derive a median redshift of
+z = 1.96 ± 0.08 and a 16th–84th percentile range of z = 1.44−2.70. The
+redshift distributions from the submm flux-limited survey (AS2UDS, blue
+histogram), the AS2UDS sample with radio detection and K < 22.5 mag limits
+applied to mimic the selection of our sample (green histogram), and finally the
+radio-identified SMG sample (yellow histogram) from Chapman et al. (2005)
+are also shown for comparison.
+11
+The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January
+Hyun et al.
+
+neighbors in z–M* space from the large sample of K-detected field
+galaxies in the UKIRT Infrared Deep Sky Survey Ultra Deep
+Survey (UDS) from Dudzevičiūtė et al. (2020). This matching
+perturbed each SMG 10 times by its uncertainties in redshift and
+stellar mass and selected the closest match from the field sample
+within a redshift window of δz = 0.1. Then, these UDS galaxies
+were plotted on the SFR–M* plane, and the best-fit linear
+correlation was derived, which is given in Equation (10):
+=
+-
+*
+M
+M
+log SFR
+0.57 log
+4.55.
+10
+(
+)
+(
+)
+(
+)
+
+This best-fit trend for the field population is shown in Figure 13
+and we compared it to the SFR–M* relations of “main-sequence”
+star-forming galaxies at z = 1–5 from Pearson et al. (2018),
+finding that it corresponds well to the track for z ∼ 2. This shows
+that our SMG sample have higher SFRs than “typical” galaxies of
+their mass at their respective redshifts, by a factor of ∼6,
+indicating that our SMGs are examples of the most massive,
+highest-SFR end of the galaxy populations at z ∼ 1–3.
+5.4. Dust Mass Estimates
+Finally, we take advantage of the fact that the 850 μm
+continuum brightness is tightly correlated to the cold dust mass of
+galaxies across a wide range of redshifts. In turn, the cold dust
+mass of galaxies is believed to be a good indicator of the mass of
+their cool gas reservoirs (e.g., Scoville et al. 2016). We derived
+estimates of the cold dust masses of our SMGs from the
+MAGPHYS fits and compared these to those for the similar radio-
+detected
+and
+K < 22.5
+sample
+of
+SMGs
+from
+AS2UDS
+(Dudzevičiūtė et al. 2020) in Figure 14. Both samples show
+similar ranges in dust mass, Md ∼ 2–20 × 108 Me, and the
+S2TDF sample shows a median dust mass of 5.9 ± 0.7 ×
+108 Me (Md = 4.4 ± 0.8 × 108 Me when including the supple-
+mentary S/N = 3.5–4.0 subset) suggesting typical gas masses of
+Mg ∼ 2–3 × 1010 Me, for gas-to-dust mass ratios of ∼60 (Birkin
+et al. 2021).
+In Figure 14, we also compare the cold dust masses for our
+SMGs with those inferred from MAGPHYS SED fitting to more
+typical “main-sequence” galaxies, using the K-band “field”
+selected sample (with a K < 22.5 cut) from Dudzevičiūtė et al.
+(2020).
+The
+median
+dust
+mass
+for
+these
+galaxies
+is
+Md „ 108 Me, considerably lower than that of the SMGs,
+suggesting a similar difference in their cool gas masses. Such a
+difference would then provide a natural explanation for the
+Figure 12. Photometric redshift vs. flux density and flux ratio for SCUBA-2 SMGs. S2TDF sources at S/N > 4.0 and at S/N = 3.5–4.0 are presented with filled red
+circles and with filled pink circles, respectively. The median error of each parameter is drawn in each panel (green). (Left) 850 μm flux density vs. photometric redshift
+for our radio-identified SMGs with optical/NIR counterparts. For comparison we show a radio and K-band limited subsample of SMGs from AS2UDS (Dudzevičiūtė
+et al. 2020), and the fitted trend for the full AS2UDS survey from Stach et al. (2019). There is broad agreement between the redshift–flux distributions in the JWST-
+TDF and these previous works. (Right) The variation in submm-to-radio flux ratio and photometric redshift for our sample, again compared to a radio/K-band-limited
+subset of the SMGs in AS2UDS (converted from 1.4 to 3 GHz assuming a radio spectral index of −0.7). We also show the expected trend for the composite SMG
+SED from Dudzevičiūtė et al. (2020), which broadly reproduces the behavior we see.
+Figure 13. SFR–M* distribution for the S2TDF, compared to a K < 22.5
+limited, radio-selected subsample of SMGs from AS2UDS (Dudzevičiūtė
+et al. 2020). We show the median error at the bottom left corner in green. The
+solid line is the linear trend derived from a sample of K-band-detected field
+galaxies that has been matched to the S2TDF SMGs in redshift and stellar mass
+(see the main text), and the shaded area indicates the 20% uncertainty in the
+normalization. We also show the SFR–stellar mass relation for the “main
+sequence” of star-forming galaxies at various redshifts derived in Pearson et al.
+(2018). Both of these demonstrate that the S2TDF SMGs have SFRs above
+those typical for similar-mass field galaxies at their redshifts.
+12
+The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January
+Hyun et al.
+
+much higher levels of star formation activity seen in the SMGs,
+as a consequence of their much more massive reservoirs of
+cool gas.
+The ratio of dust mass to stellar mass in galaxies is a
+potentially useful parameter that links the growth of stellar
+populations in galaxies with the associated formation of metals
+in stars and dust in their atmospheres and the broader
+interstellar medium (ISM). Critically this ratio may provide
+constraints on the processes that destroy dust grains within the
+ISM (e.g., Santini et al. 2010; Dunne et al. 2011; Calura et al.
+2017; Donevski et al. 2020; Dudzevičiūtė et al. 2021).
+We show in Figure 14 the variation in the DSR for our
+SMGs and the radio/K-band-matched AS2UDS sample from
+Dudzevičiūtė et al. (2020). We also show the median DSRs for
+local and high-redshift galaxies. These comprise low-redshift
+spirals and ULIRGs, higher-redshift FIR-selected galaxies,
+SMGs, and a small number of very-high-redshift QSOs
+(Kennicutt et al. 2003, 2011; Greve et al. 2005; Clements
+et al. 2010; Santini et al. 2010; Lutz et al. 2011; Calura et al.
+2014, 2017).
+For
+the
+S2TDF
+main
+sample,
+the
+SMG
+DSR
+is
+=
+*
+M
+M
+log
+d
+10(
+)
+–2.5 ± 0.1 (–2.5 ± 0.2, S/N > 3.5), com-
+pared to −2.3 ± 0.1 for the AS2UDS matched sample, with a
+Kolmogorov–Smirnov two-sample test showing that the two
+populations are indistinguishable, PKS = 0.6. Our median
+estimates agree well with earlier crude estimates for SMGs
+(Greve et al. 2005; Santini et al. 2010) and appear to reflect a
+gradual increase in DSR from z = 0 to z ∼ 2 (which may in turn
+reflect the rising gas fraction in galaxies over this period, e.g.,
+Geach et al. 2005; Tacconi et al. 2018; Dudzevičiūtė et al.
+2020, 2021; Birkin et al. 2021).
+We also plot two sets of theoretical models on Figure 14 for
+the “main-sequence” and “starburst” models from Davé et al.
+(2019) and Pantoni et al. (2019). The former cosmological
+models by Davé et al. (2019) suggest little variation in the DSR
+as a function of SFR/M* (which distinguishes the two model
+families), but these fail to reproduce the range in DSR: our
+SMGs span a 16th–84th percentile range of DSRs of
+~
+*
+M
+M
+log
+d
+10(
+)
+−2.0 to −2.9, which is approximately twice
+the estimated uncertainty, indicating a possible intrinsic
+dispersion in the DSR of the population. In contrast, the
+simple analytic models from Pantoni et al. (2019) do broadly
+span the range in DSR in the SMGs. This likely reflects the
+relative paucity of large numbers of rapidly evolving massive,
+gas- and dust-rich galaxies at high redshifts in typical
+cosmological simulations.
+6. Summary and Conclusion
+We have conducted a SCUBA-2 850 μm survey of the
+JWST-TDF, which will be a prime deep extragalactic field
+benefiting from deep multi-epoch observations with the James
+Webb Space Telescope. The main results of our study are as
+follows.
+Our JCMT SCUBA-2 850 μm survey of the JWST-TDF
+identified 83 and 31 submm sources, at S/N > 4 and
+S/N = 3.5–4, respectively, in a survey area of ∼0.087 deg2.
+The mean 1σ sensitivity across the survey area is 1.0 mJy beam−1
+from the 41.3 hr of observation, with the deepest region achieving
+0.8 mJy beam−1. The latter is comparable to the 850 μm
+confusion limit of the JCMT (σc = 0.7 mJy beam−1).
+To derive the survey completeness and the boosting factor,
+we performed jackknife simulations using number counts
+reflecting the expected source density in the survey. The 50%
+and 80% completeness limits are 3.0 mJy and 4.0 mJy,
+respectively. The false detection rates derived from the
+simulations are 1% and 8% at S/N > 4.0 and S/N > 3.5.
+The purity of the sample approaches 100% at S/N > 4.25.
+We also analyzed sensitive new VLA 3 GHz observations of
+the JWST-TDF. These observations have a synthesized beam
+of a 0 7 FWHM and cover a 24′ field of view, reaching a 1σ
+sensitivity of 1 μJy at the field center. We have presented a
+Figure 14. In both panels, red and pink filled circles show the S2TDF sources
+at S/N > 4.0 and at S/N = 3.5–4.0, respectively. The median errors are also
+presented (green). (Top) The variation in cold dust mass from MAGPHYS for
+our SMGs with redshift. We compare this to that of a similarly radio-detected
+and K < 22.5 sample of SMGs from AS2UDS (Dudzevičiūtė et al. 2020). The
+details of the symbols are the same as those in Figure 12. In addition, we show
+a K-band-selected field galaxy sample with K < 22.5 (gray points) and the
+trend (yellow solid line) of estimated dust masses from Dudzevičiūtė et al.
+(2020). This illustrates significantly higher cold dust masses, and by
+implication higher cold gas masses, for the SMGs, compared to typical star-
+forming galaxies at similar redshifts. (Bottom) DSR as a function of redshift for
+radio-identified SMGs in this work and in AS2UDS (Dudzevičiūtė et al. 2020).
+For comparison we present the median DSR of various samples collated by
+Calura et al. (2017) with open diamond symbols: blue, cyan, deep pink, brown,
+and purple colors represent spiral galaxies from SINGS (Kennicutt et al. 2003),
+spiral galaxies from KINGFISH (Kennicutt et al. 2011), a local ULIRG sample
+(Clements et al. 2010), FIR-selected galaxies in the COSMOS/GOODS
+surveys (Calura et al. 2017; Lutz et al. 2011), and higher-redshift SMGs (Greve
+et al. 2005; Santini et al. 2010), respectively. We also show the model
+predictions of “main-sequence” and starburst dusty star-forming galaxies from
+Davé et al. (2019) and Pantoni et al. (2019), respectively.
+13
+The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January
+Hyun et al.
+
+AS2UDS(Radio-detectedandK<22.5&S870
+m>2.2mJV
+OS2TDF(SNR=3.5-4.0）
+●S2TDF(SNR>4.0)
+1.00
+Ma[10° Ma
+0.10
+0.01
+0
+1
+2
+3
+4catalog of 756 sources detected above a flux density limit of
+S/N = 5 in this field.
+The SCUBA-2 sources were crossmatched with the radio
+sources from our deep VLA 3 GHz map. We found 85 radio
+counterparts to 66 SCUBA-2 sources. There are nine SCUBA-
+2 sources with two radio counterparts and five with three radio
+counterparts. In the case of the main sample, 54 out of 83
+SCUBA-2 sources matched with 70 radio sources and there are
+five/six triple/double-counterpart cases. For the supplementary
+sample, 12 out of 31 SCUBA-2 sources matched with 15 radio
+sources including three double counterparts. A catalog of the
+S2TDF submm sources is presented, including those that were
+matched with radio and optical/NIR sources.
+We investigated the cumulative and differential SMG
+number counts as a function of the flux densities. Overall,
+the trends are similar to the results from other studies, but there
+is a slight excess at higher flux densities (S850 > 10 mJy), which
+arises from four bright sources in our relatively small
+survey area.
+There are 51 (61) radio-located S2TDF SMGs with optical or
+NIR counterparts at S/N > 4 (S/N > 3.5). SED fitting was
+performed to these sources to derive their properties. We found
+that their estimated photometric redshifts have a median of
+z = 2.22 ± 0.12
+(z = 1.96 ± 0.08),
+their
+median
+SFR
+is
+300 ± 40 Me yr−1 (240 ± 40 Me yr−1), and they have stellar
+masses of 1.7 ± 0.3 × 1011 Me (1.6 ± 0.3 × 1011 Me) for the
+sample at S/N > 4.0 (S/N > 3.5). This demonstrates that these
+galaxies lie above the main sequence at their redshifts, and that
+they represent the high-mass end of the star-forming popula-
+tion, suggesting that they are progenitors of massive early-type
+galaxies today.
+We estimated dust masses and DSRs for our SMG sample,
+finding large dust masses, Md = 5.9 ± 0.7 × 108 Me, which
+imply correspondingly high cold gas masses, Mg ∼ 2–3 ×
+1010 Me, much higher than expected for typical field galaxies.
+These large reservoirs of gas are the fuel that drives the intense
+activity we see in the SMG population.
+Our study provides a submm data set that is critical for
+investigating the obscured star formation activity of galaxies
+out to high redshifts within the JWST-TDF. Together with the
+anticipated JWST data, we expect that the S2TDF will provide
+valuable insights into the physical mechanisms that triggered
+vigorous star formation activity in the early universe, by
+combining the morphological information from JWST and the
+extreme galaxies identified by the S2TDF.
+We thank Rick Perley and Ken Kellermann for their help with
+the VLA observations and their reduction and analysis. This work
+was supported by National Research Foundation of Korea (NRF)
+grants, No. 2020R1A2C3011091 and No. 2021M3F7A1084525,
+funded by the Korean government (MSIT). M.H. acknowledges
+support from a Korea Astronomy and Space Science Institute
+grant
+funded
+by
+the
+Korean
+government
+(MSIT)
+(No.
+2022183005) and support from the Global PhD Fellowship
+Program through the NRF funded by the Ministry of Education
+(NRF-2013H1A2A1033110). I.R.S. acknowledges support from
+STFC (ST/T000244/1). R.A.W., S.H.C., and R.A.J. acknowl-
+edge support from NASA JWST Interdisciplinary Scientist grants
+NAG5-12460, NNX14AN10G, and 80NSSC18K0200 from
+GSFC. C.N.A.W. acknowledges funding from Hubble Space
+Telescope grant GO-15278 and the NIRCam Development
+Contract NAS5-02105 from NASA to the University of Arizona.
+H.S. acknowledges support from NRF grant No. 2021R1A2
+C4002725 and No. 2022R1A4A3031306. Y.M. acknowledges
+support from JSPS KAKENHI grant Nos. 17KK0098, 19H00697,
+20H01953, 21H01133, 21H04489, and 22H01273. The James
+Clerk Maxwell Telescope is operated by the East Asian
+Observatory on behalf of The National Astronomical Observatory
+of Japan; Academia Sinica Institute of Astronomy and Astro-
+physics; the Korea Astronomy and Space Science Institute; Center
+for Astronomical Mega-Science (as well as the National Key
+R&D Program of China with No. 2017YFA0402700). Additional
+funding support is provided by the Science and Technology
+Facilities Council of the United Kingdom and participating
+universities in the United Kingdom and Canada. Additional funds
+for the construction of SCUBA-2 were provided by the Canada
+Foundation for Innovation. The National Radio Astronomy
+Observatory is a facility of the National Science Foundation
+operated under cooperative agreement by Associated Universities,
+Inc. Based in part on data collected at the Subaru Telescope and
+retrieved from the HSC data archive system, which is operated by
+the Subaru Telescope and Astronomy Data Center at the National
+Astronomical Observatory of Japan. Observations reported here
+were obtained at the MMT Observatory, a joint facility of the
+Smithsonian Institution and the University of Arizona.
+Facilities: JCMT, VLA, Subaru, MMT.
+Appendix A
+VLA Observations of the JWST-TDF
+A.1. Introduction
+The VLA observations presented here were obtained as part
+of an ongoing radio program using the VLA and the Very Long
+Baseline Array (VLBA) to generate a list of extragalactic radio
+sources in the JWST-TDF using the VLA, and then identify
+those containing a significant AGN component using the
+VLBA. The VLBA can observe multiple targets in the primary
+beam of the antennas; however, it is impractical to image the
+entire beam so accurate positions of the targets must be known
+in advance. To this end the VLA observations of the JWST-
+TDF region are centered on the very compact, 200 mJy quasar
+J1723+6547, for subsequent use as a phase reference for the
+VLBA observations (the chosen pointing contains no other
+strong radio sources).
+However, these VLA observations also provide sensitive
+high-resolution radio coverage of the entire JWST-TDF as
+shown in Figure 1. This provides an excellent resource for
+studying the radio properties of other populations uncovered in
+this field; in particular these high-resolution observations can
+be used to identify likely counterparts to the submm sources
+detected in our low-resolution SCUBA-2 map of this field.
+A.2. Observations and Data Reduction
+The observations were undertaken using the VLA “S band”
+(ν = 1.989–4.013 GHz), which is optimal for this project as it
+provides a good compromise between high sensitivity and a
+wide field of view. Since the majority of galaxies observed
+have steep spectra, the relatively low frequency also helps
+detectability. The single pointing gives the lowest detection
+level for a given total exposure near the pointing center, but at
+the cost of the sensitivity dropping away from this field center.
+We used the VLA to observe a field centered on J1723
++6547 (17 23 14.1381, +65 47 46.179) using the S-band
+receivers, with the 1.989–4.013 GHz frequency range divided
+14
+The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January
+Hyun et al.
+
+into 16 contiguous subbands each of width Δν = 128 MHz.
+Each subband was further divided into 64, 2 MHz channels.
+Observations consisted of 2 hr blocks, one or more of which
+were observed on a given day. All data taken on a given
+calendar day were processed together. The observations are
+summarized in Table 2, which gives the date, VLA configura-
+tion, and total duration of the observations. Each 2 hr block
+included the astrometric calibrator J1800+7828 and one of the
+photometric/bandpass/polarization
+calibrators
+J1331+3030
+(3C 286) and J0137+3309 (3C 48).
+A.2.1. Calibration
+Calibration and imaging used the OBIT package (Cotton
+2008).15 Due to the strong source at the field center, special
+care was needed to minimize artifacts. Calibration and editing
+was done on each day’s data independently and consisted of the
+steps described below. As the target field was dominated by a
+strong, compact source it was included as a calibrator. In each
+step deviant calibration solutions were detected and flagged
+along with the corresponding data. Standard structural and
+spectral models for J1331+3030 and J0137+3309 (Perley &
+Butler 2013a) were used as appropriate. Processing consisted
+of the following:
+1. Fixed flagging: Frequency ranges known to contain
+strong, persistent radio frequency interference (RFI) were
+flagged.
+2. Initial flagging: Running medians in time and frequency
+were used on the data to identify and flag RFI.
+3. Initial amplitude calibration: Amplitude corrections were
+determined from the “switch power” calibration signals
+injected into each data stream.
+4. Delay calibration: Residual group delays were deter-
+mined for all calibrators and the target field.
+5. Bandpass calibration: Amplitude and phase correction
+spectra were determined from the bandpass calibrator.
+6. Amplitude and phase calibration: Complex gain solutions
+for the calibrators were determined for each calibrator and
+the target field. The daily flux density of the phase
+reference source was determined by a comparison of gain
+solutions with the photometric standard(s) and was used
+to correct the amplitudes of the solutions for it and the
+target. This value is given in Table 2.
+7. Flagging of calibrated data: Flagging operations for
+which calibrated data were needed were done.
+8. Repeat: The flaggings from the above steps were kept and
+the calibration was repeated.
+9. Polarization calibration: Instrumental polarization was
+determined from the phase reference calibrator when an
+adequate range of parallactic angles was available;
+solutions were done in 16 MHz blocks. The cross-hand
+delay and phase were determined from J1331+3030
+(preferred) or J0137+3309 (Perley & Butler 2013b).
+All data sets were then subjected to a baseline-dependent time
+averaging to reduce the data volume. This averaging was
+subject to the constraint that the amplitudes were reduced by no
+more than 1% to a radius of 12′ and averaging was no more
+than 0.35 minutes.
+A.2.2. Imaging
+As the observations were made over an extended period and
+the central source is mildly variable (see calibration values in
+Table 2), data from each day’s observations were imaged and
+the contribution of the central source was subtracted. Initial
+imaging included a phase+delay self-calibration followed by
+an amplitude and phase self-calibration. The imaging used the
+OBIT task MFIMAGE, which is described in more detail in
+Cotton et al. (2018). Significant artifacts survived this process
+and additional filtering was needed to suppress them.
+A.2.3. Image Artifacts
+The imaging artifacts arising from the strong central source
+included the following pathologies, given with their remedia-
+tion (when available):
+1. “Fingerprints”—Spiral patterns resembling fingerprints
+were traced to minor residual group delay errors when the
+group delay corrections were determined solely from the
+calibrators. Including the target in the group delay
+calibration and doing phase and delay self-calibration in
+the imaging largely eliminated these.
+2. “Black stripes”—Initial imaging resulted in dark hor-
+izontal bands through and near the central source. These
+were traced to the periods of time when the fringe rate on
+a given baseline went through zero allowing greater
+sensitivity to RFI. As these are near u = 0 the resulting
+artifacts appear as horizontal stripes at the field center.
+These artifacts were largely suppressed by subtracting the
+image sky model from the data, averaging the data, and
+clipping the data above a given level.
+3. “Bowls”—All of the “A” configuration data sets but
+neither of the “B” data sets show a negative bowl around
+the position of the central source with a maximum depth
+of ∼150 μJy, even after the source was subtracted. The
+source of this artifact has not been determined but it only
+extends a few beams from the central source. As the
+central source is completely unresolved, this bowl is not
+the feature commonly seen around very extended, bright
+features.
+Table 2
+Log of the VLA Observations of the JWST-TDF
+Date
+Config.
+Timeobs
+SJ1800
+δSJ1800
+SJ1723
+δSJ1723
+(hr)
+(Jy)
+(Jy)
+(Jy)
+(Jy)
+2017 Nov 29
+B
+2
+1.810
+0.011
+0.214
+0.001
+2017 Dec 18
+B
+2
+1.680
+0.013
+0.194
+0.001
+2018 Jun 01
+A
+2
+2.188
+0.007
+0.212
+0.001
+2018 Jun 02
+A
+4
+2.144
+0.007
+0.219
+0.001
+2018 Jun 03
+A
+4
+2.138
+0.005
+0.219
+0.001
+2018 Jun 04
+A
+4
+2.136
+0.002
+0.222
+0.001
+2018 Jun 05
+A
+10
+2.152
+0.004
+0.227
+0.001
+2018 Jun 06
+A
+4
+2.178
+0.002
+0.228
+0.001
+2018 Jun 09
+A
+4
+2.164
+0.004
+0.235
+0.001
+2018 Jun 10
+A
+6
+2.138
+0.004
+0.231
+0.001
+2018 Jun 11
+A
+6
+2.099
+0.009
+0.226
+0.001
+Note. We list (1) the date of the observations, (2) the VLA configuration
+employed, (3) the total observing time, (4–5) the calibration flux density
+adopted for J1800+7828 and its uncertainty, and (6–7) the flux density of
+J1723+6547 and its uncertainty.
+15 http://www.cv.nrao.edu/~bcotton/Obit.html
+15
+The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January
+Hyun et al.
+
+A.2.4. Combined Image
+After each individual day’s data sets were imaged, the core
+was subtracted, and the artifact filtering was run, the data were
+combined into a single set and imaged with no further self-
+calibration. CLEANing proceeded to a minimum residual of
+5 μJy beam−1 or a maximum of 20,000 components. Imaging
+used a robust factor of 0.0 and the resulting images used a
+circular Gaussian restoring beam of a 0 7 FWHM. The rms in
+quiet places in the resultant image is 0.95 μJy beam−1 and the
+brightest pixel is 3.1 mJy beam−1.
+A.2.5. Source Catalog
+The construction of the source list used the OBIT task
+FNDSOU,
+which
+fits
+elliptical
+Gaussians
+to
+components
+identified in islands of emission. The derived list was compared
+with the image and entries corresponding to clear artifacts or
+multiple entries from an extended source were removed. Error
+analysis followed the development of Condon (1997). The field
+image FITS file and the full source list for 756 sources with a
+peak brightness >5 times the local rms and with a primary-
+beam gain of at least 10% at 3 GHz (within a radius of
+¢
+12 ) are
+given online with a sample shown in Table 3.
+Some sources are sufficiently extended to not be modeled by
+a simple Gaussian. The flux densities of these were determined
+by an integral of the image pixels bounded by a rectangular
+box. The LAS of the source was derived from the diagonal
+extent of this rectangle. These sources are indicated in Table 3
+by the LAS given in the “Size” column.
+A.3. Summary
+Our VLA 3 GHz observations of the JWST-TDF reach an
+rms of 1 μJy beam−1 in the region around the bright radio
+source J1723+6547 at a resolution of 0 7. The catalog of
+sources above 5σ generated from this map contains 756
+sources, which we used to identify the counterparts to the
+SCUBA-2 sources in this region. The full catalog of 3 GHz
+sources will also be employed in our ongoing VLBA
+observations of sources in this field.
+Appendix B
+Jackknife Simulation and Flux Distribution
+As we explained in Section 3.2, we performed 1000
+jackknife simulations to determine the amount of flux boosting
+and the completeness of the S2TDF. Here, we present two-
+dimensional density plots showing the simulation results
+(Figure 15).
+Figure 16 presents the flux density distribution of the
+observed flux and the deboosted flux from the empirical
+recovery method.
+Table 3
+VLA 3 GHz Source Catalog
+ID
+R.A.
+δR.A.
+Decl.
+δDecl.
+S3GHz
+δS3GHz
+Size
+δSize
+Minor
+δMinor
+P.A.
+δP.A.
+(J2000)
+(″)
+(J2000)
+(″)
+(μJy)
+(μJy)
+(″)
+(″)
+(″)
+(″)
+(deg)
+(deg)
+1
+17 21 18.6268
+0.0084
++65 47 38.086
+0.055
+63.7
+8.0
+<0.66
+L
+L
+L
+L
+L
+2
+17 21 20.0334
+0.0129
++65 50 54.965
+0.095
+45.8
+9.1
+<0.90
+L
+L
+L
+L
+L
+3
+17 21 24.2805
+0.0170
++65 48 50.988
+0.082
+88.9
+15.6
+1.00
+0.24
+<0.79
+L
+−57
+12
+4
+17 21 26.4499
+0.0216
++65 47 57.197
+0.079
+1521.6
+15.1
+LAS 16.2
+L
+L
+L
+L
+L
+5
+17 21 28.3552
+0.0151
++65 46 59.518
+0.075
+36.2
+6.1
+<1.10
+L
+L
+L
+L
+L
+Note. The central source, J1723+6547, was added to the catalog in spite of being removed from the image. We list (1) the source ID; (2–5) the J2000 positions of each
+source and their uncertainties; (6–7) the 3 GHz flux density, corrected for the primary beam, and its uncertainty; and (8–13) the deconvolved angular size and/or upper
+limit or largest angular size (LAS) and the position angle.
+(This table is available in its entirety in machine-readable form.)
+16
+The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January
+Hyun et al.
+
+Appendix C
+Optical Spectroscopy
+The spectroscopic data were obtained using Binospec
+(Fabricant et al. 2019) at the MMT Observatory. These were
+obtained using the 270 line mm−1 grating, which allows
+coverage
+of approximately
+4000–9000 Å with a typical
+dispersion of 1.30 Å pixel−1 and a resolution of 1300. The
+sample of galaxies was constructed by combining a preliminary
+catalog from the HEROES Subaru HSC imaging (G. Hasinger,
+private communication) and a catalog derived from MMT/
+MMIRS NIR imaging (C. N. A. Willmer et al. 2022, in
+preparation), limited at r ∼ 23. The slit assignment was
+prioritized for sources with X-ray, VLA data, or submm
+counterparts in preliminary catalogs coming from members of
+the collaboration (private communication from W. P. Maksym,
+R. Windhorst, or I. R. Smail, respectively).
+The data were reduced using a specially designed pipeline
+that produces wavelength- and flux-calibrated one-dimensional
+spectra (Kansky et al. 2019). Redshifts were measured using
+custom-written code using a combination of real-space cross-
+correlation and emission line fits. All spectra were visually
+inspected (by C. N. A. Willmer) and a redshift quality was
+assigned using the same criteria adopted by the DEEP2 survey
+(Newman et al. 2013), where a redshift quality of 4 is likely to
+be correct at a 95% level or better and quality 3 is correct at a
+level of ∼90%. Data qualities less than 3 were not considered
+in the analyses. The output from each of the observed fields
+was then merged into a single list using the positions of the
+NIR imaging, which are tied to the Gaia Data Release 2
+reference frame.
+Appendix D
+The Best-fit MAGPHYS Model SEDs for the S2TDF SMGs
+Figure 17 displays the best-fit MAGPHYS model SEDs for the
+85 SMGs.
+Figure 15. Two-dimensional density plots showing the results of 1000 jackknife simulations performed for our survey. (Left) A plot of injected sources as a function
+of input flux density and instrumental noise. (Middle) A density plot of completeness, the ratio of the number of recovered sources to that of injected sources, as a
+function of input flux density and instrumental noise. The contours are labeled with the completeness fraction. (Right) The plot shows the average boosting factor of
+the output flux density with the given instrumental noise. The dashed line shows the S/N = 3.5 limit. The contour labels indicate the flux boost factor.
+Figure 16. Flux density distribution of observed flux (black curve) and deboosted flux (red curve) for example sources derived from the empirical recovery method.
+Labels with red and black colors show the median value of the corrected flux and observed flux, respectively. All results assume Gaussian uncertainties of σinst = 1.0
+mJy, which is the median instrumental noise of the final map of the TDF. For bright sources, the effect of flux boosting is minimal. But, at the limit of our catalog, it
+can be considerable (;30%−50%).
+17
+The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January
+Hyun et al.
+
+Figure 17. The best-fit MAGPHYS model SEDs for the 85 SMGs in this work that have 3 GHz radio counterparts, including the 61 sources for which we have
+sufficient optical and/or NIR photometry data to derive useful constraints on their SED properties. The bold frame represents the main targets at S/N > 4. We list the
+desired photometric redshifts and if available any spectroscopic values. The sources range within z = 0.8–3.8 and all sources show a strong FIR peak in their SEDs
+traced by the SCUBA-2 850 μm emission.
+18
+The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January
+Hyun et al.
+
+Appendix E
+S2TDF Source Catalog and Stamp Images
+Figure 18 provides, in an online-only format, postage-stamp
+images for all the main and supplementary SCUBA-2 sources
+detected in the JWST-TDF. The first 10 online-only figures are
+of the main set; the last four online-only figures are of the
+supplement. Table 4 provides a detailed description of the full
+version of JWST-TDF SCUBA-2 850 μm. The catalog is provided
+in FITS format and includes both the main and supplementary
+sources, which are differentiated by the MAIN Boolean column.
+Figure 18. The 32″ × 32″ images of the main 83 SCUBA-2 sources (S/N > 4) detected in the JWST-TDF. If the source has a radio counterpart, the label is black;
+otherwise, it is red. The positions of the radio counterparts are shown with blue crosses on the JCMT image and square marks on the HSC G and MMIRS Y and K
+images. If there is more than one counterpart lighter blue marks are also added to the figure in orders of distances. The matching radius of 7 5 is indicated with four
+yellow marks. The complete set of postage stamps (14 images) is available online. The first 10 images are of the main sources; the last four are of the supplementary
+sources.
+(An extended version of this figure is available.)
+19
+The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January
+Hyun et al.
+
+Table 4
+Description of the Full Version of JWST-TDF SCUBA-2 850 μm Source Catalogs
+Index
+Label
+Units
+Description
+1
+index
+L
+Index
+2
+JCMT_index
+L
+JCMT index
+3
+JCMT_ID
+L
+IAU position–based name, S2TDFJHHMMSS+/–DDMMSS
+4
+JCMT_SHORT_ID
+L
+Short name, S2TDF.JCMT_index
+5
+ID
+L
+Identifier with VLA multiplicity index, JCMT_SHORT_ID.[0, 1, 2]
+6
+MAIN
+Boolean
+True = main (JCMT_S/N > 4); False = supplementary (JCMT_S/N = 3.5–4.0)
+7
+Got_VLA_ID
+Boolean
+True = VLA match
+8
+NOT_Got_VLA_ID
+Boolean
+True = no VLA match
+9
+Closest_ID
+Boolean
+True = closest matched object
+10
+Multiple_IDs
+Boolean
+True = multiple VLA matches
+11
+N(S>S)
+L
+Radio source number density S3GHz > 10 μJy
+12
+Ni
+L
+Radio source number density S3GHz > Si
+13
+Pc
+L
+p
+=
+P
+r N
+c
+s
+T
+2
+, Equation (8)
+14
+P*
+L
+p
+=
+*
+P
+r N
+i
+i
+2
+, Equation (9)
+15
+p
+L
+Corrected Poissonian probability, Equation (6)
+16
+n_radio_counterpart
+L
+Number of radio counterparts
+17
+closest_rank
+L
+Rank of the radio-matched object based on the separated distance [1, 2, 3]
+18
+JCMT_FLUX_PEAK
+mJy
+Peak JCMT 850 μm flux
+19
+JCMT_RMS_PEAK
+mJy
+rms uncertainty in peak flux
+20
+JCMT_FLUX_MODEL
+mJy
+Model JCMT 850 μm flux
+21
+JCMT_RMS_MODEL
+mJy
+rms uncertainty in model flux
+22
+JCMT_S/N
+L
+S/N, JCMT 850 μm
+23
+JCMT_BLEND
+L
+[0, 1], 1 = blend
+24
+JCMT_FLUX_DEBOOST
+mJy
+Deboosted JCMT 850 μm flux
+25
+JCMT_FLUX_DEBOOST_ERRLO
+mJy
+Lower uncertainty in JCMT_FLUX_DEBOOST
+26
+JCMT_FLUX_DEBOOST_ERRHI
+mJy
+Upper uncertainty in JCMT_FLUX_DEBOOST
+27
+JCMT_TOTAL_ERRLO
+mJy
+Lower total uncertainty in JCMT_FLUX_DEBOOST
+28
+JCMT_TOTAL_ERRHI
+mJy
+Upper total uncertainty in JCMT_FLUX_DEBOOST
+29
+JCMT_RA_PEAK_DEG_FIN
+deg
+R.A., peak flux, decimal degrees
+30
+JCMT_DEC_PEAK_DEG_FIN
+deg
+Decl., peak flux, decimal degrees
+31
+JCMT_RA_MODEL_DEG_FIN
+deg
+R.A., peak model flux, decimal degrees
+32
+JCMT_DEC_MODEL_DEG_FIN
+deg
+Decl., peak model flux, decimal degrees
+33
+ID_VLA
+L
+Identifier from VLA catalog (Appendix A)
+34
+RA_VLA
+deg
+R.A., VLA source, decimal degrees
+35
+DEC_VLA
+deg
+Decl., VLA source, decimal degrees
+36
+radio_flux
+μJy
+3 GHz radio flux, S3GHz
+37
+dS3GHz
+μJy
+Uncertainty in 3 GHz flux
+38
+angular_distance
+arcsec
+Angular separation between VLA and JCMT
+39
+sed_group
+L
+1: K detection + opt./NIR data; 2: no K detection; 3: VLA+submm data only
+40
+GMAG_APER_300
+mag
+G-band aperture magnitude from HSC (Bosch et al. 2018; Miyazaki et al. 2018)
+41
+GMAGERR_APER_300
+mag
+Uncertainty in GMAG_APER_300
+42
+IMAG_APER_300
+mag
+I-band aperture magnitude from HSC (Bosch et al. 2018; Miyazaki et al. 2018)
+43
+IMAGERR_APER_300
+mag
+Uncertainty in IMAG_APER_300
+44
+ZMAG_APER_300
+mag
+Z-band aperture magnitude from HSC (Bosch et al. 2018; Miyazaki et al. 2018)
+45
+ZMAGERR_APER_300
+mag
+Uncertainty in ZMAG_APER_300
+46
+YMAG_APER_3
+mag
+Y-band aperture magnitude from MMIRS (McLeod et al. 2012)
+47
+YMAGERR_APER_3
+mag
+Uncertainty in YMAG_APER_3
+48
+JMAG_APER_3
+mag
+J-band aperture magnitude from MMIRS (McLeod et al. 2012)
+49
+JMAGERR_APER_3
+mag
+Uncertainty in JMAG_APER_3
+50
+HMAG_APER_3
+mag
+H-band aperture magnitude from MMIRS (McLeod et al. 2012)
+51
+HMAGERR_APER_3
+mag
+Uncertainty in HMAG_APER_3
+52
+KMAG_APER_3
+mag
+K-band aperture magnitude from MMIRS (McLeod et al. 2012)
+53
+KMAGERR_APER_3
+mag
+Uncertainty in KMAG_APER_3
+54
+w1_ab
+mag
+WISE band 1 AB magnitude (Wright et al. 2010)
+55
+dw1
+mag
+Uncertainty in w1_ab
+56
+w2_ab
+mag
+WISE band 2 AB magnitude (Wright et al. 2010)
+57
+dw2
+mag
+Uncertainty in w2_ab
+58
+w3_ab
+mag
+WISE band 3 AB magnitude (Wright et al. 2010)
+59
+w3msigmpro
+mag
+Uncertainty in w3_ab
+60
+w4_ab
+mag
+WISE band 4 AB magnitude (Wright et al. 2010)
+61
+w4msigmpro
+mag
+Uncertainty in w4_ab
+62
+Major
+arcsec
+Ellipse-fit major axis, arcseconds
+63
+dMajor
+arcsec
+Uncertainty in Major: 99 = LAS; −99 = upper limit
+20
+The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January
+Hyun et al.
+
+ORCID iDs
+Minhee Hyun
+https://orcid.org/0000-0003-4738-4251
+Myungshin Im
+https://orcid.org/0000-0002-8537-6714
+Ian R. Smail
+https://orcid.org/0000-0003-3037-257X
+William D. Cotton
+https://orcid.org/0000-0001-7363-6489
+Jack E. Birkin
+https://orcid.org/0000-0002-3272-7568
+Satoshi Kikuta
+https://orcid.org/0000-0003-3214-9128
+Hyunjin Shim
+https://orcid.org/0000-0002-4179-2628
+Christopher N. A. Willmer
+https://orcid.org/0000-0001-
+9262-9997
+James J. Condon
+https://orcid.org/0000-0003-4724-1939
+Rogier A. Windhorst
+https://orcid.org/0000-0001-
+8156-6281
+Seth H. Cohen
+https://orcid.org/0000-0003-3329-1337
+Rolf A. Jansen
+https://orcid.org/0000-0003-1268-5230
+Chun Ly
+https://orcid.org/0000-0002-4245-2318
+Yuichi Matsuda
+https://orcid.org/0000-0003-1747-2891
+Giovanni G. Fazio
+https://orcid.org/0000-0002-0670-0708
+A. M. Swinbank
+https://orcid.org/0000-0003-1192-5837
+Haojing Yan
+https://orcid.org/0000-0001-7592-7714
+References
+An, F. X., Simpson, J. M., Smail, I., et al. 2019, ApJ, 886, 48
+An, F. X., Stach, S. M., Smail, I., et al. 2018, ApJ, 862, 101
+Barger, A. J., Cowie, L. L., & Richards, E. A. 2000, AJ, 119, 2092
+Barger, A. J., Cowie, L. L., Sanders, D. B., et al. 1998, Natur, 394, 248
+Barnes, J. E., & Hernquist, L. E. 1991, ApJL, 370, L65
+Battisti, A. J., da Cunha, E., Grasha, K., et al. 2019, ApJ, 882, 61
+Baugh, C. M., Lacey, C. G., Frenk, C. S., et al. 2005, MNRAS, 356, 1191
+Birkin, J. E., Weiss, A., Wardlow, J. L., et al. 2021, MNRAS, 501, 3926
+Blain, A. W., Chapman, S. C., Smail, I., & Ivison, R. 2004, ApJ, 611, 725
+Table 4
+(Continued)
+Index
+Label
+Units
+Description
+64
+Minor
+arcsec
+Ellipse-fit minor axis, arcseconds
+65
+dMinor
+arcsec
+Uncertainty in Minor: −99 = upper limit
+66
+PA
+deg
+Ellipse-fit position angle
+67
+dPA
+deg
+Uncertainty in PA
+68
+chi
+L
+χ2 of the SED fitting result
+69
+Mstar_l
+log(Me)
+Lower bound on Mstar
+70
+Mstar
+log(Me)
+Stellar mass in log
+71
+Mstar_u
+log(Me)
+Upper bound on Mstar
+72
+L_dust_l
+log(Le)
+Lower bound on L_dust
+73
+L_dust
+log(Le)
+Dust luminosity, Ldust, in log
+74
+L_dust_u
+log(Le)
+Upper bound on L_dust
+75
+Tdust_l
+K
+Lower bound on Tdust
+76
+Tdust
+K
+Kinetic dust temperature, Tdust
+77
+Tdust_u
+K
+Upper bound on Tdust
+78
+SFR_l
+log(Me yr–1)
+Lower bound on SFR
+79
+SFR
+log(Me yr–1)
+Star formation rate in log
+80
+SFR_u
+log(Me yr–1)
+Upper bound on SFR
+81
+Mdust_l
+log(Me)
+Lower bound on Mdust
+82
+Mdust
+log(Me)
+Dust mass in log
+83
+Mdust_u
+log(Me)
+Upper bound on Mdust
+84
+Age_l
+log(yr)
+Lower bound on Age
+85
+Age
+log(yr)
+Age in log
+86
+Age_u
+log(yr)
+Upper bound on Age
+87
+AV_l
+mag
+Lower bound on AV
+88
+AV
+mag
+V-band reddening
+89
+AV_u
+mag
+Upper bound on AV
+90
+tau_l
+Gyr
+Lower bound on tau
+91
+tau
+Gyr
+Star formation timescale parameter
+92
+tau_u
+Gyr
+Upper bound on tau
+93
+sSFR_l
+log(yr−1)
+Lower bound on sSFR
+94
+sSFR
+log(yr−1)
+Specific SFR in log
+95
+sSFR_u
+log(yr−1)
+Upper bound on sSFR
+96
+z_lower
+L
+Lower bound on z_med
+97
+z_med
+L
+Median photometric redshift
+98
+z_upper
+L
+Upper bound on z_med
+99
+eb_l
+mag
+Lower bound on eb
+100
+eb
+mag
+Color excess, E(B – V )
+101
+eb_u
+mag
+Upper bound on eb
+102
+Notes
+L
+Notes
+Note. The full version of the JWST-TDF SCUBA-2 catalog for the main (S/N > 4) sources includes VLA counterparts from the Poisson probability crossmatching
+(see Section 4.1.1). The calculated Poissonian probabilities are presented as p. If there are multiple counterparts, we list them in the order of probability. The number
+after the decimal point in “ID” indicates multiple radio counterparts to a common submm source. The full catalog, including both the main and supplementary
+(S/N = 3.5–4.0) sources, is available online in FITS format.
+(This table is available in its entirety in machine-readable form.)
+21
+The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January
+Hyun et al.
+
+Bosch, J., Armstrong, R., Bickerton, S., et al. 2018, PASJ, 70, S5
+Bothwell, M. S., Smail, I., Chapman, S. C., et al. 2013, MNRAS, 429, 3047
+Calura, F., Gilli, R., Vignali, C., et al. 2014, MNRAS, 438, 2765
+Calura, F., Pozzi, F., Cresci, G., et al. 2017, MNRAS, 465, 54
+Carilli, C. L., Daddi, E., Riechers, D., et al. 2010, ApJ, 714, 1407
+Casey, C. M., Chen, C.-C., Cowie, L. L., et al. 2013, MNRAS, 436, 1919
+Casey, C. M., Narayanan, D., & Cooray, A. 2014, PhR, 541, 45
+Chabrier, G. 2003, PASP, 115, 763
+Chapin, E. L., Berry, D. S., Gibb, A. G., et al. 2013, MNRAS, 430, 2545
+Chapman, S. C., Blain, A. W., Smail, I., & Ivison, R. J. 2005, ApJ, 622, 772
+Chen, C.-C., Cowie, L. L., Barger, A. J., et al. 2013, ApJ, 776, 131
+Chen, C.-C., Smail, I., Swinbank, A. M., et al. 2015, ApJ, 799, 194
+Clements, D. L., Dunne, L., & Eales, S. 2010, MNRAS, 403, 274
+Condon, J. J. 1992, ARA&A, 30, 575
+Condon, J. J. 1997, PASP, 109, 166
+Coppin, K., Chapin, E. L., Mortier, A. M. J., et al. 2006, MNRAS, 372, 1621
+Cotton, W. D. 2008, PASP, 120, 439
+Cotton, W. D., Condon, J. J., Kellermann, K. I., et al. 2018, ApJ, 856, 67
+Cowie, L. L., Barger, A. J., Hsu, L.-Y., et al. 2017, ApJ, 837, 139
+Cowie, L. L., Gonzalez-Lopez, J., Barger, A. J., et al. 2018, ApJ, 865, 106
+Cunha, E. d., Walter, F., Smail, I. R., et al. 2015, ApJ, 806, 110
+da Cunha, E., Charlot, S., & Elbaz, D. 2008, MNRAS, 388, 1595
+Davé, R., Angles-Alcazar, D., Narayanan, D., et al. 2019, MNRAS, 486, 2827
+Dempsey, J. T., Friberg, P., Jenness, T., et al. 2013, MNRAS, 430, 2534
+Donevski, D., Lapi, A., Małek, K., et al. 2020, A&A, 644, A144
+Driver, S. P., Andrews, S. K., Davies, L. J., et al. 2016, ApJ, 827, 108
+Dudzevičiūtė, U., Smail, I., Swinbank, A. M., et al. 2020, MNRAS, 494, 3828
+Dudzevičiūtė, U., Smail, I., Swinbank, A. M., et al. 2021, MNRAS, 500, 942
+Dunne, L., Gomez, H. L., da Cunha, E., et al. 2011, MNRAS, 417, 1510
+Eales, S., Lilly, S., Gear, W., et al. 1999, ApJ, 515, 518
+Engel, H., Tacconi, L. J., Davies, R. I., et al. 2010, ApJ, 724, 233
+Fabricant, D., Fata, R., Epps, H., et al. 2019, PASP, 131, 075004
+Farrah, D., Lonsdale, C. J., Borys, C., et al. 2006, ApJL, 641, L17
+Geach, J. E., Dunlop, J. S., Halpern, M., et al. 2017, MNRAS, 465, 1789
+Geach, J. E., Matsuda, Y., Smail, I., et al. 2005, MNRAS, 363, 1398
+Goto, T., Oi, N., Utsumi, Y., et al. 2019, PASJ, 71, 30
+Goto, T., Takagi, T., Matsuhara, H., et al. 2010, A&A, 514, A6
+Greve, T. R., Bertoldi, F., Smail, I., et al. 2005, MNRAS, 359, 1165
+Gruppioni, C., Béthermin, M., Loiacono, F., et al. 2020, A&A, 643, A8
+Hainline, L. J., Blain, A. W., Smail, I., et al. 2011, ApJ, 740, 96
+Helou, G., Soifer, B. T., & Rowan-Robinson, M. 1985, ApJL, 298, L7
+Hodge, J. A., & da Cunha, E. 2020, RSOS, 7, 200556
+Holland, W. S., Bintley, D., Chapin, E. L., et al. 2013, MNRAS, 430, 2513
+Hong, J., Im, M., Kim, M., & Ho, L. C. 2015, ApJ, 804, 34
+Hopkins, P. F., Hernquist, L., Martini, P., et al. 2005, ApJL, 625, L71
+Hsu, L.-Y., Cowie, L. L., Chen, C.-C., Barger, A. J., & Wang, W.-H. 2016,
+ApJ, 829, 25
+Hughes, D. H., Serjeant, S., Dunlop, J., et al. 1998, Natur, 394, 241
+Ikarashi, S., Kohno, K., Aguirre, J. E., et al. 2011, MNRAS, 415, 3081
+Ivison, R. J., Greve, T. R., Dunlop, J. S., et al. 2007, MNRAS, 380, 199
+Ivison, R. J., Greve, T. R., Serjeant, S., et al. 2004, ApJS, 154, 124
+Ivison, R. J., Greve, T. R., Smail, I., et al. 2002, MNRAS, 337, 1
+Ivison, R. J., Morrison, G. E., Biggs, A. D., et al. 2008, MNRAS, 390, 1117
+Ivison, R. J., Smail, I., Dunlop, J. S., et al. 2005, MNRAS, 364, 1025
+Ivison, R. J., Smail, I., Le Borgne, J.-F., et al. 1998, MNRAS, 298, 583
+Jansen, R. A., & Windhorst, R. A. 2018, PASP, 130, 124001
+Jenness, T., Berry, D. S., Cavanagh, B., et al. 2009, in ASP Conf. Ser., 411,
+Astronomical Data Analysis Software and Systems XVIII, ed. D. A. Bohlender,
+D. Durand, & P. Dowler (San Francisco, CA: ASP),418
+Kansky, J., Chilingarian, I., Fabricant, D., et al. 2019, PASP, 131, 075005
+Karim, A., Swinbank, A. M., Hodge, J. A., et al. 2013, MNRAS, 432, 2
+Kennicutt, R. C., Calzetti, D., Aniano, G., et al. 2011, PASP, 123, 1347
+Kennicutt, R. C., Jr., Armus, L., Bendo, G., et al. 2003, PASP, 115, 928
+Kim, S. J., Lee, H. M., Jeong, W.-S., et al. 2015, MNRAS, 454, 1573
+Koushan, S., Driver, S. P., Bellstedt, S., et al. 2021, MNRAS, 503, 2033
+Lilly, S. J., Eales, S. A., Gear, W. K. P., et al. 1999, ApJ, 518, 641
+Lonsdale, C. J., Farrah, D., & Smith, H. E. 2006, in Ultraluminous Infrared
+Galaxies, ed. J. W. Mason (Praxis Publishing: Chichester), 285
+Lutz, D., Poglitsch, A., Altieri, B., et al. 2011, A&A, 532, A90
+Madau, P., & Dickinson, M. 2014, ARA&A, 52, 415
+Magliocchetti, M., Silva, L., Lapi, A., et al. 2007, MNRAS, 375, 1121
+Magnelli, B., Popesso, P., Berta, S., et al. 2013, A&A, 553, A132
+McAlpine, S., Smail, I., Bower, R. G., et al. 2019, MNRAS, 488, 2440
+McLeod, B., Fabricant, D., Nystrom, G., et al. 2012, PASP, 124, 1318
+Menendez-Delmestre, K., Blain, A. W., Alexander, D. M., et al. 2007, ApJL,
+655, L65
+Michałowski, M. J., Dunlop, J. S., Cirasuolo, M., et al. 2012, A&A,
+541, A85
+Miettinen, O., Novak, M., Smolcic, V., et al. 2015, A&A, 584, A32
+Mihos, J. C., & Hernquist, L. 1996, ApJ, 464, 641
+Miller, T. B., Chapman, S. C., Aravena, M., et al. 2018, Natur, 556, 469
+Miyazaki, S., Komiyama, Y., Kawanomoto, S., et al. 2018, PASJ, 70, S1
+Newman, J. A., Cooper, M. C., Davis, M., et al. 2013, ApJS, 208, 5
+Oke, J. B., & Gunn, J. E. 1983, ApJ, 266, 713
+Pantoni, L., Lapi, A., Massardi, M., Goswami, S., & Danese, L. 2019, ApJ,
+880, 129
+Pearson, W. J., Wang, L., Hurley, P. D., et al. 2018, A&A, 615, A146
+Perley, R. A., & Butler, B. J. 2013a, ApJS, 206, 16
+Perley, R. A., & Butler, B. J. 2013b, ApJS, 204, 19
+Rickard, L. J., & Harvey, P. M. 1984, AJ, 89, 1520
+Riechers, D. A., Leung, T. K. D., Ivison, R. J., et al. 2017, ApJ, 850, 1
+Sanders, D. B., & Mirabel, I. F. 1996, ARA&A, 34, 749
+Santini, P., Maiolino, R., Magnelli, B., et al. 2010, A&A, 518, L154
+Scott, K. S., Austermann, J. E., Perera, T. A., et al. 2008, MNRAS, 385, 2225
+Scott, S. E., Fox, M. J., Dunlop, J. S., et al. 2002, MNRAS, 331, 817
+Scoville, N., Sheth, K., Aussel, H., et al. 2016, ApJ, 820, 83
+Shim, H., Kim, Y., Lee, D., et al. 2020, MNRAS, 498, 5065
+Shim, H., Lee, D., Kim, Y., et al. 2022, MNRAS, 514, 2915
+Simpson, J. M., Smail, I., Dudzeviciute, U., et al. 2020, MNRAS, 495, 3409
+Simpson, J. M., Smail, I., Swinbank, A. M., et al. 2019, ApJ, 880, 43
+Simpson, J. M., Swinbank, A. M., Smail, I., et al. 2014, ApJ, 788, 125
+Smail, I., Ivison, R. J., & Blain, A. W. 1997, ApJL, 490, L5
+Smail, I., Ivison, R. J., Owen, F. N., Blain, A. W., & Kneib, J. P. 2000, ApJ,
+528, 612
+Springel, V., Di Matteo, T., & Hernquist, L. 2005, ApJL, 620, L79
+Stach, S. M., Dudzeviciute, U., Smail, I., et al. 2019, MNRAS, 487, 4648
+Stach, S. M., Smail, I., Amvrosiadis, A., et al. 2021, MNRAS, 504, 172
+Stach, S. M., Smail, I., Swinbank, A. M., et al. 2018, ApJ, 860, 161
+Swinbank, A. M., Chapman, S. C., Smail, I., et al. 2006, MNRAS, 371, 465
+Swinbank, A. M., Smail, I., Chapman, S. C., et al. 2004, ApJ, 617, 64
+Tacconi, L. J., Genzel, R., Saintonge, A., et al. 2018, ApJ, 853, 179
+Tacconi, L. J., Genzel, R., Smail, I., et al. 2008, ApJ, 680, 246
+Warren-Smith, R. F., & Wallace, P. T. 1993, in ASP Conf. Ser. 52,
+Astronomical Data Analysis Software and Systems II, ed. R. J. Hanisch,
+R. J. V. Brissenden, & J. Barnes (San Francisco, CA: ASP), 229
+Weiß, A., Kovacs, A., Coppin, K., et al. 2009, ApJ, 707, 1201
+Wilkinson, A., Almaini, O., Chen, C.-C., et al. 2017, MNRAS, 464, 1380
+Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, AJ, 140, 1868
+Zavala, J. A., Aretxaga, I., Geach, J. E., et al. 2017, MNRAS, 464, 3369
+Zhang, Y., Zheng, X. Z., Shi, D. D., et al. 2022, MNRAS, 512, 4893
+22
+The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January
+Hyun et al.
+
diff --git a/mtE0T4oBgHgl3EQf8QKN/content/tmp_files/load_file.txt b/mtE0T4oBgHgl3EQf8QKN/content/tmp_files/load_file.txt
new file mode 100644
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+page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Durham University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' South Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content=' 520 Edgemont Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content=' University of Tsukuba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Ten-nodai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content=' Japan  6 Department of Earth Science Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Kyungpook National University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 80 Daehak-ro,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content=' Republic of Korea  7 Steward Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' University of Arizona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 933 North Cherry Avenue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content=' Arizona State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Tempe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content=' USA  9 National Astronomical Observatory of Japan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Mitaka,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Japan  10 Center for Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Harvard & Smithsonian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 60 Garden Street,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' MS-65,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' MA 02138-1516,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' USA  11 Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' University of Missouri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Columbia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' MO 65211,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' USA  Received 2022 May 20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' revised 2022 September 26;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' accepted 2022 September 26;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' published 2023 January 5  Abstract  The James Webb Space Telescope Time-Domain Field (JWST-TDF) is an ∼14′ diameter field near the North  Ecliptic Pole that will be targeted by one of the JWST Guaranteed Time Observations programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Here, we describe  our James Clerk Maxwell Telescope SCUBA-2 850 μm imaging of the JWST-TDF and present the submillimeter  source catalog and properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We also present a catalog of radio sources from Karl J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Jansky Very Large Array  3 GHz observations of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' These observations were obtained to aid JWSTʼs study of dust-obscured galaxies  that contribute significantly to cosmic star formation at high redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Our deep 850 μm map covers the JWST-  TDF at a noise level of σ850mm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 mJy beam−1, detecting 83/31 sources in the main/supplementary signal-to-  noise ratio (S/N > 4 / S/N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5–4) sample, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The 3 GHz observations cover a 24′ diameter field with  a 1σ noise of 1 μJy beam−1 at a 0 7 FWHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We identified eighty-five 3 GHz counterparts to sixty-six 850 μm  sources and then matched these with multiwavelength data from the optical to the mid-infrared wave bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We  performed spectral energy distribution fitting for 61 submillimeter galaxies (SMGs) matched with optical/near-  infrared data, and found that SMGs at S/N > 4 have a median value of zphot = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='12, star formation rates of  300 ± 40 Me yr−1 (Chabrier initial mass function), and typical cold dust masses of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='7 × 108 Me, in line  with bright SMGs from other surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The large cold dust masses indicate correspondingly large cool gas masses,  which we suggest are a key factor necessary to drive the high star formation rates seen in this population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Unified Astronomy Thesaurus concepts: Galaxy evolution (594);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' High-redshift galaxies (734);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Galaxy formation  (595);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Submillimeter astronomy (1647);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Galaxy counts (588);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Ultraluminous infrared galaxies (1735)  Supporting material: extended figure, machine-readable tables  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Introduction  Tracing the formation of galaxies and the associated star  formation across cosmic time is a key topic for understanding the  evolution of our universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Most of the star formation activity over  cosmic history is hidden by surrounding dust, making it often  difficult to observe this activity through UV and optical studies  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Madau & Dickinson 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Driver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Koushan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' As such, it is important to study this hidden star formation  in other wave bands to draw a complete picture of the evolution of  galaxies, especially those in the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Dust around young stars efficiently absorbs their UV/optical  radiation and this is reradiated in the mid-infrared (MIR) and far-  infrared (FIR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Consequently, MIR and FIR surveys of galaxies  have revealed a population of luminous infrared galaxies, whose  contribution to the cosmic star formation rate (SFR) has been  found to rise toward z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Goto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2010, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Magnelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' At higher redshifts,  z ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5, the FIR emission from galaxies is redshifted into the  submillimeter (submm) window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' As a result, surveys in the  submm are well placed to find strongly star-forming, dust-  obscured galaxies at high redshifts, which are commonly termed  submm galaxies (SMGs)—defined as galaxies that are bright in  the submm, S850mm > 1 mJy (Hodge & da Cunha 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  From their first discovery (Smail et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Barger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Hughes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Eales et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1999), SMGs have been an  important population for understanding the most obscured phases  of galaxy formation and evolution at high redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' These galaxies  are bolometrically luminous (∼1012–13 L☉;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Barger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1998) and  massive (;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1011 M☉;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Hainline et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Michałowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Dudzevičiūtė et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2020) and contain a large amount of gas  (∼1011 M☉;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Greve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Tacconi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Carilli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Bothwell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Birkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' SMGs are  considered the progenitors of elliptical or spheroidal galaxies in the  local universe (Lilly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2018) from the fact  that they reside in massive dark matter halos with M ∼ 1012−13 M☉  The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3847/1538-4365/ac9bf4  © 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The Author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Published by the American Astronomical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Original content from this work may be used under the terms  of the Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 licence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Any further  distribution of this work must maintain attribution to the author(s) and the title  of the work, journal citation and DOI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  1  (Blain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Farrah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Magliocchetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Wilkinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Stach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2021) and from their rapid  formation (Lilly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Swinbank et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Simpson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' With the benefits of the strong negative K-correction in the  submm, SMGs are relatively easy to identify even at very high  redshifts (z ∼ 6, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Riechers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Gruppioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  They are now widely studied with various facilities and in  numerous fields (Scott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Coppin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Weiß et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Ikarashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Geach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Simpson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Shim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' see also Casey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  One of the key questions about SMGs is what physical  mechanism triggers their star formation activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We might  expect that most SMGs arise from galaxy mergers based on  studies in the local universe that show that the majority of  ultraluminous infrared galaxies (ULIRGs) are merging systems  (Sanders & Mirabel 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Lonsdale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Merging  provides an effective mechanism to drive intense star formation  and  active  galactic  nucleus  (AGN)  activity  (Barnes  &  Hernquist 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Mihos & Hernquist 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Springel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' McAlpine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Many observational studies have found distorted  morphologies for SMGs (Swinbank et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Menendez-  Delmestre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Tacconi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Engel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2015) suggestive of mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Some semianalytic  models also support the scenario that the vigorous star  formation in SMGs is driven by galaxy merging (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Chen  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' However, the very strong dust obscuration, even in  the rest-frame near-infrared (NIR), makes it difficult to  distinguish  truly  interacting  systems  from  those  where  structured dust obscuration creates apparent asymmetric or  disturbed morphologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The stellar-mass morphologies of  SMGs are one area where the recently launched James Webb  Space Telescope (JWST) is expected to make significant  contributions, through high-resolution MIR imaging that is  much less sensitive to dust obscuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  The JWST Time-Domain Field (JWST-TDF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Jansen &  Windhorst 2018) is an ∼14′ diameter region in JWSTʼs  northern continuous viewing zone near the North Ecliptic Pole  (NEP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The JWST-TDF has a unique advantage as an  extragalactic survey field in that it can be observed at any  cadence with JWST and is also expected to be frequently  visited due to spacecraft housekeeping activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' In addition,  the field is free from sources brighter than mAB ∼ 16 and has  low zodiacal foreground, as well as low Galactic extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  The planned JWST observations (GTO 1176, PI: R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Windhorst)  include eight-filter 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='8–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 μm imaging with the Near-Infrared  Camera (NIRCam) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='75–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='23 μm grism spectroscopy and  2 μm direct imaging with the Near Infrared Imager and Slitless  Spectrograph (NIRISS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' These observations will detect sources  down to mAB ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 29 mag, and this sensitivity (and spatial  resolution) makes the JWST-TDF ideal for exploring the  morphologies of SMGs, especially those that are faint in the  rest-frame optical/NIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  To take advantage of the JWST-TDF for SMG studies, we  have undertaken a submm survey covering the full JWST-TDF  using the Submillimetre Common-user Bolometer Array 2  (SCUBA-2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Holland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2013) of the James Clerk Maxwell  Telescope (JCMT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We call this survey the SCUBA-2 JWST-  TDF Survey (S2TDF) and to fully exploit its data we have also  obtained sensitive, high-resolution 3 GHz observations of this  region using the Karl J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Jansky Very Large Array (VLA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Together these two observational data sets support the wider  multiwavelength studies by the JWST-TDF GTO team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' In  particular, we expect that our SCUBA-2 and VLA surveys of  the JWST-TDF will provide critical information on the  redshifted FIR and radio emission, and thus on the dust-  obscured star formation rates and AGN activity, in galaxies to  be detected by JWST and will be useful for the wider  astronomical community for years to come.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' In this paper, we  present the S2TDF, and the catalog of SCUBA-2 sources in  this field, as well as details of the VLA 3 GHz survey of this  region used to identify the counterparts to these SCUBA-2  sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Section 2 presents the SCUBA-2 850 μm observations  and the data reduction process (the corresponding description  of the 3 GHz VLA observations and their reduction and  cataloging is given in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' In Section 3, we describe  the data analysis methods used to construct the catalog of  850 μm sources, such as jackknife simulations for deriving flux  deboosting, completeness, the false detection rate, and posi-  tional accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' In Section 4, we discuss the matching of the  SCUBA-2 sources with the 3 GHz radio and optical/NIR data  to robustly identify the SMG counterparts of the SCUBA-2  sources, followed by the analysis of the spectral energy  distributions (SEDs) of these counterparts using the available  multiwavelength data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Section 5 presents the 850 μm number  counts in the JWST-TDF and examines the properties of the  radio- and optical/NIR-matched SMGs such as their redshifts,  SFRs, and dust masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' In Section 6, we summarize our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Throughout this paper, all magnitudes are given in the AB  magnitude system (Oke & Gunn 1983) and we adopt  cosmological parameters of H0 = 70 km s−1 Mpc−1, ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='7,  and ΩM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' For the derivation of SMG properties with  SED fitting, we use the Chabrier initial mass function  (Chabrier 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Observation and Data Reduction  In the following section we describe in detail the SCUBA-2  observations of the JWST-TDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The corresponding description  of the VLA 3 GHz observations is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' SCUBA-2 Observations  The SCUBA-2 850 μm data were obtained from 2018  December  to  2020  December  (programs:  M18BP026,  M19BP031, and R20XP003) with the SCUBA-2 (Holland  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2013), the submm continuum bolometer array mounted  on the JCMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We used the PONG900 mapping mode, which  generates a 15′ diameter circular map with nearly uniform  sensitivity (Chapin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' As a consequence, our data  cover the full area of the 14′ diameter JWST-TDF, centered on  17 22 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='9,  +65 49 22  (J2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Figure  1  illustrates  the  SCUBA-2 area, overlaid on the field of view of other  multiwavelength coverages of the JWST-TDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' While we  observed simultaneously at 450 and 850 μm, the weather  conditions were generally not favorable for deep 450 μm  observations12 (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Therefore, we focus only on the deep  850 μm data in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  The observation was split into 62 MSBs, each of which  corresponds to a map with an on-source exposure time of  ∼40 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The total on-source exposure time is 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3 hr  (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The sky opacity (τ225), representing the weather  12 The rms noise was ∼16 mJy beam−1 even in the deepest region of the  stacked 450 μm map using only Band 1 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The only significant source  detected is a radio-loud QSO (TDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  2  The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January  Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  conditions of the observation, ranged from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='02 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='11, with a  median value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' While we were awarded Band 3  SCUBA-2 time for the 850 μm imaging, 13% and 21% of the  data were obtained in better weather conditions in Band 1  (τ225 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='05) and Band 2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='05 < τ225 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='08).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The observation  log is summarized in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Data Reduction  The SCUBA-2 data were reduced using the Dynamical  Iterative Map Maker (DIMM) within SMURF (Sub-millimeter  User Reduction Facility) with tools from the STARLINK KAPPA  software package (Warren-Smith & Wallace 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Jenness  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2009) to deal with NDF format images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We utilized the  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Illustration of the multiwavelength data coverage in the JWST-TDF survey region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The inner dashed green and dotted dark blue circles are the NIRISS and  NIRCam survey areas, respectively (10′ and 14′ diameters, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The data coverage from other facilities is shown in different colors as indicated (for more  details, see Jansen & Windhorst 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The solid blue circle represents the JCMT SCUBA-2 850 μm survey area analyzed in this work and the VLA 3 GHz  observations, with a 24′ diameter field of view, cover the full field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The distribution of the JCMT-TDF SCUBA-2 observations undertaken in the 2018B, 2019B, and 2020X semesters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Colors represent the weather conditions  of the observations, defined in “bands” calculated from the sky opacity, τ225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The data for 2018B were obtained typically in better conditions than those in which the  data for the 2019B and 2020X semesters were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Each observation block (minimum schedulable block (MSB)) consists of an ∼40 minute exposure with the  PONG900 scanning mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The total exposure time is 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3 hr (corresponding to 62 MSBs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  3  The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January  Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  “REDUCE_SCAN_FAINT_POINT_SOURCES”  recipe  in  the  ORAC-DR data reduction pipeline for a nearly automatic  reduction of SCUBA-2 data using SMURF and KAPPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' For  more detailed information on the full data reduction process  with SMURF, see Chapin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Flux calibration was applied to convert the picowatt units in the  reduced map into units of janskys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The calibrated maps were  produced  with  the  standard  flux  conversion  factor  of  537 Jy beam−1 pW−1 during the application of the PICARD  recipes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' For more information, see Dempsey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Finally,  maps of each ∼40 minute observation were generated with 4 0  pixel sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Using all the reduced and calibrated maps, we constructed a  stacked final map by applying the PICARD recipe “MOSAIC_JCM-  T_IMAGES.” Since our main targets are faint and compact, we  used a matched filter recipe “SCUBA2_MATCHED_FILTER” to  improve their detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The matched filter process first smooths  the map using a large Gaussian kernel and then subtracts the  smoothed map from the original to eliminate large-scale residual  noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Then, the map was convolved with the SCUBA-2 beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  The flux calibration of the SCUBA-2 maps was affected by the  matched filtering, resulting in a ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='10% decline in flux (Simpson  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' To calibrate this factor, we inserted artificial bright  sources into the map and recovered their flux after filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Then,  we cropped the map within a radius of 600″ by using the PICARD  recipe “CROP_SCUBA2_IMAGES” to isolate the region where the  sensitivity is high and uniform (a mean of σ850mm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0  mJy beam−1) to detect sources, but noting that this region entirely  covers the full planned JWST-TDF footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Figure 3 shows the final signal-to-noise ratio (S/N) map of the  850 μm SCUBA-2 coverage of the JWST-TDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The noise in the  deepest regions of the map reaches σrms = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='8 mJy beam−1, which  is  close  to  the  SCUBA-2  850 μm  confusion  limit,13  σ850mm ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='7 mJy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Figure 4 presents the cumulative  area of the SCUBA-2 field as a function of instrumental noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  We plot the corresponding values for the S2CLS EGS (Zavala  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2017), the S2CLS GOODS-N (Geach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2017), the  SCUBA-2 survey in the COSMOS field (Casey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2013),  SUPER GOODS-N (Cowie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2017), SUPER GOODS-S  (Cowie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2018), the S2CLS NEP (Geach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2017), the  S2COSMOS (Simpson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2019), and the NEPSC2 (Shim  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2020), which have a variable survey area and depth, on  the figure for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Source Catalog  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Source Detection  The final SCUBA-2 flux map is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' For source  detection, we adopted a simple top-down peak-finding method,  which is widely used in submm studies of cosmological survey  fields as this approach is effective in deblending sources with a  large difference in fluxes (Geach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Simpson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Shim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Specifically we followed the approach used by  Simpson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' In the filtered map, optimized to detect faint  point sources, we searched for sources with prominent peaks in  S/N in the region with σinst < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5 mJy beam−1 (Figure 3) and  recorded their properties (such as their positions, flux densities,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=') in a “first-pass” catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' After the first detection pass, we  subtracted these sources from the map by modeling their emission  with an empirical point-spread function (PSF) made by using all  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The S/N map for the 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3 hr of SCUBA-2 observations of the JWST-  TDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The cyan dashed circle is the area (r = 10′) where we performed source  detection (σinst< 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5 mJy beam−1) and the noise levels (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 mJy)  are indicated by white contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' In the deepest region of the field, the instrumental  noise reaches 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='8 mJy beam−1, and the mean of σinst is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 mJy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Cumulative area as a function of sensitivity of the S2TDF (black solid  line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The instrumental sensitivity of our map reaches σinst = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='8 mJy beam−1 in  the deepest region of the field covering 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='087 deg2 and this is quite close to the  confusion limit of SCUBA-2 850 μm, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='7 mJy beam−1 (red dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' For  comparison, the S2CLS EGS (blue dot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Zavala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2017) covering 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='019 deg2  with a 1σ depth has σinst = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='46 mJy beam−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' the S2CLS GOODS-N (green  asterisk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Geach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2017) covering 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='07 deg2 with a 1σ depth has σinst = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1 mJy  beam−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' the SCUBA-2 survey in the COSMOS field (light green triangle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Casey  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2013) covering 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='109 deg2 with a 1σ depth has σinst = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='8 mJy beam−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  SUPER GOODS (pink cross;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Cowie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2017, 2018) covering 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='233 deg2 with a  1σ depth has σinst = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='8 mJy beam−1 and reaches σinst = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='4 mJy beam−1 in the  deepest central field covering 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='018 deg2 (brown cross);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' the S2CLS NEP (purple  diamond;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Geach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2017) covering 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='6 deg2 with a 1σ depth has σinst = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2 mJy  beam−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' the S2COSMOS (orange square;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Simpson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2019) covering 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='6 deg2  with a median noise level has σinst = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2 mJy beam−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' and the NEPSC2 (yellow  star;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Shim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2020) covering 2 deg2 with a median noise level has  σinst = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3 mJy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The horizontal cyan solid line shows the final area  (r = 600″) of this work where we performed source detection (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  13 There have been several values of the confusion limit derived from various  studies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' however we adopted the value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='7 mJy beam−1, which is presented on the  official JCMT website: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='eaobservatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='org/jcmt/instrumentation/  continuum/scuba-2/time-and-sensitivity/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  4  The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January  Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  >5σ sources in the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' If two sources lay within 40″ of each  other, we used a double-PSF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Using the map where the  “first-pass” sources had been subtracted, we repeated the detection  step for a second pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' If additional sources were detected within  7 5 of the first-pass sources, we regarded the fluxes as the same  sources as the first-pass sources and recorded the information from  the prior detections only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' This iteration went on until all sources  above a minimum threshold of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5σ were found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' More details are  given in Simpson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  In this manner, we detected 83 submm sources above a 4σ  significance limit and a total of 114 above a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5σ limit in the  850 μm map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Figure 5 shows the positions of all detected  sources in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The thicker symbols represent the sources  with S/N > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' A histogram of the fluxes of the sources is  shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Jackknife Simulation  Jackknife simulations are a widely used method to estimate  the completeness, flux deboosting rate, false detection rate, and  positional accuracy of submm surveys (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Scott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Simpson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Shim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' To construct a jackknife  map, we randomly divided our SCUBA-2 MSBs into two  groups, constructed a combined map from each group, and  subtracted one from the other to remove any real astronomical  signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Next, the map where sources had been removed was  scaled down by the square root of the total integration time to  match the noise level in the actual final map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  In this source-free “noise” map, we injected artificial sources  based on the expected source number counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We assumed the  number density distribution followed a Schechter function with  the parameters from the 850 μm number counts in Geach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  (2017), N0 = 7180, S0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5, and γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5:  ⎜  ⎟⎜  ⎟  ⎜  ⎟  ⎛  ⎝  ⎞  ⎠  ⎛  ⎝  ⎞  ⎠  ⎛  ⎝  ⎞  ⎠  = g dN  dS  N  S  S  S  S  S  exp  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  1  0  0  0  0  ( )  Based on the expected number counts and our map area we  inserted 500 sources into each noise map with 850 μm fluxes of  1–20 mJy, using the above number distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We placed the  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' SCUBA-2 flux map of JWST-TDF showing 83 detected sources at S/N > 4 and a further 31 with S/N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The color of the sources represents their  deboosted flux (see color bar) and thicker symbols represent S/N > 4 sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' By matching the 850 μm sources to the VLA 3 GHz catalog using a corrected  Poissonian probability estimate, we found that 66 submm sources have radio counterparts (box symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Circle symbols are submm sources without radio  identifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The JWST NIRISS and NIRCam survey areas are indicated by dotted and dashed circles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The deboosted 850 μm flux density distribution of the 114 detected  SCUBA-2 sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' SNR is the S/N from the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Eighty-two and  thirty-two sources have S/N > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 (red histogram) and S/N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 (pink  histogram), respectively, and every source having a deboosted flux of >2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3  mJy is in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  5  The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January  Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  sources at random positions in the jackknife map, neglecting  any clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We produced 1000 jackknife-simulated maps  resulting in a mock catalog consisting of 500,000 sources in  total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The source detection processes were the same as the  process applied to the science maps described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Finally, we compared the output source properties to those in  the original input catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Completeness  From the jackknife simulation, we estimated the survey  completeness using the recovery success rate of injected  sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Figure 7 shows the completeness as a function of the  intrinsic flux density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The 50% and 80% completeness limits  are roughly 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 mJy and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 mJy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  The completeness is not uniform across the survey area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Therefore, the completeness presented in Figure 7 is an average  value over the survey area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' To obtain a more accurate  completeness, we derived a two-dimensional completeness  map as a function of the local instrumental noise (σinst) and the  deboosted source flux (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5 for the deboosting  process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Here, the instrumental noise was taken as the standard  deviation of the jackknife noise map at the source position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The  two-dimensional completeness map is presented in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  When we measured the number counts of 850 μm sources in  Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1, we applied the two-dimensional completeness  correction to every source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' False Detection Rate  At low S/N, there is a possibility that noise fluctuations will  appear as spurious sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' To assess the reliability of the  source catalog, we performed source detection in the same way  as the real source detection on both the jackknife “noise” map  and the jackknife map with artificial sources added (termed the  “source” map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  By comparing the number of detections in the “noise” map and  that in the “source” map, we evaluated the false detection rate as a  function of S/N (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Here, the false detection rate was  defined as the ratio between the number of fake sources (false  detections) and the number of detected sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' At >3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5σ, the false  detection rate is ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='8% and it drops to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1% at >4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  A detection threshold of >3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5σ gives a good balance  between the sample size and the false detection rate for  statistical studies of the SMG population (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', counts), while a  threshold of >4σ gives a sample with high reliability for  studying their detailed properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Deboosting of Fluxes  Due to the large PSF size of the SCUBA-2 map, submm  fluxes of sources, especially those near the flux limit of the  survey, were often artificially boosted due to blending with  positive noise peaks or faint sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The correction of the  boosted fluxes is called “deboosting.”  For flux deboosting, we derived the boosting factor (B) from  the ratio of the output (observed) flux density to the input flux  density of sources in the mock catalog from the jackknife  simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The average boosting factor at a given flux is  illustrated in Figure 7, which is well fitted by a power law in  S/N (Geach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Shim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2020) as given below for  our sources:  ⎛  ⎝  ⎞  ⎠  =  +  ´ B  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3  S N  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='6  ( )  As noted earlier, in practice, the flux limits vary across the  survey field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Therefore, we derived the boosting factor as a  function of the map sensitivity as represented by the instrument  noise, σinst, and the flux density (before the boosting correction)  as described in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' All the flux values were deboosted  with the relevant boosting factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Furthermore, we derived the  additional uncertainties in the flux values that were introduced  during the deboosting process (see Appendix B), and note the  uncertainties as σdeb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Positional Uncertainty of Sources  We can estimate the positional uncertainties of the detected  sources from the jackknife simulation by comparing the input  position with the detected (recovered) position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We derived the  rms dispersion of the positional differences (σ) in the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' and  decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' coordinates from the jackknife simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The σ values  were found to be inversely proportional to the S/N of the  source as expected, and to have nearly identical values in both  coordinates as given in the equation below:  ⎛  ⎝  ⎞  ⎠  s =  \uf0b2  ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 8  S N  5  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0  ( )  Additionally, we must consider an S/N-independent intrinsic  positional error (σ0) associated with errors in the telescope  pointing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' This intrinsic positional error is independent of S/N and  can usually be described by a Gaussian distribution with a  constant rms σ0 of bright SMGs whose σ values are very small  and negligible compared to σ0 (Condon 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We estimated the  intrinsic positional error by matching high-S/N SCUBA-2  sources (S/N > 10) with VLA source positions and excluding  SCUBA-2 sources with multiple VLA source matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' This gave  five SCUBA-2 sources, and since the number of sources is small,  we used all of the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' and decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' offsets together to derive σ0  assuming the intrinsic positional errors had an identical distribu-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We found that the S/N-independent positional error is  σ0 ∼ 0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Considering the VLA positional errors (0 1) are much  smaller than this value, we ignored their contribution to σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  In sum, the positional uncertainty in both the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' and decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  directions can be taken as the equation below:  s  s  s  =  +  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  4  tot  2  0  2  ( )  In Figure 7, we show σtot in bins of S/N, with the  contributions from the S/N-dependent (Equation (3)) and  intrinsic (σ0) terms indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  To check the consistency between the positional uncertainty  based on the data and the σtot explained above, we present the  standard deviation of the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' and decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' positional offsets  between the SCUBA-2 sources and matched VLA sources and  their errors estimated from bootstrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  For the Gaussian distributions of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' and decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' errors, the  radial position offset (ρ) has a distribution that follows the  Rayleigh distribution of r s  r  s exp  2  tot  2  2  tot  2  [  (  )].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' For this  distribution, 68% and 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='6% of the radial offsets would lie  within 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5σtot and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5σtot, respectively (Condon 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Ivison  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Geach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  6  The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January  Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Astrometric Calibration Using VLA  During JCMT observations, the calibration process checks and  revises the telescopeʼs pointing, and usually, the values of the shift  are of the order of ∼1″.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' However, it is important to determine  precise absolute positions if we wish to identify the counterparts to  the submm sources and derive their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Even a slight offset  will make a significant difference in the calculation of the matching  probability with other higher-resolution data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' This astrometric  calibration is typically achieved using radio maps, for example  those from the VLA, with arcsecond resolution that is tied to the  FK5 to a better than ∼0 01 absolute astrometric precision (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=',  Ivison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1998, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  We used the sources in the VLA 3 GHz catalog of this field  described in Appendix A to improve the absolute astrometry of  our JWST-TDF SCUBA-2 map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' From the radio source catalog,  detecting 756 sources at S/N > 5, we selected sources with a 3  GHz flux density of  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 μJy, resulting in 557 VLA sources  across our survey area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We used these to search for counterparts to  our submm sources using a matching radius of 7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='14 We  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (a) Completeness of the JWST-TDF SCUBA-2 survey calculated from the ratio of the number of recovered sources with S/N > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5 to the number of input  sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The 50% and 80% completeness limits are 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 mJy (deboosted flux), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (b) The false detection rate, the ratio between the number of  sources detected in the jackknife map and that of sources detected in the flux density map (inner panel), as a function of S/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' With an S/N > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5 cut the probability  that a detected source is spurious is ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='8%, and at S/N > 4 it is ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' this drops to 0% at S/N > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (c) Average flux boosting factor as a function of S/N, showing  that the results are well fitted with a power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' However, we evaluated boosting rates for each source based on the observed flux density and the local instrumental  noise by constructing a two-dimensional parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (d) The total positional error in both R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' and decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' directions (solid line), which is the quadratic sum of the  intrinsic positional error (σ0, dashed line) and the positional difference with the bootstrapping error (σ, dotted line), as a function of S/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Red filled circles are the  standard deviation of the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' and decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' position offsets between the SCUBA-2 sources and their VLA counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  14 We set the maximum matching radius to 7 5, which is about half of the  JCMT SCUBA-2 effective beam FWHM (θ1/2 = 14 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Dempsey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2013)  at λ = 850 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' This has been widely used as a matching radius for radio  identification (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Shim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2022), corresponding to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3  times the expected total positional error (2 4 for a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0σ detection and 2 7 for a  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5σ detection) with a 14 6 FWHM (see Equation (4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  7  The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January  Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  identified possible radio counterparts for 68 submm sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' To  determine the astrometric offset of the SCUBA-2 positions  with respect to the radio source positions, we then stacked  thumbnails of the SCUBA-2 sources centered on the matched  radio source positions and made a 74″ × 74″ stacked image  (Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Through this procedure, we found a small mean  coordinate shift for the SCUBA-2 map of 0  83 ± 0 22 and  −0 34 ± 0 03 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' and decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We corrected  our cataloged SCUBA-2 source positions for this offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We  also tested the sensitivity of the derived offset to the adopted  pixel scale of the SCUBA-2 map by repeating this process with  data regridded to finer pixel scales of 0 5 and 2 0 sampling,  finding no significant differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' JWST-TDF 850 μm Source Catalog  We constructed two catalogs of the SCUBA-2 sources in the  JWST-TDF (Table 4 in Appendix D) for the main (S/N > 4)  and  supplementary  sources  (S/N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0),  comprising  eighty-three and thirty-one 850 μm sources, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The  coordinates of the sources include the astrometric corrections  derived in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  There are three different flux uncertainties given in the  catalog: the instrumental noise (σinst), the deboosting uncertainty  (σdeb), and the total uncertainty (σtotal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The total flux uncertainty  is the square root of the squared sum of the instrumental  noise, the deboosting uncertainty, and the confusion noise (σc)  resulting from faint sources below the detection flux limit within  the JCMT beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We calculated the confusion noise using  Equation (3) of Simpson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2019):  s  s  s  = 5  c  total  2  inst  2  ( )  where σtotal and σinst are derived from the standard deviation of  the 850 μm map where sources over the detection limit were  removed, and from that of the jackknife map, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The  confusion noise in this study from Equation (5) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='14 mJy  beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The S/N of the sources were not deboosted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Band Merging and SED Analysis  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Multiwavelength Matching  We matched the detected 850 μm sources in the JWST-TDF  to radio counterparts from the 3 GHz catalog described in  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' These radio counterparts are essential to provid-  ing the precise positions necessary to determine the multi-  wavelength properties of the likely SMGs from the optical to  radio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' For SMGs with radio counterparts and optical/NIR  photometric detections, we derived their redshifts and other  properties through SED fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Radio Counterparts  To reliably identify the multiwavelength counterparts to the  submm sources, we exploited the correlation between the radio  and FIR luminosities of normal or star-forming galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=',  Rickard & Harvey 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Helou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Condon 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Ivison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2004, 2005, 2008), which has been widely used in  previous submm studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Barger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Smail et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Ivison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We first crossmatched  the SCUBA-2 sources with our VLA 3 GHz radio catalog (see  Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' For matching, we used the 557 radio sources  with 3 GHz flux densities above the uniform limit of  10 μJy  (as used in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  The corrected Poissonian probability, p, used in the  matching analysis was defined as follows:  = p  E  1  exp  6  (  )  ( )  ⎧  ⎨⎩  =  + E  P  P  P  P  P  P  P  P  1  ln  7  c  c  c  c  {  (  )}  ( )  \uf085  \uf084  p  =  P  r N  8  c  s  T  2  ( )  p  = P  r N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  9  i  i  2  ( )  In line with previous work we adopted a maximum search  radius (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='7), rs, of 7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The variable ri is the  angular distance between the submm centroid and the matching  radio source (with 3 GHz flux Si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' NT and Ni are the radio  source number densities, with S3GHz > 10 μJy and S3GHz > Si,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' For each SCUBA-2 source, we calculated the  probability, p, for any radio sources within rs, and retained only  sources having radio counterparts with p \x84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='065 based on the  balance of recovery and false-positive rates in the analysis of an  Atacama Large Millimeter/submillimeter Array (ALMA)-  identified sample of SMGs by An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Figure 9 summarizes the results of the radio identifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  The SCUBA-2 sources with radio identification are listed in  Table 4 along with the corresponding probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Note that  the number after the decimal point in the source ID indicates  the radio counterpart identifications, ranked on the probability  of the match;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' it can be larger than 0 if there are multiple  matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Using a maximum search radius of 7 5, there are 89  radio identifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Four radio sources have likelihoods of  being false matches of p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='065, and removing these reduces  the total number of robust matches to 85 radio sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' There  are 52 submm sources with single radio counterparts, nine with  two radio counterparts, and five with three counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The  SCUBA-2 sources with multiple radio counterparts have an  850 μm flux density distribution that is indistinguishable from  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' A 28″ × 28″ zoomed-in image of the stacked SCUBA-2 850 μm  image at the positions of the radio counterparts matched to 67 SCUBA-2  sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The red cross is the center from the two-dimensional Gaussian fitting  of each stacked image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' This shows a small mean coordinate shift of  0 83 ± 0 22 and −0 34 ± 0 03 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' and decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', respectively, which we  corrected for in our SCUBA-2 catalog positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  8  The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January  Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  that of the parent population of all SCUBA-2 sources with  radio counterparts (S850μm = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3 mJy versus S850μm =  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='7 mJy), while the fluxes of the SCUBA-2 sources  lacking radio counterparts are somewhat fainter on average:  S850μm = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2 mJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' For the main sample, 54 out of 83  SCUBA-2 sources matched with 70 radio sources including  five/six triple/double-counterpart cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' For the supplementary  sample, 12 out of 31 SCUBA-2 sources matched with 15 radio  sources, among which there are three double-counterpart cases  and no triple-counterpart cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  To summarize, 85 VLA radio sources are matched to 66 of the  114 SCUBA-2 sources, and these matches are typically sources  that are at the brighter end of the 850 μm flux distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The  overall matched fraction corresponds to an identification rate of  58% (65% and 39% for the main and the supplementary sample),  which is broadly consistent with that seen in other radio-identified  SMG samples at these depths (60%–80%, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Ivison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Casey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Miettinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Optical/NIR Counterparts  For those SCUBA-2 sources with radio counterparts, we used  the precise radio positions and our multiwavelength coverage to  characterize the properties of these galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We used the radio  positions and matched these to the ground-based band-merged  optical and NIR catalog of the JWST-TDF (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Willmer  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2022, in preparation) with a search radius of 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' This radius  is small enough that we expected no false matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  The band-merged catalog comprises optical photometry in  the g, i, and z bands based on source detections in the combined  i+z image (appropriate for detecting red galaxies such as  SMGs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' These data were obtained with Hyper Suprime-Cam  (HSC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Bosch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Miyazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2018), on the Subaru  telescope, and have 3σ aperture magnitude limits of g = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0,  i = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2, and z = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1 in 3 0 diameter apertures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The 3 0  aperture magnitude is roughly identical to total magnitudes for  compact/faint galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  This optical catalog has been merged with NIR catalogs in  the Y, J, H, and K bands from the Magellan Infrared  Spectrograph (MMIRS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' McLeod et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2012) on the Multi-  Mirror Telescope (MMT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The 3σ aperture magnitude limits of  the MMIRS images are Y = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='7, J = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='7, H = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0, and  K = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5, again in 3 0 diameter apertures, to provide total  magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' This NIR coverage has been supplemented by the  WISE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='6–22 μm counterparts, matched to the radio sources  using a 1 0 search radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  There are 61 radio-identified SMGs matched to the merged  optical/NIR catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Of these 61, 54 are detected above 3σ in  the NIR (16 are detected with WISE), and the remaining seven  have optical HSC detections and upper limits in the other  bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' SED Fitting Analysis  We performed an SED analysis of the 61 radio-identified,  optical/NIR-matched SMGs to derive both photometric red-  shifts and other physical properties (we similarly modeled a  further 24 sources that had radio and submm constraints but no  optical or NIR detections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We employed the MAGPHYS galaxy  SED-fitting code (da Cunha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Cunha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Battisti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  MAGPHYS uses an energy balance  technique to combine observations from the optical/NIR,  MIR/FIR, and submm wave bands to constrain the physical  properties of sources, including the stellar population and dust  content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The “high-redshift” version of the code, MAGPHYS  +PHOTOZ, also allows the redshift of the source to be estimated  (Battisti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The ability of the code to include  information from the dust-reprocessed emission seen in the  MIR/FIR and submm is important when attempting to model  SMGs, which are frequently faint or undetected in the optical  and NIR wave bands used by other photometric modeling  codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' MAGPHYS+PHOTOZ has been used to model samples of  SMGs, for example by Dudzevičiūtė et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2020), who  provided an extensive discussion of the calibration and testing  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' VLA 3 GHz flux densities of 89 radio sources matched to the 68 SCUBA-2 source positions, as a function of corrected Poissonian probability, p (left), and  as a function of the angular separation between the radio sources and the SCUBA-2 sources (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' SCUBA-2 sources with S/N > 4 are marked with blue filled  circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' There are 37 multiple radio counterpart matches to 16 SCUBA-2 sources, and these are indicated with red diamond marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' After the probability cut p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='065  is applied, 33 multiple matches to 14 SCUBA-2 sources remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The red vertical line is the probability cut adopted from the analysis of ALMA identifications of  SMGs by An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' With that criterion, four radio sources are removed from the radio-identified sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Hence, in total 85 radio sources are matched to 66  SCUBA-2 sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  9  The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January  Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  of the code on dust-obscured galaxies, including its application  to modeling SEDs derived from radiative transfer models of  simulated galaxies from the  EAGLE simulation (see also  McAlpine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Our SED modeling used the optical/NIR/MIR photometry for  the matched counterparts, and for the SMGs, also included our  deboosted 850 μm observations and 3 GHz VLA data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' In the  analysis, we followed a similar approach to that used in  Dudzevičiūtė et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2020), first applying MAGPHYS+PHOTOZ to  the collated photometry of a sample of 264 sources, including four  SMGs and 66 VLA sources, with robust spectroscopic redshifts  (spanning zspec = 0−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='6) in our field (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Willmer, private  communication;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' see Appendix C), to check for systematic errors  in either the total magnitudes or the absolute calibration of the  photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' For each filter, we determined the distribution of  differences between the observed and model photometry for the  best-fit MAGPHYS model SEDs of the sources at their known  spectroscopic redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' This analysis confirmed that there were no  significant, >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='10 mag, systematic offsets in any of the filters, and  so we proceeded to model the galaxies allowing the code to fit for  the redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' This returned a median difference between the  predicted  photometric  and  spectroscopic  redshifts  of  Δz/  (1 + zspec) = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='02 for the 264 galaxies, indicating that  the photometric redshifts are unbiased, with a scaled median  absolute deviation of σz/(1 + zspec) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  We next applied MAGPHYS+PHOTOZ to the 61 SMGs in our  field that had 3 GHz radio detections and suitable optical and/  or NIR photometry (and also the 25 with just radio and submm  detections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The best-fit model SEDs for the sources are shown  in Figure 17 in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  MAGPHYS+PHOTOZ returns estimates of various physical  quantities of the modeled galaxies, not all of which are well  constrained by the observable properties (see the discussion in  Dudzevičiūtė et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Given the limited photometric  coverage used in our analysis, we consider the redshifts and  dust masses as the most robust output parameters, although we  also discuss the estimates of the SFR and stellar mass in  Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  The estimated 16th–84th percentile errors on the various  physical parameters from MAGPHYS+PHOTOZ are as follows:  photometric  redshift,  δzphot/(1 + zphot) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  stellar  mass,  δM* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='37 dex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' SFR, δSFR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='35 dex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' dust mass, δMd =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='24 dex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' and dust-to-stellar mass ratio (DSR), δMd/M* =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='43 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' In the case of the main sample (S/N > 4), the errors  are slightly smaller: photometric redshift, δzphot/(1 + zphot) =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' stellar mass, δM* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='36 dex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' SFR, δSFR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='34 dex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' dust  mass, δMd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='23 dex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' and DSR, δMd/M* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='43 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' These  uncertainties are typically 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2 times larger than the  equivalent values derived for the 22-band coverage in AS2UDS  by Dudzevičiūtė et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2020) reflecting the limitation of the  current photometric coverage of the TDF region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Results and Discussion  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 850 μm Number Count  The number density of sources as a function of flux density  is one of the most basic observables available for a population  and as such provides a fundamental test for galaxy formation  models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Baugh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' First, we present the number  counts of the submm sources detected in the JWST-TDF and  compare these with those of other surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Table 1 provides a summary of the number counts and  Figure 10 shows the cumulative and differential counts of 850 μm  sources in the ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='087 deg2 area of the JWST-TDF SCUBA-2  survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The counts use deboosted fluxes and are corrected using  the completeness curve constructed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The errors  reflect the uncertainties from the deboosting process, which were  derived from constructing 1000 mock catalogs of 850 μm sources  by sampling the deboosted flux distributions for the sources and  adopting the 16th and 84th percentiles of the resulting distribution  as the uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The final number counts were derived from  the mean values of these mock catalogs (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We also plot  the best Schechter function fits to the number counts in the  AS2COSMOS (Simpson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2020), S2COSMOS (Simpson  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2019), and S2CLS (Geach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2017) surveys for  comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  The JWST-TDF number counts in both panels trace the  Schechter function–like trend well, and agree with the results  from other surveys (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Karim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Hsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Geach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Stach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Simpson  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2019, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Shim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2020), especially at fainter flux  densities (S850mm < 10 mJy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' There is a slight excess at  S850mm > 10 mJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We believe that this is likely caused by  cosmic variance and reflects four bright sources with fluxes  exceeding S850mm = 10 mJy in our relatively small survey area  (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='087 deg2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' To quantify this, we randomly placed TDF-  sized regions in the much larger S2COSMOS survey field  (Simpson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2019) and found that regions containing  5  SCUBA-2  sources  brighter  than  S850mm = 10 mJy  were  detected ∼3% of the time—suggesting that the excess can be  explained by cosmic variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Redshifts  We found that the best-fit photometric redshifts for the radio-  identified and optical/NIR-detected SMGs with S/N > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0  (>3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5) in our sample give a median redshift of z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='12  (z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='08)  and  a  16th–84th  percentile  range  of  z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='50–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='10 (z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='44–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We note that there are high-  quality spectroscopic redshifts for only four of the dust-selected  SMGs (the low rate likely reflecting the relative faintness of the  galaxies in the optical/NIR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Nevertheless, in three of these  four cases, the spectroscopic redshifts and photometric redshift  estimates appear to agree with each other (the exception is  source 0056.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0, with zspec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='03, a spectroscopic redshift  that likely refers to a nearby foreground galaxy): 0023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0,  zspec = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='51, zphot = +  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='55  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='25  0031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0, zspec = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='36, zphot = +  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='44  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='05 and 0051.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0, zspec = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='51, zphot = +  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Table 1  The Number Counts of JWST-TDF SCUBA-2  S850mm  N(>S850mm)  S850mm  dN/dS850mm  (mJy)  (deg−2)  (mJy)  (deg−2 mJy−1)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 +  462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='9 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='4  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='6 +  181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='6 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='7  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='9  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1 +  252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='7 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='7  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='9 +  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='6 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='9  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='6 +  137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='9 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5 +  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5 +  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='8 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='7 +  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='4  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='9 +  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='9  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='4 +  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 +  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='7  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='9 +  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='8  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='9 +  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='9  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5 +  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='9  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2  Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Uncertainties were derived from the standard deviation in simulations for  each bin using the source deboosting and completeness corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  10  The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January  Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  The redshift distribution of our radio-identified 850 μm  sources is presented in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' For comparison, we show the  photometric redshift distribution for the purely submm-selected  AS2UDS survey from Dudzevičiūtė et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2020), and also the  equivalent distribution after applying an additional radio  detection and K < 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5 magnitude limit (to mimic the selection  of our radio-identified, mostly NIR-detected sample).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We also  show the spectroscopic redshift distribution for the similarly  radio-identified SMG sample from Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' In  contrast to the flux-limited AS2UDS sample, which shows a  tail of sources out to z ∼ 6, both of the radio/NIR-limited  samples exhibit a sharp cutoff at z ∼ 4 and median redshifts at  z ∼ 2, in reasonable agreement with those in the JWST-TDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Figure 12 shows the S850μm and S850μm/S3GHz versus the  photometric redshifts of the 51 (61) submm sources at  S/N > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 (S/N > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Also plotted are the K-band and  radio limited sample of AS2UDS, mimicking the selection of  the S2TDF sources, and the median redshifts of the submm  flux-limited sample of AS2UDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We conclude that the S2TDF  sources follow the distribution of those in AS2UDS when a  similar radio/NIR selection is applied to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' SFRs and Stellar Masses  The distribution of SFRs with stellar mass for those S2TDF  radio-identified SMGs with optical/NIR counterparts is plotted  in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The S2TDF sample shows a median SFR of  300 ± 40 Me yr−1 and 240 ± 40 Me yr−1 for SCUBA-2  sources with S/N > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 and S/N > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The  maximum SFR is ∼3000 Me yr−1 for both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  These galaxies have stellar masses with a 16th–84th percentile  range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='8–4 × 1011 Me, and a median of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3 ×  1011 Me, showing that most of them are already very massive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Adding the supplementary subset associated with SCUBA-2  sources with S/N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0, we found their stellar masses are  very similar with a median M* = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3 × 1011 Me and a  16th–84th percentile range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='6–5 × 1011 Me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' With their high  SFRs, they are likely to become massive early-type galaxies  today, as suggested by other SMG studies (Lilly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Birkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Figure 13 also compares the SFR–M*  distribution of the S2TDF SMGs to that of the K-band and radio  limited subsample of AS2UDS, which shows a similar distribu-  tion to the S2TDF SMGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' To compare the distribution to the trend  in more typical “main-sequence” star-forming galaxies, we also  show the SFR–M* relation for these galaxies in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' This  comparison sample was constructed by matching the redshifts and  stellar masses of the S2TDF SMGs to those of their nearest  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Cumulative and differential number counts of the 850 μm 114 sources (S850mm = 2–20 mJy, S/N > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5) in the JWST-TDF area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' For comparison, we show  results from previous submm surveys (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Karim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Hsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Geach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Stach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Simpson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2019, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The best-  fitting results from Simpson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2019, 2020), Shim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2020), and Geach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2017) are also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' In general, our number counts are in reasonable  agreement with those from the other surveys, although there is a slight excess at the bright end, S850mm > 10 mJy, arising from a small number of bright sources in  the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The photometric redshift distribution of radio-identified SMGs in  the JWST-TDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The red hatched histogram shows the distribution of samples  at S/N > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0, yielding a median redshift of z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='12 and a 16th–84th  percentile range of z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='50−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' If we include the sample with 850 μm  S/N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5–4 (pink shaded histogram), then we derive a median redshift of  z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='08 and a 16th–84th percentile range of z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='44−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The  redshift distributions from the submm flux-limited survey (AS2UDS, blue  histogram), the AS2UDS sample with radio detection and K < 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5 mag limits  applied to mimic the selection of our sample (green histogram), and finally the  radio-identified SMG sample (yellow histogram) from Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2005)  are also shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  11  The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January  Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  neighbors in z–M* space from the large sample of K-detected field  galaxies in the UKIRT Infrared Deep Sky Survey Ultra Deep  Survey (UDS) from Dudzevičiūtė et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' This matching  perturbed each SMG 10 times by its uncertainties in redshift and  stellar mass and selected the closest match from the field sample  within a redshift window of δz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Then, these UDS galaxies  were plotted on the SFR–M* plane, and the best-fit linear  correlation was derived, which is given in Equation (10):  = M  M  log SFR  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='57 log  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  10  (  )  (  )  (  )  \uf065  This best-fit trend for the field population is shown in Figure 13  and we compared it to the SFR–M* relations of “main-sequence”  star-forming galaxies at z = 1–5 from Pearson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2018),  finding that it corresponds well to the track for z ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' This shows  that our SMG sample have higher SFRs than “typical” galaxies of  their mass at their respective redshifts, by a factor of ∼6,  indicating that our SMGs are examples of the most massive,  highest-SFR end of the galaxy populations at z ∼ 1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Dust Mass Estimates  Finally, we take advantage of the fact that the 850 μm  continuum brightness is tightly correlated to the cold dust mass of  galaxies across a wide range of redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' In turn, the cold dust  mass of galaxies is believed to be a good indicator of the mass of  their cool gas reservoirs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Scoville et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We derived  estimates of the cold dust masses of our SMGs from the  MAGPHYS fits and compared these to those for the similar radio-  detected  and  K < 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5  sample  of  SMGs  from  AS2UDS  (Dudzevičiūtė et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2020) in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Both samples show  similar ranges in dust mass, Md ∼ 2–20 × 108 Me, and the  S2TDF sample shows a median dust mass of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='7 ×  108 Me (Md = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='8 × 108 Me when including the supple-  mentary S/N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 subset) suggesting typical gas masses of  Mg ∼ 2–3 × 1010 Me, for gas-to-dust mass ratios of ∼60 (Birkin  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  In Figure 14, we also compare the cold dust masses for our  SMGs with those inferred from MAGPHYS SED fitting to more  typical “main-sequence” galaxies, using the K-band “field”  selected sample (with a K < 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5 cut) from Dudzevičiūtė et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  The  median  dust  mass  for  these  galaxies  is  Md \x84 108 Me, considerably lower than that of the SMGs,  suggesting a similar difference in their cool gas masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Such a  difference would then provide a natural explanation for the  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Photometric redshift vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' flux density and flux ratio for SCUBA-2 SMGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' S2TDF sources at S/N > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 and at S/N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 are presented with filled red  circles and with filled pink circles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The median error of each parameter is drawn in each panel (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (Left) 850 μm flux density vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' photometric redshift  for our radio-identified SMGs with optical/NIR counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' For comparison we show a radio and K-band limited subsample of SMGs from AS2UDS (Dudzevičiūtė  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2020), and the fitted trend for the full AS2UDS survey from Stach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' There is broad agreement between the redshift–flux distributions in the JWST-  TDF and these previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (Right) The variation in submm-to-radio flux ratio and photometric redshift for our sample, again compared to a radio/K-band-limited  subset of the SMGs in AS2UDS (converted from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='4 to 3 GHz assuming a radio spectral index of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We also show the expected trend for the composite SMG  SED from Dudzevičiūtė et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2020), which broadly reproduces the behavior we see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' SFR–M* distribution for the S2TDF, compared to a K < 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5  limited, radio-selected subsample of SMGs from AS2UDS (Dudzevičiūtė  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We show the median error at the bottom left corner in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The  solid line is the linear trend derived from a sample of K-band-detected field  galaxies that has been matched to the S2TDF SMGs in redshift and stellar mass  (see the main text), and the shaded area indicates the 20% uncertainty in the  normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We also show the SFR–stellar mass relation for the “main  sequence” of star-forming galaxies at various redshifts derived in Pearson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Both of these demonstrate that the S2TDF SMGs have SFRs above  those typical for similar-mass field galaxies at their redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  12  The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January  Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  much higher levels of star formation activity seen in the SMGs,  as a consequence of their much more massive reservoirs of  cool gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  The ratio of dust mass to stellar mass in galaxies is a  potentially useful parameter that links the growth of stellar  populations in galaxies with the associated formation of metals  in stars and dust in their atmospheres and the broader  interstellar medium (ISM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Critically this ratio may provide  constraints on the processes that destroy dust grains within the  ISM (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Santini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Dunne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Calura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Donevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Dudzevičiūtė et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  We show in Figure 14 the variation in the DSR for our  SMGs and the radio/K-band-matched AS2UDS sample from  Dudzevičiūtė et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We also show the median DSRs for  local and high-redshift galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' These comprise low-redshift  spirals and ULIRGs, higher-redshift FIR-selected galaxies,  SMGs, and a small number of very-high-redshift QSOs  (Kennicutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2003, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Greve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Clements  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Santini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Lutz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Calura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  2014, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  For  the  S2TDF  main  sample,  the  SMG  DSR  is  = M  M  log  d  10(  )  –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1 (–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2, S/N > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5), com-  pared to −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1 for the AS2UDS matched sample, with a  Kolmogorov–Smirnov two-sample test showing that the two  populations are indistinguishable, PKS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Our median  estimates agree well with earlier crude estimates for SMGs  (Greve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Santini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2010) and appear to reflect a  gradual increase in DSR from z = 0 to z ∼ 2 (which may in turn  reflect the rising gas fraction in galaxies over this period, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=',  Geach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Tacconi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Dudzevičiūtė et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  2020, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Birkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  We also plot two sets of theoretical models on Figure 14 for  the “main-sequence” and “starburst” models from Davé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  (2019) and Pantoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The former cosmological  models by Davé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2019) suggest little variation in the DSR  as a function of SFR/M* (which distinguishes the two model  families), but these fail to reproduce the range in DSR: our  SMGs span a 16th–84th percentile range of DSRs of  ~ M  M  log  d  10(  )  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 to −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='9, which is approximately twice  the estimated uncertainty, indicating a possible intrinsic  dispersion in the DSR of the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' In contrast, the  simple analytic models from Pantoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2019) do broadly  span the range in DSR in the SMGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' This likely reflects the  relative paucity of large numbers of rapidly evolving massive,  gas- and dust-rich galaxies at high redshifts in typical  cosmological simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Summary and Conclusion  We have conducted a SCUBA-2 850 μm survey of the  JWST-TDF, which will be a prime deep extragalactic field  benefiting from deep multi-epoch observations with the James  Webb Space Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The main results of our study are as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Our JCMT SCUBA-2 850 μm survey of the JWST-TDF  identified 83 and 31 submm sources, at S/N > 4 and  S/N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5–4, respectively, in a survey area of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='087 deg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  The mean 1σ sensitivity across the survey area is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 mJy beam−1  from the 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3 hr of observation, with the deepest region achieving  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='8 mJy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The latter is comparable to the 850 μm  confusion limit of the JCMT (σc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='7 mJy beam−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  To derive the survey completeness and the boosting factor,  we performed jackknife simulations using number counts  reflecting the expected source density in the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The 50%  and 80% completeness limits are 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 mJy and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 mJy,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The false detection rates derived from the  simulations are 1% and 8% at S/N > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 and S/N > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  The purity of the sample approaches 100% at S/N > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  We also analyzed sensitive new VLA 3 GHz observations of  the JWST-TDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' These observations have a synthesized beam  of a 0 7 FWHM and cover a 24′ field of view, reaching a 1σ  sensitivity of 1 μJy at the field center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We have presented a  Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' In both panels, red and pink filled circles show the S2TDF sources  at S/N > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 and at S/N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The median errors are also  presented (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (Top) The variation in cold dust mass from MAGPHYS for  our SMGs with redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We compare this to that of a similarly radio-detected  and K < 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5 sample of SMGs from AS2UDS (Dudzevičiūtė et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The  details of the symbols are the same as those in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' In addition, we show  a K-band-selected field galaxy sample with K < 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5 (gray points) and the  trend (yellow solid line) of estimated dust masses from Dudzevičiūtė et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' This illustrates significantly higher cold dust masses, and by  implication higher cold gas masses, for the SMGs, compared to typical star-  forming galaxies at similar redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (Bottom) DSR as a function of redshift for  radio-identified SMGs in this work and in AS2UDS (Dudzevičiūtė et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  For comparison we present the median DSR of various samples collated by  Calura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2017) with open diamond symbols: blue, cyan, deep pink, brown,  and purple colors represent spiral galaxies from SINGS (Kennicutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2003),  spiral galaxies from KINGFISH (Kennicutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2011), a local ULIRG sample  (Clements et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2010), FIR-selected galaxies in the COSMOS/GOODS  surveys (Calura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Lutz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2011), and higher-redshift SMGs (Greve  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Santini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2010), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We also show the model  predictions of “main-sequence” and starburst dusty star-forming galaxies from  Davé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2019) and Pantoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2019), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  13  The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January  Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  AS2UDS(Radio-detectedandK<22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5&S870  m>2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2mJV  OS2TDF(SNR=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0）  S2TDF(SNR>4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='00  Ma[10° Ma  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='01  0  1  2  3  4catalog of 756 sources detected above a flux density limit of  S/N = 5 in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  The SCUBA-2 sources were crossmatched with the radio  sources from our deep VLA 3 GHz map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We found 85 radio  counterparts to 66 SCUBA-2 sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' There are nine SCUBA-  2 sources with two radio counterparts and five with three radio  counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' In the case of the main sample, 54 out of 83  SCUBA-2 sources matched with 70 radio sources and there are  five/six triple/double-counterpart cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' For the supplementary  sample, 12 out of 31 SCUBA-2 sources matched with 15 radio  sources including three double counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' A catalog of the  S2TDF submm sources is presented, including those that were  matched with radio and optical/NIR sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  We investigated the cumulative and differential SMG  number counts as a function of the flux densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Overall,  the trends are similar to the results from other studies, but there  is a slight excess at higher flux densities (S850 > 10 mJy), which  arises from four bright sources in our relatively small  survey area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  There are 51 (61) radio-located S2TDF SMGs with optical or  NIR counterparts at S/N > 4 (S/N > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' SED fitting was  performed to these sources to derive their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We found  that their estimated photometric redshifts have a median of  z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='12  (z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='08),  their  median  SFR  is  300 ± 40 Me yr−1 (240 ± 40 Me yr−1), and they have stellar  masses of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3 × 1011 Me (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3 × 1011 Me) for the  sample at S/N > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 (S/N > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' This demonstrates that these  galaxies lie above the main sequence at their redshifts, and that  they represent the high-mass end of the star-forming popula-  tion, suggesting that they are progenitors of massive early-type  galaxies today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  We estimated dust masses and DSRs for our SMG sample,  finding large dust masses, Md = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='7 × 108 Me, which  imply correspondingly high cold gas masses, Mg ∼ 2–3 ×  1010 Me, much higher than expected for typical field galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  These large reservoirs of gas are the fuel that drives the intense  activity we see in the SMG population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Our study provides a submm data set that is critical for  investigating the obscured star formation activity of galaxies  out to high redshifts within the JWST-TDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Together with the  anticipated JWST data, we expect that the S2TDF will provide  valuable insights into the physical mechanisms that triggered  vigorous star formation activity in the early universe, by  combining the morphological information from JWST and the  extreme galaxies identified by the S2TDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  We thank Rick Perley and Ken Kellermann for their help with  the VLA observations and their reduction and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' This work  was supported by National Research Foundation of Korea (NRF)  grants, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2020R1A2C3011091 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2021M3F7A1084525,  funded by the Korean government (MSIT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' acknowledges  support from a Korea Astronomy and Space Science Institute  grant  funded  by  the  Korean  government  (MSIT)  (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  2022183005) and support from the Global PhD Fellowship  Program through the NRF funded by the Ministry of Education  (NRF-2013H1A2A1033110).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' acknowledges support from  STFC (ST/T000244/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' acknowl-  edge support from NASA JWST Interdisciplinary Scientist grants  NAG5-12460, NNX14AN10G, and 80NSSC18K0200 from  GSFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' acknowledges funding from Hubble Space  Telescope grant GO-15278 and the NIRCam Development  Contract NAS5-02105 from NASA to the University of Arizona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' acknowledges support from NRF grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2021R1A2  C4002725 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2022R1A4A3031306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' acknowledges  support from JSPS KAKENHI grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 17KK0098, 19H00697,  20H01953, 21H01133, 21H04489, and 22H01273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The James  Clerk Maxwell Telescope is operated by the East Asian  Observatory on behalf of The National Astronomical Observatory  of Japan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Academia Sinica Institute of Astronomy and Astro-  physics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' the Korea Astronomy and Space Science Institute;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Center  for Astronomical Mega-Science (as well as the National Key  R&D Program of China with No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2017YFA0402700).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Additional  funding support is provided by the Science and Technology  Facilities Council of the United Kingdom and participating  universities in the United Kingdom and Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Additional funds  for the construction of SCUBA-2 were provided by the Canada  Foundation for Innovation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The National Radio Astronomy  Observatory is a facility of the National Science Foundation  operated under cooperative agreement by Associated Universities,  Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Based in part on data collected at the Subaru Telescope and  retrieved from the HSC data archive system, which is operated by  the Subaru Telescope and Astronomy Data Center at the National  Astronomical Observatory of Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Observations reported here  were obtained at the MMT Observatory, a joint facility of the  Smithsonian Institution and the University of Arizona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Facilities: JCMT, VLA, Subaru, MMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Appendix A  VLA Observations of the JWST-TDF  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Introduction  The VLA observations presented here were obtained as part  of an ongoing radio program using the VLA and the Very Long  Baseline Array (VLBA) to generate a list of extragalactic radio  sources in the JWST-TDF using the VLA, and then identify  those containing a significant AGN component using the  VLBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The VLBA can observe multiple targets in the primary  beam of the antennas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' however, it is impractical to image the  entire beam so accurate positions of the targets must be known  in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' To this end the VLA observations of the JWST-  TDF region are centered on the very compact, 200 mJy quasar  J1723+6547, for subsequent use as a phase reference for the  VLBA observations (the chosen pointing contains no other  strong radio sources).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  However, these VLA observations also provide sensitive  high-resolution radio coverage of the entire JWST-TDF as  shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' This provides an excellent resource for  studying the radio properties of other populations uncovered in  this field;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' in particular these high-resolution observations can  be used to identify likely counterparts to the submm sources  detected in our low-resolution SCUBA-2 map of this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Observations and Data Reduction  The observations were undertaken using the VLA “S band”  (ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='989–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='013 GHz), which is optimal for this project as it  provides a good compromise between high sensitivity and a  wide field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Since the majority of galaxies observed  have steep spectra, the relatively low frequency also helps  detectability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The single pointing gives the lowest detection  level for a given total exposure near the pointing center, but at  the cost of the sensitivity dropping away from this field center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  We used the VLA to observe a field centered on J1723  +6547 (17 23 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1381, +65 47 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='179) using the S-band  receivers, with the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='989–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='013 GHz frequency range divided  14  The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January  Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  into 16 contiguous subbands each of width Δν = 128 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Each subband was further divided into 64, 2 MHz channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Observations consisted of 2 hr blocks, one or more of which  were observed on a given day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' All data taken on a given  calendar day were processed together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The observations are  summarized in Table 2, which gives the date, VLA configura-  tion, and total duration of the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Each 2 hr block  included the astrometric calibrator J1800+7828 and one of the  photometric/bandpass/polarization  calibrators  J1331+3030  (3C 286) and J0137+3309 (3C 48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Calibration  Calibration and imaging used the OBIT package (Cotton  2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='15 Due to the strong source at the field center, special  care was needed to minimize artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Calibration and editing  was done on each day’s data independently and consisted of the  steps described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' As the target field was dominated by a  strong, compact source it was included as a calibrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' In each  step deviant calibration solutions were detected and flagged  along with the corresponding data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Standard structural and  spectral models for J1331+3030 and J0137+3309 (Perley &  Butler 2013a) were used as appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Processing consisted  of the following:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Fixed flagging: Frequency ranges known to contain  strong, persistent radio frequency interference (RFI) were  flagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Initial flagging: Running medians in time and frequency  were used on the data to identify and flag RFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Initial amplitude calibration: Amplitude corrections were  determined from the “switch power” calibration signals  injected into each data stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Delay calibration: Residual group delays were deter-  mined for all calibrators and the target field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Bandpass calibration: Amplitude and phase correction  spectra were determined from the bandpass calibrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Amplitude and phase calibration: Complex gain solutions  for the calibrators were determined for each calibrator and  the target field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The daily flux density of the phase  reference source was determined by a comparison of gain  solutions with the photometric standard(s) and was used  to correct the amplitudes of the solutions for it and the  target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' This value is given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Flagging of calibrated data: Flagging operations for  which calibrated data were needed were done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Repeat: The flaggings from the above steps were kept and  the calibration was repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Polarization calibration: Instrumental polarization was  determined from the phase reference calibrator when an  adequate range of parallactic angles was available;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  solutions were done in 16 MHz blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The cross-hand  delay and phase were determined from J1331+3030  (preferred) or J0137+3309 (Perley & Butler 2013b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  All data sets were then subjected to a baseline-dependent time  averaging to reduce the data volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' This averaging was  subject to the constraint that the amplitudes were reduced by no  more than 1% to a radius of 12′ and averaging was no more  than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='35 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Imaging  As the observations were made over an extended period and  the central source is mildly variable (see calibration values in  Table 2), data from each day’s observations were imaged and  the contribution of the central source was subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Initial  imaging included a phase+delay self-calibration followed by  an amplitude and phase self-calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The imaging used the  OBIT task MFIMAGE, which is described in more detail in  Cotton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Significant artifacts survived this process  and additional filtering was needed to suppress them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Image Artifacts  The imaging artifacts arising from the strong central source  included the following pathologies, given with their remedia-  tion (when available):  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' “Fingerprints”—Spiral patterns resembling fingerprints  were traced to minor residual group delay errors when the  group delay corrections were determined solely from the  calibrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Including the target in the group delay  calibration and doing phase and delay self-calibration in  the imaging largely eliminated these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' “Black stripes”—Initial imaging resulted in dark hor-  izontal bands through and near the central source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' These  were traced to the periods of time when the fringe rate on  a given baseline went through zero allowing greater  sensitivity to RFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' As these are near u = 0 the resulting  artifacts appear as horizontal stripes at the field center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  These artifacts were largely suppressed by subtracting the  image sky model from the data, averaging the data, and  clipping the data above a given level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' “Bowls”—All of the “A” configuration data sets but  neither of the “B” data sets show a negative bowl around  the position of the central source with a maximum depth  of ∼150 μJy, even after the source was subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The  source of this artifact has not been determined but it only  extends a few beams from the central source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' As the  central source is completely unresolved, this bowl is not  the feature commonly seen around very extended, bright  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Table 2  Log of the VLA Observations of the JWST-TDF  Date  Config.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Timeobs  SJ1800  δSJ1800  SJ1723  δSJ1723  (hr)  (Jy)  (Jy)  (Jy)  (Jy)  2017 Nov 29  B  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='810  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content='001  2017 Dec 18  B  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content='001  2018 Jun 01  A  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content='001  Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We list (1) the date of the observations, (2) the VLA configuration  employed, (3) the total observing time, (4–5) the calibration flux density  adopted for J1800+7828 and its uncertainty, and (6–7) the flux density of  J1723+6547 and its uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  15 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='cv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='nrao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='edu/~bcotton/Obit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='html  15  The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January  Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Combined Image  After each individual day’s data sets were imaged, the core  was subtracted, and the artifact filtering was run, the data were  combined into a single set and imaged with no further self-  calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' CLEANing proceeded to a minimum residual of  5 μJy beam−1 or a maximum of 20,000 components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Imaging  used a robust factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0 and the resulting images used a  circular Gaussian restoring beam of a 0 7 FWHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The rms in  quiet places in the resultant image is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='95 μJy beam−1 and the  brightest pixel is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1 mJy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Source Catalog  The construction of the source list used the OBIT task  FNDSOU,  which  fits  elliptical  Gaussians  to  components  identified in islands of emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The derived list was compared  with the image and entries corresponding to clear artifacts or  multiple entries from an extended source were removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Error  analysis followed the development of Condon (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The field  image FITS file and the full source list for 756 sources with a  peak brightness >5 times the local rms and with a primary-  beam gain of at least 10% at 3 GHz (within a radius of  ¢  12 ) are  given online with a sample shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Some sources are sufficiently extended to not be modeled by  a simple Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The flux densities of these were determined  by an integral of the image pixels bounded by a rectangular  box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The LAS of the source was derived from the diagonal  extent of this rectangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' These sources are indicated in Table 3  by the LAS given in the “Size” column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Summary  Our VLA 3 GHz observations of the JWST-TDF reach an  rms of 1 μJy beam−1 in the region around the bright radio  source J1723+6547 at a resolution of 0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The catalog of  sources above 5σ generated from this map contains 756  sources, which we used to identify the counterparts to the  SCUBA-2 sources in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The full catalog of 3 GHz  sources will also be employed in our ongoing VLBA  observations of sources in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Appendix B  Jackknife Simulation and Flux Distribution  As we explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2, we performed 1000  jackknife simulations to determine the amount of flux boosting  and the completeness of the S2TDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Here, we present two-  dimensional density plots showing the simulation results  (Figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Figure 16 presents the flux density distribution of the  observed flux and the deboosted flux from the empirical  recovery method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Table 3  VLA 3 GHz Source Catalog  ID  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  δR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content='  Decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content='0151  +65 46 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='518  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='075  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='2  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1  <1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='10  L  L  L  L  L  Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The central source, J1723+6547, was added to the catalog in spite of being removed from the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We list (1) the source ID;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (2–5) the J2000 positions of each  source and their uncertainties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (6–7) the 3 GHz flux density, corrected for the primary beam, and its uncertainty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' and (8–13) the deconvolved angular size and/or upper  limit or largest angular size (LAS) and the position angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  (This table is available in its entirety in machine-readable form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=')  16  The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January  Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Appendix C  Optical Spectroscopy  The spectroscopic data were obtained using Binospec  (Fabricant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2019) at the MMT Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' These were  obtained using the 270 line mm−1 grating, which allows  coverage  of approximately  4000–9000 Å with a typical  dispersion of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='30 Å pixel−1 and a resolution of 1300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The  sample of galaxies was constructed by combining a preliminary  catalog from the HEROES Subaru HSC imaging (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Hasinger,  private communication) and a catalog derived from MMT/  MMIRS NIR imaging (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Willmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2022, in  preparation), limited at r ∼ 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The slit assignment was  prioritized for sources with X-ray, VLA data, or submm  counterparts in preliminary catalogs coming from members of  the collaboration (private communication from W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Maksym,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Windhorst, or I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Smail, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  The data were reduced using a specially designed pipeline  that produces wavelength- and flux-calibrated one-dimensional  spectra (Kansky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Redshifts were measured using  custom-written code using a combination of real-space cross-  correlation and emission line fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' All spectra were visually  inspected (by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Willmer) and a redshift quality was  assigned using the same criteria adopted by the DEEP2 survey  (Newman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2013), where a redshift quality of 4 is likely to  be correct at a 95% level or better and quality 3 is correct at a  level of ∼90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Data qualities less than 3 were not considered  in the analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The output from each of the observed fields  was then merged into a single list using the positions of the  NIR imaging, which are tied to the Gaia Data Release 2  reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Appendix D  The Best-fit MAGPHYS Model SEDs for the S2TDF SMGs  Figure 17 displays the best-fit MAGPHYS model SEDs for the  85 SMGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Two-dimensional density plots showing the results of 1000 jackknife simulations performed for our survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (Left) A plot of injected sources as a function  of input flux density and instrumental noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (Middle) A density plot of completeness, the ratio of the number of recovered sources to that of injected sources, as a  function of input flux density and instrumental noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The contours are labeled with the completeness fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' (Right) The plot shows the average boosting factor of  the output flux density with the given instrumental noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The dashed line shows the S/N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5 limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The contour labels indicate the flux boost factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Flux density distribution of observed flux (black curve) and deboosted flux (red curve) for example sources derived from the empirical recovery method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Labels with red and black colors show the median value of the corrected flux and observed flux, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' All results assume Gaussian uncertainties of σinst = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0  mJy, which is the median instrumental noise of the final map of the TDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' For bright sources, the effect of flux boosting is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' But, at the limit of our catalog, it  can be considerable (;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='30%−50%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  17  The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January  Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The best-fit MAGPHYS model SEDs for the 85 SMGs in this work that have 3 GHz radio counterparts, including the 61 sources for which we have  sufficient optical and/or NIR photometry data to derive useful constraints on their SED properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The bold frame represents the main targets at S/N > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' We list the  desired photometric redshifts and if available any spectroscopic values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The sources range within z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='8–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='8 and all sources show a strong FIR peak in their SEDs  traced by the SCUBA-2 850 μm emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  18  The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January  Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Appendix E  S2TDF Source Catalog and Stamp Images  Figure 18 provides, in an online-only format, postage-stamp  images for all the main and supplementary SCUBA-2 sources  detected in the JWST-TDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The first 10 online-only figures are  of the main set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' the last four online-only figures are of the  supplement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Table 4 provides a detailed description of the full  version of JWST-TDF SCUBA-2 850 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The catalog is provided  in FITS format and includes both the main and supplementary  sources, which are differentiated by the MAIN Boolean column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The 32″ × 32″ images of the main 83 SCUBA-2 sources (S/N > 4) detected in the JWST-TDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' If the source has a radio counterpart, the label is black;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  otherwise, it is red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The positions of the radio counterparts are shown with blue crosses on the JCMT image and square marks on the HSC G and MMIRS Y and K  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' If there is more than one counterpart lighter blue marks are also added to the figure in orders of distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The matching radius of 7 5 is indicated with four  yellow marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The complete set of postage stamps (14 images) is available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The first 10 images are of the main sources;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' the last four are of the supplementary  sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  (An extended version of this figure is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=')  19  The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January  Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Table 4  Description of the Full Version of JWST-TDF SCUBA-2 850 μm Source Catalogs  Index  Label  Units  Description  1  index  L  Index  2  JCMT_index  L  JCMT index  3  JCMT_ID  L  IAU position–based name, S2TDFJHHMMSS+/–DDMMSS  4  JCMT_SHORT_ID  L  Short name, S2TDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='JCMT_index  5  ID  L  Identifier with VLA multiplicity index, JCMT_SHORT_ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' [0, 1, 2]  6  MAIN  Boolean  True = main (JCMT_S/N > 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' False = supplementary (JCMT_S/N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0)  7  Got_VLA_ID  Boolean  True = VLA match  8  NOT_Got_VLA_ID  Boolean  True = no VLA match  9  Closest_ID  Boolean  True = closest matched object  10  Multiple_IDs  Boolean  True = multiple VLA matches  11  N(S>S)  L  Radio source number density S3GHz > 10 μJy  12  Ni  L  Radio source number density S3GHz > Si  13  Pc  L  p  =  P  r N  c  s  T  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Equation (8)  14  P*  L  p  = P  r N  i  i  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Equation (9)  15  p  L  Corrected Poissonian probability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Equation (6)  16  n_radio_counterpart  L  Number of radio counterparts  17  closest_rank  L  Rank of the radio-matched object based on the separated distance [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 3]  18  JCMT_FLUX_PEAK  mJy  Peak JCMT 850 μm flux  19  JCMT_RMS_PEAK  mJy  rms uncertainty in peak flux  20  JCMT_FLUX_MODEL  mJy  Model JCMT 850 μm flux  21  JCMT_RMS_MODEL  mJy  rms uncertainty in model flux  22  JCMT_S/N  L  S/N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' JCMT 850 μm  23  JCMT_BLEND  L  [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1 = blend  24  JCMT_FLUX_DEBOOST  mJy  Deboosted JCMT 850 μm flux  25  JCMT_FLUX_DEBOOST_ERRLO  mJy  Lower uncertainty in JCMT_FLUX_DEBOOST  26  JCMT_FLUX_DEBOOST_ERRHI  mJy  Upper uncertainty in JCMT_FLUX_DEBOOST  27  JCMT_TOTAL_ERRLO  mJy  Lower total uncertainty in JCMT_FLUX_DEBOOST  28  JCMT_TOTAL_ERRHI  mJy  Upper total uncertainty in JCMT_FLUX_DEBOOST  29  JCMT_RA_PEAK_DEG_FIN  deg  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', peak flux, decimal degrees  30  JCMT_DEC_PEAK_DEG_FIN  deg  Decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', peak flux, decimal degrees  31  JCMT_RA_MODEL_DEG_FIN  deg  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', peak model flux, decimal degrees  32  JCMT_DEC_MODEL_DEG_FIN  deg  Decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', peak model flux, decimal degrees  33  ID_VLA  L  Identifier from VLA catalog (Appendix A)  34  RA_VLA  deg  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', VLA source, decimal degrees  35  DEC_VLA  deg  Decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', VLA source, decimal degrees  36  radio_flux  μJy  3 GHz radio flux, S3GHz  37  dS3GHz  μJy  Uncertainty in 3 GHz flux  38  angular_distance  arcsec  Angular separation between VLA and JCMT  39  sed_group  L  1: K detection + opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='/NIR data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2: no K detection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 3: VLA+submm data only  40  GMAG_APER_300  mag  G-band aperture magnitude from HSC (Bosch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Miyazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2018)  41  GMAGERR_APER_300  mag  Uncertainty in GMAG_APER_300  42  IMAG_APER_300  mag  I-band aperture magnitude from HSC (Bosch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Miyazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2018)  43  IMAGERR_APER_300  mag  Uncertainty in IMAG_APER_300  44  ZMAG_APER_300  mag  Z-band aperture magnitude from HSC (Bosch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Miyazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2018)  45  ZMAGERR_APER_300  mag  Uncertainty in ZMAG_APER_300  46  YMAG_APER_3  mag  Y-band aperture magnitude from MMIRS (McLeod et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2012)  47  YMAGERR_APER_3  mag  Uncertainty in YMAG_APER_3  48  JMAG_APER_3  mag  J-band aperture magnitude from MMIRS (McLeod et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2012)  49  JMAGERR_APER_3  mag  Uncertainty in JMAG_APER_3  50  HMAG_APER_3  mag  H-band aperture magnitude from MMIRS (McLeod et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2012)  51  HMAGERR_APER_3  mag  Uncertainty in HMAG_APER_3  52  KMAG_APER_3  mag  K-band aperture magnitude from MMIRS (McLeod et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2012)  53  KMAGERR_APER_3  mag  Uncertainty in KMAG_APER_3  54  w1_ab  mag  WISE band 1 AB magnitude (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2010)  55  dw1  mag  Uncertainty in w1_ab  56  w2_ab  mag  WISE band 2 AB magnitude (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2010)  57  dw2  mag  Uncertainty in w2_ab  58  w3_ab  mag  WISE band 3 AB magnitude (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2010)  59  w3msigmpro  mag  Uncertainty in w3_ab  60  w4_ab  mag  WISE band 4 AB magnitude (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2010)  61  w4msigmpro  mag  Uncertainty in w4_ab  62  Major  arcsec  Ellipse-fit major axis, arcseconds  63  dMajor  arcsec  Uncertainty in Major: 99 = LAS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' −99 = upper limit  20  The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January  Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  ORCID iDs  Minhee Hyun  https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content=' Fazio  https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Swinbank  https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='org/0000-0003-1192-5837  Haojing Yan  https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='org/0000-0001-7592-7714  References  An, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Simpson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2019, ApJ, 886, 48  An, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Stach, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2018, ApJ, 862, 101  Barger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Cowie, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', & Richards, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2000, AJ, 119, 2092  Barger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Cowie, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Sanders, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1998, Natur, 394, 248  Barnes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', & Hernquist, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1991, ApJL, 370, L65  Battisti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', da Cunha, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Grasha, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2019, ApJ, 882, 61  Baugh, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Lacey, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Frenk, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2005, MNRAS, 356, 1191  Birkin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Weiss, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Wardlow, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2021, MNRAS, 501, 3926  Blain, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Chapman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', & Ivison, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2004,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' ApJ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 611,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 725  Table 4  (Continued)  Index  Label  Units  Description  64  Minor  arcsec  Ellipse-fit minor axis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' arcseconds  65  dMinor  arcsec  Uncertainty in Minor: −99 = upper limit  66  PA  deg  Ellipse-fit position angle  67  dPA  deg  Uncertainty in PA  68  chi  L  χ2 of the SED fitting result  69  Mstar_l  log(Me)  Lower bound on Mstar  70  Mstar  log(Me)  Stellar mass in log  71  Mstar_u  log(Me)  Upper bound on Mstar  72  L_dust_l  log(Le)  Lower bound on L_dust  73  L_dust  log(Le)  Dust luminosity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Ldust,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' in log  74  L_dust_u  log(Le)  Upper bound on L_dust  75  Tdust_l  K  Lower bound on Tdust  76  Tdust  K  Kinetic dust temperature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Tdust  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='77  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content='K  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='Upper bound on Tdust  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='78  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content='79  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='SFR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content='80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content='Upper bound on SFR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='81  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='Mdust_l  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content='Lower bound on Mdust  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='82  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content='83  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content='84  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content='Lower bound on Age  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='85  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='Age  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content='86  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content='87  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='AV_l  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='mag  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='Lower bound on AV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='88  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='AV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='mag  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='V-band reddening  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='89  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content='mag  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='Upper bound on AV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='90  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='tau_l  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content='tau  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='Gyr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content='92  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content='Gyr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content='94  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content='log(yr−1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content='95  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content='96  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='z_lower  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='L  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='Lower bound on z_med  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='97  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content='L  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='Median photometric redshift  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='98  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+page_content='L  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='Upper bound on z_med  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='99  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='eb_l  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='mag  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='Lower bound on eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='mag  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='Color excess,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' E(B – V )  101  eb_u  mag  Upper bound on eb  102  Notes  L  Notes  Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The full version of the JWST-TDF SCUBA-2 catalog for the main (S/N > 4) sources includes VLA counterparts from the Poisson probability crossmatching  (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The calculated Poissonian probabilities are presented as p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' If there are multiple counterparts, we list them in the order of probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The number  after the decimal point in “ID” indicates multiple radio counterparts to a common submm source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' The full catalog, including both the main and supplementary  (S/N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='5–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='0) sources, is available online in FITS format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  (This table is available in its entirety in machine-readable form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=')  21  The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January  Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='  Bosch, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Armstrong, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Bickerton, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2018, PASJ, 70, S5  Bothwell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Chapman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2013, MNRAS, 429, 3047  Calura, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Gilli, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Vignali, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2014, MNRAS, 438, 2765  Calura, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Pozzi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Cresci, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2017, MNRAS, 465, 54  Carilli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Daddi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Riechers, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2010, ApJ, 714, 1407  Casey, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Cowie, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2013, MNRAS, 436, 1919  Casey, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Narayanan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', & Cooray, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2014, PhR, 541, 45  Chabrier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2003, PASP, 115, 763  Chapin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Berry, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Gibb, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2013, MNRAS, 430, 2545  Chapman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Blain, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', & Ivison, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2005, ApJ, 622, 772  Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Cowie, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Barger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2013, ApJ, 776, 131  Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Swinbank, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2015, ApJ, 799, 194  Clements, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Dunne, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', & Eales, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2010, MNRAS, 403, 274  Condon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1992, ARA&A, 30, 575  Condon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1997, PASP, 109, 166  Coppin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Chapin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Mortier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2006, MNRAS, 372, 1621  Cotton, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2008, PASP, 120, 439  Cotton, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Condon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Kellermann, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2018, ApJ, 856, 67  Cowie, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Barger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Hsu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2017, ApJ, 837, 139  Cowie, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Gonzalez-Lopez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Barger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2018, ApJ, 865, 106  Cunha, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Walter, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2015, ApJ, 806, 110  da Cunha, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Charlot, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', & Elbaz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2008, MNRAS, 388, 1595  Davé, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Angles-Alcazar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Narayanan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2019, MNRAS, 486, 2827  Dempsey, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Friberg, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Jenness, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2013, MNRAS, 430, 2534  Donevski, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Lapi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Małek, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2020, A&A, 644, A144  Driver, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Andrews, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Davies, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2016, ApJ, 827, 108  Dudzevičiūtė, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Swinbank, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2020, MNRAS, 494, 3828  Dudzevičiūtė, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Swinbank, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2021, MNRAS, 500, 942  Dunne, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Gomez, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', da Cunha, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2011, MNRAS, 417, 1510  Eales, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Lilly, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Gear, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1999, ApJ, 515, 518  Engel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Tacconi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Davies, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2010, ApJ, 724, 233  Fabricant, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Fata, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Epps, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2019, PASP, 131, 075004  Farrah, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Lonsdale, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Borys, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2006, ApJL, 641, L17  Geach, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Dunlop, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Halpern, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2017, MNRAS, 465, 1789  Geach, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Matsuda, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2005, MNRAS, 363, 1398  Goto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Oi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Utsumi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2019, PASJ, 71, 30  Goto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Takagi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Matsuhara, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2010, A&A, 514, A6  Greve, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Bertoldi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2005, MNRAS, 359, 1165  Gruppioni, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Béthermin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Loiacono, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2020, A&A, 643, A8  Hainline, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Blain, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2011, ApJ, 740, 96  Helou, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Soifer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', & Rowan-Robinson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1985, ApJL, 298, L7  Hodge, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', & da Cunha, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2020, RSOS, 7, 200556  Holland, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Bintley, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Chapin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2013, MNRAS, 430, 2513  Hong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Im, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Kim, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', & Ho, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2015, ApJ, 804, 34  Hopkins, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Hernquist, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Martini, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2005, ApJL, 625, L71  Hsu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Cowie, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Barger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', & Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2016,  ApJ, 829, 25  Hughes, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Serjeant, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Dunlop, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1998, Natur, 394, 241  Ikarashi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Kohno, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Aguirre, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2011, MNRAS, 415, 3081  Ivison, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Greve, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Dunlop, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2007, MNRAS, 380, 199  Ivison, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Greve, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Serjeant, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2004, ApJS, 154, 124  Ivison, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Greve, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2002, MNRAS, 337, 1  Ivison, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Morrison, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Biggs, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2008, MNRAS, 390, 1117  Ivison, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Dunlop, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2005, MNRAS, 364, 1025  Ivison, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Le Borgne, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1998, MNRAS, 298, 583  Jansen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', & Windhorst, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2018, PASP, 130, 124001  Jenness, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Berry, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Cavanagh, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2009, in ASP Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', 411,  Astronomical Data Analysis Software and Systems XVIII, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Bohlender,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Durand, & P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Dowler (San Francisco, CA: ASP),418  Kansky, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Chilingarian, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Fabricant, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2019, PASP, 131, 075005  Karim, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Swinbank, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Hodge, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2013, MNRAS, 432, 2  Kennicutt, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Calzetti, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Aniano, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2011, PASP, 123, 1347  Kennicutt, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Armus, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Bendo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2003, PASP, 115, 928  Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Lee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Jeong, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2015, MNRAS, 454, 1573  Koushan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Driver, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Bellstedt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2021, MNRAS, 503, 2033  Lilly, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Eales, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Gear, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1999, ApJ, 518, 641  Lonsdale, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Farrah, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', & Smith, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2006, in Ultraluminous Infrared  Galaxies, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Mason (Praxis Publishing: Chichester), 285  Lutz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Poglitsch, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Altieri, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2011, A&A, 532, A90  Madau, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', & Dickinson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2014, ARA&A, 52, 415  Magliocchetti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Silva, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Lapi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2007, MNRAS, 375, 1121  Magnelli, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Popesso, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Berta, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2013, A&A, 553, A132  McAlpine, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Bower, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2019, MNRAS, 488, 2440  McLeod, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Fabricant, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Nystrom, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2012, PASP, 124, 1318  Menendez-Delmestre, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Blain, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Alexander, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2007, ApJL,  655, L65  Michałowski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Dunlop, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Cirasuolo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2012, A&A,  541, A85  Miettinen, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Novak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Smolcic, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2015, A&A, 584, A32  Mihos, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', & Hernquist, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1996, ApJ, 464, 641  Miller, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Chapman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Aravena, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2018, Natur, 556, 469  Miyazaki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Komiyama, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Kawanomoto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2018, PASJ, 70, S1  Newman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Cooper, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Davis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2013, ApJS, 208, 5  Oke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', & Gunn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1983, ApJ, 266, 713  Pantoni, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Lapi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Massardi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Goswami, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', & Danese, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2019, ApJ,  880, 129  Pearson, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Hurley, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2018, A&A, 615, A146  Perley, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', & Butler, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2013a, ApJS, 206, 16  Perley, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', & Butler, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2013b, ApJS, 204, 19  Rickard, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', & Harvey, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1984, AJ, 89, 1520  Riechers, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Leung, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Ivison, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2017, ApJ, 850, 1  Sanders, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', & Mirabel, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1996, ARA&A, 34, 749  Santini, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Maiolino, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Magnelli, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2010, A&A, 518, L154  Scott, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Austermann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Perera, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2008, MNRAS, 385, 2225  Scott, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Fox, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Dunlop, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2002, MNRAS, 331, 817  Scoville, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Sheth, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Aussel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2016, ApJ, 820, 83  Shim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Kim, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Lee, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2020, MNRAS, 498, 5065  Shim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Lee, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Kim, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2022, MNRAS, 514, 2915  Simpson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Dudzeviciute, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2020, MNRAS, 495, 3409  Simpson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Swinbank, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2019, ApJ, 880, 43  Simpson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Swinbank, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2014, ApJ, 788, 125  Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Ivison, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', & Blain, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1997, ApJL, 490, L5  Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Ivison, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Owen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Blain, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', & Kneib, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2000, ApJ,  528, 612  Springel, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Di Matteo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', & Hernquist, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2005, ApJL, 620, L79  Stach, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Dudzeviciute, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2019, MNRAS, 487, 4648  Stach, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Amvrosiadis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2021, MNRAS, 504, 172  Stach, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Swinbank, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2018, ApJ, 860, 161  Swinbank, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Chapman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2006, MNRAS, 371, 465  Swinbank, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Chapman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2004, ApJ, 617, 64  Tacconi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Genzel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Saintonge, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2018, ApJ, 853, 179  Tacconi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Genzel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2008, ApJ, 680, 246  Warren-Smith, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', & Wallace, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 1993, in ASP Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 52,  Astronomical Data Analysis Software and Systems II, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Hanisch,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Brissenden, & J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Barnes (San Francisco, CA: ASP), 229  Weiß, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Kovacs, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Coppin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2009, ApJ, 707, 1201  Wilkinson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Almaini, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2017, MNRAS, 464, 1380  Wright, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Eisenhardt, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Mainzer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2010, AJ, 140, 1868  Zavala, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Aretxaga, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Geach, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2017, MNRAS, 464, 3369  Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Zheng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', Shi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
+page_content=' 2022, MNRAS, 512, 4893  22  The Astrophysical Journal Supplement Series, 264:19 (22pp), 2023 January  Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE0T4oBgHgl3EQf8QKN/content/2301.02786v1.pdf'}
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+arXiv:2301.13561v1  [math.ST]  31 Jan 2023
+General weighted cumulative residual (past)
+extropy of minimum (maximum) ranked set
+sampling with unequal samples
+Santosh Kumar Chaudhary*, Nitin Gupta**
+Department of Mathematics, Indian Institute of Technology
+Kharagpur, West Bengal 721302, India
+Abstract
+The general weighted cumulative residual extropy (GWCRJ) and gen-
+eral weighted cumulative past extropy (GWCPJ) are introduced in this
+paper. There are some results in relation to GWCPJ and GWCRJ. We
+take into account GWCRJ-based uncertainty measures for the minimal
+ranked set sampling technique with unequal samples (minRSSU). Ad-
+ditionally, we take into account GWCPJ-based uncertainty measures
+for the maximum ranked set sampling technique with unequal sam-
+ples (maxRSSU). Stochastic comparison for Simple random sampling
+(SRS) is discussed. We looked at the monotone properties of minRSSU
+and maxRSSU as well as stochastic comparison. Finally, two empirical
+estimators of GWCPJ and GWCRJ are obtained.
+Key Words: Extropy, General weighted cumulative past extropy, Gen-
+eral weighted cumulative residual extropy, Ranked set sampling, Simple
+random sampling
+Mathematical Subject Classification: 62B10; 62D05; 60E15.
+1
+Introduction
+The method of enhancing the accuracy of the sample mean as an approxima-
+tion of the population means, McIntyre (1952) proposed ranked set sampling
+(RSS). The RSS process involves selecting n sets of units, each of size n, at
+random from a population. The units in each set are then ranked visually
+* Corresponding author E-mail: skchaudhary1994@kgpian.iitkgp.ac.in
+** E-mail: nitin.gupta@maths.iitkgp.ac.in
+
+2
+S.K.Chaudhary, N. Gupta
+or using some low-cost techniques. Pick the unit with the lowest rank out
+of the first n units. Select the unit with the second-lowest rank from the
+second group of n units. Up until selecting the unit with the highest rank in
+the n-th set, the process is repeated. Then, we measure the relevant variable
+using the units we have chosen. In many practical aspects related to applica-
+tions in industrial statistics, environmental and ecological studies, statistical
+genetics, etc., RSS schemes have been playing a significant role. We assume
+that XSRS = {Xi, i = 1, . . . , n} denotes a simple random sampling (SRS)
+of size n from a continuous distribution with probability density function
+(pdf) f and cumulative distribution function (cdf) F. In comparison to SRS,
+the RSS is a more effective sampling method for estimating the population
+mean. As a helpful improvement to the RSS technique, Qiu and Eftekhar-
+ian (2021) investigated the extropy of the minimum and maximum ranked
+set sampling procedure with unequal samples (minRSSU and maxRSSU). In
+the minRSSU and maxRSSU, we draw n simple random samples, in which
+the size of the i−th samples is i, i = 1, . . . n. The one-cycle minRSSU or
+maxRSSU involves an initial ranking of n samples of size n.
+We collect
+Zi = X(1:i)i for all i = 1, 2, . . . , n, and Yi = X(i:i)i for all i = 1, 2, . . . , n,
+where X(i:j)j denotes the i-th order statistic from the j-th SRS of size j. The
+resulting sample is called one-cycle minRSSU and maxRSSU of size n and de-
+noted by ZminRSSU = {Zi, i = 1, . . . , n} and YmaxRSSU = {Yi, i = 1, . . . , n},
+respectively (see Biradar and Santosha (2014)).
+Measures of information for RSS and its variations have been devel-
+oped by several authors (see Eskandarzadeh et al.(2018), Raqab and Qiu
+(2019), Qiu and Raqab (2022) and Gupta and Chaudhary (2022) etc.). Qiu
+and Eftekharian (2021) considered the information content of minRSSU and
+maxRSSU in terms of extropy and also compared the extropy of these two
+sampling data with that of SRS and RSS data. Kazemi et al. (2021) consid-
+ered uncertainty measures of minimum ranked set sampling procedure with
+unequal samples (MinRSSU) in terms of cumulative residual extropy and its
+dynamic version and they also compared the uncertainty and information
+content of cumulative residual extropy based on MinRSSU and SRS designs.
+Chacko and George (2022) considered the extropy properties of the RSS
+when ranking is not perfect and also obtained upper and lower bounds of
+the extropy of RSS.
+Extropy was introduced by Lad et al.(2015) as a measure of uncertainty
+as
+J(X) = −1
+2
+� ∞
+−∞
+f 2(x)dx.
+(1.1)
+Jahanshahi et al. (2019) defined Cumulative residual extropy as
+ξ(X) = −1
+2
+� ∞
+−∞
+¯F 2(x)dx.
+(1.2)
+
+S.K.Chaudhary, N. Gupta
+3
+Cumulative past extropy is defined as
+¯ξ(X) = −1
+2
+� ∞
+−∞
+F 2(x)dx.
+(1.3)
+Extropy and its various form have been studied by several authors (see
+Bansal and Gupta (2021), Gupta and Chaudhary (2022), and Kazemi et
+al.(2022) etc).
+Bansal and Gupta (2021) studied weighted extropies and
+past extropy of order statistics and k-record values. Gupta and Chaudhary
+(2022) studied the general weighted extropy of RSS where they defined gen-
+eral weighted extropy (GWJ) with weight w(x) ≥ 0 as
+Jw(X) = −1
+2
+� ∞
+−∞
+w(x)f 2(x)dx
+= −1
+2E(δw
+X(U)),
+where δw
+X(u) = w(F −1(u))f(F −1(u)) and U is uniformly distributed random
+variable on (0, 1), i.e., U ∼ Uniform(0, 1). This motivated us to generalize
+cumulative residual and past extropy and study it for ranked set sampling.
+Here, we defined general weighted cumulative past extropy (GWCPJ) of
+absolutely continuous random variable X with weight w(x) ≥ 0 as
+¯ξw(X) = −1
+2
+� ∞
+−∞
+w(x)F 2(x)dx
+= −1
+2
+� 1
+0
+w(F −1(u))
+u2
+f(F −1(u))du
+= −1
+2E(Λw
+X(U)),
+(1.4)
+where Λw
+X(u) = u2w(F −1(u))
+f(F −1(u)) .
+Also, we define general weighted cumulative residual extropy (GWCRJ) of
+absolutely continuous random variable X with weight w(x) ≥ 0 as
+ξw(X) = −1
+2
+� ∞
+−∞
+w(x) ¯F 2(x)dx
+= −1
+2
+� 1
+0
+w(F −1(u)) (1 − u)2
+f(F −1(u))du
+= −1
+2E(∆w
+X(U)),
+(1.5)
+where ∆w
+X(u) = (1−u)2w(F −1(u))
+f(F −1(u))
+.
+
+4
+S.K.Chaudhary, N. Gupta
+This paper is organized as follows: section 2 discusses results on GWCPJ
+and GWCRJ. Section 3 provides examples and results for SRS of GWCPJ
+and GWCRJ. MinRSSU and maxRSSU for GWCRJ and GWCPJ, respec-
+tively are studied in section 4.
+Stochastic comparison for MinRSSU and
+maxRSSU for GWCRJ and GWCPJ, respectively are obtained in section
+5.
+Section 6 is the monotone properties of MinRSSU and maxRSSU for
+GWCRJ and GWCPJ, respectively.
+In section 7, we obtained empirical
+estimators of GWCRJ and GWCPJ. Finally, section 8 concludes this paper.
+2
+Results on GWCPJ and GWCRJ
+In the following theorem, we will study the behaviour of GWCPJ with re-
+spect to weight functions w1 and w2 and the dispersive order of random
+variables X and Y .
+Theorem 1 Let X and Y be nonnegative random variables with pdf’s f and
+g, cdf’s F and G, respectively having uX = uY < ∞.
+(a) If w1 is decreasing, w1(x) ≤ w2(x) and X ≤disp Y , then ¯ξw1(X) ≥
+¯ξw2(Y ).
+(b) If w1 is decreasing, w1(x) ≥ w2(x) and X ≥disp Y , then ¯ξw1(X) ≤
+¯ξw2(Y ).
+Proof (a) Since X ≤disp Y , therefore we have f(F −1(u)) ≥ g(G−1(u)) for
+u ∈ (0, 1).
+Then using Theorem 3.B.13(b) of Shaked and Shanthikumar
+(2007), X ≤disp Y implies that X ≥st Y . Hence F −1(u) ≥ G−1(u) for all
+u ∈ (0, 1). Since w1 is decreasing and w1(x) ≤ w2(x) , then w1(F −1(u)) ≤
+w1(G−1(u)) ≤ w2(G−1(u)). Hence,
+Λw1
+X (u) = u2 w1(F −1(u))
+f(F −1(u))
+≤ u2 w2(G−1(u))
+g(G−1(u))
+=Λw2
+Y (u).
+Now using (1.4),
+¯ξw1(X) = −1
+2E
+�
+Λw1
+X (U)
+�
+≥ −1
+2E
+�
+Λw2
+Y (U)
+�
+= ¯ξw2(Y ).
+
+S.K.Chaudhary, N. Gupta
+5
+(b) Proof is Similar to part (a).
+■
+If we take w1(x) = w2(x) = w(x) in the above theorem, then we have
+the following corollary.
+Corollary 1 Let X and Y be nonnegative random variables with pdf’s f and
+g, cdf’s F and G, respectively having uX = uY < ∞. Let w is decreasing.
+(a) If X ≤disp Y , then ¯ξw(X) ≥ ¯ξw(Y ).
+(b) If X ≥disp Y , then ¯ξw(X) ≤ ¯ξw(Y ).
+In the following theorem, we will study the behaviour of GWCRJ with
+respect to weight functions w1 and w2 and the dispersive order of random
+variables X and Y .
+Theorem 2 Let X and Y be nonnegative random variables with pdf’s f and
+g, cdf’s F and G, respectively having uX = uY < ∞.
+(a) If w1 is decreasing, w1(x) ≤ w2(x) and X ≤disp Y , then ξw1(X) ≥
+ξw2(Y ).
+(b) If w1 is decreasing, w1(x) ≥ w2(x) and X ≥disp Y , then ξw1(X) ≤
+ξw2(Y ).
+Proof (a) Since X ≤disp Y , therefore we have f(F −1(u)) ≥ g(G−1(u)) for
+u ∈ (0, 1).
+Then using Theorem 3.B.13(b) of Shaked and Shanthikumar
+(2007), X ≤disp Y implies that X ≥st Y . Hence F −1(u) ≥ G−1(u) for all
+u ∈ (0, 1). Since w1 is decreasing and w1(x) ≤ w2(x) , then w1(F −1(u)) ≤
+w1(G−1(u)) ≤ w2(G−1(u)). Hence
+∆w1
+X (u) = (1 − u)2w1(F −1(u))
+f(F −1(u))
+≤ (1 − u)2w2(G−1(u))
+g(G−1(u))
+=∆w2
+Y (u).
+(2.1)
+Now using (1.5),
+ξw1(X) = −1
+2E
+�
+∆w1
+X (U)
+�
+≥ −1
+2E
+�
+∆w2
+Y (U)
+�
+= ξw2(Y ).
+(b) Proof is Similar to part (a).
+■
+If we take w1(x) = w2(x) = w(x) in the above theorem, then we have
+the following corollary.
+
+6
+S.K.Chaudhary, N. Gupta
+Corollary 2 Let X and Y be nonnegative random variables with pdf’s f and
+g, cdf’s F and G, respectively having uX = uY < ∞. Let w is decreasing.
+(a) If X ≤disp Y , then ξw(X) ≥ ξw(Y ).
+(b) If X ≥disp Y , then ξw(X) ≤ ξw(Y ).
+3
+Simple Random Sampling
+Let X be a random variable with finite mean µ and variance σ2. For XSRS =
+{Xi, i = 1, . . . , n}, the joint pdf is �n
+i=1 f(xi), as Xi’s, i = 1, . . . , n are
+independent and identically distributed (i.i.d.). Hence the GWCPJ of X(n)
+SRS
+can be defined as
+¯ξw(X(n)
+SRS) = −1
+2
+n
+�
+i=1
+�� ∞
+−∞
+w(xi)F 2(xi)dxi
+�
+= −1
+2
+�
+−2¯ξw(X)
+�n
+= −1
+2 (E(Λw
+X(U)))n .
+(3.1)
+Also, the GWCRJ of X(n)
+SRS can be defined as
+ξw(X(n)
+SRS) = −1
+2
+n
+�
+i=1
+�� ∞
+−∞
+w(xi) ¯F 2
+X(xi)dxi
+�
+= −1
+2 (−2ξw(X))n
+= −1
+2 (E(∆w
+X(U)))n ,
+where Λw
+X(U) and ∆w
+X(U) are given in section 1.
+Example 1 If U is a uniform random variable on interval (0,1) and w(x) =
+xm, m > 0 then
+E[Λw
+X(U)] = E
+�
+U 2w(U)
+�
+=
+1
+m + 3,
+E[∆w
+X(U)] = E
+�
+(1 − U)2w(U)
+�
+=
+1
+(m + 1) −
+2
+(m + 2) +
+1
+(m + 3),
+¯ξw(X(n)
+SRS) = −1
+2
+�
+1
+m + 3
+�n
+,
+ξw(X(n)
+SRS) = −1
+2
+�
+1
+(m + 1) −
+2
+(m + 2) +
+1
+(m + 3)
+�n
+.
+
+S.K.Chaudhary, N. Gupta
+7
+Theorem 3 Let X be a non-negative absolutely continuous random vari-
+able with pdf f and cdf F. Assume ψ(x) is an increasing function and
+w(ψ(x))ψ′(x) ≤ (≥)w(x) and ψ(0) = 0. If Y = ψ(X), then ¯ξw(X(n)
+SRS) ≤ (≥
+)¯ξw(Y(n)
+SRS).
+Proof Let the random variable Y has pdf g and cdf G. Then
+Λw
+Y (u) = u2w
+�
+G−1(u)
+�
+g (G−1(u))
+= u2w(ψ(F −1(u)))ψ′(F −1(u))
+f(F −1(u))
+∀
+0 < u < 1.
+Note that ψ(x) ≥ ψ(0), ∀ x ≥ 0. Since w(ψ(x))ψ′(x) ≤ (≥)w(x). Hence for
+0 < u < 1, we have
+Λw
+Y (u) = u2w(ψ(F −1(u)))ψ′(F −1(u))
+f(F −1(u))
+≤ (≥)u2w((F −1(u)))
+f(F −1(u))
+= Λw
+X(u).
+Therefore ¯ξw(X(n)
+SRS) ≥ (≤)¯ξw(Y(n)
+SRS) using equation (3.1).
+■
+Example 2 Let ψ(x) = ex − 1,
+w(x) = x, x ≥ 0. It is easy to verify that
+ψ(x) is an increasing function and ψ(0) = 0. Let h(x) = w(ψ(x))ψ′(x) −
+w(x). That is, h(x) = e2x − ex − x. Note that h(0) = h
+′(0) = 0. Moreover,
+h
+′(x) ≥ 0 ∀ x ≥ 0. Thus, h(x) is increasing function on x ∈ [0, ∞]. So,
+h(x) ≥ h(0) = 0, that is, w(ψ(x))ψ′(x) ≥ w(x) ∀x ≥ 0. Therefore, using
+Theorem 3 for Y = ψ(X), we have,
+¯ξw(X(n)
+SRS) ≥ ¯ξw(Y(n)
+SRS).
+4
+MinRSSU and maxRSSU for GWCRJ and GWCPJ
+Let Xi:i represent i-th order statistics from a sample of size i and X1:i rep-
+resent 1-st order statistics from a sample of size i. Let Fi:i denote cdf of i-th
+order statistics from a sample of size i and F1:i denotes cdf of 1-th order
+statistics from a sample of size i.
+
+8
+S.K.Chaudhary, N. Gupta
+The GWCPJ of X(n)
+maxRSSU can be defined as
+¯ξw(X(n)
+maxRSSU) = −1
+2
+n
+�
+i=1
+�
+−2¯ξw(Xi:i)
+�
+= −1
+2
+n
+�
+i=1
+�� ∞
+−∞
+w(x)F 2
+i:i(x)dx
+�
+= −1
+2
+n
+�
+i=1
+�� ∞
+−∞
+w(x)F 2i
+X (x)dx
+�
+= −1
+2
+n
+�
+i=1
+�� 1
+0
+u2iw(F −1(u))
+f(F −1(u))
+du
+�
+= −1
+2
+n
+�
+i=1
+E
+�U 2iw(F −1(U))
+f(F −1(U))
+�
+= −1
+2
+n
+�
+i=1
+E [Ψw
+X(U)] , ,
+(4.1)
+where Ψw
+X(u) = u2iw(F −1(u))
+f(F −1(u)) .
+The GWCRJ of X(n)
+minRSSU can be defined as
+ξw(X(n)
+minRSSU) = −1
+2
+n
+�
+i=1
+[−2ξw(X1:i)]
+= −1
+2
+n
+�
+i=1
+�� ∞
+−∞
+w(x) ¯F 2
+1:i(x)dx
+�
+= −1
+2
+n
+�
+i=1
+�� ∞
+−∞
+w(x) ¯F 2i(x)dx
+�
+= −1
+2
+n
+�
+i=1
+�� 1
+0
+(1 − u)2iw(F −1(u))
+f(F −1(u))
+du
+�
+= −1
+2
+n
+�
+i=1
+E
+�
+(1 − U)2iw(F −1(U))
+f(F −1(U))
+�
+= −1
+2
+n
+�
+i=1
+E [Φw
+X(U)] , ,
+(4.2)
+where Φw
+X(u) = (1−u)2iw(F −1(u))
+f(F −1(u))
+.
+
+S.K.Chaudhary, N. Gupta
+9
+Let us see some examples to illustrate the above measures.
+Example 3 If U is a uniform random variable on interval (0,1) and w(x) =
+xm, m > 0 then
+¯ξw(X(n)
+maxRSSU) = −1
+2
+n
+�
+i=1
+�
+1
+2i + m + 1
+�
+,
+ξw(X(n)
+minRSSU) = −(Γ(m + 1))2
+2
+n
+�
+i=1
+Γ(2i + 1)
+Γ(2i + m + 2),
+where
+Γ(α) =
+� ∞
+0
+e−xxα−1dx
+.
+Example 4 Let X be an exponential distribution with cdf F(x) = 1 −
+e−λx, λ > 0, x > 0. Let w(x) = xm, m > 0, x > 0, then
+ξw(X(n)
+minRSSU) = −1
+2
+n
+�
+i=1
+�� 1
+0
+(1 − u)2iw(F −1(u))
+f(F −1(u))
+du
+�
+= −1
+2
+n
+�
+i=1
+(−1)m
+λm+1
+� 1
+0
+(1 − u)2i−1(log(1 − u))mdu.
+Taking 1 − u = e−x in above expression, we get
+ξw(X(n)
+minRSSU) = −1
+2
+n
+�
+i=1
+1
+λm+1
+� ∞
+0
+xme−2ixdx
+= −1
+2
+n
+�
+i=1
+1
+λm+1
+Γ(m + 1)
+(2i)m+1 dx
+= −1
+2
+�Γ(m + 1)
+λm+1
+�n
+n
+�
+i=1
+� 1
+2i
+�m+1
+= −1
+2
+�Γ(m + 1)
+(2λ)m+1
+�n � 1
+n!
+�m+1
+.
+Example 5 Let a random variable X have survival function ¯F(x) = (1 −
+
+10
+S.K.Chaudhary, N. Gupta
+x)b,
+0 < x < 1,
+b > 0. If w(x) = xm,
+x > 0,
+m > 0 then
+ξw(X(n)
+minRSSU) = −1
+2
+n
+�
+i=1
+�� 1
+0
+w(x) ¯F 2i(x)dx
+�
+= −1
+2
+n
+�
+i=1
+�� 1
+0
+xm(1 − x)2ibdx
+�
+= −1
+2
+n
+�
+i=1
+B(m + 1, 2ib + 1),
+where B(α, β) is beta function defined as
+B(α, β) =
+� 1
+0
+xα−1(1 − x)1−βdx.
+Theorem 4 Let X be a non-negative absolutely continuous random vari-
+able with pdf f and cdf F. Assume ψ(x) is an increasing function and
+w(ψ(x))ψ′(x) ≤ (≥)w(x) and ψ(0) = 0. If Y = ψ(X), then ¯ξw(X(n)
+maxRSSU) ≤
+(≥)¯ξw(Y(n)
+maxRSSU).
+Proof Let random variable Y has pdf g and cdf G. Since w(ψ(x))ψ′(x) ≤
+(≥)w(x) and ψ(0) = 0, therefore,
+Ψw
+Y (u) = u2iw
+�
+G−1(u)
+�
+g (G−1(u))
+= u2iw(ψ(F −1(u)))ψ′(F −1(u))
+f(F −1(u))
+≤ (≥)u2iw(F −1(u))
+f(F −1(u))
+= Ψw
+X(u).
+Hence, From (4.1) we obtain
+¯ξw(X(n)
+maxRSSU) ≥ (≤)¯ξw(Y(n)
+maxRSSU).
+■
+Example 6 Let ψ(x) = ex − 1,
+w(x) = x, x ≥ 0. Note that from example
+2, ψ(x) is an increasing function and w(ψ(x))ψ′(x) ≥ w(x) and ψ(0) = 0.
+Using Theorem 3 for Y = ψ(X), we have,
+¯ξw(X(n)
+maxRSSU) ≥ ¯ξw(Y(n)
+maxRSSU)
+.
+
+S.K.Chaudhary, N. Gupta
+11
+Theorem 5 Let X be a non-negative absolutely continuous random vari-
+able with pdf f and cdf F. Assume ψ(x) is an increasing function and
+w(ψ(x))ψ′(x) ≤ (≥)w(x) and ψ(0) = 0. If Y = ψ(X), then ξw(X(n)
+minRSSU) ≤
+(≥)ξw(Y(n)
+minRSSU).
+Proof Let random variable Y has pdf g and cdf G. Since w(ψ(x))ψ′(x) ≤
+(≥)w(x) and ψ(0) = 0, therefore,
+Φw
+Y (u) = (1 − u)2iw
+�
+G−1(u)
+�
+g (G−1(u))
+= (1 − u)2iw(ψ(F −1(u)))ψ′(F −1(u))
+f(F −1(u))
+≤ (≥)(1 − u)2iw(F −1(u))
+f(F −1(u))
+= Φw
+X(u).
+Hence, From (4.2) we obtain
+ξw(X(n)
+minRSSU) ≥ (≤)ξw(Y(n)
+minRSSU).
+■
+Example 7 Let ψ(x) = ex − 1,
+w(x) = x, x ≥ 0. Note that from Example
+2, ψ(x) is an increasing function and w(ψ(x))ψ′(x) ≥ w(x) and ψ(0) = 0.
+Using Theorem 3 for Y = ψ(X), we have,
+ξw(X(n)
+minRSSU) ≥ ξw(Y(n)
+minRSSU)
+.
+Theorem 6 Let X be an absolutely continuous random variable with pdf f
+and cdf F. Then for n ≥ 2,
+¯ξw(X(n)
+maxRSSU) ≥ ¯ξw(X(n)
+SRS).
+
+12
+S.K.Chaudhary, N. Gupta
+Proof Consider
+¯ξw(X(n)
+maxRSSU) = −1
+2
+n
+�
+i=1
+E
+�U 2iw(F −1(U))
+f(F −1(U))
+�
+= −1
+2
+n
+�
+i=1
+�� 1
+0
+u2iw(F −1(u))
+f(F −1(u))
+du
+�
+≥ −1
+2
+n
+�
+i=1
+�� 1
+0
+u2w(F −1(u))
+f(F −1(u)) du
+�
+= −1
+2
+n
+�
+i=1
+E
+�U 2w(F −1(U))
+f(F −1(U))
+�
+= ¯ξw(X(n)
+SRS).
+Hence the result.
+■
+Theorem 7 Let X be an absolutely continuous random variable with pdf f
+and cdf F. Then for n ≥ 2,
+ξw(X(n)
+minRSSU) ≥ ξw(X(n)
+SRS).
+Proof Consider
+ξw(X(n)
+minRSSU) = −1
+2
+n
+�
+i=1
+E
+�
+(1 − U)2iw(F −1(U))
+f(F −1(U))
+�
+= −1
+2
+n
+�
+i=1
+�� 1
+0
+(1 − u)2iw(F −1(u))
+f(F −1(u))
+du
+�
+≥ −1
+2
+n
+�
+i=1
+�� 1
+0
+(1 − u)2w(F −1(u))
+f(F −1(u))
+du
+�
+= −1
+2
+n
+�
+i=1
+E
+�
+(1 − U)2w(F −1(U))
+f(F −1(U))
+�
+= ξw(X(n)
+SRS).
+Hence the result.
+■
+
+S.K.Chaudhary, N. Gupta
+13
+5
+Stochastic Comparision
+The following lemma will be used in deriving Theorem 9 and Theorem 11.
+Lemma 1 is also used by Qiu and Raqab (2022) and Gupta and Chaudhary
+(2022) to derive their results.
+Lemma 1 [Ahmed et al. (1986)] Let X and Y be nonnegative random vari-
+ables with pdf’s f and g, respectively, satisfying f(0) ≥ g(0) > 0. If X ≤su Y
+(or X ≤∗ Y or X ≤c Y ), then X ≤disp Y .
+In the following theorem, we will study the behaviour maxRSSU of
+GWCPJ with respect to weight functions w1 and w2 and dispersive order of
+random variables X and Y .
+Theorem 8 Let X and Y be nonnegative random variables with pdf’s f and
+g, cdf’s F and G, respectively having uX = uY < ∞.
+(a) If w1 is decreasing, w1(x) ≤ w2(x) and X ≤disp Y , then ¯ξw1(X(n)
+maxRSSU) ≥
+¯ξw2(Y(n)
+maxRSSU).
+(b) If w1 is decreasing, w1(x) ≥ w2(x) and X ≥disp Y , then ¯ξw1(X(n)
+maxRSSU) ≤
+¯ξw2(Y(n)
+maxRSSU).
+Proof (a) Since X ≤disp Y , therefore we have f(F −1(u)) ≥ g(G−1(u)) for
+u ∈ (0, 1).
+Then using Theorem 3.B.13(b) of Shaked and Shanthikumar
+(2007), X ≤disp Y implies that X ≥st Y . Hence, F −1(u) ≥ G−1(u) for all
+u ∈ (0, 1). Since w1 is decreasing and w1(x) ≤ w2(x) , then w1(F −1(u)) ≤
+w1(G−1(u)) ≤ w2(G−1(u)). Hence,
+Ψw
+X(u) = u2i w1(F −1(u))
+f(F −1(u)) ≤ u2i w2(G−1(u))
+g(G−1(u)) = Ψw
+Y (u)
+Now using (4.1),
+¯ξw1(X(n)
+maxRSSU) ≥ ¯ξw2(Y(n)
+maxRSSU).
+(b) Proof is Similar to part (a).
+■
+If we take w1(x) = w2(x) = w(x) in the above theorem, then we have
+the following corollary.
+
+14
+S.K.Chaudhary, N. Gupta
+Corollary 3 Let X and Y be nonnegative random variables with pdf’s f and
+g, cdf’s F and G, respectively having uX = uY < ∞. Let w is decreasing.
+(a) If X ≤disp Y , then ¯ξw(X(n)
+maxRSSU) ≥ ¯ξw(Y(n)
+maxRSSU).
+(b) If X ≥disp Y , then ¯ξw(X(n)
+maxRSSU) ≤ ¯ξw(Y(n)
+maxRSSU).
+One may refer Shaked and Shantikumar (2007) for details of convex
+transform order (≤c), star order (≤⋆), super additive order (≤su), and dis-
+persive order (≤disp). In view of Theorem 8 and Lemma 1, the following
+result is obtained.
+Theorem 9 Let X and Y be nonnegative random variables with pdf’s f and
+g, cdf’s F and G, respectively having uX = uY < ∞.
+(a) If w1 is decreasing, w1(x) ≤ w2(x) and X ≤su Y ( or X ≤∗ Y or
+X ≤c Y ), then ¯ξw1(X(n)
+maxRSSU) ≥ ¯ξw2(Y(n)
+maxRSSU).
+(b) If w1 is decreasing, w1(x) ≥ w2(x) and X ≥su Y (or X ≥∗ Y or X ≥c Y
+), then ¯ξw1(X(n)
+maxRSSU) ≤ ¯ξw2(Y(n)
+maxRSSU).
+If we take w1(x) = w2(x) = w(x) in the above theorem, then we have
+the following corollary.
+Corollary 4 Let X and Y be nonnegative random variables with pdf’s f
+and g, cdf’s F and G, respectively having uX = uY < ∞.
+(a) If w is decreasing and X ≤su Y ( or X ≤∗ Y or X ≤c Y ), then
+¯ξw(X(n)
+maxRSSU) ≥ ¯ξw(Y(n)
+maxRSSU).
+(b) If w is decreasing and X ≥su Y (or X ≥∗ Y or X ≥c Y ), then
+¯ξw(X(n)
+maxRSSU) ≤ ¯ξw(Y(n)
+maxRSSU).
+In the following theorem, we will study the behaviour of minRSSU of
+GWCRJ with respect to weight functions w1 and w2 and the dispersive
+order of random variables X and Y .
+Theorem 10 Let X and Y be nonnegative random variables with pdf’s f
+and g, cdf’s F and G, respectively having uX = uY < ∞.
+(a) If w1 is decreasing, w1(x) ≤ w2(x) and X ≤disp Y , then ξw1(X(n)
+minRSSU) ≥
+ξw2(Y(n)
+minRSSU).
+
+S.K.Chaudhary, N. Gupta
+15
+(b) If w1 is decreasing, w1(x) ≥ w2(x) and X ≥disp Y , then ξw1(X(n)
+minRSSU) ≤
+ξw2(Y(n)
+minRSSU).
+Proof (a) Since X ≤disp Y , therefore we have f(F −1(u)) ≥ g(G−1(u)) for
+u ∈ (0, 1).
+Then using Theorem 3.B.13(b) of Shaked and Shanthikumar
+(2007), X ≤disp Y implies that X ≥st Y . Hence F −1(u) ≥ G−1(u) for all
+u ∈ (0, 1). Since w1 is decreasing and w1(x) ≤ w2(x) , then w1(F −1(u)) ≤
+w1(G−1(u)) ≤ w2(G−1(u)). Hence
+Φw
+X(u) = (1 − u)2i w1(F −1(u))
+f(F −1(u)) ≤ (1 − u)2i w2(G−1(u))
+g(G−1(u)) = Φw
+Y (u)
+Now using (4.2),
+ξw1(X(n)
+minRSSU) ≥ ξw2(Y(n)
+minRSSU).
+(b) Proof is Similar to part (a).
+■
+If we take w1(x) = w2(x) = w(x) in the above theorem, then we have
+the following corollary.
+Corollary 5 Let X and Y be nonnegative random variables with pdf’s f and
+g, cdf’s F and G, respectively having uX = uY < ∞. Let w is decreasing.
+(a) If X ≤disp Y , then ξw(X(n)
+minRSSU) ≥ ξw(Y(n)
+minRSSU).
+(b) If X ≥disp Y , then ξw(X(n)
+minRSSU) ≤ ξw(Y(n)
+minRSSU).
+In view of Theorem 10 and Lemma 1, the following result is obtained.
+Theorem 11 Let X and Y be nonnegative random variables with pdf’s f
+and g, cdf’s F and G, respectively having uX = uY < ∞.
+(a) If w1 is decreasing, w1(x) ≤ w2(x) and X ≤su Y (or X ≤∗ Y or
+X ≤c Y ), then ξw1(X(n)
+minRSSU) ≥ ξw2(Y(n)
+minRSSU).
+(b) If w1 is decreasing, w1(x) ≥ w2(x) and X ≥su Y (or X ≥∗ Y or
+X ≥c Y ), then ξw1(X(n)
+minRSSU) ≤ ξw2(Y(n)
+minRSSU).
+If we take w1(x) = w2(x) = w(x) in the above theorem, then we have
+the following corollary.
+
+16
+S.K.Chaudhary, N. Gupta
+Corollary 6 Let X and Y be nonnegative random variables with pdf’s f
+and g, cdf’s F and G, respectively having uX = uY < ∞.
+(a) If w is decreasing and X ≤su Y (or X ≤∗ Y or X ≤c Y ), then
+ξw(X(n)
+minRSSU) ≥ ξw(Y(n)
+minRSSU).
+(b) If w is decreasing and X ≥su Y (or X ≥∗ Y or X ≥c Y ), then
+ξw(X(n)
+minRSSU) ≤ ξw(Y(n)
+minRSSU).
+6
+Monotone Properties
+The following result gives the conditions under which the ξw(X(n)
+minRSSU) will
+increase with respect to n.
+Theorem 12 If w(F −1(u)
+f(F −1(u)) ≤ 1, 0 < u < 1, then ξw(X(n)
+minRSSU) is increas-
+ing in n ≥ 1.
+Proof Consider
+ξw(X(n+1)
+minRSSU)
+ξw(X(n)
+minRSSU)
+=
+� 1
+0
+(1 − u)2n+2w(F −1(u))
+f(F −1(u))
+du ≤
+1
+2n + 3 ≤ 1 ∀ n ≥ 1.
+Hence,
+ξw(X(n+1)
+minRSSU) ≥ ξw(X(n)
+minRSSU).
+The following result gives the conditions under which the ¯ξw(X(n)
+maxRSSU)
+will increase with respect to n.
+Theorem 13 If w(F −1(u)
+f(F −1(u)) ≤ 1, 0 < u < 1, then ¯ξw(X(n)
+maxRSSU) is increas-
+ing in n ≥ 1.
+Proof Consider
+¯ξw(X(n+1)
+maxRSSU)
+¯ξw(X(n)
+maxRSSU)
+=
+� 1
+0
+u2n+2w(F −1(u))
+f(F −1(u))
+du ≤
+1
+2n + 3 ≤ 1 ∀ n ≥ 1.
+Hence,
+¯ξw(X(n+1)
+maxRSSU) ≥ ¯ξw(X(n)
+maxRSSU).
+
+S.K.Chaudhary, N. Gupta
+17
+7
+Empirical measures of GWCPJ and GWCRJ
+The empirical measure of F is defined as
+F
+�
+n(x) =
+
+
+
+
+
+0,
+x < X1:n
+i
+n,
+Xi:n ≤ x < Xi+1:n,
+i = 1, 2, . . . , n − 1.
+1,
+x ≥ Xn:n
+where X1:n ≤ X2:n ≤ X3:n ≤ · · · ≤ Xn:n denote order statistics of random
+sample X1, X2, . . . , Xn.
+The empirical measure of GWCPJ for w(x) = xm, x > 0, m > 0 is
+¯ξm
+�
+(Fn) = −1
+2
+� ∞
+−∞
+xmF 2
+n
+�
+(x)dx = −1
+2
+n−1
+�
+i=1
+� xi+1:n
+xi:n
+xm
+� i
+n
+�2
+dx
+= −
+1
+2(m + 1)
+n−1
+�
+i=1
+�
+(xm+1
+i+1:n − xm+1
+i:n )
+� i
+n
+�2�
+.
+The empirical measure of GWCRJ for w(x) = xm, x > 0, m > 0 is
+ξm
+�
+(Fn) = −1
+2
+� ∞
+−∞
+xm ¯F 2
+n
+�
+(x)dx = −1
+2
+n−1
+�
+i=1
+� xi+1:n
+xi:n
+xm
+� i
+n
+�2
+dx
+= −
+1
+2(m + 1)
+n−1
+�
+i=1
+�
+(xm+1
+i+1:n − xm+1
+i:n )
+�
+1 − i
+n
+�2�
+.
+Analogous to Theorem 9 of Rao et al. (2004), Theorem 5.1 of Tahmasebi
+et al. (2020) and Theorem 4.1 of Tahmasebi and Toomaj (2020), we state
+the following theorems and the proof follows similarly.
+Theorem 14 Let X be a non negative absolutely continuous random vari-
+able with cdf F. Then for any random X in Lp for some p > 2, ¯ξm
+�
+(Fn)
+converges almost surely to ¯ξm(X) as n → ∞.
+Theorem 15 Let X be a non negative absolutely continuous random vari-
+able with cdf F. Then for any random X in Lp for some p > 2, ξm
+�
+(Fn)
+converges almost surely to ξm(X) as n → ∞.
+
+18
+S.K.Chaudhary, N. Gupta
+We create another estimator based on the kernel-smoothed estimator of
+the distribution function because smoothed estimators perform better than
+non-smoothed estimators.
+The second estimator of GWCPJ for w(x) = xm, x > 0, m > 0 is
+¯ξm
+�
+(Fh) = −1
+2
+� ∞
+−∞
+xmF 2
+h(x)dx = −1
+2
+n−1
+�
+i=1
+� xi+1:n
+xi:n
+xm �
+F 2
+h(xi)
+�2 dx
+= −
+1
+2(m + 1)
+n−1
+�
+i=1
+�
+(xm+1
+i+1:n − xm+1
+i:n )
+�
+F 2
+h(xi)
+�2�
+.
+The second estimator of GWCRJ for w(x) = xm, x > 0, m > 0 is
+ξm
+�
+(Fh) = −1
+2
+� ∞
+−∞
+xm ¯F 2
+h(x)dx = −1
+2
+n−1
+�
+i=1
+� xi+1:n
+xi:n
+xm � ¯F 2
+h(xi)
+�
+dx
+= −
+1
+2(m + 1)
+n−1
+�
+i=1
+�
+(xm+1
+i+1:n − xm+1
+i:n )
+� ¯F 2
+h(xi)
+��
+,
+where ¯F = 1 − F and Fh is defined by Nadaraya (1964) as
+Fh(x) = 1
+n
+n
+�
+i=1
+L
+�x − Xi
+h
+�
+.
+Here, L(x) =
+� x
+−∞ K(t)dt, that is, L is cdf of positive kernel K and h is a
+bandwidth parameter (see Sarda (1993) and Jahanshahi (2020)).
+8
+Conclusion
+Generalized versions of cumulative past and residual extropy have been de-
+fined. The general GWCRJ and GWCPJ uncertainty measures based on
+the minRSSU and maxRSSU data, respectively, have been introduced in
+this study.
+Additionally, this work presented the GWCRJ and GWCPJ-
+based uncertainty measures of SRS. For minRSSU, maxRSSU, and SRS
+data, some results of the GWCPJ and GWCRJ measures were obtained, in-
+cluding stochastic orders. With help of empirical measure of the distribution
+function, we obtained two estimators of GWCRJ and GWCPJ.
+
+S.K.Chaudhary, N. Gupta
+19
+Funding
+Santosh Kumar Chaudhary would like to thank the council of scientific
+and industrial research (CSIR), Government of India (File Number 09/0081
+(14002)/2022-EMR-I) for financial assistance.
+Conflict of interest
+The authors declare no conflict of interest.
+References
+Al-Nasser, A. D. (2007). L ranked set sampling:
+A generalization
+procedure for robust visual sampling. Communications in Statis-
+tics—Simulation and Computation®, 36(1), 33-43.
+Biradar, B., Santosha, C. (2014) Estimation of the Mean of the
+Exponential Distribution Using Maximum Ranked Set Sampling with
+Unequal Samples. Open Journal of Statistics, 4(8), 641-649. doi:
+10.4236/ojs.2014.48060.
+Nadaraya, E. A. (1964). Some new estimates for distribution functions.
+Theory of Probability and Its Applications, 9(3), 497-500.
+Eskandarzadeh, M., Crescenzo, A. D., Tahmasebi, S. (2018). Measures
+of information for maximum ranked set sampling with unequal samples.
+Communications in Statistics-Theory and Methods, 47(19), 4692-4709.
+Gupta,
+N.,
+Chaudhary,
+S. K. (2023). Some characterizations of
+continuous symmetric distributions based on extropy of record values.
+Statistical Papers,
+1-18. DOI: https://doi.org/10.1007/s00362-022-
+01392-y.
+Gupta, N., Chaudhary, S. K. (2022). On general weighted extropy of
+ranked set sampling. arXiv preprint arXiv:2207.02003. (Communicated
+to journal)
+Qiu,
+G.,
+Eftekharian,
+A. (2021). Extropy information of maxi-
+mum and minimum ranked set sampling with unequal samples.
+
+20
+S.K.Chaudhary, N. Gupta
+Communications in Statistics-Theory and Methods, 50(13), 2979-2995.
+Jahanshahi, S. M. A.,
+Zarei,
+H.,
+Khammar, A. H. (2020). On
+cumulative residual extropy. Probability in the Engineering and
+Informational Sciences, 34(4), 605-625.
+Kazemi, M. R., Tahmasebi, S., Cal`ı, C., Longobardi, M. (2021).
+Cumulative residual extropy of minimum ranked set sampling with
+unequal samples. Results in Applied Mathematics, 10, 100156.
+Kazemi, M. R., Hashempour, M., Longobardi, M. (2022). Weighted
+Cumulative Past Extropy and Its Inference. Entropy, 24(10), 1444.
+Lad, F., Sanfilippo, G., Agro, G. (2015). Extropy:
+Complemen-
+tary dual of entropy.
+McIntyre,
+G. A. (1952). A method for unbiased selective sam-
+pling, using ranked sets. Australian journal of agricultural research,
+3(4), 385-390.
+Qiu, G.,
+Raqab, M. Z. (2022). On weighted extropy of ranked
+set sampling and its comparison with simple random sampling counter-
+part. Communications in Statistics-Theory and Methods, 1-18. DOI:
+10.1080/03610926.2022.2082478
+Raqab, M. Z., Qiu, G. (2019). On extropy properties of ranked
+set sampling. Statistics, 53(1), 210-226.
+Rao, M., Chen, Y., Vemuri, B. C., Wang, F. (2004). Cumulative
+residual entropy: a new measure of information. IEEE transactions on
+Information Theory, 50(6), 1220-1228.
+Shaked,
+M.,
+Shanthikumar,
+J. G. (Eds.). (2007). Stochastic or-
+ders. New York, NY: Springer New York.
+Sarda, P. (1993). Smoothing parameter selection for smooth dis-
+tribution functions. Journal of Statistical Planning and Inference,
+35(1), 65-75.
+Shannon,
+C. E. (1948).
+A mathematical
+theory of communica-
+
+S.K.Chaudhary, N. Gupta
+21
+tion. The Bell system technical journal, 27(3), 379-423.
+Tahmasebi,
+S. (2020). Weighted extensions of generalized cumu-
+lative residual entropy and their applications. Communications in
+Statistics-Theory and Methods, 49(21), 5196-5219.
+Wolfe, D. A. (2004). Ranked set sampling:
+An approach to more
+efficient data collection. Statistical Science 19 (4):636–43.
+Santosh Kumar Chaudhary
+Department of Mathematics,
+Indian Institute of Technology Kharagpur
+Kharagpur-721302, INDIA
+E-mail: skchaudhary1994@kgpian.iitkgp.ac.in
+Nitin Gupta
+Department of Mathematics,
+Indian Institute of Technology Kharagpur
+Kharagpur-721302, INDIA
+E-mail: nitin.gupta@maths.iitkgp.ac.in
+
diff --git a/n9FRT4oBgHgl3EQfbjfy/content/tmp_files/load_file.txt b/n9FRT4oBgHgl3EQfbjfy/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf,len=494
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='13561v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='ST]  31 Jan 2023  General weighted cumulative residual (past)  extropy of minimum (maximum) ranked set  sampling with unequal samples  Santosh Kumar Chaudhary*, Nitin Gupta**  Department of Mathematics, Indian Institute of Technology  Kharagpur, West Bengal 721302, India  Abstract  The general weighted cumulative residual extropy (GWCRJ) and gen-  eral weighted cumulative past extropy (GWCPJ) are introduced in this  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' There are some results in relation to GWCPJ and GWCRJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' We  take into account GWCRJ-based uncertainty measures for the minimal  ranked set sampling technique with unequal samples (minRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Ad-  ditionally, we take into account GWCPJ-based uncertainty measures  for the maximum ranked set sampling technique with unequal sam-  ples (maxRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Stochastic comparison for Simple random sampling  (SRS) is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' We looked at the monotone properties of minRSSU  and maxRSSU as well as stochastic comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Finally, two empirical  estimators of GWCPJ and GWCRJ are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Key Words: Extropy, General weighted cumulative past extropy, Gen-  eral weighted cumulative residual extropy, Ranked set sampling, Simple  random sampling  Mathematical Subject Classification: 62B10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' 62D05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' 60E15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  1  Introduction  The method of enhancing the accuracy of the sample mean as an approxima-  tion of the population means, McIntyre (1952) proposed ranked set sampling  (RSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' The RSS process involves selecting n sets of units, each of size n, at  random from a population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' The units in each set are then ranked visually  Corresponding author E-mail: skchaudhary1994@kgpian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
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+page_content='in  ** E-mail: nitin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='gupta@maths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
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+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='Chaudhary, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Gupta  or using some low-cost techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Pick the unit with the lowest rank out  of the first n units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Select the unit with the second-lowest rank from the  second group of n units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Up until selecting the unit with the highest rank in  the n-th set, the process is repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Then, we measure the relevant variable  using the units we have chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' In many practical aspects related to applica-  tions in industrial statistics, environmental and ecological studies, statistical  genetics, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=', RSS schemes have been playing a significant role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' We assume  that XSRS = {Xi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' , n} denotes a simple random sampling (SRS)  of size n from a continuous distribution with probability density function  (pdf) f and cumulative distribution function (cdf) F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' In comparison to SRS,  the RSS is a more effective sampling method for estimating the population  mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' As a helpful improvement to the RSS technique, Qiu and Eftekhar-  ian (2021) investigated the extropy of the minimum and maximum ranked  set sampling procedure with unequal samples (minRSSU and maxRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' In  the minRSSU and maxRSSU, we draw n simple random samples, in which  the size of the i−th samples is i, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' The one-cycle minRSSU or  maxRSSU involves an initial ranking of n samples of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  We collect  Zi = X(1:i)i for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' , n, and Yi = X(i:i)i for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' , n,  where X(i:j)j denotes the i-th order statistic from the j-th SRS of size j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' The  resulting sample is called one-cycle minRSSU and maxRSSU of size n and de-  noted by ZminRSSU = {Zi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' , n} and YmaxRSSU = {Yi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' , n},  respectively (see Biradar and Santosha (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Measures of information for RSS and its variations have been devel-  oped by several authors (see Eskandarzadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (2018), Raqab and Qiu  (2019), Qiu and Raqab (2022) and Gupta and Chaudhary (2022) etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Qiu  and Eftekharian (2021) considered the information content of minRSSU and  maxRSSU in terms of extropy and also compared the extropy of these two  sampling data with that of SRS and RSS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Kazemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (2021) consid-  ered uncertainty measures of minimum ranked set sampling procedure with  unequal samples (MinRSSU) in terms of cumulative residual extropy and its  dynamic version and they also compared the uncertainty and information  content of cumulative residual extropy based on MinRSSU and SRS designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Chacko and George (2022) considered the extropy properties of the RSS  when ranking is not perfect and also obtained upper and lower bounds of  the extropy of RSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Extropy was introduced by Lad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (2015) as a measure of uncertainty  as  J(X) = −1  2  � ∞  −∞  f 2(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='1)  Jahanshahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (2019) defined Cumulative residual extropy as  ξ(X) = −1  2  � ∞  −∞  ¯F 2(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='2)  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='Chaudhary, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Gupta  3  Cumulative past extropy is defined as  ¯ξ(X) = −1  2  � ∞  −∞  F 2(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='3)  Extropy and its various form have been studied by several authors (see  Bansal and Gupta (2021), Gupta and Chaudhary (2022), and Kazemi et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (2022) etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Bansal and Gupta (2021) studied weighted extropies and  past extropy of order statistics and k-record values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Gupta and Chaudhary  (2022) studied the general weighted extropy of RSS where they defined gen-  eral weighted extropy (GWJ) with weight w(x) ≥ 0 as  Jw(X) = −1  2  � ∞  −∞  w(x)f 2(x)dx  = −1  2E(δw  X(U)),  where δw  X(u) = w(F −1(u))f(F −1(u)) and U is uniformly distributed random  variable on (0, 1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=', U ∼ Uniform(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' This motivated us to generalize  cumulative residual and past extropy and study it for ranked set sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Here, we defined general weighted cumulative past extropy (GWCPJ) of  absolutely continuous random variable X with weight w(x) ≥ 0 as  ¯ξw(X) = −1  2  � ∞  −∞  w(x)F 2(x)dx  = −1  2  � 1  0  w(F −1(u))  u2  f(F −1(u))du  = −1  2E(Λw  X(U)),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='4)  where Λw  X(u) = u2w(F −1(u))  f(F −1(u)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Also, we define general weighted cumulative residual extropy (GWCRJ) of  absolutely continuous random variable X with weight w(x) ≥ 0 as  ξw(X) = −1  2  � ∞  −∞  w(x) ¯F 2(x)dx  = −1  2  � 1  0  w(F −1(u)) (1 − u)2  f(F −1(u))du  = −1  2E(∆w  X(U)),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='5)  where ∆w  X(u) = (1−u)2w(F −1(u))  f(F −1(u))  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  4  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='Chaudhary, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Gupta  This paper is organized as follows: section 2 discusses results on GWCPJ  and GWCRJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Section 3 provides examples and results for SRS of GWCPJ  and GWCRJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' MinRSSU and maxRSSU for GWCRJ and GWCPJ, respec-  tively are studied in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Stochastic comparison for MinRSSU and  maxRSSU for GWCRJ and GWCPJ, respectively are obtained in section  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Section 6 is the monotone properties of MinRSSU and maxRSSU for  GWCRJ and GWCPJ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  In section 7, we obtained empirical  estimators of GWCRJ and GWCPJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Finally, section 8 concludes this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  2  Results on GWCPJ and GWCRJ  In the following theorem, we will study the behaviour of GWCPJ with re-  spect to weight functions w1 and w2 and the dispersive order of random  variables X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Theorem 1 Let X and Y be nonnegative random variables with pdf’s f and  g, cdf’s F and G, respectively having uX = uY < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (a) If w1 is decreasing, w1(x) ≤ w2(x) and X ≤disp Y , then ¯ξw1(X) ≥  ¯ξw2(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (b) If w1 is decreasing, w1(x) ≥ w2(x) and X ≥disp Y , then ¯ξw1(X) ≤  ¯ξw2(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Proof (a) Since X ≤disp Y , therefore we have f(F −1(u)) ≥ g(G−1(u)) for  u ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Then using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='13(b) of Shaked and Shanthikumar  (2007), X ≤disp Y implies that X ≥st Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Hence F −1(u) ≥ G−1(u) for all  u ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Since w1 is decreasing and w1(x) ≤ w2(x) , then w1(F −1(u)) ≤  w1(G−1(u)) ≤ w2(G−1(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Hence,  Λw1  X (u) = u2 w1(F −1(u))  f(F −1(u))  ≤ u2 w2(G−1(u))  g(G−1(u))  =Λw2  Y (u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Now using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='4),  ¯ξw1(X) = −1  2E  �  Λw1  X (U)  �  ≥ −1  2E  �  Λw2  Y (U)  �  = ¯ξw2(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='Chaudhary, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Gupta  5  (b) Proof is Similar to part (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  ■  If we take w1(x) = w2(x) = w(x) in the above theorem, then we have  the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Corollary 1 Let X and Y be nonnegative random variables with pdf’s f and  g, cdf’s F and G, respectively having uX = uY < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Let w is decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (a) If X ≤disp Y , then ¯ξw(X) ≥ ¯ξw(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (b) If X ≥disp Y , then ¯ξw(X) ≤ ¯ξw(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  In the following theorem, we will study the behaviour of GWCRJ with  respect to weight functions w1 and w2 and the dispersive order of random  variables X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Theorem 2 Let X and Y be nonnegative random variables with pdf’s f and  g, cdf’s F and G, respectively having uX = uY < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (a) If w1 is decreasing, w1(x) ≤ w2(x) and X ≤disp Y , then ξw1(X) ≥  ξw2(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (b) If w1 is decreasing, w1(x) ≥ w2(x) and X ≥disp Y , then ξw1(X) ≤  ξw2(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Proof (a) Since X ≤disp Y , therefore we have f(F −1(u)) ≥ g(G−1(u)) for  u ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Then using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='13(b) of Shaked and Shanthikumar  (2007), X ≤disp Y implies that X ≥st Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Hence F −1(u) ≥ G−1(u) for all  u ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Since w1 is decreasing and w1(x) ≤ w2(x) , then w1(F −1(u)) ≤  w1(G−1(u)) ≤ w2(G−1(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Hence  ∆w1  X (u) = (1 − u)2w1(F −1(u))  f(F −1(u))  ≤ (1 − u)2w2(G−1(u))  g(G−1(u))  =∆w2  Y (u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='1)  Now using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='5),  ξw1(X) = −1  2E  �  ∆w1  X (U)  �  ≥ −1  2E  �  ∆w2  Y (U)  �  = ξw2(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (b) Proof is Similar to part (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  ■  If we take w1(x) = w2(x) = w(x) in the above theorem, then we have  the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  6  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='Chaudhary, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Gupta  Corollary 2 Let X and Y be nonnegative random variables with pdf’s f and  g, cdf’s F and G, respectively having uX = uY < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Let w is decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (a) If X ≤disp Y , then ξw(X) ≥ ξw(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (b) If X ≥disp Y , then ξw(X) ≤ ξw(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  3  Simple Random Sampling  Let X be a random variable with finite mean µ and variance σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' For XSRS =  {Xi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' , n}, the joint pdf is �n  i=1 f(xi), as Xi’s, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' , n are  independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Hence the GWCPJ of X(n)  SRS  can be defined as  ¯ξw(X(n)  SRS) = −1  2  n  �  i=1  �� ∞  −∞  w(xi)F 2(xi)dxi  �  = −1  2  �  −2¯ξw(X)  �n  = −1  2 (E(Λw  X(U)))n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='1)  Also, the GWCRJ of X(n)  SRS can be defined as  ξw(X(n)  SRS) = −1  2  n  �  i=1  �� ∞  −∞  w(xi) ¯F 2  X(xi)dxi  �  = −1  2 (−2ξw(X))n  = −1  2 (E(∆w  X(U)))n ,  where Λw  X(U) and ∆w  X(U) are given in section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Example 1 If U is a uniform random variable on interval (0,1) and w(x) =  xm, m > 0 then  E[Λw  X(U)] = E  �  U 2w(U)  �  =  1  m + 3,  E[∆w  X(U)] = E  �  (1 − U)2w(U)  �  =  1  (m + 1) −  2  (m + 2) +  1  (m + 3),  ¯ξw(X(n)  SRS) = −1  2  �  1  m + 3  �n  ,  ξw(X(n)  SRS) = −1  2  �  1  (m + 1) −  2  (m + 2) +  1  (m + 3)  �n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='Chaudhary, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Gupta  7  Theorem 3 Let X be a non-negative absolutely continuous random vari-  able with pdf f and cdf F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Assume ψ(x) is an increasing function and  w(ψ(x))ψ′(x) ≤ (≥)w(x) and ψ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' If Y = ψ(X), then ¯ξw(X(n)  SRS) ≤ (≥  )¯ξw(Y(n)  SRS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Proof Let the random variable Y has pdf g and cdf G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Then  Λw  Y (u) = u2w  �  G−1(u)  �  g (G−1(u))  = u2w(ψ(F −1(u)))ψ′(F −1(u))  f(F −1(u))  ∀  0 < u < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Note that ψ(x) ≥ ψ(0), ∀ x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Since w(ψ(x))ψ′(x) ≤ (≥)w(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Hence for  0 < u < 1, we have  Λw  Y (u) = u2w(ψ(F −1(u)))ψ′(F −1(u))  f(F −1(u))  ≤ (≥)u2w((F −1(u)))  f(F −1(u))  = Λw  X(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Therefore ¯ξw(X(n)  SRS) ≥ (≤)¯ξw(Y(n)  SRS) using equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  ■  Example 2 Let ψ(x) = ex − 1,  w(x) = x, x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' It is easy to verify that  ψ(x) is an increasing function and ψ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Let h(x) = w(ψ(x))ψ′(x) −  w(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' That is, h(x) = e2x − ex − x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Note that h(0) = h  ′(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Moreover,  h  ′(x) ≥ 0 ∀ x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Thus, h(x) is increasing function on x ∈ [0, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' So,  h(x) ≥ h(0) = 0, that is, w(ψ(x))ψ′(x) ≥ w(x) ∀x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Therefore, using  Theorem 3 for Y = ψ(X), we have,  ¯ξw(X(n)  SRS) ≥ ¯ξw(Y(n)  SRS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  4  MinRSSU and maxRSSU for GWCRJ and GWCPJ  Let Xi:i represent i-th order statistics from a sample of size i and X1:i rep-  resent 1-st order statistics from a sample of size i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Let Fi:i denote cdf of i-th  order statistics from a sample of size i and F1:i denotes cdf of 1-th order  statistics from a sample of size i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  8  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='Chaudhary, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Gupta  The GWCPJ of X(n)  maxRSSU can be defined as  ¯ξw(X(n)  maxRSSU) = −1  2  n  �  i=1  �  −2¯ξw(Xi:i)  �  = −1  2  n  �  i=1  �� ∞  −∞  w(x)F 2  i:i(x)dx  �  = −1  2  n  �  i=1  �� ∞  −∞  w(x)F 2i  X (x)dx  �  = −1  2  n  �  i=1  �� 1  0  u2iw(F −1(u))  f(F −1(u))  du  �  = −1  2  n  �  i=1  E  �U 2iw(F −1(U))  f(F −1(U))  �  = −1  2  n  �  i=1  E [Ψw  X(U)] , ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='1)  where Ψw  X(u) = u2iw(F −1(u))  f(F −1(u)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  The GWCRJ of X(n)  minRSSU can be defined as  ξw(X(n)  minRSSU) = −1  2  n  �  i=1  [−2ξw(X1:i)]  = −1  2  n  �  i=1  �� ∞  −∞  w(x) ¯F 2  1:i(x)dx  �  = −1  2  n  �  i=1  �� ∞  −∞  w(x) ¯F 2i(x)dx  �  = −1  2  n  �  i=1  �� 1  0  (1 − u)2iw(F −1(u))  f(F −1(u))  du  �  = −1  2  n  �  i=1  E  �  (1 − U)2iw(F −1(U))  f(F −1(U))  �  = −1  2  n  �  i=1  E [Φw  X(U)] , ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='2)  where Φw  X(u) = (1−u)2iw(F −1(u))  f(F −1(u))  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='Chaudhary, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Gupta  9  Let us see some examples to illustrate the above measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Example 3 If U is a uniform random variable on interval (0,1) and w(x) =  xm, m > 0 then  ¯ξw(X(n)  maxRSSU) = −1  2  n  �  i=1  �  1  2i + m + 1  �  ,  ξw(X(n)  minRSSU) = −(Γ(m + 1))2  2  n  �  i=1  Γ(2i + 1)  Γ(2i + m + 2),  where  Γ(α) =  � ∞  0  e−xxα−1dx  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Example 4 Let X be an exponential distribution with cdf F(x) = 1 −  e−λx, λ > 0, x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Let w(x) = xm, m > 0, x > 0, then  ξw(X(n)  minRSSU) = −1  2  n  �  i=1  �� 1  0  (1 − u)2iw(F −1(u))  f(F −1(u))  du  �  = −1  2  n  �  i=1  (−1)m  λm+1  � 1  0  (1 − u)2i−1(log(1 − u))mdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Taking 1 − u = e−x in above expression, we get  ξw(X(n)  minRSSU) = −1  2  n  �  i=1  1  λm+1  � ∞  0  xme−2ixdx  = −1  2  n  �  i=1  1  λm+1  Γ(m + 1)  (2i)m+1 dx  = −1  2  �Γ(m + 1)  λm+1  �n  n  �  i=1  � 1  2i  �m+1  = −1  2  �Γ(m + 1)  (2λ)m+1  �n � 1  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  �m+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Example 5 Let a random variable X have survival function ¯F(x) = (1 −  10  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='Chaudhary, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Gupta  x)b,  0 < x < 1,  b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' If w(x) = xm,  x > 0,  m > 0 then  ξw(X(n)  minRSSU) = −1  2  n  �  i=1  �� 1  0  w(x) ¯F 2i(x)dx  �  = −1  2  n  �  i=1  �� 1  0  xm(1 − x)2ibdx  �  = −1  2  n  �  i=1  B(m + 1, 2ib + 1),  where B(α, β) is beta function defined as  B(α, β) =  � 1  0  xα−1(1 − x)1−βdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Theorem 4 Let X be a non-negative absolutely continuous random vari-  able with pdf f and cdf F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Assume ψ(x) is an increasing function and  w(ψ(x))ψ′(x) ≤ (≥)w(x) and ψ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' If Y = ψ(X), then ¯ξw(X(n)  maxRSSU) ≤  (≥)¯ξw(Y(n)  maxRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Proof Let random variable Y has pdf g and cdf G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Since w(ψ(x))ψ′(x) ≤  (≥)w(x) and ψ(0) = 0, therefore,  Ψw  Y (u) = u2iw  �  G−1(u)  �  g (G−1(u))  = u2iw(ψ(F −1(u)))ψ′(F −1(u))  f(F −1(u))  ≤ (≥)u2iw(F −1(u))  f(F −1(u))  = Ψw  X(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Hence, From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='1) we obtain  ¯ξw(X(n)  maxRSSU) ≥ (≤)¯ξw(Y(n)  maxRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  ■  Example 6 Let ψ(x) = ex − 1,  w(x) = x, x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Note that from example  2, ψ(x) is an increasing function and w(ψ(x))ψ′(x) ≥ w(x) and ψ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Using Theorem 3 for Y = ψ(X), we have,  ¯ξw(X(n)  maxRSSU) ≥ ¯ξw(Y(n)  maxRSSU)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='Chaudhary, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Gupta  11  Theorem 5 Let X be a non-negative absolutely continuous random vari-  able with pdf f and cdf F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Assume ψ(x) is an increasing function and  w(ψ(x))ψ′(x) ≤ (≥)w(x) and ψ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' If Y = ψ(X), then ξw(X(n)  minRSSU) ≤  (≥)ξw(Y(n)  minRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Proof Let random variable Y has pdf g and cdf G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Since w(ψ(x))ψ′(x) ≤  (≥)w(x) and ψ(0) = 0, therefore,  Φw  Y (u) = (1 − u)2iw  �  G−1(u)  �  g (G−1(u))  = (1 − u)2iw(ψ(F −1(u)))ψ′(F −1(u))  f(F −1(u))  ≤ (≥)(1 − u)2iw(F −1(u))  f(F −1(u))  = Φw  X(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Hence, From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='2) we obtain  ξw(X(n)  minRSSU) ≥ (≤)ξw(Y(n)  minRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  ■  Example 7 Let ψ(x) = ex − 1,  w(x) = x, x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Note that from Example  2, ψ(x) is an increasing function and w(ψ(x))ψ′(x) ≥ w(x) and ψ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Using Theorem 3 for Y = ψ(X), we have,  ξw(X(n)  minRSSU) ≥ ξw(Y(n)  minRSSU)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Theorem 6 Let X be an absolutely continuous random variable with pdf f  and cdf F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Then for n ≥ 2,  ¯ξw(X(n)  maxRSSU) ≥ ¯ξw(X(n)  SRS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  12  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='Chaudhary, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Gupta  Proof Consider  ¯ξw(X(n)  maxRSSU) = −1  2  n  �  i=1  E  �U 2iw(F −1(U))  f(F −1(U))  �  = −1  2  n  �  i=1  �� 1  0  u2iw(F −1(u))  f(F −1(u))  du  �  ≥ −1  2  n  �  i=1  �� 1  0  u2w(F −1(u))  f(F −1(u)) du  �  = −1  2  n  �  i=1  E  �U 2w(F −1(U))  f(F −1(U))  �  = ¯ξw(X(n)  SRS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Hence the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  ■  Theorem 7 Let X be an absolutely continuous random variable with pdf f  and cdf F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Then for n ≥ 2,  ξw(X(n)  minRSSU) ≥ ξw(X(n)  SRS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Proof Consider  ξw(X(n)  minRSSU) = −1  2  n  �  i=1  E  �  (1 − U)2iw(F −1(U))  f(F −1(U))  �  = −1  2  n  �  i=1  �� 1  0  (1 − u)2iw(F −1(u))  f(F −1(u))  du  �  ≥ −1  2  n  �  i=1  �� 1  0  (1 − u)2w(F −1(u))  f(F −1(u))  du  �  = −1  2  n  �  i=1  E  �  (1 − U)2w(F −1(U))  f(F −1(U))  �  = ξw(X(n)  SRS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Hence the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  ■  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='Chaudhary, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Gupta  13  5  Stochastic Comparision  The following lemma will be used in deriving Theorem 9 and Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Lemma 1 is also used by Qiu and Raqab (2022) and Gupta and Chaudhary  (2022) to derive their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Lemma 1 [Ahmed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (1986)] Let X and Y be nonnegative random vari-  ables with pdf’s f and g, respectively, satisfying f(0) ≥ g(0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' If X ≤su Y  (or X ≤∗ Y or X ≤c Y ), then X ≤disp Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  In the following theorem, we will study the behaviour maxRSSU of  GWCPJ with respect to weight functions w1 and w2 and dispersive order of  random variables X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Theorem 8 Let X and Y be nonnegative random variables with pdf’s f and  g, cdf’s F and G, respectively having uX = uY < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (a) If w1 is decreasing, w1(x) ≤ w2(x) and X ≤disp Y , then ¯ξw1(X(n)  maxRSSU) ≥  ¯ξw2(Y(n)  maxRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (b) If w1 is decreasing, w1(x) ≥ w2(x) and X ≥disp Y , then ¯ξw1(X(n)  maxRSSU) ≤  ¯ξw2(Y(n)  maxRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Proof (a) Since X ≤disp Y , therefore we have f(F −1(u)) ≥ g(G−1(u)) for  u ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Then using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='13(b) of Shaked and Shanthikumar  (2007), X ≤disp Y implies that X ≥st Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Hence, F −1(u) ≥ G−1(u) for all  u ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Since w1 is decreasing and w1(x) ≤ w2(x) , then w1(F −1(u)) ≤  w1(G−1(u)) ≤ w2(G−1(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Hence,  Ψw  X(u) = u2i w1(F −1(u))  f(F −1(u)) ≤ u2i w2(G−1(u))  g(G−1(u)) = Ψw  Y (u)  Now using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='1),  ¯ξw1(X(n)  maxRSSU) ≥ ¯ξw2(Y(n)  maxRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (b) Proof is Similar to part (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  ■  If we take w1(x) = w2(x) = w(x) in the above theorem, then we have  the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  14  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='Chaudhary, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Gupta  Corollary 3 Let X and Y be nonnegative random variables with pdf’s f and  g, cdf’s F and G, respectively having uX = uY < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Let w is decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (a) If X ≤disp Y , then ¯ξw(X(n)  maxRSSU) ≥ ¯ξw(Y(n)  maxRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (b) If X ≥disp Y , then ¯ξw(X(n)  maxRSSU) ≤ ¯ξw(Y(n)  maxRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  One may refer Shaked and Shantikumar (2007) for details of convex  transform order (≤c), star order (≤⋆), super additive order (≤su), and dis-  persive order (≤disp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' In view of Theorem 8 and Lemma 1, the following  result is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Theorem 9 Let X and Y be nonnegative random variables with pdf’s f and  g, cdf’s F and G, respectively having uX = uY < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (a) If w1 is decreasing, w1(x) ≤ w2(x) and X ≤su Y ( or X ≤∗ Y or  X ≤c Y ), then ¯ξw1(X(n)  maxRSSU) ≥ ¯ξw2(Y(n)  maxRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (b) If w1 is decreasing, w1(x) ≥ w2(x) and X ≥su Y (or X ≥∗ Y or X ≥c Y  ), then ¯ξw1(X(n)  maxRSSU) ≤ ¯ξw2(Y(n)  maxRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  If we take w1(x) = w2(x) = w(x) in the above theorem, then we have  the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Corollary 4 Let X and Y be nonnegative random variables with pdf’s f  and g, cdf’s F and G, respectively having uX = uY < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (a) If w is decreasing and X ≤su Y ( or X ≤∗ Y or X ≤c Y ), then  ¯ξw(X(n)  maxRSSU) ≥ ¯ξw(Y(n)  maxRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (b) If w is decreasing and X ≥su Y (or X ≥∗ Y or X ≥c Y ), then  ¯ξw(X(n)  maxRSSU) ≤ ¯ξw(Y(n)  maxRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  In the following theorem, we will study the behaviour of minRSSU of  GWCRJ with respect to weight functions w1 and w2 and the dispersive  order of random variables X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Theorem 10 Let X and Y be nonnegative random variables with pdf’s f  and g, cdf’s F and G, respectively having uX = uY < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (a) If w1 is decreasing, w1(x) ≤ w2(x) and X ≤disp Y , then ξw1(X(n)  minRSSU) ≥  ξw2(Y(n)  minRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='Chaudhary, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Gupta  15  (b) If w1 is decreasing, w1(x) ≥ w2(x) and X ≥disp Y , then ξw1(X(n)  minRSSU) ≤  ξw2(Y(n)  minRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Proof (a) Since X ≤disp Y , therefore we have f(F −1(u)) ≥ g(G−1(u)) for  u ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Then using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='13(b) of Shaked and Shanthikumar  (2007), X ≤disp Y implies that X ≥st Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Hence F −1(u) ≥ G−1(u) for all  u ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Since w1 is decreasing and w1(x) ≤ w2(x) , then w1(F −1(u)) ≤  w1(G−1(u)) ≤ w2(G−1(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Hence  Φw  X(u) = (1 − u)2i w1(F −1(u))  f(F −1(u)) ≤ (1 − u)2i w2(G−1(u))  g(G−1(u)) = Φw  Y (u)  Now using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='2),  ξw1(X(n)  minRSSU) ≥ ξw2(Y(n)  minRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (b) Proof is Similar to part (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  ■  If we take w1(x) = w2(x) = w(x) in the above theorem, then we have  the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Corollary 5 Let X and Y be nonnegative random variables with pdf’s f and  g, cdf’s F and G, respectively having uX = uY < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Let w is decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (a) If X ≤disp Y , then ξw(X(n)  minRSSU) ≥ ξw(Y(n)  minRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (b) If X ≥disp Y , then ξw(X(n)  minRSSU) ≤ ξw(Y(n)  minRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  In view of Theorem 10 and Lemma 1, the following result is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Theorem 11 Let X and Y be nonnegative random variables with pdf’s f  and g, cdf’s F and G, respectively having uX = uY < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (a) If w1 is decreasing, w1(x) ≤ w2(x) and X ≤su Y (or X ≤∗ Y or  X ≤c Y ), then ξw1(X(n)  minRSSU) ≥ ξw2(Y(n)  minRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (b) If w1 is decreasing, w1(x) ≥ w2(x) and X ≥su Y (or X ≥∗ Y or  X ≥c Y ), then ξw1(X(n)  minRSSU) ≤ ξw2(Y(n)  minRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  If we take w1(x) = w2(x) = w(x) in the above theorem, then we have  the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  16  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='Chaudhary, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Gupta  Corollary 6 Let X and Y be nonnegative random variables with pdf’s f  and g, cdf’s F and G, respectively having uX = uY < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (a) If w is decreasing and X ≤su Y (or X ≤∗ Y or X ≤c Y ), then  ξw(X(n)  minRSSU) ≥ ξw(Y(n)  minRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  (b) If w is decreasing and X ≥su Y (or X ≥∗ Y or X ≥c Y ), then  ξw(X(n)  minRSSU) ≤ ξw(Y(n)  minRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  6  Monotone Properties  The following result gives the conditions under which the ξw(X(n)  minRSSU) will  increase with respect to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Theorem 12 If w(F −1(u)  f(F −1(u)) ≤ 1, 0 < u < 1, then ξw(X(n)  minRSSU) is increas-  ing in n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Proof Consider  ξw(X(n+1)  minRSSU)  ξw(X(n)  minRSSU)  =  � 1  0  (1 − u)2n+2w(F −1(u))  f(F −1(u))  du ≤  1  2n + 3 ≤ 1 ∀ n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Hence,  ξw(X(n+1)  minRSSU) ≥ ξw(X(n)  minRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  The following result gives the conditions under which the ¯ξw(X(n)  maxRSSU)  will increase with respect to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Theorem 13 If w(F −1(u)  f(F −1(u)) ≤ 1, 0 < u < 1, then ¯ξw(X(n)  maxRSSU) is increas-  ing in n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Proof Consider  ¯ξw(X(n+1)  maxRSSU)  ¯ξw(X(n)  maxRSSU)  =  � 1  0  u2n+2w(F −1(u))  f(F −1(u))  du ≤  1  2n + 3 ≤ 1 ∀ n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Hence,  ¯ξw(X(n+1)  maxRSSU) ≥ ¯ξw(X(n)  maxRSSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='Chaudhary, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Gupta  17  7  Empirical measures of GWCPJ and GWCRJ  The empirical measure of F is defined as  F  �  n(x) =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  0,  x < X1:n  i  n,  Xi:n ≤ x < Xi+1:n,  i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' , n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  1,  x ≥ Xn:n  where X1:n ≤ X2:n ≤ X3:n ≤ · · · ≤ Xn:n denote order statistics of random  sample X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' , Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  The empirical measure of GWCPJ for w(x) = xm, x > 0, m > 0 is  ¯ξm  �  (Fn) = −1  2  � ∞  −∞  xmF 2  n  �  (x)dx = −1  2  n−1  �  i=1  � xi+1:n  xi:n  xm  � i  n  �2  dx  = −  1  2(m + 1)  n−1  �  i=1  �  (xm+1  i+1:n − xm+1  i:n )  � i  n  �2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  The empirical measure of GWCRJ for w(x) = xm, x > 0, m > 0 is  ξm  �  (Fn) = −1  2  � ∞  −∞  xm ¯F 2  n  �  (x)dx = −1  2  n−1  �  i=1  � xi+1:n  xi:n  xm  � i  n  �2  dx  = −  1  2(m + 1)  n−1  �  i=1  �  (xm+1  i+1:n − xm+1  i:n )  �  1 − i  n  �2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Analogous to Theorem 9 of Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (2004), Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='1 of Tahmasebi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (2020) and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='1 of Tahmasebi and Toomaj (2020), we state  the following theorems and the proof follows similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Theorem 14 Let X be a non negative absolutely continuous random vari-  able with cdf F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Then for any random X in Lp for some p > 2, ¯ξm  �  (Fn)  converges almost surely to ¯ξm(X) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Theorem 15 Let X be a non negative absolutely continuous random vari-  able with cdf F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Then for any random X in Lp for some p > 2, ξm  �  (Fn)  converges almost surely to ξm(X) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  18  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='Chaudhary, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Gupta  We create another estimator based on the kernel-smoothed estimator of  the distribution function because smoothed estimators perform better than  non-smoothed estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  The second estimator of GWCPJ for w(x) = xm, x > 0, m > 0 is  ¯ξm  �  (Fh) = −1  2  � ∞  −∞  xmF 2  h(x)dx = −1  2  n−1  �  i=1  � xi+1:n  xi:n  xm �  F 2  h(xi)  �2 dx  = −  1  2(m + 1)  n−1  �  i=1  �  (xm+1  i+1:n − xm+1  i:n )  �  F 2  h(xi)  �2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  The second estimator of GWCRJ for w(x) = xm, x > 0, m > 0 is  ξm  �  (Fh) = −1  2  � ∞  −∞  xm ¯F 2  h(x)dx = −1  2  n−1  �  i=1  � xi+1:n  xi:n  xm � ¯F 2  h(xi)  �  dx  = −  1  2(m + 1)  n−1  �  i=1  �  (xm+1  i+1:n − xm+1  i:n )  � ¯F 2  h(xi)  ��  ,  where ¯F = 1 − F and Fh is defined by Nadaraya (1964) as  Fh(x) = 1  n  n  �  i=1  L  �x − Xi  h  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Here, L(x) =  � x  −∞ K(t)dt, that is, L is cdf of positive kernel K and h is a  bandwidth parameter (see Sarda (1993) and Jahanshahi (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  8  Conclusion  Generalized versions of cumulative past and residual extropy have been de-  fined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' The general GWCRJ and GWCPJ uncertainty measures based on  the minRSSU and maxRSSU data, respectively, have been introduced in  this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Additionally, this work presented the GWCRJ and GWCPJ-  based uncertainty measures of SRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' For minRSSU, maxRSSU, and SRS  data, some results of the GWCPJ and GWCRJ measures were obtained, in-  cluding stochastic orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' With help of empirical measure of the distribution  function, we obtained two estimators of GWCRJ and GWCPJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='Chaudhary, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Gupta  19  Funding  Santosh Kumar Chaudhary would like to thank the council of scientific  and industrial research (CSIR), Government of India (File Number 09/0081  (14002)/2022-EMR-I) for financial assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Conflict of interest  The authors declare no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  References  Al-Nasser, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' L ranked set sampling:  A generalization  procedure for robust visual sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Communications in Statis-  tics—Simulation and Computation®, 36(1), 33-43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Biradar, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=', Santosha, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (2014) Estimation of the Mean of the  Exponential Distribution Using Maximum Ranked Set Sampling with  Unequal Samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Open Journal of Statistics, 4(8), 641-649.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='4236/ojs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='48060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Nadaraya, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (1964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Some new estimates for distribution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Theory of Probability and Its Applications, 9(3), 497-500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Eskandarzadeh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=', Crescenzo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=', Tahmasebi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Measures  of information for maximum ranked set sampling with unequal samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Communications in Statistics-Theory and Methods, 47(19), 4692-4709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Gupta,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=',  Chaudhary,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Some characterizations of  continuous symmetric distributions based on extropy of record values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Statistical Papers,  1-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' DOI: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='1007/s00362-022-  01392-y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Gupta, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=', Chaudhary, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' On general weighted extropy of  ranked set sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' arXiv preprint arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='02003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (Communicated  to journal)  Qiu,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=',  Eftekharian,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Extropy information of maxi-  mum and minimum ranked set sampling with unequal samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  20  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='Chaudhary, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Gupta  Communications in Statistics-Theory and Methods, 50(13), 2979-2995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Jahanshahi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=',  Zarei,  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=',  Khammar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' On  cumulative residual extropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Probability in the Engineering and  Informational Sciences, 34(4), 605-625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Kazemi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=', Tahmasebi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=', Cal`ı, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=', Longobardi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Cumulative residual extropy of minimum ranked set sampling with  unequal samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Results in Applied Mathematics, 10, 100156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Kazemi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=', Hashempour, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=', Longobardi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Weighted  Cumulative Past Extropy and Its Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Entropy, 24(10), 1444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Lad, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=', Sanfilippo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=', Agro, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Extropy:  Complemen-  tary dual of entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  McIntyre,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (1952).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' A method for unbiased selective sam-  pling, using ranked sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Australian journal of agricultural research,  3(4), 385-390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Qiu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=',  Raqab, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' On weighted extropy of ranked  set sampling and its comparison with simple random sampling counter-  part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Communications in Statistics-Theory and Methods, 1-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' DOI:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='1080/03610926.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='2082478  Raqab, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=', Qiu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' On extropy properties of ranked  set sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Statistics, 53(1), 210-226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Rao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=', Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=', Vemuri, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=', Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Cumulative  residual entropy: a new measure of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' IEEE transactions on  Information Theory, 50(6), 1220-1228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Shaked,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=',  Shanthikumar,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Stochastic or-  ders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' New York, NY: Springer New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Sarda, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Smoothing parameter selection for smooth dis-  tribution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Journal of Statistical Planning and Inference,  35(1), 65-75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Shannon,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (1948).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  A mathematical  theory of communica-  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='Chaudhary, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Gupta  21  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' The Bell system technical journal, 27(3), 379-423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Tahmasebi,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Weighted extensions of generalized cumu-  lative residual entropy and their applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Communications in  Statistics-Theory and Methods, 49(21), 5196-5219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Wolfe, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Ranked set sampling:  An approach to more  efficient data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content=' Statistical Science 19 (4):636–43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='  Santosh Kumar Chaudhary  Department of Mathematics,  Indian Institute of Technology Kharagpur  Kharagpur-721302, INDIA  E-mail: skchaudhary1994@kgpian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='iitkgp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='in  Nitin Gupta  Department of Mathematics,  Indian Institute of Technology Kharagpur  Kharagpur-721302, INDIA  E-mail: nitin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='gupta@maths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
+page_content='iitkgp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FRT4oBgHgl3EQfbjfy/content/2301.13561v1.pdf'}
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+Dynamical mean-field theory for the Hubbard-Holstein model on a quantum device
+Steffen Backes1,2,3,∗ Yuta Murakami2, Shiro Sakai2, and Ryotaro Arita1,2
+1Research Center for Advanced Science and Technology,
+University of Tokyo, Komaba, Tokyo 153-8904, Japan
+2Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan and
+3CPHT, CNRS, ´Ecole polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France
+(Dated: January 6, 2023)
+Recent developments in quantum hardware and quantum algorithms have made it possible to
+utilize the capabilities of current noisy intermediate-scale quantum devices for addressing problems
+in quantum chemistry and condensed matter physics. Here we report a demonstration of solving
+the dynamical mean-field theory (DMFT) impurity problem for the Hubbard-Holstein model on
+the IBM 27-qubit Quantum Falcon Processor Kawasaki, including self-consistency of the DMFT
+equations. This opens up the possibility to investigate strongly correlated electron systems coupled
+to bosonic degrees of freedom and impurity problems with frequency-dependent interactions. The
+problem involves both fermionic and bosonic degrees of freedom to be encoded on the quantum
+device, which we solve using a recently proposed Krylov variational quantum algorithm to obtain
+the impurity Green’s function. We find the resulting spectral function to be in good agreement with
+the exact result, exhibiting both correlation and plasmonic satellites and significantly surpassing
+the accuracy of standard Trotter-expansion approaches. Our results provide an essential building
+block to study electronic correlations and plasmonic excitations on future quantum computers with
+modern ab initio techniques.
+I.
+INTRODUCTION
+Recent developments on quantum and quantum-
+classical hybrid algorithms as well as in the continuous
+increase in computational capabilities have enabled sig-
+nificant progress in simulating interacting fermionic sys-
+tems on quantum computers[1–8]. Such simulations have
+a fundamental importance in condensed matter physics,
+quantum chemistry, and material science, since the expo-
+nentially increasing dimension of the Hilbert space has
+limited classical computations to rather small systems
+unless crude approximations are made. The algorithms
+developed for these classical computations, however, may
+benefit from an implementation on quantum computers.
+An example of such algorithms is the dynamical mean-
+field theory (DMFT)[9–12].
+Feasibility of employing
+quantum computers for DMFT has already been demon-
+strated by simulations and actual implementations on
+noisy intermediate-scale quantum (NISQ) devices[13–17].
+However, the accuracy of an implementation on quantum
+devices is limited by the current NISQ devices, which
+allow only for shallow circuits and exhibit a significant
+level of noise due to possible bit-flips or phase-flips[18–
+22]. This places a significant constraint on methods such
+as the Trotter expansion, which often requires deep cir-
+cuits with several controlled gate operations[13, 14, 23].
+To reduce circuit complexity compared to Trotter expan-
+sion approaches and improve the accuracy of the Green’s
+function obtained on quantum devices, different algo-
+rithms have been proposed. Real-time approaches such
+as the variational quantum simulation[24–28] or varia-
+tional Hamiltonian ansatz (VHA)[29–31] are based on
+∗ steffen-backes@g.ecc.u-tokyo.ac.jp
+obtaining a variational form of the time evolution opera-
+tor, while the Lehmann-representation[15, 28], quantum
+embedding[32, 33], or Krylov techniques[34, 35] obtain
+the Green’s function directly in the frequency represen-
+tation. The recently proposed Krylov variational quan-
+tum algorithm (KVQA)[34, 35] has been shown in sim-
+ulations to be a promising candidate for solving the im-
+purity problem related to DMFT on quantum hardware,
+as it requires significantly shallower circuits than other
+approaches.
+At the same time, the DMFT framework has been
+extended to include nonlocal interactions and screening
+effects[36–40] in terms of a frequency dependent effective
+interaction[41].
+This model with a dynamical interac-
+tion can be represented in terms of a Hubbard-Holstein
+impurity model, where the impurity electrons are cou-
+pled to bosonic degrees of freedom.
+These extensions
+of DMFT pose an even greater computational challenge,
+and thus can be promising candidates to benefit from uti-
+lizing quantum algorithms implemented on future quan-
+tum computers. Although the implementation of these
+methods is a necessary step to reach future applications
+to more realistic Hamiltonians, it has not yet been done
+so far.
+Here we present, to the best of our knowledge, 1) the
+first implementation of the KVQA on a current NISQ
+device, the IBM 27-qubit Quantum Falcon Processor
+Kawasaki, and 2) the first implementation of the DMFT
+impurity problem for the Hubbard-Holstein model on a
+quantum computer to obtain the impurity Green’s func-
+tion for an electronic system coupled to bosonic degrees
+of freedom. Our investigation paves the way for possi-
+ble applications of ab initio computational methods for
+strongly correlated electron systems on future quantum
+computing devices.
+arXiv:2301.01860v1  [quant-ph]  5 Jan 2023
+
+2
+The manuscript is structured as follows:
+we first
+present the model subject to our study and the formalism
+to obtain the impurity Green’s function on the quantum
+device.
+Next, we present our results obtained on the
+Kawasaki quantum processor and compare to the exact
+solution. The last section of this manuscript concludes
+with summary of the main points of our work.
+II.
+MODEL & FORMALISM
+The DMFT approximation is based on representing
+the local Green’s function of a lattice system of interact-
+ing electrons by a single-site impurity model, coupled to
+noninteracting bath degrees of freedom. The main chal-
+lenge lies in solving the interacting impurity problem,
+which despite the reduction to a local model remains a
+formidable many-body problem. Here we study a two-
+site DMFT impurity problem of the Hubbard-Holstein
+model at half-filling, consisting of one interacting impu-
+rity site coupled to a bosonic degree of freedom, and one
+non-interacting bath-site at zero energy which can ex-
+change electrons with the impurity site. The system is
+given by the Hamiltonian
+H =Un↑n↓ + V
+�
+σ
+(c†
+σdσ + d†
+σcσ) − µ(n↑ + n↓)
++ ω0b†b + λ(b† + b)(n↑ + n↓),
+(1)
+where c†, c and d†, d correspond to the impurity and
+bath electronic creation/annihilation operators, respec-
+tively, with the density nσ = c†
+σcσ for the spin σ. µ is
+the impurity potential, b†, b represent the bosonic cre-
+ation/annihilation operators, U is the local Coulomb in-
+teraction, V the hybridization amplitude, ω0 the energy
+of the bosonic mode, and λ the coupling strength to the
+fermionic degrees of freedom.
+Using the Jordan-Wigner transformation[4, 42], we
+represent the electronic creation and annihilation opera-
+tors by corresponding Pauli operators X, Y, Z via
+c†
+i = Z0 ⊗ ...Zi−1 ⊗ (Xi − iYi)/2,
+(2)
+ci = Z0 ⊗ ...Zi−1 ⊗ (Xi + iYi)/2,
+(3)
+where
+the
+index
+i
+labels
+the
+different
+flavors
+(bath/impurity site and possible spin). Introducing the
+Pauli Z operators acting on all flavors j < i ensures
+the fermionic commutation relations {ci, c†
+j} = δij. For
+the bosonic degrees of freedom, which in general involve
+an infinite number of possible excitations, one has to
+introduce a cutoff in practice. This cutoff depends on
+the number of qubits available for encoding the bosonic
+states, and the level of noise of the quantum hardware, as
+an increasing number of bosonic excitations lead to more
+complex quantum circuits.
+Different types of bosonic
+encoding have been discussed, which aim at either reduc-
+ing gate complexity or the number of qubits[3, 43, 44],
+and have been also applied to simulating systems with
+both fermionic and bosonic degrees of freedom such as
+the Holstein polaron problem[45, 46]. Here we restrict
+our simulation to one possible boson, represented by one
+additional qubit, which is an approximation justified for
+large bosonic frequency ω0.
+In this case the different
+encodings become equivalent. This leads to the following
+representation for the bosonic operators
+b†b = (I − Z)/2,
+(4)
+b† + b = X,
+(5)
+where I is the identity operator.
+Due to the bosonic
+commutation relations no additional padding with Z op-
+erators is needed.
+With these transformations we map the Hamiltonian in
+Eq.(1) on five qubits (q1, q2, q3, q4, q5), using the ordering
+(↑imp, ↓imp, ↑bath, ↓bath, B), i.e. the first two qubits rep-
+resent the impurity spin up/down component, the third
+and fourth qubits represent the bath spin up/down com-
+ponent, and the last qubit represents the boson.
+The
+resulting Hamiltonian takes the form
+H =(U/4 − µ + ω0/2)IIIII + U
+4 ZZIII
+− V
+2 (XZXII + Y ZY II + IXZXI + IY ZY I)
+− ω0
+2 IIIIZ + λIIIIX.
+(6)
+Terms with vanishing expectation value at half filling
+such as ZIIII have been dropped. To obtain the approx-
+imate ground state for this Hamiltonian we use the varia-
+tional quantum eigensolver (VQE) approach[5, 6, 47, 48]
+with a hardware-efficient ansatz[49] that exploits the
+symmetry properties of the wave function to reduce the
+number of gates as much as possible (see further expla-
+nation in the results section), as shown in Fig.1.
+To obtain the impurity Green’s function we make use of
+ibm kawasaki, an IBM Quantum System One with a 27-
+qubit Falcon R5.1 processor, and the recently proposed
+Krylov variational quantum algorithm (KVQA)[34, 35].
+Compared to other approaches such as the Trotter de-
+composition or variational algorithms that obtain the
+Green’s function on the time axis, or the Lehmann repre-
+sentation that requires the calculation of excited states,
+the KVQA has been shown in simulations to require
+significantly shallower circuits with less controlled gate
+operations.
+The procedure gives the retarded Green’s
+function via a continuous fraction expansion, which for
+the half-filled particle-hole symmetric case can be written
+as[11, 34]
+G(z) = 1
+2
+1
+z − a0 −
+b2
+1
+z−a1−
+b2
+2
+...
+,
+(7)
+for each electron and hole part. The coefficients an, bn
+
+3
+0
+π/2
+π
+3π/2
+2π
+0
+π/2
+π
+3π/2
+2π
+-2
+ 0
+ 2
+ 4
+ 6
+(b)
+θ0
+θ1
+-2
+ 0
+ 2
+ 4
+ 6
+FIG. 1. (a) The variational quantum eigensolver circuit for
+generating the ground state wave function of the Hubbard-
+Holstein DMFT impurity problem from Eq.(6). The ansatz
+wave function is parameterized by two rotation angles θ0 and
+θ1, indicated by the two red framed RY rotation gates. (b)
+The energy landscape of the expectation value of the Hamilto-
+nian operator measured on the Kawasaki quantum computer,
+depending on the parameters θ0 and θ1. The minimum around
+(θ0, θ1) ≈ (1.0, 5.7) (indicated by the green dot) corresponds
+to the approximate ground state energy E0 = −2.43, which
+is close to the exact value of E0,exact = −2.62.
+are generated by the Krylov algorithm as
+b2
+n = ⟨χn−1|H2|χn−1⟩ − a2
+n − b2
+n−1,
+(8)
+|χn⟩ = 1
+bn
+(H|χn−1⟩ − an−1|χn−1⟩ − bn−1|χn−2⟩) , (9)
+an = ⟨χn|H|χn⟩,
+(10)
+where we use |χ0⟩− = c|GS⟩ (|χ0⟩+ = c†|GS⟩) for the oc-
+cupied(unoccupied) part of the spectral function, given
+by −ImG(ω + iδ + E0)/π, with δ > 0 being a small con-
+vergence parameter. These expressions can be efficiently
+evaluated on a quantum computer if generating circuits
+for the Krylov basis states |χn⟩ can be found. Here we
+use the approach outlined in Ref.[34], which is based on
+a variational procedure to find the optimal circuit that
+generates a quantum state maximizing the overlap with
+|χn⟩.
+III.
+RESULTS
+In the first step we will discuss the results for the impu-
+rity Green’s function obtained for a fixed set of parame-
+ters, and in the second step we will demonstrate that the
+approach is robust enough to obtain a reliable DMFT
+self-consistent solution on the Bethe lattice.
+We first generate the ground state of the impurity
+problem using the VQE circuit as shown in Fig.1 (a)
+for the parameters U = 4, ω0 = 5, λ = 1.5 and V = 0.8,
+which are representative for the self-consistent DMFT so-
+lution, as discussed below. By exploiting the symmetry
+properties of the impurity problem at half filling, we re-
+duce the VQE circuit to a simplified ansatz for the ground
+state wave function of the following form
+|ψ⟩ = [sin θ0 (| ↑↓, 0⟩ + |0, ↑↓⟩) + cos θ0 (| ↑, ↓⟩ − | ↓, ↑⟩)]
+⊗ (cos θ1|0B⟩ + sin θ1|1B⟩) .
+(11)
+This form is only approximately able to represent the true
+ground state wave function but allows for a parametriza-
+tion of the ansatz wave function and all resulting quanti-
+ties such as the total energy in terms of only two parame-
+ters θ0, θ1. This enables us to resort to a two-dimensional
+scan of the energy landscape to reliably find the low-
+est energy and thus the best ground state approxima-
+tion without relying on an optimization procedure that
+is hampered by local minima and barren plateaus[50–52].
+The resulting energy potential landscape obtained on the
+Kawasaki quantum processor is shown in Fig.1 (b). We
+find an energy minimum of E0 = −2.43 and thus the
+best possible approximation to the ground state around
+(θ0, θ1) = (1.0, 5.7), which is close to the exact ground
+state energy E0,exact = −2.62 for the system with the
+same bosonic cutoff. The resulting ground state energy
+using the noise-free Qiskit simulator environment[53], ob-
+tained as E0,sim = −2.58, is in close agreement with
+the exact value, indicating that the overestimation of the
+ground state energy on Kawasaki is induced to a large
+degree by the noise of the device rather than the two-
+parameter approximation used in Eq.(11). Using error
+mitigation techniques[54] should thus be a promising way
+for further improvements on the ground state energy.
+We use the same approach of scanning the two-
+dimensional parameter space for obtaining the Krylov
+states and eventually the retarded Green’s function and
+spectral function of the impurity site.
+First, we find
+the parameters for a circuit that approximates the state
+|χ0⟩ = c↑,imp|GS⟩, which is composed of the state | ↓
+, 0, 0B⟩, |0, ↓, 0B⟩, | ↓, 0, 1B⟩, |0, ↓, 1B⟩ by using a similar
+hardware efficient ansatz parametrized by two rotation
+angles. The final parameters are given by the ones that
+maximize the overlap with |χ0⟩. The best candidate state
+that we find has an overlap of approximately 0.95 (over-
+lap of 1 signals exact representation) with |χ0⟩, but we
+point out that this error is composed both of the error
+in finding the right circuit parameters, and in the mea-
+surement of the overlap itself, i.e. even an exact repre-
+
+(a)
+Ry
+ob
+Impurity
+3元/2
+Ry
+Ry
+q1
+00
+T
+Ry
+q2
+Bath
+T
+R
+q3
+T
+Boson
+Ry
+q4
+014
+ 0
+ 0.5
+ 1
+ 1.5
+ 2
+ 2.5
+ 3
+ 3.5
+-8
+-6
+-4
+-2
+ 0
+ 2
+ 4
+ 6
+ 8
+A(ω)
+ω
+exact
+simulation
+Kawasaki
+-8
+-6
+-4
+FIG. 2. The spectral function of the impurity site obtained on
+the Kawasaki quantum computer (red solid line), compared to
+the noise-free simulation (blue line) and the exact result with
+the same boson-cutoff (black dashed line), using a broadening
+of δ = 0.1. The renormalized bonding-antibonding splitting
+and the correlation induced satellites at energies around ±3
+are well reproduced.
+A small plasmon peak is observed at
+higher energies, with the underestimated weight being a result
+of the limitation of the ground state ansatz wave function.
+sentation will not result in an overlap of 1 because of the
+measurement error on the quantum device.
+Then we progressively measure the expectation val-
+ues for an, b2
+n, and construct the circuit for next Krylov
+state |χn⟩, as outlined in Ref.[34]. Exploiting particle-
+hole symmetry at half-filling, we can obtain the unoccu-
+pied part of the spectrum from the same an, b2
+n param-
+eters obtained for the occupied part. We find that we
+can reliably obtain the Krylov states up to n = 2 on the
+quantum computer. With further iterations the signal-
+to-noise ratio for b2
+n becomes too small, often resulting in
+small negative values. Therefore, we obtain the Green’s
+function from the parameters for the first two Krylov it-
+erations, which generates up to three possible peaks in
+each the occupied and unoccupied part of the spectrum.
+In Fig.2 we show the resulting impurity Green’s func-
+tion on real frequencies, obtained on the Kawasaki quan-
+tum computer.
+The spectrum contains renormalized
+bonding-antibonding states close to the Fermi level, cor-
+relation satellites at intermediate energies and a small
+plasmonic satellite at higher energies, correctly reproduc-
+ing the qualitative features of the exact spectral function.
+Because we calculate the Krylov states up to n = 2,
+only first plasmonic satellite can be accessed, while the
+exact solution has infinitely many satellites of exponen-
+tially vanishing weight. The gap around the Fermi level
+is correctly reproduced on the quantum device, with a
+splitting of ∆ = 0.9, underestimating the exact value of
+∆exact = 1.25 by a minor degree.
+The correlation in-
+duced satellites at around ±3 are well reproduced by the
+KVQA, albeit their energetic position is overestimated
+(±3.8 compared to the exact ±2.8). Furthermore, on the
+Kawasaki quantum computer we are able to observe the
+ 0.2
+ 0.3
+ 0.4
+ 0.5
+ 0.6
+ 0.7
+ 0.8
+ 0.9
+ 1
+ 0.5
+ 0.6
+ 0.7
+ 0.8
+ 0.9
+ 1
+Z
+V
+V2
+exact NB=1
+exact (no cutoff)
+Kawasaki
+FIG. 3. The quasiparticle weight Z =
+�
+1 − Re ∂Σ
+∂ω |ω=0
+�−1 ob-
+tained from the solution of the Hubbard-Holstein DMFT im-
+purity model for different values of the hybridization strength
+V . DMFT self-consistency is obtained at V 2 = Z, i.e. at the
+intersection of the two lines indicated by the arrows. The re-
+sult on the quantum device shows a self-consistency solution
+around V = 0.84, close to the exact result with one boson
+cutoff (0.79) and no boson cutoff (0.81).
+first plasmonic satellite at higher energies in the spec-
+tral function at around ±6.5. Even when the agreement
+with the exact position and weight is not precise (ener-
+getic position overestimated by 1 and about 25% of the
+correct weight), this result demonstrates that the KVQA
+is able to resolve spectral features with relative spectral
+weight of less than one percent on current NISQ devices.
+We want to point out that such a feature would be es-
+pecially difficult to observe with approaches that obtain
+the Green’s function in real time, like the Trotter ex-
+pansion or variational methods. The Fourier transform
+from time to frequency space requires long time scales
+and fine temporal resolution, which makes it difficult to
+obtain on current NISQ devices. We thus conclude that
+the KVQA is especially well suited for obtaining subtle
+spectral features on noisy quantum devices, as possible
+errors manifest mostly in the position and weight of the
+spectral features and thus retain most of the qualitative
+aspects of the true spectrum.
+We now focus on obtaining a solution to the self-
+consistent DMFT equations for the Hubbard-Holstein
+model on the Bethe lattice.
+We set the bandwidth
+W = 4, for which the second energy moment of the
+non-interacting density of states is M2 = 1. The self-
+consistency condition is given by Gimp = Gloc, i.e. the
+impurity Green’s function has to be equal to the local lat-
+tice Green’s function. For the two-site DMFT impurity
+model at half filling, the bath energy is fixed to the Fermi
+level, and the self-consistency condition is simplified to a
+condition for the hybridization amplitude[55]
+V 2 = ZM2,
+(12)
+
+5
+where Z =
+�
+1 − Re ∂Σ
+∂ω |ω=0
+�−1 is the quasiparticle weight,
+obtained from the derivative of the self-energy Σ = G−1
+0 −
+G−1 on real frequencies. For the parameters considered
+here, the self-energy shows Fermi liquid behavior[55], and
+the quasiparticle weight Z is well defined[56]. Though,
+as both the interacting Green’s function G and non-
+interacting Green’s function G0 approach zero at ω = 0,
+the derivative of Σ(ω) strongly depends on a precise can-
+cellation of two divergent terms.
+Similar to previous
+reports[16] we find this method to be unreliable due to
+the finite noise level on the quantum device. We therefore
+obtained Z by integrating the weight of the two peaks
+closest to the Fermi level. The resulting spectral weight
+as a function of the hybridization strength V is shown
+in Fig.3. The analytical result with a one boson cutoff
+results in a self-consistent solution at V = 0.79, which
+is close to the exact result V = 0.81 without any cutoff
+on the number of bosons. Comparing to the quasiparti-
+cle weight obtained on the Kawasaki quantum computer,
+we observe a good qualitative agreement with the exact
+solution, albeit the quasiparticle weight is overestimated
+by about 10%. A self-consistent solution can be reliably
+observed at around V = 0.84, in reasonable agreement
+with the exact result.
+For comparison we also show the resulting Green’s
+function in the time domain obtained on the Kawasaki
+quantum processor from a Trotter decomposition ap-
+proach with one Trotter step in Fig.4. The Fourier trans-
+form of the KVQA Green’s function from frequency space
+to time shows a qualitative agreement with the exact re-
+sult.
+In contrast, the Trotter decomposition is signifi-
+cantly less accurate and mostly fluctuates around zero,
+i.e. it shows no clear oscillatory behavior and thus pro-
+vides almost no information about possible spectral fea-
+tures. We identified two main reasons for this: First, the
+Trotter decomposition converges very slowly for the cur-
+rent system with the number of Trotter steps, therefore,
+only one step is not sufficient to reproduce the Green’s
+function, as already the simulation without any gate
+noise is far from the exact result. Second, the Trotter
+decomposition requires more controlled gate operations
+than the KVQA which introduce significantly more noise,
+and thus reduces the quality of the obtained data. There-
+fore, going beyond one Trotter step and consequently in-
+creasing the circuit size would lead to further loss of ac-
+curacy. This result shows that the KVQA is significantly
+more robust and can provide a more accurate result for
+the Green’s function of the DMFT impurity problem for
+the same quantum computing device.
+IV.
+CONCLUSION
+We have presented an implementation of the Krylov
+variational quantum algorithm on the IBM Kawasaki 27-
+qubit quantum computer to obtain the Green’s function
+for the Hubbard-Holstein two-site impurity model, and
+demonstrated that DMFT self-consistency can be reli-
+-1.5
+-1
+-0.5
+ 0
+ 0.5
+ 1
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+Im[Gimp(t)]
+time t
+KVQA Simulation
+KVQA Kawasaki
+Trotter Simulation
+Trotter Kawasaki
+FIG. 4. The imaginary part of the impurity Green’s function
+obtained from a standard Trotter expansion simulation with
+one Trotter step (green), compared to the Trotter expansion
+performed on the Kawasaki quantum processor (blue solid
+curve), and the KVQA result (red and black curve) for V =
+1.0, λ = 1.5, ω0 = 5.
+The Trotter expansion provides a
+much poorer result than the KVQA for two main reasons: The
+expansion converges slowly with the number of Trotter steps,
+and it requires much more controlled gate operations than the
+KVQA, and hence introduces significantly more noise.
+ably obtained. This model extends the Hubbard model
+and couples the electrons with bosonic degrees of free-
+dom, and is the essential building block for impurity
+models with frequency dependent interactions such as
+in extended DMFT, GW+DMFT and further nonlocal
+extensions of DMFT which include nonlocal interactions
+and dynamical screening effects.
+We have presented a
+hardware efficient ansatz for the ground state wave func-
+tion that exploits the symmetry of the wave function
+and allows a parametrization in terms of only two pa-
+rameters. This enabled us to perform a scan of the full
+two-dimensional parameter space to reliably obtain the
+ground state and Krylov basis states instead of relying
+on numerical minimization techniques. The obtained im-
+purity Green’s function is in good qualitative agreement
+with the exact result, exhibiting all major spectral fea-
+tures of the bonding-antibonding, correlation and plas-
+monic satellites. The approach was shown to be robust
+enough to reliably obtain the self-consistent solution of
+the DMFT equations. We find that the accuracy greatly
+surpasses previously employed approaches such as the
+Trotter expansion. Our work forms the basis of future
+studies of electron-boson coupled systems and nonlocal
+extensions of DMFT on near-term quantum computers,
+which are not only important for real materials calcu-
+lations, but also are computationally very intensive on
+classical computers, and thus are promising candidates
+for harnessing the computational capabilities of future
+quantum computers.
+
+6
+FIG.
+5.
+The
+qubits
+used
+in
+this
+work
+on
+the IBM 27-qubit Quantum Falcon Processor Kawasaki.
+The
+figure
+is
+based
+on
+the
+schematic
+provided
+by
+https://quantum-computing.ibm.com/services/resources.
+ACKNOWLEDGMENTS
+We thank E. L¨otstedt, T. Nishi, and K. Yamanouchi
+for fruitful discussions and support in utilizing the
+Kawasaki quantum computer at the University of Tokyo.
+This work is partly supported by the UTokyo Quantum
+Initiative.
+Appendix A: Implementation details
+For this work we employed the IBM Qiskit environ-
+ment v.
+0.39.2[53] to implement the quantum circuits
+and measurements on the IBM 27-qubit Quantum Falcon
+Processor Kawasaki. For all measurements we employed
+the Sampler primitive, and used the maximum number
+of shots possible N = 32000. For the layout we chose the
+qubits that showed the lowest readout assignment error
+and were connected with the smallest CNOT error as
+much as possible, as shown in Fig.5.
+To find the Krylov states |χn⟩ in the KVQA we em-
+ployed the variational method as outlined in Ref.[34]: To
+obtain the parameter set {θn} that generates the state
+|χn⟩ = U(θn)|0⟩ via the set of unitary gate operations
+U(θn) we minimize the three functions
+ϵn0({θ}) =
+�|⟨0|U †(θ)HU(θn−1)|0⟩|
+|bn|
+− 1
+�2
+,
+(A1)
+ϵn1({θ}) = |⟨0|U †(θ)U(θn−1)|0⟩|2,
+(A2)
+ϵn2({θ}) = |⟨0|U †(θ)U(θn−2)|0⟩|2,
+(A3)
+which at the minimum ϵni
+=
+0 ∀i satisfy that
+⟨χn|H|χn−1⟩ = bn, ⟨χn|χn−1⟩ = 0 and ⟨χn|χn−2⟩ = 0.
+Specifically we made use of the Trotter-like expansion
+discussed in the supplementary material I of Ref.[34] to
+evaluate the cost function ϵn0, which circumvents the in-
+troduction of another controlled ancilla qubit. We used
+10 steps between t = 0.01...0.3 to linearly extrapolate the
+value to t = 0. In practice we minimize the sum of these
+three functions, which is the quantity shown in Fig.6 (c)
+and (d).
+Reducing the ansatz wave function to a circuit
+0
+π/2
+π
+3π/2
+2π
+0
+π/2
+π
+3π/2
+2π
+-2
+ 0
+ 2
+ 4
+ 6
+(a)
+θ0
+θ1
+-2
+ 0
+ 2
+ 4
+ 6
+0
+π/2
+π
+3π/2
+2π
+0
+π/2
+π
+3π/2
+2π
+ 0
+ 0.2
+ 0.4
+ 0.6
+ 0.8
+ 1
+(b)
+θ0
+θ1
+ 0
+ 0.2
+ 0.4
+ 0.6
+ 0.8
+ 1
+0
+π/2
+π
+3π/2
+2π
+0
+π/2
+π
+3π/2
+2π
+ 0
+ 0.5
+ 1
+ 1.5
+ 2
+ 2.5
+(c)
+θ0
+θ1
+ 0
+ 0.5
+ 1
+ 1.5
+ 2
+ 2.5
+0
+π/2
+π
+3π/2
+2π
+0
+π/2
+π
+3π/2
+2π
+ 0
+ 1
+ 2
+ 3
+(d)
+θ0
+θ1
+ 0
+ 1
+ 2
+ 3
+FIG. 6.
+The optimization surfaces for the circuit shown in
+Fig.1 to find (a) the minimal energy to determine the ground
+state energy E0 , (b) the maximum overlap to determine the
+first Krylov vector |χ0⟩, (c) and (d) the minimum of the error
+function �
+i ϵni(θ) to find the next Krylov vectors |χ1⟩ and
+|χ2⟩, respectively (see Eq.(A1)-(A3) ).
+parametrized by two parameters as shown in Eq.(11) al-
+lowed us to use a two-dimensional scan of the param-
+eter space to determine the approximate ground state
+and Krylov vectors to obtain the impurity Green’s func-
+tion. In Fig.6 we show the corresponding optimization
+surfaces obtained on the Kawasaki quantum processor.
+The energy surface in Fig.6 (a) showed a clear minimum
+indicating a unique solution for the approximate ground
+state.
+Also the obtained overlap to represent the first
+Krylov state |χ0⟩ = c|GS⟩ showed a clear maximum with
+an overlap of more than 0.9 (Fig.6 (b) ). The optimiza-
+tion surface in Fig.6 (c) for the remaining Krylov states
+showed two main local minima, corresponding to two pos-
+sible Krylov vectors. Choosing either local minima leads
+to the other minima becoming the only global minima in
+the next iteration, as can be seen in Fig.6 (d). The precise
+location of the minima was noticeably affected by noise,
+but we found that the resulting spectral function did not
+strongly depend on the exact choice of the parameters.
+While the peak positions and weights of mainly the two
+outermost peaks (correlation induced and plasmon satel-
+lite) at energies above ±3 were affected to a small degree,
+the qualitative features of the spectral function stayed the
+same.
+Appendix B: Trotterization and variational approach
+As discussed in the main text, we found that the
+Trotter expansion or the variational Hamiltonian ansatz
+(VHA)[29–31], using McLachlan’s variational principle
+as detailed in Ref. [31], for the two-site DMFT impu-
+rity problem show a very poor agreement with the exact
+result, and converge very slowly with increasing the num-
+ber of Trotterization steps. To demonstrate this, we show
+in Fig.7 the resulting impurity Green’s function obtained
+from the Trotter expansion and VHA obtained from a
+classical simulation, for NT = 1 and NT = 2 Trotteriza-
+tion steps. The agreement with the exact result is very
+
+Qubit:
+Readout assignment error
+Avg 1.134e-2
+min 3.600e-3
+max 2.490e-2
+Connection:
+CNOT error
+Avg 7.162e-1
+20
+min 3.370e-3
+max 1.000e+07
+-1.5
+-1
+-0.5
+ 0
+ 0.5
+ 1
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+(a)
+NT=1
+Im[Gimp(t)]
+time t
+exact
+Trotter
+VHA
+-1.5
+-1
+-0.5
+ 0
+ 0.5
+ 1
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+(b)
+NT=2
+Im[Gimp(t)]
+time t
+exact
+Trotter
+VHA
+FIG. 7.
+The imaginary part of the impurity Green’s function as a function of time obtained from a Trotter expansion (red
+line) and variational Hamiltonian ansatz (VHA, blue line) on a classical computer, compared to the exact result for the same
+boson cutoff NB = 1 at V = 1.0 (other parameters as in the main text). (a) and (b) show the results for NT = 1 and NT = 2
+Trotterization steps, respectively. Both results are significantly less accurate than the KVQA one shown in Fig.4. Increasing
+the number of Trotter steps to NT = 2 results in improvement at small times for the Trotter expansion, but actually worsens
+the agreement with the exact solution for VHA.
+poor, and quickly deviates even at small time scales. In-
+creasing the number of Trotter steps improves the result
+for the Trotter expansion at small times, but we have
+found that at least NT ∼ 8−10 steps are needed in order
+to obtain a reasonable agreement up to Tmax = 10. Such
+a large circuit is currently not feasible for present NISQ
+devices. In contrast to that, the VHA obtains an even
+worse result when increasing the Trotterization steps to
+NT = 2. We found that the result strongly depends on
+the operator ordering in the expansion of the time evo-
+lution operator
+UV HA =
+NT
+�
+i=1
+� NH
+�
+m=1
+eiθm(t)Pm
+�
+,
+(B1)
+where Pm are the Pauli operators of the Hamiltonian
+after the Jordan-Wigner transformation, with the total
+number of operator terms NH. For most orderings the
+ansatz did not converge to the correct result when in-
+creasing the number of Trotter steps.
+[1] D. S. Abrams and S. Lloyd, Simulation of many-body
+fermi systems on a universal quantum computer, Phys.
+Rev. Lett. 79, 2586 (1997).
+[2] G. Ortiz, J. E. Gubernatis, E. Knill, and R. Laflamme,
+Quantum algorithms for fermionic simulations, Phys.
+Rev. A 64, 022319 (2001).
+[3] R. D. Somma, G. Ortiz, E. H. Knill, and J. Gubernatis,
+Quantum simulations of physics problems, in Quan-
+tum Information and Computation, Vol. 5105, edited by
+E. Donkor, A. R. Pirich, and H. E. Brandt, International
+Society for Optics and Photonics (SPIE, 2003) pp. 96 –
+103.
+[4] J. D. Whitfield, J. Biamonte, and A. Aspuru-Guzik,
+Simulation of electronic structure hamiltonians using
+quantum computers, Molecular Physics 109, 735 (2011),
+https://doi.org/10.1080/00268976.2011.552441.
+[5] A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q.
+Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien,
+A variational eigenvalue solver on a photonic quantum
+processor, Nature Communications 5, 4213 (2014).
+[6] J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-
+Guzik,
+The
+theory
+of
+variational
+hybrid
+quantum-
+classical algorithms, New Journal of Physics 18, 023023
+(2016).
+[7] F. A. ope Kunal Arya, R. Babbush, D. Bacon, J. C.
+Bardin, R. Barends, S. Boixo, M. Broughton, B. B.
+Buckley, D. A. Buell, B. Burkett, N. Bushnell, Y. Chen,
+Z. Chen, B. Chiaro, R. Collins, W. Courtney, S. De-
+mura, A. Dunsworth, E. Farhi, A. Fowler, B. Foxen,
+C. Gidney, M. Giustina, R. Graff, S. Habegger, M. P.
+Harrigan, A. Ho, S. Hong, T. Huang, W. J. Huggins,
+L. Ioffe, S. V. Isakov, E. Jeffrey, Z. Jiang, C. Jones,
+D. Kafri, K. Kechedzhi, J. Kelly, S. Kim, P. V. Klimov,
+A. Korotkov, F. Kostritsa, D. Landhuis, P. Laptev,
+M. Lindmark, E. Lucero, O. Martin, J. M. Martinis, J. R.
+McClean, M. McEwen, A. Megrant, X. Mi, M. Mohseni,
+W. Mruczkiewicz, J. Mutus, O. Naaman, M. Neeley,
+C. Neill, H. Neven, M. Y. Niu, T. E. O’Brien, E. Ostby,
+A. Petukhov, H. Putterman, C. Quintana, P. Roushan,
+N. C. Rubin, D. Sank, K. J. Satzinger, V. Smelyanskiy,
+D. Strain, K. J. Sung, M. Szalay, T. Y. Takeshita,
+
+8
+A. Vainsencher, T. White, N. Wiebe, Z. J. Yao, P. Yeh,
+and A. Zalcman, Hartree-fock on a superconducting
+qubit quantum computer, Science 369, 1084 (2020),
+https://www.science.org/doi/pdf/10.1126/science.abb9811.
+[8] W. J. Huggins, B. A. O’Gorman, N. C. Rubin, D. R.
+Reichman, R. Babbush, and J. Lee, Unbiasing fermionic
+quantum monte carlo with a quantum computer, Nature
+603, 416 (2022).
+[9] W. Metzner and D. Vollhardt, Correlated lattice fermions
+in d = ∞ dimensions, Phys. Rev. Lett. 62, 324 (1989).
+[10] A. Georges and G. Kotliar, Hubbard model in infinite
+dimensions, Phys. Rev. B 45, 6479 (1992).
+[11] A. Georges, G. Kotliar, W. Krauth, and M. J. Rozen-
+berg, Dynamical mean-field theory of strongly correlated
+fermion systems and the limit of infinite dimensions, Rev.
+Mod. Phys. 68, 13 (1996).
+[12] D. Vollhardt, K. Byczuk, and M. Kollar, Dynamical
+mean-field theory, in Strongly Correlated Systems: The-
+oretical Methods, edited by A. Avella and F. Mancini
+(Springer Berlin Heidelberg, Berlin, Heidelberg, 2012)
+pp. 203–236.
+[13] B. Bauer, D. Wecker, A. J. Millis, M. B. Hastings, and
+M. Troyer, Hybrid quantum-classical approach to corre-
+lated materials, Phys. Rev. X 6, 031045 (2016).
+[14] J. M. Kreula, S. R. Clark, and D. Jaksch, Non-linear
+quantum-classical scheme to simulate non-equilibrium
+strongly correlated fermionic many-body dynamics, Sci-
+entific Reports 6, 32940 (2016).
+[15] I. Rungger, N. Fitzpatrick, H. Chen, C. H. Alderete,
+H. Apel, A. Cowtan, A. Patterson, D. M. Ramo, Y. Zhu,
+N. H. Nguyen, E. Grant, S. Chretien, L. Wossnig, N. M.
+Linke, and R. Duncan, Dynamical mean field theory al-
+gorithm and experiment on quantum computers (2019).
+[16] T. Keen,
+T. Maier,
+S. Johnston, and P. Lougov-
+ski, Quantum-classical simulation of two-site dynamical
+mean-field theory on noisy quantum hardware, Quantum
+Science and Technology 5, 035001 (2020).
+[17] B. Jaderberg, A. Agarwal, K. Leonhardt, M. Kiffner, and
+D. Jaksch, Minimum hardware requirements for hybrid
+quantum–classical dmft, Quantum Science and Technol-
+ogy 5, 034015 (2020).
+[18] P. W. Shor, Scheme for reducing decoherence in quantum
+computer memory, Phys. Rev. A 52, R2493 (1995).
+[19] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N.
+Cleland, Surface codes:
+Towards practical large-scale
+quantum computation, Phys. Rev. A 86, 032324 (2012).
+[20] J. Preskill, Quantum Computing in the NISQ era and
+beyond, Quantum 2, 79 (2018).
+[21] W. Huang, C. H. Yang, K. W. Chan, T. Tanttu,
+B. Hensen, R. C. C. Leon, M. A. Fogarty, J. C. C. Hwang,
+F. E. Hudson, K. M. Itoh, A. Morello, A. Laucht, and
+A. S. Dzurak, Fidelity benchmarks for two-qubit gates in
+silicon, Nature 569, 532 (2019).
+[22] e. a. Chen, Zijun, Exponential suppression of bit or
+phase errors with cyclic error correction, Nature 595, 383
+(2021).
+[23] A. Chiesa, F. Tacchino, M. Grossi, P. Santini, I. Tav-
+ernelli, D. Gerace, and S. Carretta, Quantum hardware
+simulating four-dimensional inelastic neutron scattering,
+Nature Physics 15, 455 (2019).
+[24] Y. Li and S. C. Benjamin, Efficient variational quantum
+simulator incorporating active error minimization, Phys.
+Rev. X 7, 021050 (2017).
+[25] X. Yuan, S. Endo, Q. Zhao, Y. Li, and S. C. Benjamin,
+Theory of variational quantum simulation, Quantum 3,
+191 (2019).
+[26] S. McArdle, T. Jones, S. Endo, Y. Li, S. C. Benjamin,
+and X. Yuan, Variational ansatz-based quantum simula-
+tion of imaginary time evolution, npj Quantum Informa-
+tion 5, 75 (2019).
+[27] K. Heya, K. M. Nakanishi, K. Mitarai, and K. Fujii, Sub-
+space variational quantum simulator (2019).
+[28] S. Endo, I. Kurata, and Y. O. Nakagawa, Calculation of
+the green’s function on near-term quantum computers,
+Phys. Rev. Research 2, 033281 (2020).
+[29] D. Wecker, M. B. Hastings, and M. Troyer, Progress to-
+wards practical quantum variational algorithms, Phys.
+Rev. A 92, 042303 (2015).
+[30] J.-M.
+Reiner,
+F.
+Wilhelm-Mauch,
+G.
+Sch¨on,
+and
+M. Marthaler, Finding the ground state of the hubbard
+model by variational methods on a quantum computer
+with gate errors, Quantum Science and Technology 4,
+035005 (2019).
+[31] F. Libbi, J. Rizzo, F. Tacchino, N. Marzari, and I. Tav-
+ernelli, Effective calculation of the green’s function in
+the time domain on near-term quantum processors, Phys.
+Rev. Research 4, 043038 (2022).
+[32] C. Lupo, F. Jamet, W. H. T. Tse, I. Rungger, and C. We-
+ber, Maximally localized dynamical quantum embedding
+for solving many-body correlated systems, Nature Com-
+putational Science 1, 410 (2021).
+[33] C. Vorwerk, N. Sheng, M. Govoni, B. Huang, and
+G. Galli, Quantum embedding theories to simulate con-
+densed systems on quantum computers, Nature Compu-
+tational Science 2, 424 (2022).
+[34] F. Jamet, A. Agarwal, C. Lupo, D. E. Browne, C. Weber,
+and I. Rungger, Krylov variational quantum algorithm
+for first principles materials simulations (2021).
+[35] F. Jamet, A. Agarwal, and I. Rungger, Quantum sub-
+space expansion algorithm for green’s functions (2022).
+[36] P. Sun and G. Kotliar, Extended dynamical mean-field
+theory and GW method, Phys. Rev. B 66, 085120 (2002).
+[37] S. Biermann, F. Aryasetiawan, and A. Georges, First-
+principles approach to the electronic structure of strongly
+correlated systems: Combining the GW approximation
+and dynamical mean-field theory, Phys. Rev. Lett. 90,
+086402 (2003).
+[38] S. Biermann, Dynamical screening effects in correlated
+electron materials - a progress report on combined many-
+body perturbation and dynamical mean field theory:
+GW+DMFT, Journal of Physics: Condensed Matter 26,
+173202 (2014).
+[39] F. Nilsson, L. Boehnke, P. Werner, and F. Aryasetiawan,
+Multitier self-consistent gw + EDMFT, Phys. Rev. Ma-
+terials 1, 043803 (2017).
+[40] A. Rubtsov, M. Katsnelson, and A. Lichtenstein, Dual
+boson approach to collective excitations in correlated
+fermionic systems, Annals of Physics 327, 1320 (2012).
+[41] F. Aryasetiawan, M. Imada, A. Georges, G. Kotliar,
+S.
+Biermann,
+and
+A.
+I.
+Lichtenstein,
+Frequency-
+dependent local interactions and low-energy effective
+models from electronic structure calculations, Phys. Rev.
+B 70, 195104 (2004).
+[42] P.
+Jordan
+and
+E.
+Wigner,
+¨Uber
+das
+paulische
+¨Aquivalenzverbot, Zeitschrift f¨ur Physik 47, 631 (1928).
+
+9
+[43] L. Veis, J. Viˇsˇn´ak, H. Nishizawa, H. Nakai, and J. Pit-
+tner, Quantum chemistry beyond born–oppenheimer
+approximation
+on
+a
+quantum
+computer:
+A
+simu-
+lated
+phase
+estimation
+study,
+International
+Jour-
+nal
+of
+Quantum
+Chemistry
+116,
+1328
+(2016),
+https://onlinelibrary.wiley.com/doi/pdf/10.1002/qua.25176.
+[44] S. McArdle, A. Mayorov, X. Shan, S. Benjamin, and
+X. Yuan, Digital quantum simulation of molecular vi-
+brations, Chem. Sci. 10, 5725 (2019).
+[45] A.
+Macridin,
+P.
+Spentzouris,
+J.
+Amundson,
+and
+R. Harnik, Digital quantum computation of fermion-
+boson interacting systems, Phys. Rev. A 98, 042312
+(2018).
+[46] A.
+Macridin,
+P.
+Spentzouris,
+J.
+Amundson,
+and
+R. Harnik, Electron-phonon systems on a universal quan-
+tum computer, Phys. Rev. Lett. 121, 110504 (2018).
+[47] M. H. Yung, J. Casanova, A. Mezzacapo, J. McClean,
+L. Lamata, A. Aspuru-Guzik, and E. Solano, From tran-
+sistor to trapped-ion computers for quantum chemistry,
+Scientific Reports 4, 3589 (2014).
+[48] J. Tilly, H. Chen, S. Cao, D. Picozzi, K. Setia, Y. Li,
+E. Grant, L. Wossnig, I. Rungger, G. H. Booth, and
+J. Tennyson, The variational quantum eigensolver: A re-
+view of methods and best practices, Physics Reports 986,
+1 (2022), the Variational Quantum Eigensolver: a review
+of methods and best practices.
+[49] A. Kandala, A. Mezzacapo, K. Temme, M. Takita,
+M. Brink, J. M. Chow, and J. M. Gambetta, Hardware-
+efficient
+variational
+quantum
+eigensolver
+for
+small
+molecules and quantum magnets, Nature 549, 242
+(2017).
+[50] J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Bab-
+bush, and H. Neven, Barren plateaus in quantum neural
+network training landscapes, Nature Communications 9,
+4812 (2018).
+[51] S. Wang, E. Fontana, M. Cerezo, K. Sharma, A. Sone,
+L. Cincio, and P. J. Coles, Noise-induced barren plateaus
+in variational quantum algorithms, Nature Communica-
+tions 12, 6961 (2021).
+[52] M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin,
+S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan,
+L. Cincio, and P. J. Coles, Variational quantum algo-
+rithms, Nature Reviews Physics 3, 625 (2021).
+[53] A. tA-v et al., Qiskit: An open-source framework for
+quantum computing (2021).
+[54] P. D. Nation, H. Kang, N. Sundaresan, and J. M.
+Gambetta, Scalable mitigation of measurement errors on
+quantum computers, PRX Quantum 2, 040326 (2021).
+[55] M. Potthoff, Two-site dynamical mean-field theory, Phys.
+Rev. B 64, 165114 (2001).
+[56] The gap in A(ω) in Fig.2 results from the bonding-
+antibonding splitting of the two-site model and is not
+due to a singular self-energy.
+
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+page_content=' Japan  2Center for Emergent Matter Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' RIKEN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Wako,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Saitama 351-0198,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Japan and  3CPHT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' ´Ecole polytechnique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Institut Polytechnique de Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' 91120 Palaiseau,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' France  (Dated: January 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' 2023)  Recent developments in quantum hardware and quantum algorithms have made it possible to  utilize the capabilities of current noisy intermediate-scale quantum devices for addressing problems  in quantum chemistry and condensed matter physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Here we report a demonstration of solving  the dynamical mean-field theory (DMFT) impurity problem for the Hubbard-Holstein model on  the IBM 27-qubit Quantum Falcon Processor Kawasaki, including self-consistency of the DMFT  equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' This opens up the possibility to investigate strongly correlated electron systems coupled  to bosonic degrees of freedom and impurity problems with frequency-dependent interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The  problem involves both fermionic and bosonic degrees of freedom to be encoded on the quantum  device, which we solve using a recently proposed Krylov variational quantum algorithm to obtain  the impurity Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' We find the resulting spectral function to be in good agreement with  the exact result, exhibiting both correlation and plasmonic satellites and significantly surpassing  the accuracy of standard Trotter-expansion approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Our results provide an essential building  block to study electronic correlations and plasmonic excitations on future quantum computers with  modern ab initio techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  INTRODUCTION  Recent developments on quantum and quantum-  classical hybrid algorithms as well as in the continuous  increase in computational capabilities have enabled sig-  nificant progress in simulating interacting fermionic sys-  tems on quantum computers[1–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Such simulations have  a fundamental importance in condensed matter physics,  quantum chemistry, and material science, since the expo-  nentially increasing dimension of the Hilbert space has  limited classical computations to rather small systems  unless crude approximations are made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The algorithms  developed for these classical computations, however, may  benefit from an implementation on quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  An example of such algorithms is the dynamical mean-  field theory (DMFT)[9–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Feasibility of employing  quantum computers for DMFT has already been demon-  strated by simulations and actual implementations on  noisy intermediate-scale quantum (NISQ) devices[13–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  However, the accuracy of an implementation on quantum  devices is limited by the current NISQ devices, which  allow only for shallow circuits and exhibit a significant  level of noise due to possible bit-flips or phase-flips[18–  22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' This places a significant constraint on methods such  as the Trotter expansion, which often requires deep cir-  cuits with several controlled gate operations[13, 14, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  To reduce circuit complexity compared to Trotter expan-  sion approaches and improve the accuracy of the Green’s  function obtained on quantum devices, different algo-  rithms have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Real-time approaches such  as the variational quantum simulation[24–28] or varia-  tional Hamiltonian ansatz (VHA)[29–31] are based on  ∗ steffen-backes@g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='ecc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='u-tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='jp  obtaining a variational form of the time evolution opera-  tor, while the Lehmann-representation[15, 28], quantum  embedding[32, 33], or Krylov techniques[34, 35] obtain  the Green’s function directly in the frequency represen-  tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The recently proposed Krylov variational quan-  tum algorithm (KVQA)[34, 35] has been shown in sim-  ulations to be a promising candidate for solving the im-  purity problem related to DMFT on quantum hardware,  as it requires significantly shallower circuits than other  approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  At the same time, the DMFT framework has been  extended to include nonlocal interactions and screening  effects[36–40] in terms of a frequency dependent effective  interaction[41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  This model with a dynamical interac-  tion can be represented in terms of a Hubbard-Holstein  impurity model, where the impurity electrons are cou-  pled to bosonic degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  These extensions  of DMFT pose an even greater computational challenge,  and thus can be promising candidates to benefit from uti-  lizing quantum algorithms implemented on future quan-  tum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Although the implementation of these  methods is a necessary step to reach future applications  to more realistic Hamiltonians, it has not yet been done  so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Here we present, to the best of our knowledge, 1) the  first implementation of the KVQA on a current NISQ  device, the IBM 27-qubit Quantum Falcon Processor  Kawasaki, and 2) the first implementation of the DMFT  impurity problem for the Hubbard-Holstein model on a  quantum computer to obtain the impurity Green’s func-  tion for an electronic system coupled to bosonic degrees  of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Our investigation paves the way for possi-  ble applications of ab initio computational methods for  strongly correlated electron systems on future quantum  computing devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='01860v1  [quant-ph]  5 Jan 2023  2  The manuscript is structured as follows:  we first  present the model subject to our study and the formalism  to obtain the impurity Green’s function on the quantum  device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Next, we present our results obtained on the  Kawasaki quantum processor and compare to the exact  solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The last section of this manuscript concludes  with summary of the main points of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  MODEL & FORMALISM  The DMFT approximation is based on representing  the local Green’s function of a lattice system of interact-  ing electrons by a single-site impurity model, coupled to  noninteracting bath degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The main chal-  lenge lies in solving the interacting impurity problem,  which despite the reduction to a local model remains a  formidable many-body problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Here we study a two-  site DMFT impurity problem of the Hubbard-Holstein  model at half-filling, consisting of one interacting impu-  rity site coupled to a bosonic degree of freedom, and one  non-interacting bath-site at zero energy which can ex-  change electrons with the impurity site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The system is  given by the Hamiltonian  H =Un↑n↓ + V  �  σ  (c†  σdσ + d†  σcσ) − µ(n↑ + n↓)  + ω0b†b + λ(b† + b)(n↑ + n↓),  (1)  where c†, c and d†, d correspond to the impurity and  bath electronic creation/annihilation operators, respec-  tively, with the density nσ = c†  σcσ for the spin σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' µ is  the impurity potential, b†, b represent the bosonic cre-  ation/annihilation operators, U is the local Coulomb in-  teraction, V the hybridization amplitude, ω0 the energy  of the bosonic mode, and λ the coupling strength to the  fermionic degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Using the Jordan-Wigner transformation[4, 42], we  represent the electronic creation and annihilation opera-  tors by corresponding Pauli operators X, Y, Z via  c†  i = Z0 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='Zi−1 ⊗ (Xi − iYi)/2,  (2)  ci = Z0 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='Zi−1 ⊗ (Xi + iYi)/2,  (3)  where  the  index  i  labels  the  different  flavors  (bath/impurity site and possible spin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Introducing the  Pauli Z operators acting on all flavors j < i ensures  the fermionic commutation relations {ci, c†  j} = δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' For  the bosonic degrees of freedom, which in general involve  an infinite number of possible excitations, one has to  introduce a cutoff in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' This cutoff depends on  the number of qubits available for encoding the bosonic  states, and the level of noise of the quantum hardware, as  an increasing number of bosonic excitations lead to more  complex quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Different types of bosonic  encoding have been discussed, which aim at either reduc-  ing gate complexity or the number of qubits[3, 43, 44],  and have been also applied to simulating systems with  both fermionic and bosonic degrees of freedom such as  the Holstein polaron problem[45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Here we restrict  our simulation to one possible boson, represented by one  additional qubit, which is an approximation justified for  large bosonic frequency ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  In this case the different  encodings become equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' This leads to the following  representation for the bosonic operators  b†b = (I − Z)/2,  (4)  b† + b = X,  (5)  where I is the identity operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Due to the bosonic  commutation relations no additional padding with Z op-  erators is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  With these transformations we map the Hamiltonian in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' (1) on five qubits (q1, q2, q3, q4, q5), using the ordering  (↑imp, ↓imp, ↑bath, ↓bath, B), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' the first two qubits rep-  resent the impurity spin up/down component, the third  and fourth qubits represent the bath spin up/down com-  ponent, and the last qubit represents the boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  The  resulting Hamiltonian takes the form  H =(U/4 − µ + ω0/2)IIIII + U  4 ZZIII  − V  2 (XZXII + Y ZY II + IXZXI + IY ZY I)  − ω0  2 IIIIZ + λIIIIX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  (6)  Terms with vanishing expectation value at half filling  such as ZIIII have been dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' To obtain the approx-  imate ground state for this Hamiltonian we use the varia-  tional quantum eigensolver (VQE) approach[5, 6, 47, 48]  with a hardware-efficient ansatz[49] that exploits the  symmetry properties of the wave function to reduce the  number of gates as much as possible (see further expla-  nation in the results section), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  To obtain the impurity Green’s function we make use of  ibm kawasaki, an IBM Quantum System One with a 27-  qubit Falcon R5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='1 processor, and the recently proposed  Krylov variational quantum algorithm (KVQA)[34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Compared to other approaches such as the Trotter de-  composition or variational algorithms that obtain the  Green’s function on the time axis, or the Lehmann repre-  sentation that requires the calculation of excited states,  the KVQA has been shown in simulations to require  significantly shallower circuits with less controlled gate  operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  The procedure gives the retarded Green’s  function via a continuous fraction expansion, which for  the half-filled particle-hole symmetric case can be written  as[11, 34]  G(z) = 1  2  1  z − a0 −  b2  1  z−a1−  b2  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  ,  (7)  for each electron and hole part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The coefficients an, bn  3  0  π/2  π  3π/2  2π  0  π/2  π  3π/2  2π  2  0  2  4  6  (b)  θ0  θ1  2  0  2  4  6  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' (a) The variational quantum eigensolver circuit for  generating the ground state wave function of the Hubbard-  Holstein DMFT impurity problem from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The ansatz  wave function is parameterized by two rotation angles θ0 and  θ1, indicated by the two red framed RY rotation gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' (b)  The energy landscape of the expectation value of the Hamilto-  nian operator measured on the Kawasaki quantum computer,  depending on the parameters θ0 and θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The minimum around  (θ0, θ1) ≈ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='0, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='7) (indicated by the green dot) corresponds  to the approximate ground state energy E0 = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='43, which  is close to the exact value of E0,exact = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  are generated by the Krylov algorithm as  b2  n = ⟨χn−1|H2|χn−1⟩ − a2  n − b2  n−1,  (8)  |χn⟩ = 1  bn  (H|χn−1⟩ − an−1|χn−1⟩ − bn−1|χn−2⟩) , (9)  an = ⟨χn|H|χn⟩,  (10)  where we use |χ0⟩− = c|GS⟩ (|χ0⟩+ = c†|GS⟩) for the oc-  cupied(unoccupied) part of the spectral function, given  by −ImG(ω + iδ + E0)/π, with δ > 0 being a small con-  vergence parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' These expressions can be efficiently  evaluated on a quantum computer if generating circuits  for the Krylov basis states |χn⟩ can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Here we  use the approach outlined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' [34], which is based on  a variational procedure to find the optimal circuit that  generates a quantum state maximizing the overlap with  |χn⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  RESULTS  In the first step we will discuss the results for the impu-  rity Green’s function obtained for a fixed set of parame-  ters, and in the second step we will demonstrate that the  approach is robust enough to obtain a reliable DMFT  self-consistent solution on the Bethe lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  We first generate the ground state of the impurity  problem using the VQE circuit as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='1 (a)  for the parameters U = 4, ω0 = 5, λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='5 and V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='8,  which are representative for the self-consistent DMFT so-  lution, as discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' By exploiting the symmetry  properties of the impurity problem at half filling, we re-  duce the VQE circuit to a simplified ansatz for the ground  state wave function of the following form  |ψ⟩ = [sin θ0 (| ↑↓, 0⟩ + |0, ↑↓⟩) + cos θ0 (| ↑, ↓⟩ − | ↓, ↑⟩)]  ⊗ (cos θ1|0B⟩ + sin θ1|1B⟩) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  (11)  This form is only approximately able to represent the true  ground state wave function but allows for a parametriza-  tion of the ansatz wave function and all resulting quanti-  ties such as the total energy in terms of only two parame-  ters θ0, θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' This enables us to resort to a two-dimensional  scan of the energy landscape to reliably find the low-  est energy and thus the best ground state approxima-  tion without relying on an optimization procedure that  is hampered by local minima and barren plateaus[50–52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  The resulting energy potential landscape obtained on the  Kawasaki quantum processor is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='1 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' We  find an energy minimum of E0 = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='43 and thus the  best possible approximation to the ground state around  (θ0, θ1) = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='0, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='7), which is close to the exact ground  state energy E0,exact = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='62 for the system with the  same bosonic cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The resulting ground state energy  using the noise-free Qiskit simulator environment[53], ob-  tained as E0,sim = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='58, is in close agreement with  the exact value, indicating that the overestimation of the  ground state energy on Kawasaki is induced to a large  degree by the noise of the device rather than the two-  parameter approximation used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='(11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Using error  mitigation techniques[54] should thus be a promising way  for further improvements on the ground state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  We use the same approach of scanning the two-  dimensional parameter space for obtaining the Krylov  states and eventually the retarded Green’s function and  spectral function of the impurity site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  First, we find  the parameters for a circuit that approximates the state  |χ0⟩ = c↑,imp|GS⟩, which is composed of the state | ↓  , 0, 0B⟩, |0, ↓, 0B⟩, | ↓, 0, 1B⟩, |0, ↓, 1B⟩ by using a similar  hardware efficient ansatz parametrized by two rotation  angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The final parameters are given by the ones that  maximize the overlap with |χ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The best candidate state  that we find has an overlap of approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='95 (over-  lap of 1 signals exact representation) with |χ0⟩, but we  point out that this error is composed both of the error  in finding the right circuit parameters, and in the mea-  surement of the overlap itself, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' even an exact repre-  (a)  Ry  ob  Impurity  3元/2  Ry  Ry  q1  00  T  Ry  q2  Bath  T  R  q3  T  Boson  Ry  q4  014  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='5  8  6  4  2  0  2  4  6  8  A(ω)  ω  exact  simulation  Kawasaki  8  6  4  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The spectral function of the impurity site obtained on  the Kawasaki quantum computer (red solid line), compared to  the noise-free simulation (blue line) and the exact result with  the same boson-cutoff (black dashed line), using a broadening  of δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The renormalized bonding-antibonding splitting  and the correlation induced satellites at energies around ±3  are well reproduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  A small plasmon peak is observed at  higher energies, with the underestimated weight being a result  of the limitation of the ground state ansatz wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  sentation will not result in an overlap of 1 because of the  measurement error on the quantum device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Then we progressively measure the expectation val-  ues for an, b2  n, and construct the circuit for next Krylov  state |χn⟩, as outlined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='[34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Exploiting particle-  hole symmetry at half-filling, we can obtain the unoccu-  pied part of the spectrum from the same an, b2  n param-  eters obtained for the occupied part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' We find that we  can reliably obtain the Krylov states up to n = 2 on the  quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' With further iterations the signal-  to-noise ratio for b2  n becomes too small, often resulting in  small negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Therefore, we obtain the Green’s  function from the parameters for the first two Krylov it-  erations, which generates up to three possible peaks in  each the occupied and unoccupied part of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='2 we show the resulting impurity Green’s func-  tion on real frequencies, obtained on the Kawasaki quan-  tum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  The spectrum contains renormalized  bonding-antibonding states close to the Fermi level, cor-  relation satellites at intermediate energies and a small  plasmonic satellite at higher energies, correctly reproduc-  ing the qualitative features of the exact spectral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Because we calculate the Krylov states up to n = 2,  only first plasmonic satellite can be accessed, while the  exact solution has infinitely many satellites of exponen-  tially vanishing weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The gap around the Fermi level  is correctly reproduced on the quantum device, with a  splitting of ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='9, underestimating the exact value of  ∆exact = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='25 by a minor degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  The correlation in-  duced satellites at around ±3 are well reproduced by the  KVQA, albeit their energetic position is overestimated  (±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='8 compared to the exact ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Furthermore, on the  Kawasaki quantum computer we are able to observe the  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='9  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='9  1  Z  V  V2  exact NB=1  exact (no cutoff)  Kawasaki  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The quasiparticle weight Z =  �  1 − Re ∂Σ  ∂ω |ω=0  �−1 ob-  tained from the solution of the Hubbard-Holstein DMFT im-  purity model for different values of the hybridization strength  V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' DMFT self-consistency is obtained at V 2 = Z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' at the  intersection of the two lines indicated by the arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The re-  sult on the quantum device shows a self-consistency solution  around V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='84, close to the exact result with one boson  cutoff (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='79) and no boson cutoff (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  first plasmonic satellite at higher energies in the spec-  tral function at around ±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Even when the agreement  with the exact position and weight is not precise (ener-  getic position overestimated by 1 and about 25% of the  correct weight), this result demonstrates that the KVQA  is able to resolve spectral features with relative spectral  weight of less than one percent on current NISQ devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  We want to point out that such a feature would be es-  pecially difficult to observe with approaches that obtain  the Green’s function in real time, like the Trotter ex-  pansion or variational methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The Fourier transform  from time to frequency space requires long time scales  and fine temporal resolution, which makes it difficult to  obtain on current NISQ devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' We thus conclude that  the KVQA is especially well suited for obtaining subtle  spectral features on noisy quantum devices, as possible  errors manifest mostly in the position and weight of the  spectral features and thus retain most of the qualitative  aspects of the true spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  We now focus on obtaining a solution to the self-  consistent DMFT equations for the Hubbard-Holstein  model on the Bethe lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  We set the bandwidth  W = 4, for which the second energy moment of the  non-interacting density of states is M2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The self-  consistency condition is given by Gimp = Gloc, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' the  impurity Green’s function has to be equal to the local lat-  tice Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' For the two-site DMFT impurity  model at half filling, the bath energy is fixed to the Fermi  level, and the self-consistency condition is simplified to a  condition for the hybridization amplitude[55]  V 2 = ZM2,  (12)  5  where Z =  �  1 − Re ∂Σ  ∂ω |ω=0  �−1 is the quasiparticle weight,  obtained from the derivative of the self-energy Σ = G−1  0 −  G−1 on real frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' For the parameters considered  here, the self-energy shows Fermi liquid behavior[55], and  the quasiparticle weight Z is well defined[56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Though,  as both the interacting Green’s function G and non-  interacting Green’s function G0 approach zero at ω = 0,  the derivative of Σ(ω) strongly depends on a precise can-  cellation of two divergent terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Similar to previous  reports[16] we find this method to be unreliable due to  the finite noise level on the quantum device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' We therefore  obtained Z by integrating the weight of the two peaks  closest to the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The resulting spectral weight  as a function of the hybridization strength V is shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The analytical result with a one boson cutoff  results in a self-consistent solution at V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='79, which  is close to the exact result V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='81 without any cutoff  on the number of bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Comparing to the quasiparti-  cle weight obtained on the Kawasaki quantum computer,  we observe a good qualitative agreement with the exact  solution, albeit the quasiparticle weight is overestimated  by about 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' A self-consistent solution can be reliably  observed at around V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='84, in reasonable agreement  with the exact result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  For comparison we also show the resulting Green’s  function in the time domain obtained on the Kawasaki  quantum processor from a Trotter decomposition ap-  proach with one Trotter step in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The Fourier trans-  form of the KVQA Green’s function from frequency space  to time shows a qualitative agreement with the exact re-  sult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  In contrast, the Trotter decomposition is signifi-  cantly less accurate and mostly fluctuates around zero,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' it shows no clear oscillatory behavior and thus pro-  vides almost no information about possible spectral fea-  tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' We identified two main reasons for this: First, the  Trotter decomposition converges very slowly for the cur-  rent system with the number of Trotter steps, therefore,  only one step is not sufficient to reproduce the Green’s  function, as already the simulation without any gate  noise is far from the exact result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Second, the Trotter  decomposition requires more controlled gate operations  than the KVQA which introduce significantly more noise,  and thus reduces the quality of the obtained data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' There-  fore, going beyond one Trotter step and consequently in-  creasing the circuit size would lead to further loss of ac-  curacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' This result shows that the KVQA is significantly  more robust and can provide a more accurate result for  the Green’s function of the DMFT impurity problem for  the same quantum computing device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  CONCLUSION  We have presented an implementation of the Krylov  variational quantum algorithm on the IBM Kawasaki 27-  qubit quantum computer to obtain the Green’s function  for the Hubbard-Holstein two-site impurity model, and  demonstrated that DMFT self-consistency can be reli-  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='5  1  0  2  4  6  8  10  Im[Gimp(t)]  time t  KVQA Simulation  KVQA Kawasaki  Trotter Simulation  Trotter Kawasaki  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The imaginary part of the impurity Green’s function  obtained from a standard Trotter expansion simulation with  one Trotter step (green), compared to the Trotter expansion  performed on the Kawasaki quantum processor (blue solid  curve), and the KVQA result (red and black curve) for V =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='0, λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='5, ω0 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  The Trotter expansion provides a  much poorer result than the KVQA for two main reasons: The  expansion converges slowly with the number of Trotter steps,  and it requires much more controlled gate operations than the  KVQA, and hence introduces significantly more noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  ably obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' This model extends the Hubbard model  and couples the electrons with bosonic degrees of free-  dom, and is the essential building block for impurity  models with frequency dependent interactions such as  in extended DMFT, GW+DMFT and further nonlocal  extensions of DMFT which include nonlocal interactions  and dynamical screening effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  We have presented a  hardware efficient ansatz for the ground state wave func-  tion that exploits the symmetry of the wave function  and allows a parametrization in terms of only two pa-  rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' This enabled us to perform a scan of the full  two-dimensional parameter space to reliably obtain the  ground state and Krylov basis states instead of relying  on numerical minimization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The obtained im-  purity Green’s function is in good qualitative agreement  with the exact result, exhibiting all major spectral fea-  tures of the bonding-antibonding, correlation and plas-  monic satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The approach was shown to be robust  enough to reliably obtain the self-consistent solution of  the DMFT equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' We find that the accuracy greatly  surpasses previously employed approaches such as the  Trotter expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Our work forms the basis of future  studies of electron-boson coupled systems and nonlocal  extensions of DMFT on near-term quantum computers,  which are not only important for real materials calcu-  lations, but also are computationally very intensive on  classical computers, and thus are promising candidates  for harnessing the computational capabilities of future  quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  6  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  The  qubits  used  in  this  work  on  the IBM 27-qubit Quantum Falcon Processor Kawasaki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  The  figure  is  based  on  the  schematic  provided  by  https://quantum-computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='com/services/resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  We thank E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' L¨otstedt, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Nishi, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Yamanouchi  for fruitful discussions and support in utilizing the  Kawasaki quantum computer at the University of Tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  This work is partly supported by the UTokyo Quantum  Initiative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Appendix A: Implementation details  For this work we employed the IBM Qiskit environ-  ment v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='2[53] to implement the quantum circuits  and measurements on the IBM 27-qubit Quantum Falcon  Processor Kawasaki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' For all measurements we employed  the Sampler primitive, and used the maximum number  of shots possible N = 32000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' For the layout we chose the  qubits that showed the lowest readout assignment error  and were connected with the smallest CNOT error as  much as possible, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  To find the Krylov states |χn⟩ in the KVQA we em-  ployed the variational method as outlined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' [34]: To  obtain the parameter set {θn} that generates the state  |χn⟩ = U(θn)|0⟩ via the set of unitary gate operations  U(θn) we minimize the three functions  ϵn0({θ}) =  �|⟨0|U †(θ)HU(θn−1)|0⟩|  |bn|  − 1  �2  ,  (A1)  ϵn1({θ}) = |⟨0|U †(θ)U(θn−1)|0⟩|2,  (A2)  ϵn2({θ}) = |⟨0|U †(θ)U(θn−2)|0⟩|2,  (A3)  which at the minimum ϵni  =  0 ∀i satisfy that  ⟨χn|H|χn−1⟩ = bn, ⟨χn|χn−1⟩ = 0 and ⟨χn|χn−2⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Specifically we made use of the Trotter-like expansion  discussed in the supplementary material I of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' [34] to  evaluate the cost function ϵn0, which circumvents the in-  troduction of another controlled ancilla qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' We used  10 steps between t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='3 to linearly extrapolate the  value to t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' In practice we minimize the sum of these  three functions, which is the quantity shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='6 (c)  and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Reducing the ansatz wave function to a circuit  0  π/2  π  3π/2  2π  0  π/2  π  3π/2  2π  2  0  2  4  6  (a)  θ0  θ1  2  0  2  4  6  0  π/2  π  3π/2  2π  0  π/2  π  3π/2  2π  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='8  1  (b)  θ0  θ1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='8  1  0  π/2  π  3π/2  2π  0  π/2  π  3π/2  2π  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='5  (c)  θ0  θ1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='5  0  π/2  π  3π/2  2π  0  π/2  π  3π/2  2π  0  1  2  3  (d)  θ0  θ1  0  1  2  3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  The optimization surfaces for the circuit shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='1 to find (a) the minimal energy to determine the ground  state energy E0 , (b) the maximum overlap to determine the  first Krylov vector |χ0⟩, (c) and (d) the minimum of the error  function �  i ϵni(θ) to find the next Krylov vectors |χ1⟩ and  |χ2⟩, respectively (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' (A1)-(A3) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  parametrized by two parameters as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' (11) al-  lowed us to use a two-dimensional scan of the param-  eter space to determine the approximate ground state  and Krylov vectors to obtain the impurity Green’s func-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='6 we show the corresponding optimization  surfaces obtained on the Kawasaki quantum processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  The energy surface in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='6 (a) showed a clear minimum  indicating a unique solution for the approximate ground  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Also the obtained overlap to represent the first  Krylov state |χ0⟩ = c|GS⟩ showed a clear maximum with  an overlap of more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='9 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='6 (b) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The optimiza-  tion surface in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='6 (c) for the remaining Krylov states  showed two main local minima, corresponding to two pos-  sible Krylov vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Choosing either local minima leads  to the other minima becoming the only global minima in  the next iteration, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='6 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The precise  location of the minima was noticeably affected by noise,  but we found that the resulting spectral function did not  strongly depend on the exact choice of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  While the peak positions and weights of mainly the two  outermost peaks (correlation induced and plasmon satel-  lite) at energies above ±3 were affected to a small degree,  the qualitative features of the spectral function stayed the  same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Appendix B: Trotterization and variational approach  As discussed in the main text, we found that the  Trotter expansion or the variational Hamiltonian ansatz  (VHA)[29–31], using McLachlan’s variational principle  as detailed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' [31], for the two-site DMFT impu-  rity problem show a very poor agreement with the exact  result, and converge very slowly with increasing the num-  ber of Trotterization steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' To demonstrate this, we show  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='7 the resulting impurity Green’s function obtained  from the Trotter expansion and VHA obtained from a  classical simulation, for NT = 1 and NT = 2 Trotteriza-  tion steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' The agreement with the exact result is very  Qubit:  Readout assignment error  Avg 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='134e-2  min 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='600e-3  max 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='490e-2  Connection:  CNOT error  Avg 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='162e-1  20  min 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='370e-3  max 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='000e+07  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='5  1  0  2  4  6  8  10  (a)  NT=1  Im[Gimp(t)]  time t  exact  Trotter  VHA  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='5  1  0  2  4  6  8  10  (b)  NT=2  Im[Gimp(t)]  time t  exact  Trotter  VHA  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  The imaginary part of the impurity Green’s function as a function of time obtained from a Trotter expansion (red  line) and variational Hamiltonian ansatz (VHA, blue line) on a classical computer, compared to the exact result for the same  boson cutoff NB = 1 at V = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='0 (other parameters as in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' (a) and (b) show the results for NT = 1 and NT = 2  Trotterization steps, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Both results are significantly less accurate than the KVQA one shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Increasing  the number of Trotter steps to NT = 2 results in improvement at small times for the Trotter expansion, but actually worsens  the agreement with the exact solution for VHA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  poor, and quickly deviates even at small time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' In-  creasing the number of Trotter steps improves the result  for the Trotter expansion at small times, but we have  found that at least NT ∼ 8−10 steps are needed in order  to obtain a reasonable agreement up to Tmax = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Such  a large circuit is currently not feasible for present NISQ  devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' In contrast to that, the VHA obtains an even  worse result when increasing the Trotterization steps to  NT = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' We found that the result strongly depends on  the operator ordering in the expansion of the time evo-  lution operator  UV HA =  NT  �  i=1  � NH  �  m=1  eiθm(t)Pm  �  ,  (B1)  where Pm are the Pauli operators of the Hamiltonian  after the Jordan-Wigner transformation, with the total  number of operator terms NH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' For most orderings the  ansatz did not converge to the correct result when in-  creasing the number of Trotter steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Abrams and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Lloyd, Simulation of many-body  fermi systems on a universal quantum computer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' 79, 2586 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [2] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Ortiz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Gubernatis, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Knill, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Laflamme,  Quantum algorithms for fermionic simulations, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' A 64, 022319 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Somma, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Ortiz, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Knill, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Gubernatis,  Quantum simulations of physics problems, in Quan-  tum Information and Computation, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' 5105, edited by  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Donkor, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Pirich, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Brandt, International  Society for Optics and Photonics (SPIE, 2003) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' 96 –  103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Whitfield, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Biamonte, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Aspuru-Guzik,  Simulation of electronic structure hamiltonians using  quantum computers, Molecular Physics 109, 735 (2011),  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='1080/00268976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='552441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Peruzzo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' McClean, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Shadbolt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Yung, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Zhou, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Love, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Aspuru-Guzik, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' O’Brien,  A variational eigenvalue solver on a photonic quantum  processor, Nature Communications 5, 4213 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' McClean, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Romero, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Babbush, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Aspuru-  Guzik,  The  theory  of  variational  hybrid  quantum-  classical algorithms, New Journal of Physics 18, 023023  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [7] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' ope Kunal Arya, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Babbush, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Bacon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Bardin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
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+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Harrigan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
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+page_content=' Huggins,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
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+page_content=' Isakov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
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+page_content=' Klimov,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
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+page_content=' Laptev,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Lindmark, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Lucero, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Martin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Martinis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  McClean, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' McEwen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Megrant, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Mi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Mohseni,  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Mruczkiewicz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Mutus, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Naaman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Neeley,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Neill, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Neven, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Niu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' O’Brien, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Ostby,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Petukhov, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Putterman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Quintana, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Roushan,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Rubin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Sank, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Satzinger, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Smelyanskiy,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Strain, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Sung, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Szalay, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Takeshita,  8  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Vainsencher, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' White, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Wiebe, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Yao, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Yeh,  and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Zalcman, Hartree-fock on a superconducting  qubit quantum computer, Science 369, 1084 (2020),  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='org/doi/pdf/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='abb9811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [8] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Huggins, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' O’Gorman, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Rubin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Reichman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Babbush, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Lee, Unbiasing fermionic  quantum monte carlo with a quantum computer, Nature  603, 416 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [9] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Metzner and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Vollhardt, Correlated lattice fermions  in d = ∞ dimensions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' 62, 324 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Georges and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Kotliar, Hubbard model in infinite  dimensions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' B 45, 6479 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Georges, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Kotliar, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Krauth, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Rozen-  berg, Dynamical mean-field theory of strongly correlated  fermion systems and the limit of infinite dimensions, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' 68, 13 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [12] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Vollhardt, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Byczuk, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Kollar, Dynamical  mean-field theory, in Strongly Correlated Systems: The-  oretical Methods, edited by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Avella and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Mancini  (Springer Berlin Heidelberg, Berlin, Heidelberg, 2012)  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' 203–236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [13] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Bauer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Wecker, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Millis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Hastings, and  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Troyer, Hybrid quantum-classical approach to corre-  lated materials, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' X 6, 031045 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [14] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Kreula, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Clark, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Jaksch, Non-linear  quantum-classical scheme to simulate non-equilibrium  strongly correlated fermionic many-body dynamics, Sci-  entific Reports 6, 32940 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [15] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Rungger, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Fitzpatrick, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Alderete,  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Apel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Cowtan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Patterson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Ramo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Zhu,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Nguyen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Grant, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Chretien, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Wossnig, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Linke, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Duncan, Dynamical mean field theory al-  gorithm and experiment on quantum computers (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [16] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Keen,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Maier,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Johnston, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Lougov-  ski, Quantum-classical simulation of two-site dynamical  mean-field theory on noisy quantum hardware, Quantum  Science and Technology 5, 035001 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [17] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Jaderberg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Agarwal, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Leonhardt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Kiffner, and  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Jaksch, Minimum hardware requirements for hybrid  quantum–classical dmft, Quantum Science and Technol-  ogy 5, 034015 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [18] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Shor, Scheme for reducing decoherence in quantum  computer memory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' A 52, R2493 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Fowler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Mariantoni, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Martinis, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Cleland, Surface codes:  Towards practical large-scale  quantum computation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' A 86, 032324 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Preskill, Quantum Computing in the NISQ era and  beyond, Quantum 2, 79 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [21] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Huang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Yang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Chan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Tanttu,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Hensen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Leon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Fogarty, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Hwang,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Hudson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Itoh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Morello, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Laucht, and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Dzurak, Fidelity benchmarks for two-qubit gates in  silicon, Nature 569, 532 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [22] e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Chen, Zijun, Exponential suppression of bit or  phase errors with cyclic error correction, Nature 595, 383  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [23] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Chiesa, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Tacchino, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Grossi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Santini, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Tav-  ernelli, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Gerace, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Carretta, Quantum hardware  simulating four-dimensional inelastic neutron scattering,  Nature Physics 15, 455 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [24] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Li and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Benjamin, Efficient variational quantum  simulator incorporating active error minimization, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' X 7, 021050 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [25] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Yuan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Endo, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Li, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Benjamin,  Theory of variational quantum simulation, Quantum 3,  191 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [26] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' McArdle, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Jones, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Endo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Benjamin,  and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Yuan, Variational ansatz-based quantum simula-  tion of imaginary time evolution, npj Quantum Informa-  tion 5, 75 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [27] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Heya, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Nakanishi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Mitarai, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Fujii, Sub-  space variational quantum simulator (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [28] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Endo, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Kurata, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Nakagawa, Calculation of  the green’s function on near-term quantum computers,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Research 2, 033281 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [29] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Wecker, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Hastings, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Troyer, Progress to-  wards practical quantum variational algorithms, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' A 92, 042303 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [30] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Reiner,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Wilhelm-Mauch,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Sch¨on,  and  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Marthaler, Finding the ground state of the hubbard  model by variational methods on a quantum computer  with gate errors, Quantum Science and Technology 4,  035005 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [31] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Libbi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Rizzo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Tacchino, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Marzari, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Tav-  ernelli, Effective calculation of the green’s function in  the time domain on near-term quantum processors, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Research 4, 043038 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [32] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Lupo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Jamet, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Tse, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Rungger, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' We-  ber, Maximally localized dynamical quantum embedding  for solving many-body correlated systems, Nature Com-  putational Science 1, 410 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [33] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Vorwerk, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Sheng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Govoni, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Huang, and  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Galli, Quantum embedding theories to simulate con-  densed systems on quantum computers, Nature Compu-  tational Science 2, 424 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [34] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Jamet, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Agarwal, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Lupo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Browne, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Weber,  and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Rungger, Krylov variational quantum algorithm  for first principles materials simulations (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [35] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Jamet, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Agarwal, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Rungger, Quantum sub-  space expansion algorithm for green’s functions (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [36] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Sun and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Kotliar, Extended dynamical mean-field  theory and GW method, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' B 66, 085120 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [37] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Biermann, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Aryasetiawan, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Georges, First-  principles approach to the electronic structure of strongly  correlated systems: Combining the GW approximation  and dynamical mean-field theory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' 90,  086402 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [38] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Biermann, Dynamical screening effects in correlated  electron materials - a progress report on combined many-  body perturbation and dynamical mean field theory:  GW+DMFT, Journal of Physics: Condensed Matter 26,  173202 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [39] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Nilsson, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Boehnke, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Werner, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Aryasetiawan,  Multitier self-consistent gw + EDMFT, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Ma-  terials 1, 043803 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [40] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Rubtsov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Katsnelson, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Lichtenstein, Dual  boson approach to collective excitations in correlated  fermionic systems, Annals of Physics 327, 1320 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [41] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Aryasetiawan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Imada, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Georges, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Kotliar,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Biermann,  and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Lichtenstein,  Frequency-  dependent local interactions and low-energy effective  models from electronic structure calculations, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  B 70, 195104 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [42] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Jordan  and  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Wigner,  ¨Uber  das  paulische  ¨Aquivalenzverbot, Zeitschrift f¨ur Physik 47, 631 (1928).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  9  [43] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Veis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Viˇsˇn´ak, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Nishizawa, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Nakai, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Pit-  tner, Quantum chemistry beyond born–oppenheimer  approximation  on  a  quantum  computer:  A  simu-  lated  phase  estimation  study,  International  Jour-  nal  of  Quantum  Chemistry  116,  1328  (2016),  https://onlinelibrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='wiley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='com/doi/pdf/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='1002/qua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='25176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [44] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' McArdle, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Mayorov, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Shan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Benjamin, and  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Yuan, Digital quantum simulation of molecular vi-  brations, Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' 10, 5725 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [45] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Macridin,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Spentzouris,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Amundson,  and  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Harnik, Digital quantum computation of fermion-  boson interacting systems, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' A 98, 042312  (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [46] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Macridin,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Spentzouris,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Amundson,  and  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Harnik, Electron-phonon systems on a universal quan-  tum computer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' 121, 110504 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [47] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Yung, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Casanova, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Mezzacapo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' McClean,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Lamata, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Aspuru-Guzik, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Solano, From tran-  sistor to trapped-ion computers for quantum chemistry,  Scientific Reports 4, 3589 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [48] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Tilly, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Cao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Picozzi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Setia, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Li,  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Grant, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Wossnig, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Rungger, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Booth, and  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Tennyson, The variational quantum eigensolver: A re-  view of methods and best practices, Physics Reports 986,  1 (2022), the Variational Quantum Eigensolver: a review  of methods and best practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [49] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Kandala, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Mezzacapo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Temme, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Takita,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Brink, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Chow, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Gambetta, Hardware-  efficient  variational  quantum  eigensolver  for  small  molecules and quantum magnets, Nature 549, 242  (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [50] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' McClean, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Boixo, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Smelyanskiy, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Bab-  bush, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Neven, Barren plateaus in quantum neural  network training landscapes, Nature Communications 9,  4812 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [51] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Wang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Fontana, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Cerezo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Sharma, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Sone,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Cincio, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Coles, Noise-induced barren plateaus  in variational quantum algorithms, Nature Communica-  tions 12, 6961 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [52] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Cerezo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Arrasmith, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Babbush, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Benjamin,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Endo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Fujii, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' McClean, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Mitarai, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Yuan,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Cincio, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Coles, Variational quantum algo-  rithms, Nature Reviews Physics 3, 625 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [53] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' tA-v et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=', Qiskit: An open-source framework for  quantum computing (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [54] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Nation, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Kang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Sundaresan, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Gambetta, Scalable mitigation of measurement errors on  quantum computers, PRX Quantum 2, 040326 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [55] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' Potthoff, Two-site dynamical mean-field theory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content=' B 64, 165114 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='  [56] The gap in A(ω) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
+page_content='2 results from the bonding-  antibonding splitting of the two-site model and is not  due to a singular self-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQf5f5n/content/2301.01860v1.pdf'}
diff --git a/o9E1T4oBgHgl3EQfiAR_/content/tmp_files/2301.03247v1.pdf.txt b/o9E1T4oBgHgl3EQfiAR_/content/tmp_files/2301.03247v1.pdf.txt
new file mode 100644
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+Reflection by two level system: phase singularities on the Poincar´e hypersphere
+Ben Lang1, Edmund Harbord2 and Ruth Oulton2
+1George Green Institute for Electromagnetics Research,
+Faculty of Engineering, The University of Nottingham, NG7 2RD, UK
+2Quantum Engineering Technology Labs, H. H. Wills Physics Laboratory and
+Department of Electrical & Electronic Engineering, University of Bristol, BS8 1FD, UK
+We consider the reflection of a photon by a two-level system in a quasi-one-dimensional waveguide.
+This is important in part because it forms the backdrop for more complicated proposals where
+many emitters are coupled to the waveguide: leading to super and subradiant coupling even when
+the emitters are distant. The incorporation of chiral effects, for example unidirectional emission of
+dipole emitters, has already led to rich physics such as dimer coupling. However, chirality is not
+the only effect of the dipole, as we explore from a phase singularity perspective. We demonstrate
+that control of the dipole allows a rich variety of control of the phase and amplitude of the scattered
+light in both directions. This expands the scope for the physics of 1D chains of emitters.
+The exploration of quantum emitter systems coupled
+to “one-dimensional” waveguide-like photonic structures
+has developed into a wide field over the past few years.
+The modified photonic density of states allows near-
+perfect coupling between quantum emitters in the waveg-
+uide that does not decay with distance. Studies of arrays
+of atoms or quantum emitters in such systems predict a
+rich variety of physics. For instance, carefully positioned
+emitters are predicted to show superradiance effects [1].
+Unidirectional emission and scattering leads to symme-
+try breaking in the coupling of arrays of atoms, and the
+formation of “dimers” of many-body dark states [2]. In
+those studies, one exploits the interference between back-
+scattered and forward-scattered light.
+However, these
+studies make a priori assumptions, for example that the
+phase change on reflection is always π [3], or only con-
+sidering the specific case of forward scattering for chiral
+emitters at specific points in the waveguide.
+In this work we explore generalised complex dipole sys-
+tems in a waveguide with generalised polarization tex-
+ture. The result is that one has signficant control over
+the phase and amplitude of the scattered light, particu-
+larly in reflection. Such generalized dipole properties will
+diversify the capabilites of one-dimensional atom-chain
+systems, and relax the positioning requirements for those
+emitters.
+For simplicity we consider the reflection of a photon
+from just one two level system (TLS), a classic prob-
+lem in 1D quantum optics [4]. A photon incident on the
+TLS can be reflected, transmitted or scattered out of the
+waveguide, ie. lost (although in a 1D system these losses
+are assumed to be small). These possibilities are sum-
+marised using complex reflection and transmission coef-
+ficients, r, t, with |r|2 + |t|2 ≤ 1 with equality occurring
+only at zero loss. The situation is depicted in fig.1,(a).
+The generalised nature of our study is that it includes
+both the waveguide polarisation at the TLS location
+(which, even for simple waveguide is very sensitive to
+location, fig.1(b)) and the electric dipole of the TLS. Re-
+cently there has been interest in so called “chiral” cou-
+r
+Ef
+d
+t
+in
+loss
+(a)
+(b)
+FIG. 1. (a) Schematic cartoon of the scattering process. The
+input is split between reflection, transmission and loss, with
+the coefficients determined by the waveguide electric field at
+the location of the TLS (red), and the transition dipole of the
+TLS (black). (b) Polarisation structure in a simple waveg-
+uide, consisting of a high-index rectangular block (grey) in
+air (white).
+Polarisation ellipses are shown through a cut,
+with light propagating up the page (frame arrow).
+pling, which exploits the longitudinal component of the
+waveguide electric field to make the TLS couple asym-
+metrically to the waveguide modes in either direction
+[5–9]. Chiral coupling enables systems where the reflec-
+tion/transmission coefficients are different for light inci-
+dent on the TLS from either side. For example a TLS
+decoupled from the forwards mode will have t = +1 for
+a photon injected forwards, but a photon incident in the
+backward direction may have t anywhere between +1 and
+−1 depending on the coupling strengths to the backward
+and loss modes (the applications of t = 0 are discussed
+in [10, 11] and t = −1 in [12, 13]).
+The complex coefficients r and t depend on both the
+exact location in the waveguide and the properties of the
+dipole. In nanophotonic systems the electric E-field po-
+larization at any specific point r varies such that one can
+describe a polarization ellipse at each point r [12]. The
+arXiv:2301.03247v1  [physics.optics]  9 Jan 2023
+
+2
+ellipse gives the relative magnitudes of the Ex and Ey
+components of the field and their relative phase. It is
+common to see purely linear E-field points, purely cir-
+cular (C) points, where the Ex and Ey components are
+equal and out of phase by π/2, and arbitrary elliptical
+points in between those two extremes.
+Unidirectional
+chiral coupling occurs when a circular dipole emitter (ie
+a linear combination of dx and dy dipole out of phase
+by π/2) is placed at a C-point (the direction of emission
+is dictated by the sign of the phase between the dipole
+components). However one can construct the arbitrary
+complex dipole d = αdx + βdy. Our previous work [14],
+has highlighted that even when emitters are placed at an
+elliptical point, emission may be made unidirectional by
+matching the helicity and eccentricity of the ellipse and
+the dipole, but putting the long axes of the two ellipses
+orthogonal. This negates the backscattered component
+so that only forward scattering can occur.
+We now explore the case where reflection is desired,
+by considering an arbitrary dipole where the linear com-
+ponent is able to couple to the backscattered direction.
+What is interesting here is the phase of the reflection.
+While input-output models give only a π phase shift to
+the backscattered light, we show here that the orienta-
+tion of the ellipse governs the phase of the reflected light,
+giving rise to a rich structure. This is an extremely useful
+property as it allows one to tune the phase delay when
+considering a chain of atoms.
+This means that one is
+no longer constrained to precise positions of the emit-
+ters in a 1D chain. Also, by dynamically controlling the
+dipole ellipse orientation, one may control the reflectivity
+in-situ.
+We consider electric dipole interactions between the
+TLS and the waveguide, and assume a narrow-band pho-
+ton (narrow in frequency, long in time). The crucial pa-
+rameters determining the single photon t, and r coeffi-
+cients are the local polarisation of the forwards (back-
+wards) waveguide mode at the TLS position Ef(b) and
+the electric dipole of the TLS transition d.
+Both Ef
+and d are complex vectors in space.
+Without loss of
+generality we assume a basis in which both are 2D. The
+time-reversal symmetry of Maxwell’s equations requires
+that Ef = E∗
+b, which amounts to reversing the ellipse
+arrowheads.
+The r, t coefficients are given by [14]:
+t = 1 −
+d · E∗
+f d∗ · Ef
+D
+,
+(1)
+r = −d · E∗
+b d∗ · Ef
+D
+,
+(2)
+with
+D =1
+2
+�
+|d · E∗
+f|2 + |d · E∗
+b|2�
++ ζ (d · Gl · d∗ + iℏϵ0δ) .
+(3)
+(a)
+(b)
+Δx
+2Δx
+2λΔθ
+Δθ
+FIG. 2. (a) Michelson interferometer phase from mirror mo-
+tion. (b) Comparable effect from rotating a reflecting dipole
+to interact with a delayed polarisation component.
+We will assume resonance, and thus δ, the detuning
+between the incident photon and the TLS transition fre-
+quency, is set to zero. Gl is the Green’s function control-
+ling the interaction of the TLS with non-guided modes
+[15], and the parameter ζ characterises the inverse of the
+waveguide mode coupling strength to the TLS. For sim-
+plicity we set ζd · Gl · d∗ = L, with L a scalar control-
+ling loss. If the TLS is prepared in its excited state the
+fraction of the radiated intensity (photon probability) to
+enter the guided modes is given by β = W/(W + L) with
+W = 1
+2(|d·E∗
+f|2 +|d·E∗
+b|2). By making loss independent
+of d we are implicitly assuming at least two loss modes,
+with polarisations orthogonal at the TLS location.
+First, we consider the case of a linear dipole.
+Here
+it is well-known that the TLS acts as a mirror, with |r|
+approaching unity for low loss [4]. However, the phase of
+the reflection has received less attention.
+It should first be clarified what is meant by “the phase
+of the reflection”. Naturally this phase must be deter-
+mined relative to the phase of the input and at some
+particular location in space. The location is important
+because the input signal (travelling forwards) will have
+a phase that changes spatially like eikx, while the back-
+wards travelling reflection will instead evolve as e−ikx.
+As the distance between the phase reference point and
+the location of the reflecting TLS is increased the phase
+of the reflection will change according to the additional
+round trip distance - the mechanism of a Michelson in-
+terferometer.
+However, if some arbitrary (but fixed) location is
+picked at which to compare the input and output phases
+it is then possible to calculate how changing other pa-
+rameters alters the phase of the reflected light.
+Consider a TLS at a point of circular polarisation,
+Ef = (1, i). Here the y component of the electric field
+is a quarter-wave delayed relative to the x component.
+Thus a linear dipole aligned along the y axis is effectively
+a quarter wavelength “further away” than one along x.
+Here rotating a dipole some angle ∆θ, transforms the re-
+flectivity as r = r0 exp(2i∆θ), with r0 the reflectivity for
+
+3
+(a)
+(b)
+(c)
+(d)
+π
+-π
+0
+S2
+S1
+S3
+FIG. 3. r as a function of d. First column: hue (opacity)
+indicates the phase (amplitude) of the reflection from the
+dipole on that point of the Poincar´e sphere, for the polar-
+isation depicted on the left.
+These polarisations are Ef =
+[i cos(nπ/12), sin(nπ/12)] with (a,b,c,d) having n = (3, 4, 5, 6)
+respectively. Second column: streamlines indicating the phase
+gradient. In (d) the phase is constant so streamlines cannot
+be plotted. Instead a key indicates the locations of cardinal
+dipoles on the sphere. S1−3 mark the axes of the the Stokes
+Parameters [16].
+∆θ = 0. The mechanism is reminiscent of a Michelson
+interferometer, where moving the mirror by distance ∆x
+delays the phase by the additional round-trip distance,
+2∆x/λ, depicted in fig.2. Here we effectively move the
+mirror by rotating it.
+This phase is physically meaningful. One could build a
+interferometer with a rotating dipole replacing the mov-
+ing mirror and measure the effect. The phase may also
+be important in other contexts. For example, if multi-
+ple TLSs are coupled to a single waveguide their spac-
+ing is of critical importance, as it determines whether
+emission and scattering adds constructively or destruc-
+tively, thereby controlling the superradiance [17], dipole-
+dipole frequency shifts [18, 19], reflectivity [20] and inter-
+atom entanglement [21, 22]. But, as we have just moti-
+vated, the phase delay is also dependent on the transition
+dipoles, so that the effective emitter spacing depends on
+how the TLS dipoles are oriented. Thus, the realisation
+of proposals that depend on the phase delay between ad-
+jacent TLSs [23], is not determined entirely by the TLS
+separation.
+The ability of both distance and dipole to control phase
+motivates a mention of superconducting giant atoms.
+While chirality normally exploits the relative phase be-
+tween two polarisation components at a single location,
+these giant atoms achieve a similar effect by reaching spa-
+tially to exploit the relative phase between two locations
+in the waveguide [24].
+Requiring that |Ef| = |d| = 1 for simplicity and ne-
+glecting global phases both vectors can be written in the
+form: [cos(θ), sin(θ) exp(iφ)].
+The angles θ, φ can be
+used to represent the vectors as points on the surface of
+the Poincar´e sphere. In the general case a dipole can be
+picked from anywhere on the surface of this sphere. The
+aforementioned linear dipoles comprise the equator, with
+circular dipoles at the poles and ellipses elsewhere.
+In fig.3,(a) we fix a circular polarisation, and calculate
+r for all possible dipoles. Each dipole, d, corresponds to
+a point on the sphere with its own value of r. In the first
+column of spheres the hue indicates the phase ∠r, with
+the opacity indicating |r| [with r = |r| exp(i∠r)]. In the
+second column the phase gradient is plotted. Following
+any line in the arrowhead direction the phase changes
+by −2π in a complete cycle. The Michelson-like phase
+motivated above appears on the equator. We have set
+|Ef| = |d| = 1 and L = 0.01 to consider the situation
+where coupling to loss modes is weak compared to typical
+waveguide coupling. Such low loss is appropriate to, for
+example, photonic crystal waveguide systems [25]. Due
+to this low loss |r| is close to unity almost everywhere on
+the sphere, becoming significantly less only when the dot
+products |d∗ · Ef|2 or |d∗ · Eb|2 are comparable to L.
+Plotting the data on a sphere highlights important
+topological restrictions that are not obvious from inspec-
+tion of equ.(1). The phase swirling about the equator di-
+rectly requires that in each hemisphere there is a dipole
+such that r = 0. This can be seen from the figure, af-
+ter fixing the equator one cannot find a smooth function
+without at least one zero in each hemisphere.
+The zero points are phase singularities, a ubiquitous
+wave phenomenon that occur where a complex scalar field
+takes value 0 at some location, with the phase angle vary-
+ing by 2πm in a circuit of the zero point [26, 27]. The
+total phase change along a closed curve is equal to the
+�
+n 2πmn with n counting over the singularities enclosed.
+
+4
+d
+Ef
+FIG. 4. Each small sphere denotes r as a function of d as in
+fig.3. For each the polarisation, Ef used is indicated by its
+location on the larger (wire frame) sphere.
+Thus, our 2π variation along the equator is enough to
+ensure that both hemispheres contain at least one phase
+singularity with r = 0.
+With circular polarisation these points are the poles.
+At one d∗ · Ef = 0 and the dipole decouples from the
+forwards mode, so the TLS cannot in any way interact
+with the input photon. At the other d∗ · Eb = 0 so that
+whatever the TLS does to the photon it cannot involve
+any scatting to the backward mode (hence r = 0).
+Moving to parts (b, c, d) of the figure we vary the
+polarisation of the waveguide. This deforms the phase
+structures continuously, preserving the two singularities
+until they mutually annihilate on the equator for a linear
+polarisation.
+This highlights that for any polarisation
+there is a dipole that decouples from the forward mode
+and another the backward one (the singularities), with
+the two coinciding only for a linear polarisation [14].
+The phase gradient can be considered a vector field.
+As this field lives on the sphere it is subject to the “hairy
+ball theorem” which requires that it cannot be smooth
+and nonzero everywhere. The theorem name refers to a
+consequence of this, that a hairy ball cannot be combed
+to have all the hair lie flat. More precisely the theorem re-
+quires the total Poincar´e-Hopf indices of the vector field’s
+zero points equal the sphere’s Euler characteristic of +2
+[28]. The vector fields depicted by streamlines in fig.3
+each have two singular points where the vector winds
+in a circle (the phase singularities), such circles have an
+index of +1 irrespective of the arrowhead directions, so
+that the pair has the required +2 total.
+The polarisation is able to explore its own Poincar´e
+sphere of possible values, such that the space of Ef ⊗ d
+has the form S2 ⊗ S2, a “sphere of spheres” which we
+term a Poincar´e hypersphere. This full space is depicted
+in fig.4, with the larger sphere indicating the polarisation
+and the smaller ones the dipole. One sees that (starting
+from the north pole of the larger sphere) stretching our
+initially circular polarisation brings the polar phase sin-
+gularities closer to one another, and that the line of lat-
+itude moved along is determined by the latitude line on
+the big sphere. Opposite points on the Poincar´e sphere
+correspond to orthogonal polarisations, so that one phase
+singularity on each sub-sphere points to the centre of the
+larger sphere. This corresponds to the d∗ · Ef = 0 case.
+The other phase singularity’s location on each sub-sphere
+is given by reflecting the first through the equator to set
+d∗ · Eb = 0.
+Taking a tangent, it is interesting to consider the ge-
+ometry. Ef and d are each 2D, giving us a total of 4
+dimensions. A phase singularity has 2 dimensions fewer
+than the space it is embedded in, so that the singular-
+ities on our 2-spheres were pointlike (0D), and in 3D
+space singularities represent lines of darkness or silence in
+fields [29]. Our singularities in the 4D space are 2D sur-
+faces. There are two such surfaces, each corresponding
+to d∗ · (Ef or Eb) = 0. These surfaces map to spherical
+shells, and the intersection of the two maps to a circle.
+On this circle both dipole and polarisation are linear and
+are orthongonal to one another, for example resembling a
+“×” in real space. The circular nature of the intersection
+relates to the fact that the “×” can be rotated freely, eg.
+“+”.
+The transmission, t can be equally assessed on the
+sphere (or hypersphere). However t lacks the complex
+structure seen in r. For the most part, a t equivalent of
+fig.3 simply shows a phase of ∠t = π in one hemisphere
+and ∠t = 0 in the other, the two separated by a |t| = 0
+equator. This |t| = 0 equator can be considered a phase
+singularity by re-introducing the detuning, δ. For a fixed
+Ef, within the 3D space defined by d ⊗ δ it takes the
+form of a 1D line phase singularity, looped into a circle
+(near the equator) in the δ = 0 slice.
+In conclusion, we demonstrate a rich behaviour of the
+reflection coefficient, r, of a complex dipole TLS in a one-
+dimensional photonic waveguide. r is a complex field sup-
+porting phase singularities that live on the Poincar´e (hy-
+per)sphere. The existence of two dipoles for each polari-
+sation such that r = 0 and the existence of a Michelson-
+interferometer like phase gradient on the equator can
+both be motivated by physical arguments. We showed
+that the two effects are intimately linked, due to the
+topological constraints faced by phase singularities liv-
+ing on the surface of a sphere. The rich dependence of
+r on dipole and waveguide polarisation for a single TLS
+will offer new avenues for exploitation of chains of TLSs
+in waveguides.
+Acknowledgement: We acknowledge support from EP-
+SRC grants no. EP/N003381/1 “One-dimensional quan-
+tum emitters and photons for quantum technologies” and
+EP/M024156/1 “Spin Space” and PHD funding DTA-
+1407622.
+
+5
+[1] A. Gonz´alez-Tudela and D. Porras, Phys. Rev. Lett. 110,
+080502 (2013).
+[2] T. Ramos, H. Pichler, A. J. Daley, and P. Zoller, Phys.
+Rev. Lett. 113, 237203 (2014).
+[3] P. Lodahl, S. Mahmoodian, S. Stobbe, P. Schneeweiss,
+J. Volz, A. Rauschenbeutel, H. Pichler,
+and P. Zoller,
+Nature 541, 473 (2017).
+[4] J. T. Shen and S. Fan, Opt. Lett. 30, 2001 (2005).
+[5] J. Petersen, J. Volz,
+and A. Rauschenbeutel, Science
+346, 67 (2014).
+[6] F. J. Rodr´ıguez-Fortu˜no,
+G. Marino,
+P. Ginzburg,
+D. O’Connor, A. Mart´ınez, G. A. Wurtz,
+and A. V.
+Zayats, Science 340, 328 (2013).
+[7] A. Aiello, P. Banzer, M. Neugebauer,
+and G. Leuchs,
+Nat. Photon. 9, 789 (2015).
+[8] R. J. Coles,
+D. M. Price,
+J. E. Dixon,
+B. Roy-
+all,
+E.
+Clarke,
+P.
+Kok,
+M.
+S.
+Skolnick,
+A.
+M.
+Fox,
+and M. N. Makhonin, Nat. Commun. 7 (2016),
+10.1038/ncomms11183.
+[9] M. F. Picardi, A. V. Zayats,
+and F. J. Rodr´ıguez-
+Fortu˜no, Phys. Rev. Lett. 120, 117402 (2018).
+[10] C. Gonzalez-Ballestero, E. Moreno, F. J. Garcia-Vidal,
+and A. Gonzalez-Tudela, Phys. Rev. A 94, 063817 (2016).
+[11] C. Sayrin, C. Junge, R. Mitsch, B. Albrecht, D. O’Shea,
+P. Schneeweiss, J. Volz,
+and A. Rauschenbeutel, Phys.
+Rev. X 5, 041036 (2015).
+[12] A. B. Young, A. C. T. Thijssen, D. M. Beggs, P. An-
+drovitsaneas, L. Kuipers, J. G. Rarity, S. Hughes,
+and
+R. Oulton, Phys. Rev. Lett. 115, 153901 (2015).
+[13] I. S¨ollner, S. Mahmoodian, S. L. Hansen, L. Midolo,
+A. Javadi, G. Kirˇsansk˙e, T. Pregnolato, H. El-Ella, E. H.
+Lee, J. D. Song, S. Stobbe,
+and P. Lodahl, Nat. Nan-
+otechnol. 10, 775 (2015).
+[14] B. Lang, D. P. S. McCutcheon, E. Harbord, A. B. Young,
+and R. Oulton, Phys. Rev. Lett. 128, 073602 (2022).
+[15] V. S. C. Manga Rao and S. Hughes, Phys. Rev. B 75,
+205437 (2007).
+[16] E. Collett, Field Guide to Polarization (SPIE press,
+2005).
+[17] R. Jones, G. Buonaiuto, B. Lang, I. Lesanovsky,
+and
+B. Olmos, Phys. Rev. Lett. 124, 093601 (2020).
+[18] D. Dzsotjan, J. K¨astel, and M. Fleischhauer, Phys. Rev.
+B 84, 075419 (2011).
+[19] R. Jones, J. A. Needham, I. Lesanovsky, F. Intravaia,
+and B. Olmos, Phys. Rev. A 97, 053841 (2018).
+[20] Y. Zhou, Z. Chen,
+and J.-T. Shen, Phys. Rev. A 101,
+043831 (2020).
+[21] I. M. Mirza and J. C. Schotland, Phys. Rev. A 94, 012302
+(2016).
+[22] H. Pichler, T. Ramos, A. J. Daley, and P. Zoller, Phys.
+Rev. A 91, 042116 (2015).
+[23] R. Holzinger, R. Guti´errez-J´auregui, T. H¨onigl-Decrinis,
+G. Kirchmair, A. Asenjo-Garcia,
+and H. Ritsch, Phys.
+Rev. Lett. 129, 253601 (2022).
+[24] X. Wang, T. Liu, A. F. Kockum, H.-R. Li, and F. Nori,
+Phys. Rev. Lett. 126, 043602 (2021).
+[25] L. Scarpelli, B. Lang, F. Masia, D. M. Beggs, E. A. Mul-
+jarov, A. B. Young, R. Oulton, M. Kamp, S. H¨ofling,
+C. Schneider,
+and W. Langbein, Phys. Rev. B 100,
+035311 (2019).
+[26] J. F. Nye and M. V. Berry, Proc. Royal Soc. A 336, 165
+(1974).
+[27] M. Berry, Nature 403, 21 (2000).
+[28] M. V. Berry, M. R. Dennis, and R. L. Lee, New Journal
+of Physics 6, 162 (2004).
+[29] M. J. Padgett, K. O’Holleran, R. P. King,
+and M. R.
+Dennis, Contemporary Physics 52, 265 (2011).
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf,len=430
+page_content='Reflection by two level system: phase singularities on the Poincar´e hypersphere  Ben Lang1, Edmund Harbord2 and Ruth Oulton2  1George Green Institute for Electromagnetics Research,  Faculty of Engineering, The University of Nottingham, NG7 2RD, UK  2Quantum Engineering Technology Labs, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Wills Physics Laboratory and  Department of Electrical & Electronic Engineering, University of Bristol, BS8 1FD, UK  We consider the reflection of a photon by a two-level system in a quasi-one-dimensional waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  This is important in part because it forms the backdrop for more complicated proposals where  many emitters are coupled to the waveguide: leading to super and subradiant coupling even when  the emitters are distant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' The incorporation of chiral effects, for example unidirectional emission of  dipole emitters, has already led to rich physics such as dimer coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' However, chirality is not  the only effect of the dipole, as we explore from a phase singularity perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' We demonstrate  that control of the dipole allows a rich variety of control of the phase and amplitude of the scattered  light in both directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' This expands the scope for the physics of 1D chains of emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  The exploration of quantum emitter systems coupled  to “one-dimensional” waveguide-like photonic structures  has developed into a wide field over the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  The modified photonic density of states allows near-  perfect coupling between quantum emitters in the waveg-  uide that does not decay with distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Studies of arrays  of atoms or quantum emitters in such systems predict a  rich variety of physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' For instance, carefully positioned  emitters are predicted to show superradiance effects [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  Unidirectional emission and scattering leads to symme-  try breaking in the coupling of arrays of atoms, and the  formation of “dimers” of many-body dark states [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' In  those studies, one exploits the interference between back-  scattered and forward-scattered light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  However, these  studies make a priori assumptions, for example that the  phase change on reflection is always π [3], or only con-  sidering the specific case of forward scattering for chiral  emitters at specific points in the waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  In this work we explore generalised complex dipole sys-  tems in a waveguide with generalised polarization tex-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' The result is that one has signficant control over  the phase and amplitude of the scattered light, particu-  larly in reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Such generalized dipole properties will  diversify the capabilites of one-dimensional atom-chain  systems, and relax the positioning requirements for those  emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  For simplicity we consider the reflection of a photon  from just one two level system (TLS), a classic prob-  lem in 1D quantum optics [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' A photon incident on the  TLS can be reflected, transmitted or scattered out of the  waveguide, ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' lost (although in a 1D system these losses  are assumed to be small).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' These possibilities are sum-  marised using complex reflection and transmission coef-  ficients, r, t, with |r|2 + |t|2 ≤ 1 with equality occurring  only at zero loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' The situation is depicted in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='1,(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  The generalised nature of our study is that it includes  both the waveguide polarisation at the TLS location  (which, even for simple waveguide is very sensitive to  location, fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='1(b)) and the electric dipole of the TLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Re-  cently there has been interest in so called “chiral” cou-  r  Ef  d  t  in  loss  (a)  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' (a) Schematic cartoon of the scattering process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' The  input is split between reflection, transmission and loss, with  the coefficients determined by the waveguide electric field at  the location of the TLS (red), and the transition dipole of the  TLS (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' (b) Polarisation structure in a simple waveg-  uide, consisting of a high-index rectangular block (grey) in  air (white).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  Polarisation ellipses are shown through a cut,  with light propagating up the page (frame arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  pling, which exploits the longitudinal component of the  waveguide electric field to make the TLS couple asym-  metrically to the waveguide modes in either direction  [5–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Chiral coupling enables systems where the reflec-  tion/transmission coefficients are different for light inci-  dent on the TLS from either side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' For example a TLS  decoupled from the forwards mode will have t = +1 for  a photon injected forwards, but a photon incident in the  backward direction may have t anywhere between +1 and  −1 depending on the coupling strengths to the backward  and loss modes (the applications of t = 0 are discussed  in [10, 11] and t = −1 in [12, 13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  The complex coefficients r and t depend on both the  exact location in the waveguide and the properties of the  dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' In nanophotonic systems the electric E-field po-  larization at any specific point r varies such that one can  describe a polarization ellipse at each point r [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' The  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='03247v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='optics]  9 Jan 2023  2  ellipse gives the relative magnitudes of the Ex and Ey  components of the field and their relative phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' It is  common to see purely linear E-field points, purely cir-  cular (C) points, where the Ex and Ey components are  equal and out of phase by π/2, and arbitrary elliptical  points in between those two extremes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  Unidirectional  chiral coupling occurs when a circular dipole emitter (ie  a linear combination of dx and dy dipole out of phase  by π/2) is placed at a C-point (the direction of emission  is dictated by the sign of the phase between the dipole  components).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' However one can construct the arbitrary  complex dipole d = αdx + βdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Our previous work [14],  has highlighted that even when emitters are placed at an  elliptical point, emission may be made unidirectional by  matching the helicity and eccentricity of the ellipse and  the dipole, but putting the long axes of the two ellipses  orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' This negates the backscattered component  so that only forward scattering can occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  We now explore the case where reflection is desired,  by considering an arbitrary dipole where the linear com-  ponent is able to couple to the backscattered direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  What is interesting here is the phase of the reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  While input-output models give only a π phase shift to  the backscattered light, we show here that the orienta-  tion of the ellipse governs the phase of the reflected light,  giving rise to a rich structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' This is an extremely useful  property as it allows one to tune the phase delay when  considering a chain of atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  This means that one is  no longer constrained to precise positions of the emit-  ters in a 1D chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Also, by dynamically controlling the  dipole ellipse orientation, one may control the reflectivity  in-situ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  We consider electric dipole interactions between the  TLS and the waveguide, and assume a narrow-band pho-  ton (narrow in frequency, long in time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' The crucial pa-  rameters determining the single photon t, and r coeffi-  cients are the local polarisation of the forwards (back-  wards) waveguide mode at the TLS position Ef(b) and  the electric dipole of the TLS transition d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  Both Ef  and d are complex vectors in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  Without loss of  generality we assume a basis in which both are 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' The  time-reversal symmetry of Maxwell’s equations requires  that Ef = E∗  b, which amounts to reversing the ellipse  arrowheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  The r, t coefficients are given by [14]:  t = 1 −  d · E∗  f d∗ · Ef  D  ,  (1)  r = −d · E∗  b d∗ · Ef  D  ,  (2)  with  D =1  2  �  |d · E∗  f|2 + |d · E∗  b|2�  + ζ (d · Gl · d∗ + iℏϵ0δ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  (3)  (a)  (b)  Δx  2Δx  2λΔθ  Δθ  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' (a) Michelson interferometer phase from mirror mo-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' (b) Comparable effect from rotating a reflecting dipole  to interact with a delayed polarisation component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  We will assume resonance, and thus δ, the detuning  between the incident photon and the TLS transition fre-  quency, is set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Gl is the Green’s function control-  ling the interaction of the TLS with non-guided modes  [15], and the parameter ζ characterises the inverse of the  waveguide mode coupling strength to the TLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' For sim-  plicity we set ζd · Gl · d∗ = L, with L a scalar control-  ling loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' If the TLS is prepared in its excited state the  fraction of the radiated intensity (photon probability) to  enter the guided modes is given by β = W/(W + L) with  W = 1  2(|d·E∗  f|2 +|d·E∗  b|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' By making loss independent  of d we are implicitly assuming at least two loss modes,  with polarisations orthogonal at the TLS location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  First, we consider the case of a linear dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  Here  it is well-known that the TLS acts as a mirror, with |r|  approaching unity for low loss [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' However, the phase of  the reflection has received less attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  It should first be clarified what is meant by “the phase  of the reflection”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Naturally this phase must be deter-  mined relative to the phase of the input and at some  particular location in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' The location is important  because the input signal (travelling forwards) will have  a phase that changes spatially like eikx, while the back-  wards travelling reflection will instead evolve as e−ikx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  As the distance between the phase reference point and  the location of the reflecting TLS is increased the phase  of the reflection will change according to the additional  round trip distance - the mechanism of a Michelson in-  terferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  However, if some arbitrary (but fixed) location is  picked at which to compare the input and output phases  it is then possible to calculate how changing other pa-  rameters alters the phase of the reflected light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  Consider a TLS at a point of circular polarisation,  Ef = (1, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Here the y component of the electric field  is a quarter-wave delayed relative to the x component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  Thus a linear dipole aligned along the y axis is effectively  a quarter wavelength “further away” than one along x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  Here rotating a dipole some angle ∆θ, transforms the re-  flectivity as r = r0 exp(2i∆θ), with r0 the reflectivity for  3  (a)  (b)  (c)  (d)  π  π  0  S2  S1  S3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' r as a function of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' First column: hue (opacity)  indicates the phase (amplitude) of the reflection from the  dipole on that point of the Poincar´e sphere, for the polar-  isation depicted on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  These polarisations are Ef =  [i cos(nπ/12), sin(nπ/12)] with (a,b,c,d) having n = (3, 4, 5, 6)  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Second column: streamlines indicating the phase  gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' In (d) the phase is constant so streamlines cannot  be plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Instead a key indicates the locations of cardinal  dipoles on the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' S1−3 mark the axes of the the Stokes  Parameters [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  ∆θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' The mechanism is reminiscent of a Michelson  interferometer, where moving the mirror by distance ∆x  delays the phase by the additional round-trip distance,  2∆x/λ, depicted in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Here we effectively move the  mirror by rotating it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  This phase is physically meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' One could build a  interferometer with a rotating dipole replacing the mov-  ing mirror and measure the effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' The phase may also  be important in other contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' For example, if multi-  ple TLSs are coupled to a single waveguide their spac-  ing is of critical importance, as it determines whether  emission and scattering adds constructively or destruc-  tively, thereby controlling the superradiance [17], dipole-  dipole frequency shifts [18, 19], reflectivity [20] and inter-  atom entanglement [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' But, as we have just moti-  vated, the phase delay is also dependent on the transition  dipoles, so that the effective emitter spacing depends on  how the TLS dipoles are oriented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Thus, the realisation  of proposals that depend on the phase delay between ad-  jacent TLSs [23], is not determined entirely by the TLS  separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  The ability of both distance and dipole to control phase  motivates a mention of superconducting giant atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  While chirality normally exploits the relative phase be-  tween two polarisation components at a single location,  these giant atoms achieve a similar effect by reaching spa-  tially to exploit the relative phase between two locations  in the waveguide [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  Requiring that |Ef| = |d| = 1 for simplicity and ne-  glecting global phases both vectors can be written in the  form: [cos(θ), sin(θ) exp(iφ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  The angles θ, φ can be  used to represent the vectors as points on the surface of  the Poincar´e sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' In the general case a dipole can be  picked from anywhere on the surface of this sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' The  aforementioned linear dipoles comprise the equator, with  circular dipoles at the poles and ellipses elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='3,(a) we fix a circular polarisation, and calculate  r for all possible dipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Each dipole, d, corresponds to  a point on the sphere with its own value of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' In the first  column of spheres the hue indicates the phase ∠r, with  the opacity indicating |r| [with r = |r| exp(i∠r)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' In the  second column the phase gradient is plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Following  any line in the arrowhead direction the phase changes  by −2π in a complete cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' The Michelson-like phase  motivated above appears on the equator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' We have set  |Ef| = |d| = 1 and L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='01 to consider the situation  where coupling to loss modes is weak compared to typical  waveguide coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Such low loss is appropriate to, for  example, photonic crystal waveguide systems [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Due  to this low loss |r| is close to unity almost everywhere on  the sphere, becoming significantly less only when the dot  products |d∗ · Ef|2 or |d∗ · Eb|2 are comparable to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  Plotting the data on a sphere highlights important  topological restrictions that are not obvious from inspec-  tion of equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' The phase swirling about the equator di-  rectly requires that in each hemisphere there is a dipole  such that r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' This can be seen from the figure, af-  ter fixing the equator one cannot find a smooth function  without at least one zero in each hemisphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  The zero points are phase singularities, a ubiquitous  wave phenomenon that occur where a complex scalar field  takes value 0 at some location, with the phase angle vary-  ing by 2πm in a circuit of the zero point [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' The  total phase change along a closed curve is equal to the  �  n 2πmn with n counting over the singularities enclosed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  4  d  Ef  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Each small sphere denotes r as a function of d as in  fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' For each the polarisation, Ef used is indicated by its  location on the larger (wire frame) sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  Thus, our 2π variation along the equator is enough to  ensure that both hemispheres contain at least one phase  singularity with r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  With circular polarisation these points are the poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  At one d∗ · Ef = 0 and the dipole decouples from the  forwards mode, so the TLS cannot in any way interact  with the input photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' At the other d∗ · Eb = 0 so that  whatever the TLS does to the photon it cannot involve  any scatting to the backward mode (hence r = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  Moving to parts (b, c, d) of the figure we vary the  polarisation of the waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' This deforms the phase  structures continuously, preserving the two singularities  until they mutually annihilate on the equator for a linear  polarisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  This highlights that for any polarisation  there is a dipole that decouples from the forward mode  and another the backward one (the singularities), with  the two coinciding only for a linear polarisation [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  The phase gradient can be considered a vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  As this field lives on the sphere it is subject to the “hairy  ball theorem” which requires that it cannot be smooth  and nonzero everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' The theorem name refers to a  consequence of this, that a hairy ball cannot be combed  to have all the hair lie flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' More precisely the theorem re-  quires the total Poincar´e-Hopf indices of the vector field’s  zero points equal the sphere’s Euler characteristic of +2  [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' The vector fields depicted by streamlines in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='3  each have two singular points where the vector winds  in a circle (the phase singularities), such circles have an  index of +1 irrespective of the arrowhead directions, so  that the pair has the required +2 total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  The polarisation is able to explore its own Poincar´e  sphere of possible values, such that the space of Ef ⊗ d  has the form S2 ⊗ S2, a “sphere of spheres” which we  term a Poincar´e hypersphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' This full space is depicted  in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='4, with the larger sphere indicating the polarisation  and the smaller ones the dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' One sees that (starting  from the north pole of the larger sphere) stretching our  initially circular polarisation brings the polar phase sin-  gularities closer to one another, and that the line of lat-  itude moved along is determined by the latitude line on  the big sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Opposite points on the Poincar´e sphere  correspond to orthogonal polarisations, so that one phase  singularity on each sub-sphere points to the centre of the  larger sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' This corresponds to the d∗ · Ef = 0 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  The other phase singularity’s location on each sub-sphere  is given by reflecting the first through the equator to set  d∗ · Eb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  Taking a tangent, it is interesting to consider the ge-  ometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Ef and d are each 2D, giving us a total of 4  dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' A phase singularity has 2 dimensions fewer  than the space it is embedded in, so that the singular-  ities on our 2-spheres were pointlike (0D), and in 3D  space singularities represent lines of darkness or silence in  fields [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Our singularities in the 4D space are 2D sur-  faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' There are two such surfaces, each corresponding  to d∗ · (Ef or Eb) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' These surfaces map to spherical  shells, and the intersection of the two maps to a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  On this circle both dipole and polarisation are linear and  are orthongonal to one another, for example resembling a  “×” in real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' The circular nature of the intersection  relates to the fact that the “×” can be rotated freely, eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  “+”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  The transmission, t can be equally assessed on the  sphere (or hypersphere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' However t lacks the complex  structure seen in r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' For the most part, a t equivalent of  fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='3 simply shows a phase of ∠t = π in one hemisphere  and ∠t = 0 in the other, the two separated by a |t| = 0  equator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' This |t| = 0 equator can be considered a phase  singularity by re-introducing the detuning, δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' For a fixed  Ef, within the 3D space defined by d ⊗ δ it takes the  form of a 1D line phase singularity, looped into a circle  (near the equator) in the δ = 0 slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  In conclusion, we demonstrate a rich behaviour of the  reflection coefficient, r, of a complex dipole TLS in a one-  dimensional photonic waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' r is a complex field sup-  porting phase singularities that live on the Poincar´e (hy-  per)sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' The existence of two dipoles for each polari-  sation such that r = 0 and the existence of a Michelson-  interferometer like phase gradient on the equator can  both be motivated by physical arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' We showed  that the two effects are intimately linked, due to the  topological constraints faced by phase singularities liv-  ing on the surface of a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' The rich dependence of  r on dipole and waveguide polarisation for a single TLS  will offer new avenues for exploitation of chains of TLSs  in waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  Acknowledgement: We acknowledge support from EP-  SRC grants no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' EP/N003381/1 “One-dimensional quan-  tum emitters and photons for quantum technologies” and  EP/M024156/1 “Spin Space” and PHD funding DTA-  1407622.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  5  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Gonz´alez-Tudela and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Porras, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' 110,  080502 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [2] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Ramos, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Pichler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Daley, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Zoller, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' 113, 237203 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [3] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Lodahl, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Mahmoodian, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Stobbe, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Schneeweiss,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Volz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Rauschenbeutel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Pichler,  and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Zoller,  Nature 541, 473 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Shen and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Fan, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' 30, 2001 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Petersen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Volz,  and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Rauschenbeutel, Science  346, 67 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [6] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Rodr´ıguez-Fortu˜no,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Marino,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Ginzburg,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' O’Connor, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Mart´ınez, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Wurtz,  and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  Zayats, Science 340, 328 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Aiello, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Banzer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Neugebauer,  and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Leuchs,  Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' 9, 789 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [8] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Coles,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Price,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Dixon,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Roy-  all,  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  Clarke,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  Kok,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  Skolnick,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  Fox,  and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Makhonin, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' 7 (2016),  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='1038/ncomms11183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Picardi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Zayats,  and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Rodr´ıguez-  Fortu˜no, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' 120, 117402 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [10] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Gonzalez-Ballestero, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Moreno, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Garcia-Vidal,  and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Gonzalez-Tudela, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' A 94, 063817 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [11] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Sayrin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Junge, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Mitsch, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Albrecht, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' O’Shea,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Schneeweiss, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Volz,  and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Rauschenbeutel, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' X 5, 041036 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Young, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Thijssen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Beggs, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' An-  drovitsaneas, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Kuipers, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Rarity, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Hughes,  and  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Oulton, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' 115, 153901 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [13] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' S¨ollner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Mahmoodian, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Hansen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Midolo,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Javadi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Kirˇsansk˙e, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Pregnolato, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' El-Ella, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Song, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Stobbe,  and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Lodahl, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Nan-  otechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' 10, 775 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [14] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Lang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' McCutcheon, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Harbord, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Young,  and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Oulton, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' 128, 073602 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [15] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Manga Rao and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Hughes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' B 75,  205437 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [16] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Collett, Field Guide to Polarization (SPIE press,  2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [17] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Jones, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Buonaiuto, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Lang, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Lesanovsky,  and  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Olmos, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' 124, 093601 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [18] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Dzsotjan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' K¨astel, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Fleischhauer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  B 84, 075419 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [19] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Jones, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Needham, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Lesanovsky, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Intravaia,  and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Olmos, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' A 97, 053841 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [20] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Zhou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Chen,  and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Shen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' A 101,  043831 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [21] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Mirza and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Schotland, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' A 94, 012302  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [22] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Pichler, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Ramos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Daley, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Zoller, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' A 91, 042116 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [23] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Holzinger, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Guti´errez-J´auregui, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' H¨onigl-Decrinis,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Kirchmair, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Asenjo-Garcia,  and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Ritsch, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' 129, 253601 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [24] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Wang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Liu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Kockum, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Li, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Nori,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' 126, 043602 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [25] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Scarpelli, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Lang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Masia, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Beggs, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Mul-  jarov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Young, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Oulton, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Kamp, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' H¨ofling,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Schneider,  and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Langbein, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' B 100,  035311 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [26] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Nye and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Berry, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Royal Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' A 336, 165  (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Berry, Nature 403, 21 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [28] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Berry, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Dennis, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Lee, New Journal  of Physics 6, 162 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  [29] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' Padgett, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' O’Holleran, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' King,  and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
+page_content='  Dennis, Contemporary Physics 52, 265 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfiAR_/content/2301.03247v1.pdf'}
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+A Generalized Estimating Equation Approach to
+Network Regression
+Riddhi Pratim Ghosh1, Jukka-Pekka Onnela2, and Ian Barnett3
+1Department of Mathematics and Statistics, Bowling Green State University
+2Department of Biostatistics, Harvard University
+3Department of Biostatistics, University of Pennsylvania
+Abstract
+Regression models applied to network data where node attributes are the depen-
+dent variables poses a methodological challenge. As has been well studied, naive re-
+gression neither properly accounts for community structure, nor does it account for
+the dependent variable acting as both model outcome and covariate. To address this
+methodological gap, we propose a network regression model motivated by the important
+observation that controlling for community structure can, when a network is modu-
+lar, significantly account for meaningful correlation between observations induced by
+network connections. We propose a generalized estimating equation (GEE) approach
+to learn model parameters based on clusters defined through any single-membership
+community detection algorithm applied to the observed network. We provide a nec-
+essary condition on the network size and edge formation probabilities to establish the
+asymptotic normality of the model parameters under the assumption that the graph
+structure is a stochastic block model. We evaluate the performance of our approach
+through simulations and apply it to estimate the joint impact of baseline covariates
+1
+arXiv:2301.04804v1  [stat.ME]  12 Jan 2023
+
+and network effects on COVID-19 incidence rate among countries connected by a net-
+work of commercial airline traffic. We find that during the beginning of the pandemic
+the network effect has some influence, the percentage of urban population has more
+influence on the incidence rate compared to the network effect after the travel ban was
+in effect.
+Keywords: network regression; transportation networks; generalized estimating equations;
+COVID-19
+2
+
+1
+Introduction
+Network data provide quantitative information to study and unveil the pattern of interac-
+tions among various objects from individuals to countries. The inferential task in scientific
+applications requires learning about the influence that an individual has on others, which
+is further complicated as the individuals are often nested in communities. Two important
+methodological challenges of network regression are (1) accounting for correlation induced
+by network structure, and (2) allowing for node attributes to appear in the data both as
+outcome as well as covariates in the design matrix. For example, network regression was
+applied in the study of the nature and extent of the person-to-person spread of obesity,
+Christakis & Fowler (2007) found that obesity appears to spread through social ties, a phe-
+nomenon that may in part be explained by the notion of homophily – that birds of a feather
+flock together (Shrum et al., 1988; Igarashi et al., 2005; Tifferet, 2019). Sensitivity analyses
+suggest that contagion effects for obesity and smoking cessation are reasonably robust to
+possible latent homophily or environmental confounding; those for happiness and loneliness
+are somewhat less so (VanderWeele, 2011). To further investigate the causal relationship,
+Shalizi & Thomas (2011) provided three factors underlying such interaction: homophily, or
+the formation of social ties due to matching individual traits; social contagion; the causal
+effect of an individual’s covariates on his or her measurable responses; and an individual’s
+response. This has led to the development of two types of models: (1) community detection
+models that aim to find distinct communities or clusters of similar individuals (Holland et
+al., 1983; Newman, 2018), and (2) a more general framework of network regression that
+can model such phenomenon using a direct link between observed individual attributes or
+covariates and the network interactions (Holland & Leinhardt, 1981; P. D. Hoff et al., 2002;
+P. Hoff, 2021). Some approaches that aim at merging the above two models exploit distri-
+butional assumptions to incorporate covariate information in detecting communities in the
+network (Binkiewicz et al., 2017; Mu et al., 2022). These examples highlight the need for
+regression techniques that are robust to non-trivial network structure because in all these
+3
+
+studies community labels are known which makes the evaluation of covariate effects easy.
+However, in a more realistic situation, community labels are unknown (unobserved) with
+differences in covariates existing through the unobserved communities making the inference
+challenging.
+In this article, we aim to develop a network regression model motivated by the important
+observation that the differences between the communities in a network can be attributed to
+the differences in the influences of covariates responsible for an edge formation. Our model
+is based on a flexible network effect assumption, and it allows us to perform tests for the
+model parameters in addition to the estimation. Aside from detecting communities solely
+based on degree, the degree corrected stochastic block model was one of the first model
+that allows within community heterogeniety (Qin & Rohe, 2013). There has been a surge
+of interests among psychologists and social scientists to study such behavior- Aukett et al.
+(1988) shows that gender difference plays an important role in friendship patterns: women
+shows a preference for a few, closer, intimate same-sex friendships based on sharing emo-
+tions whereas men build up friendship based on the activities they do together. Staber (1993)
+studies how women and men form entrepreneurial relationships, concluding that women’s
+networks are wider with more strangers and higher proportion of cross-sex ties. In each of
+these network regression examples, community labels are known which alleviates the infer-
+ential task of finding the effects of covariates and learning community structures. However,
+in many other scenarios, community labels are unknown which makes the inferential task
+more challenging. For example, here we will consider the network regression problem of
+seeing the impact of air travel network flow between countries on COVID-19 incidence rates,
+where network structure is unknown and must be estimated. The current literature lacks
+powerful methods that can address this important issue. Our focus is to bridge this gap by
+leveraging covariate-assisted information to propose an effective tool which also accounts for
+the network modeling through adjacency matrix.
+Generalized estimating equations (GEE) (Liang & Zeger, 1986; Zeger, 1986) provide a
+4
+
+popular approach that is often used to analyze longitudinal and other types of correlated data
+(Burton et al., 1998; Diggle et al., 2002; Mandel et al., 2021) We propose first performing
+community detection in order to estimate community membership before then using a GEE
+regression model to account for the resulting estimated community structure. This approach
+is general and agnostic to the community detection algorithm used so long as each node can
+belong to at most one community.
+Given that network regression needs to account for
+correlation between attributes from nodes that are connected, a GEE allows for arbitrary
+correlation between nodes within the same community. Of note, this approach is best suited
+for highly modular networks so that the GEE assumption of independence between different
+communities is more accurate, though we explore its performance in less modular networks
+with more between-community mixing.
+The rest of this article is organized as follows. In Section 2 we introduce our network
+regression model along with our GEE extension, and we describe the transportation network
+we use to model country-specific COVID-19 incidence rates. In Section 3 we present the
+theoretical results followed by extensive simulation results and real data analysis. Finally,
+Section 4 concludes the article with a discussion.
+2
+Methods
+In this section, we introduce our network regression model and describe the air travel new-
+torks for each month of the first quarter of 2020 (the beginning of the COVID-19 pandemic)
+with country-specific information such as COVID-19 incidence rate, GDP, and population
+size. Next, we present a generalized estimating equation (GEE) approach to network regres-
+sion that accounts for network-induced correlation between observations.
+5
+
+2.1
+Model and notation
+We consider a directed network of n nodes and an n × n adjacency matrix A = (aij) with
+0’s on the diagonal (i.e. no self-loops). We denote the feature variable of the ith node by
+yi, the l-dimensional vector of covariates by α, and the corresponding design matrix by xi.
+Denoting the coefficient of interest by β, the network regression model for node attribute yi
+we consider:
+yi = α⊤xi + β ·
+�
+j̸=i
+Ajiyj/(n − 1) + ϵi,
+(1)
+where ϵi
+iid
+∼ N(0, σ2) and α ∈ Rl.
+One can note that the above model is reminiscent
+of the first-order autoregressive spatial model of Kelejian & Prucha (1998) which frequently
+contains a spatial lag of the dependent variable as a covariate that is spatially autoregressive.
+Using vector and matrix notation, the above model can also be written as
+y = X⊤α + β · Ay/(n − 1) + ϵ,
+where y is the concatenation of the yis in a vector of length n, and X = [x1 : x2 : ... : xn],
+is the matrix of covariates. Note, model (1) can be trivially adapted to generalized linear
+models with non-linear link functions and non-Gaussian data in the scope of GEE models.
+2.2
+Data description
+With 622 million confirmed cases and 6.5 million deaths globally as of October 01, 2022,
+the COVID-19 pandemic has had a tremendous impact on the world, shrinking the global
+economy by 5.2%, the largest recession in the history post World War II (Bank, 2020). The
+travel bans in places worldwide have severely affected the tourism industry, with estimated
+losses of 900 billion to 1.2 trillion USD and tourism down 58%-78% (Le et al., 2022). The
+6
+
+airline industry has also suffered heavily, with 43 airlines declaring bankruptcy and 193 of
+740 European airlines at risk of closing. Here we focus on the start of the pandemic covering
+the transition to travel bans across the world to study the relative effectiveness of travel bans
+for controlling and contributing to COVID-19 incidence rates.
+We use pandemic data from the Johns Hopkins University coronavirus data repository
+through April 30, 2020, (COVID, 19). Flight data are from the Official Airline Guide (OAG)
+(Strohmeier et al., 2021). Because only data for January and February 2020 are available
+from OAG, we used the estimated fight data for other time periods using the OpenSky
+Network database (Sch¨afer et al., 2014; Strohmeier et al., 2021). This database tracks the
+number of fights from one country to another over time, which we use to estimate country-
+to-country flight data for other months. We include as covariates the GDP, total population,
+and percentage of the urban population for each country in our network. Constructing a
+network based on by the flight data and incorporating the above country-specific attributes
+such as GDP, population, etc.
+as covariates through α in (1), we aim to estimate the
+effectiveness of travel bans through β in model (1).
+2.3
+Generalized estimation equation (GEE) approach
+Out network contains K communities where community k is defined by
+Ek = {i : gi = k},
+where index gi represents the community membership of node i, |Ek| = nk (number of nodes
+in community k) and � nk = n. Let yk and ϵk denote the concatenated vectors of yijs
+and ϵijs respectively, and Xk = [x1, x2, ..., xnk] denote the l × nk sub-matrix of covariates
+corresponding to cluster k.
+To fit our network regression model in the GEE framework, we use communities as
+clusters and use the following equation to model the node attributes of members of the kth
+7
+
+cluster:
+yk = X⊤
+k α + βZk + ϵk,
+(2)
+where Ak is the nk ×nk sub-matrix of A pertaining to the cluster k, and Zk = Akyk/(n−1).
+The marginal mean µk of yk has the form:
+µk = E(yk|Xk) = (Ink − βAk/(n − 1))−1X⊤
+k α,
+k = 1, 2, ..., K,
+where Ink is the identity matrix of order nk.
+Adopting a GEE approach (Liang & Zeger, 1986), the resulting estimating equation is
+given by
+K
+�
+k=1
+D⊤
+k V −1
+k
+(yk − µk) = 0,
+k = 1, 2, ..., K,
+(3)
+where Dk =
+∂µk
+∂(β,α)⊤ is of dimension nk × (l + 1) and Vk is the nk × nk working covariance
+matrix of yk. The explicit form for Dk is
+Dk = [− (Ink − (β/(n − 1))Ak)−1Ak(Ink − (β/(n − 1))Ak)−1X⊤α
+�
+��
+�
+nk×1
+:
+(Ink − (β/(n − 1))Ak)−1X⊤
+k
+�
+��
+�
+nk×l
+].
+One can note that Dk consists of two partitioned matrices where the first one corresponds
+to the network parameter β and the second one is due to the covariate α. We can solve
+for ˆα and ˆβ in equation (3) through iterative reweighted least squares, and use the robust
+sandwich covariance estimator to perform inference on α and β.
+8
+
+3
+Results
+3.1
+Theoretical results
+In this section, we prove the asymptotic normality of the resulting GEE estimator for β
+and α jointly. Towards this, we assume constant probabilities of edge formation between
+and within communities where we denote these quantities by p and q respectively as in a
+stochastic block model (Holland et al., 1983). Our proof of asymptotic normality hinges on
+Theorem 2 of Liang & Zeger (1986) which establishes the asymptotic normality of the re-
+gression parameter in the classical GEE approach under the assumption that the correlation
+parameter appropriately scaled by the number of communitites is consistent. Our primary
+distinction from this approach is that we must account for probabilities of edge formation
+instead of correlation between observations. Therefore, we first establish the consistency of
+p and q following Proposition 1 of Chen et al. (2021) and subsequently show asymptotic
+normality of (ˆβ, ˆα) from the estimating equation (3).
+3.1.1
+Consistency of ˆp and ˆq
+Proposition 3.1. Consider a network generated from a stochastic block model of K com-
+munities of size m so that total number of nodes is n = Km. Assume that mγp → p∗ and
+mγq → q∗ as n → ∞, where p∗ and q∗ are positive fixed constants, and γ ∈ [0, 2), then
+m1+γ/2(ˆp − p) and m1+γ/2(ˆq − q) are both oP(1).
+Proof. See the appendix in Section 5.1.
+Guided by the above proposition, we establish the asymptotic normality of the model
+parameters in the following theorem. The key difference with the classical formula is the
+inclusion of the community size in the covariance coming from the consistency of p and q.
+9
+
+3.1.2
+Asymptotic normality of ˆβ
+Theorem 3.2. Under the conditions of Proposition 3.1 K1/2m1+γ/2�
+(ˆβ, ˆα) − (β, α)
+�⊤ is
+asymptotically multivariate normal with zero mean and covariance given by
+V = lim
+K→∞Km2+γ�
+K
+�
+k=1
+D⊤
+k V −1
+k
+Dk
+�−1�
+K
+�
+k=1
+D⊤
+k V −1
+k
+cov(Yk)V −1
+k
+Dk
+��
+K
+�
+k=1
+D⊤
+k V −1
+k
+Dk
+�−1
+.
+Proof. See the appendix in Section 5.2.
+Remark: The variance formula involves the term m2+γ, where m is the community size.
+If m is large, then the covariance will increase at a rate m2+γ. This is reminiscent of the well-
+known fact that sandwich estimator ˆV of the covariance of (β, α)⊤ is not stable if m is large
+relative to K. In essence, if m grows at a similar rate to K the sandwich estimator becomes
+and unstable estimator of covariance. In practice this implies that the GEE approach works
+best when the network contains many smaller communities rather than only a few larger
+communities.
+3.2
+Simulations
+We simulate a network of n nodes having balanced communities of size m = 10 via a
+stochastic block model and vary n in {200, 400} with the number of communities (K) being
+20 and 40 for both values of n, respectively. Let p and q denote the within community and
+between community probabilities of an edge formation as in a stochastic block model. In
+each setting, we vary (p, q) ∈ {(0.8, 0), (0.7, 0.1), (0.6, 0.2), (0.5, 0.3)}.
+3.2.1
+Estimation of β and α
+The choice of true model parameters and the corresponding data-generating process are
+detailed to span networks with varying degrees of modularity. We set β0 = 0.5 and α0 =
+10
+
+(1, 1, 1, 1, 0.5, 0.5, 0.5, −0.5, −0.5, 2)⊤ (l = 10 in equation (1)). We simulate the l × n matrix
+X by a multivariate normal distribution in the following manner. The jth column of X, if
+it belongs to community k (k = 1, 2, ..., K), follows MVN((k/10)1l, 0.0001Il), where 1l and
+Il are the vector with 1s and identity matrix of dimension l, respectively. In each setting,
+the adjacency matrix A of dimension n is simulated by the stochastic block model which
+has K communities of the aforementioned size such that within and between community
+edge probabilities are p and q, respectively. Finally, the response variable yi is generated
+according to equation (1) with σ = 0.01.
+To estimate β and α, we first perform a community detection on the directed graph
+obtained from the adjacency matrix A as in Rosvall & Bergstrom (2007). Next, with the
+resulting communities we fit a GEE using the geepack R package (Halekoh et al., 2006) and
+report the estimates of bias and variance by taking the average of over B = 1000 replications.
+The squared bias and variance of the estimated β increase as the networks become less
+modular (i.e. q increases) for both our GEE approach as well as naive least squares (see
+Figure 1). The naive least squares method does not assume any community structure and
+reports the parameter estimates from assuming independence between observations using
+equation (1) directly.
+From Figure 1 we see that less modular networks are more difficult to fit in general based
+on the increase in bias for both methods. Despite this our GEE approach uniformly is less
+biased that naive least squares, which demonstrates that controlling for correlation within
+communities is effectively done by GEE. This also exposes the weakness of our GEE ap-
+proach: if the network is not modular with high degrees of mixing between communities (i.e.
+large q) the GEE framework cannot accommodate this due its assumption of independence
+between communities. However, even when q is large we still observe that GEE somewhat
+mitigates the impact of correlation induced by the network at least partially, which ex-
+plains why its bias is smaller than that of naive least squares. Essentially, even when the
+network structure suggests the GEE assumptions are incorrect, one is still generally better
+11
+
+off partially accounting for network structure using our GEE approach rather than ignore
+community structure altogether.
+The smaller standard errors demonstrated by least squares in Figure 1 reflect the inaccu-
+racy of the method. By ignoring community structure and assuming independence between
+observations in network regression settings, we expect to see anti-conservative hypothesis
+tests, too-narrow confidence intervals, and general overconfidence. This overconfidence is
+reflected in the too-small standard errors for naive least squares as well as in the anti-
+conservative Type I errors we demonstrate next.
+Figure 1: Bias squared and standard error of estimates of β with varying degrees
+of network modularity. In order to estimate β B = 1000 replications were performed for
+n = 200, 400 with the average degree 7 and 16 respectively. The values of (p, q) are varied
+over {(.8, 0), (.7, .1), (.6, .2), (.5, .3)}.
+3.2.2
+Hypothesis testing of β
+We consider an approach to testingthe hypothesis of H0 : β = 0 against the alternative
+HA : β ̸= 0. We perform a simulation study to obtain Type I error in a variety of network
+12
+
+SquaredBiasandStandardErrorofβ
+Squared bias of β
+o(β)
+1.50 
+0.003
+1.25 
+Method
+Method
+Squared bias of β
+GEE
+GEE
+0.002 
+LS
+LS
+a
+1.00
+n
+n
+200
+200
+0.001
+400
+400
+0.75 
+0.000-
+0'0
+0.1
+0.2
+0.0 
+0.1
+0.2
+0.3 
+q
+bstructures. We consider two working correlation structures for our GEE model: indepen-
+dence and exchangeable. First, we simulate our data as in Section 3.2 with β = 0. We
+obtain an empirical null distribution by replicating this procedure B = 1000 times to ob-
+tain ˆβ(1), ˆβ(2), ..., ˆβ(B). We estimate the P-value by �B
+b=1 I(|ˆβ(b)| < ˆβ)/B. We summarize
+our Type I error results at the 0.05 significance level in Table 1 which demonstrates that
+as network modularity decreases (q gets larger), the hypothesis test for H0 becomes anti-
+conservative. This is true for both least squares as well as GEE, although least squares is
+more adversely impacted. In addition, while GEE excels in the highly modular setting, as
+expected, LS is still anti-conservative. This demonstrates the efficacy of GEE as a better
+inferential tool over naive least squares in network regression settings, even when GEE as-
+sumptions of between-community independence do not hold. It is also instructive to note
+that although the P-values for independent correlation structure are generally less than those
+for exchangeable structure, the differences are not overwhelming.
+Table 1:
+Type I error for the test of H0 : β = 0. Comparison of Type-I error at the
+0.05 significance level for a network of n nodes with K balanced communities for different
+choices of within community edge probability (p) and between community edge probability
+(q) among GEE with independent and exchangeable correlation structure and naive least
+squares.
+(n, K)
+(p, q)
+GEE (independent)
+GEE (exchangeable)
+LS
+(0.8, 0)
+0.049
+0.055
+0.062
+(200, 20)
+(0.7, 0.1)
+0.050
+0.055
+0.079
+(0.6, 0.2)
+0.058
+0.061
+0.091
+(0.5, 0.3)
+0.072
+0.075
+0.095
+(0.8, 0)
+0.050
+0.050
+0.060
+(400, 40)
+(0.7, 0.1)
+0.051
+0.056
+0.075
+(0.6, 0.2)
+0.059
+0.062
+0.090
+(0.5, 0.3)
+0.070
+0.075
+0.096
+13
+
+3.3
+Real data analysis
+Air travel networks were constructed from flight data retrieved from the Official Airline
+Guide (OAG), with node attribute outcomes being the country-specific COVID-19 incidence
+rates by month available from the Johns Hopkins University coronavirus data repository
+through April 30, 2020, (COVID, 19). In addition we included country-specific GDP (WBG,
+2020a),total population (WBG, 2020b), and percentage of the urban population (WBG,
+2020c)of the countries from the website of the World Bank as covariates. We study the effec-
+tiveness of travel bans on the incidence rate of the COVID-19 using our network regression
+model by performing a month-by-month analysis from January-April, 2020 which span the
+period of before the pandemic till one month after the travel restriction was in effect. In
+addition, we study the importance of baseline covariates effects such as GDP, percentage of
+urban population, and population size of the countries vs the network effect. The GDP and
+the population of the most populated countries are in the order of 1012 and 106 respectively,
+so we scale these variables by 1012, 106, and 102, respectively, in order to stablize coefficient
+estimation.
+With these data retrieved across different continents to model the network effect through
+our porposed model in (1) we assume a directed stochastic block model framework, where
+nodes correspond to countries and each block contains countries having a large number of
+commercial flights traveling among them compared to the others. Edge formation is deter-
+mined by thresholding the population-normalized count of flights arriving at the destination
+country.
+We let the incidence rate yi of the ith country to be the number of cases per 1000
+populations, and the covariates xin×4 = (1n×1 : xi2n×1 : xi3n×1 : xi4n×1) to be a matrix
+whose first column has 1s as the intercept and the second, third and fourth columns have
+the population size, GDP, percentage of the urban population of the ith country scaled by
+106, 1012, and 102 respectively. For constructing the adjacency matrix A, we consider two
+scenarios: unweighted and weighted. For unweighted adjacency matrices, from the directed
+14
+
+Figure 2: Changes in air travel networks and the impact of incidence rates over
+time.
+Monthly average number of flights per million population over the countries and
+monthly estimate of β for Jan-Apr 2020 demonstrate a downward and an upward trend,
+respectively.
+graph dictated by the number of flights in a particular month, we first construct a count
+matrix C that counts the number of flights from one country to the other. Then we construct
+an unweighted adjacency matrix A with 0s and 1s where the entry of A is 1 if it exceeds
+the third quartile of the elements of C, and 0 otherwise. For weighted adjacency matrices
+we divide the elements of C by the total population of the destination country per million.
+The main rationale behind this scaling is that one would expect more flights traveling to
+a populated country compared to a less populated country. Therefore, dividing it by the
+population of the country will result in the elements of the weighted adjacency matrix being
+on the same scale.
+With the two adjacency matrices constructed in this way, we fit model (3) and summarize
+the results in Table 2. The estimates of β increase from February to April both for the
+weighted and unweighed cases suggesting that there is an increasing association between the
+travel and spread of the pandemic. To further investigate this behavior, we plot the average
+number of flights per million population in Figure 2 which shows that although the average
+15
+
+0.03
+150
+Averageflights permillion population
+Estimateof
+β
+0.02
+100
+0.01
+ 50 
+0.00
+Jan
+Feb
+Mar
+Apr
+Jan
+Feb
+Mar
+Apr
+Month
+Monthnumber of flights is decreasing from January to April, the estimates of β are increasing.
+This reflects the fact that while travel bans led to a decrease in the total number of flights in
+March and April, the increasing β in these months implies that each flight had an increased
+likelihood of transmitting COVID-19 to the destination country, increasing the correlation
+between the incidence rates in the two countries.
+Table 2 also demonstrates that while the overall population of a country has a negligible
+effect on the incidence rate leading to smaller values of α2, the percentage of the urban
+population plays a crucial role especially when the travel ban has already been in effect, for
+example, the value of α4 is 62 (55 for weighted case) in April compared to 0 in February.
+Moreover, the corresponding values are consistent for weighted and unweighted networks.
+Table 2 also demonstrates the utility of including baseline covariates alongside network
+effect to study the effectiveness of travel bans in mitigating the spread of COVID-19. By
+comparing the values of β and α4, one can note that for the initial months of the pandemic,
+the network effect is more compared to the effect of urban population. However, when the
+travel ban has already taken place during March (for most of the countries), the effect of
+urban population supersedes the network effect as its value increase to 62.33 drastically
+from 9.88 suggesting that urbanity is the next important factor to consider for controlling
+the spread of the pandemic after the travel bans.
+We also compare our results with naive least squares in Table 3 which shows coefficient
+estimates for both models. Of note, the April estimate of β is dramatically larger in the naive
+least squares model, which may be a manifestation of the increased bias we expect to see for
+naive least squares. Standard errors for the coefficient estimates are uniformly smaller for
+least squares as we expect, likely a representation of the overconfidence of the least squares
+model that comes from ignoring network induced correlation. This case demonstrates that
+using least squares would incorrectly dramatically increase both the magnitude and degree
+of significance of the network effects on incidence rates, particularly in the months post-
+lockdown when travel bans were in place. In contrast, the GEE model provides a more
+16
+
+realistic view.
+Table 2: Estimates of the parameters for unweighted and (weighted) networks
+under the GEE model. β, α1, α2, α3 and α4 correspond to the coefficients of the adjacency
+matrix (network effect), intercept, population, GDP, and percentage of the urban population,
+respectively.
+Month of 2020
+β
+α1
+α2
+α3
+α4
+Jan
+1.8759 (0.0038)
+-0.0002 (-0.0001)
+0.0000 (0.0000)
+0.0013 (0.0013)
+0.0003 (0.0000 )
+Feb
+2.1946 (0.0014)
+-0.0048 (-0.0037)
+0.0000 (0.0000)
+0.0555 (0.0575)
+-0.0094 (-0.0072 )
+Mar
+2.7186 (0.0030)
+-0.7846 (-0.7490)
+-0.0005 (-0.0007)
+-0.1446 (-0.0233)
+9.8844 (9.8862 )
+Apr
+4.6574 (0.0328)
+-4.7307 (-3.5655)
+-0.0085 (-0.0125)
+0.0626 (0.4685)
+62.3300 (54.9767)
+Table 3: Comparison of the naive linear regression with GEE for the weighted
+networks. The numbers reported in the parenthesis correspond to our GEE method while
+the others correspond to least squares. β, α1, α2, α3 and α4 correspond to the coefficients of
+the adjacency matrix, intercept, population, GDP, and percentage of the urban population
+respectively.
+Month of 2020
+β
+α1
+α2
+α3
+α4
+Jan
+0.0424 (0.0038)
+-0.0001 (-0.0001)
+0.0000 (0.0000)
+0.0013 (0.0013)
+0.0000 (0.0000 )
+Feb
+0.0105 (0.0014)
+-0.0037 (-0.0037)
+0.0000 (0.0000)
+0.0576 (0.0575)
+-0.0073 (-0.0072 )
+Mar
+0.0458 (0.0030)
+-0.7060 (-0.7490)
+-0.0009 (-0.0007)
+0.0017 (-0.0233)
+9.5519 (9.8862 )
+Apr
+1.2655 (0.0328)
+-5.5415 (-3.5655)
+-0.0073 (-0.0125)
+-0.2041 (0.4685)
+67.5900 (54.9767)
+4
+Discussion
+We have proposed a generalized estimating equation (GEE) approach to network regression
+model. Assuming independent community structure and using the simultaneous estimation
+of memberships, the GEE approach allows for community dependent covariate coefficient
+estimation, and thereby provides a flexible and efficient solution to the network regression
+model. Moreover, this allows us to do a hypothesis testing of the network regression pa-
+rameter β which helps us to decide the importance of including such term in our analysis.
+We provided a relevant real data example of COVID-19 cases along with baseline covari-
+ates such as GDP, population size, and percentage of urban population of countries across
+17
+
+different continents, and the number of commercial flights traveling between them to study
+the importance of travel bans to mitigate the spread of the COVID-19 pandemic. We have
+constructed the adjacency matrix from the count of flights rendered in a network of countries
+via a stochastic block model where each block contains countries having a similar number
+of flights. Our proposed model has helped us to understand the importance of the baseline
+covariates vs network effect. Since we have dealt with longitudinal data, it is also instructive
+to note that our proposed model offers us the flexibility of clustering both in the network
+space and also over time.
+One limitation of our results is the balanced community assumption made in Proposition
+3.1 which may not reflect some unbalanced networks. This assumption can be relaxed some-
+what with small departures from the balanced design. Further, under the stochastic block
+model, extremely unbalanced networks still allow for consistent estimation of p and q, how-
+ever for a more flexible network model that allows for community-specific edge probabilities,
+consistent estimation of edge probabilities requires each community to grow asymptotically
+with n.
+5
+Appendix
+5.1
+Proof of Proposition 3.1
+From the construction of the adjacency matrices, one can note that the entries Aijs are i.i.d.
+Bernoulli random variables with
+E(Aij) =
+�
+�
+�
+�
+�
+�
+�
+p,
+if i, j belong to the same community
+q,
+otherwise
+18
+
+and
+V ar(Aij) =
+�
+�
+�
+�
+�
+�
+�
+p(1 − p),
+if i, j belong to the same community
+q(1 − q),
+otherwise
+Denote by S the set {i, j : 1 ≤ i < j ≤ n, i, j belongs to the same community}, and its
+complement by S′. Therefore, for a directed graph, one can write
+E
+� �
+i,j∈S
+Aij
+�
+= 2K
+�m
+2
+�
+p,
+V ar
+� �
+i,j∈S
+Aij
+�
+= 2K
+�m
+2
+�
+p(1 − p) = s2
+mp,
+E
+� �
+i,j∈S′
+Aij
+�
+= K(K − 1)m2q, and
+V ar
+� �
+i,j∈S′
+Aij
+�
+= K(K − 1)m2q(1 − q) = s2
+mq.
+Next, one can verify Lindeberg condition to establish the central limit theorem:
+�
+i,j∈S Aij − 2K
+�m
+2
+�
+p
+�
+2K
+�m
+2
+�
+p(1 − p)
+and
+�
+i,j∈S′ Aij − K(K − 1)m2q
+�
+K(K − 1)m2q(1 − q)
+d→ N(0, 1)
+(4)
+as Km2p(1 − p) and Km2q(1 − q) → ∞.
+The Lindeberg’s condition requires us to verify
+1
+s2
+mp
+�
+i,j∈S
+E
+�
+B2
+ijI{|Bij|≥ϵsmp}
+�
+→ 0
+and
+1
+s2
+mq
+�
+i,j∈S′
+E
+�
+B2
+ijI{|Bij|≥ϵsmq }
+�
+→ 0,
+19
+
+where Bij = Aij −E(Aij). Since |Bij| ≤ 1, the above condition is satisfied when Km2p(1−p)
+and Km2q(1 − q) tend to ∞ under which ϵsmp and ϵsmq are both greater than 1.
+Dividing the numerator and denominator of (4) by 2K
+�m
+2
+�
+and K(K −1)m2 respectively,
+one obtains both
+mγ/2�
+2K
+�m
+2
+�
+�
+i,j∈S Aij/
+�
+2K(m
+2)
+�
+−p
+√
+mγp(1−p)
+and
+mγ/2�
+K(K − 1)m2
+�
+i,j∈S′ Aij/
+�
+K(K−1)m2
+�
+−q
+√
+mγq(1−q)
+both converge to N(0, 1) distribution. Since ˆp = �
+i,j∈S Aij/
+�
+K
+�m
+2
+��
+, and ˆq = �
+i,j∈S′ Aij/
+�
+K(K−
+1)m2�
+, both ˆp and ˆq are m1+γ/2 consistent.
+5.2
+Proof of Theorem 3.2
+Here we present the sketch of the proof of Theorem 3.2. One can write (3) as
+K
+�
+k=1
+Uk(α, β, p, q) =
+K
+�
+k=1
+D⊤
+k V −1
+k
+Sk = 0,
+(5)
+where Sk = yk − µk, and U is a function of the model parameters.
+Let b = (β, α)⊤ denote the vector of model parameters, and π = (p, q)⊤ denote the
+vector of model parameters, and vector of within and between edge probabilities respectively.
+Letting b fixed, the Taylor expansion yields
+�K
+k=1 Uk(b, π∗)
+K1/2m1+γ/2
+=
+�K
+k=1 Uk(b, π)
+K1/2m1+γ/2
++
+�K
+k=1
+∂Uk(b,π)
+∂π
+K1/2m1+γ/2 m1+γ/2(π∗ − π) + oP(1)
+(6)
+= ˜A + ˜B ˜C + oP(1),
+One can note that ˜B = oP(1) as ∂Uk(b, π)/∂π are linear functions of Sk’s defined in (5)
+whose means are zero, and ˜C = OP(1) thanks to Proposition 3.1. Therefore,
+�K
+k=1 Uk(b,π∗)
+K1/2m1+γ/2
+is asymptotically equivalent to
+�K
+k=1 Uk(b,π)
+K1/2m1+γ/2
+whose asymptotic distribution is multivariate
+20
+
+normal with zero mean and covariance is equal to V as defined in Theorem 3.2. The proof
+is thus complete following Liang & Zeger (1986).
+6
+Acknowledgement
+RG would like to thank Anupam Kundu, postdoctoral associate at Yale School of Public
+Health and Thien Le, postdoctoral fellow at Harvard for many useful discussions regarding
+real data processing. JP acknowledges support from R01AI138901, and IB acknowledges
+support from R01MH116884.
+References
+Aukett, R., Ritchie, J., & Mill, K. (1988). Gender differences in friendship patterns. Sex
+roles, 19(1), 57–66.
+Bank, W. (2020). Covid-19 to plunge global economy into worst recession since World War
+II. World Bank Group Washington, DC.
+Binkiewicz, N., Vogelstein, J. T., & Rohe, K. (2017). Covariate-assisted spectral clustering.
+Biometrika, 104(2), 361–377.
+Burton, P., Gurrin, L., & Sly, P. (1998). Extending the simple linear regression model to
+account for correlated responses: an introduction to generalized estimating equations and
+multi-level mixed modelling. Statistics in Medicine, 17(11), 1261–1291.
+Chen, M., Kato, K., & Leng, C. (2021). Analysis of networks via the sparse β-model. Journal
+of the Royal Statistical Society: Series B (Statistical Methodology).
+Christakis, N. A., & Fowler, J. H. (2007). The spread of obesity in a large social network
+over 32 years. New England Journal of Medicine, 357(4), 370–379.
+21
+
+COVID, G.
+(19).
+database (2021) retrieved from https://github. com/cssegisanddata.
+COVID-19.
+Diggle, P., Diggle, P. J., Heagerty, P., Liang, K.-Y., Zeger, S., et al. (2002). Analysis of
+longitudinal data. Oxford University Press.
+Halekoh, U., Højsgaard, S., & Yan, J.
+(2006).
+The r package geepack for generalized
+estimating equations. Journal of Statistical Software, 15/2, 1–11.
+Hoff, P. (2021). Additive and multiplicative effects network models. Statistical Science,
+36(1), 34–50.
+Hoff, P. D., Raftery, A. E., & Handcock, M. S. (2002). Latent space approaches to social
+network analysis. Journal of the American Statistical Association, 97(460), 1090–1098.
+Holland, P. W., Laskey, K. B., & Leinhardt, S. (1983). Stochastic blockmodels: First steps.
+Social Networks, 5(2), 109–137.
+Holland, P. W., & Leinhardt, S. (1981). An exponential family of probability distributions
+for directed graphs. Journal of the American Statistical Association, 76(373), 33–50.
+Igarashi, T., Takai, J., & Yoshida, T. (2005). Gender differences in social network devel-
+opment via mobile phone text messages: A longitudinal study. Journal of Social and
+Personal Relationships, 22(5), 691–713.
+Kelejian, H. H., & Prucha, I. R.
+(1998).
+A generalized spatial two-stage least squares
+procedure for estimating a spatial autoregressive model with autoregressive disturbances.
+The Journal of Real Estate Finance and Economics, 17(1), 99–121.
+Le, T.-M., Raynal, L., Talbot, O., Hambridge, H., Drovandi, C., Mira, A., . . . Onnela, J.-P.
+(2022). Framework for assessing and easing global covid-19 travel restrictions. Scientific
+Reports, 12(1), 1–13.
+22
+
+Liang, K.-Y., & Zeger, S. L. (1986). Longitudinal data analysis using generalized linear
+models. Biometrika, 73(1), 13–22.
+Mandel, F., Ghosh, R. P., & Barnett, I. (2021). Neural networks for clustered and longitu-
+dinal data using mixed effects models. Biometrics.
+Mu, C., Mele, A., Hao, L., Cape, J., Athreya, A., & Priebe, C. E. (2022). On spectral algo-
+rithms for community detection in stochastic blockmodel graphs with vertex covariates.
+IEEE Transactions on Network Science and Engineering.
+Newman, M. E. (2018). Network structure from rich but noisy data. Nature Physics, 14(6),
+542–545.
+Qin, T., & Rohe, K. (2013). Regularized spectral clustering under the degree-corrected
+stochastic blockmodel. Advances in Neural Information Processing Systems, 26.
+Rosvall, M., & Bergstrom, C. T. (2007). Maps of information flow reveal community structure
+in complex networks. arXiv preprint physics.soc-ph/0707.0609.
+Sch¨afer, M., Strohmeier, M., Lenders, V., Martinovic, I., & Wilhelm, M. (2014). Bringing
+up opensky: A large-scale ads-b sensor network for research. In Ipsn-14 proceedings of
+the 13th International Symposium on Information Processing in Sensor Networks (pp.
+83–94).
+Shalizi, C. R., & Thomas, A. C. (2011). Homophily and contagion are generically confounded
+in observational social network studies. Sociological Methods & Research, 40(2), 211–239.
+Shrum, W., Cheek Jr, N. H., & MacD, S. (1988). Friendship in school: Gender and racial
+homophily. Sociology of Education, 227–239.
+Staber, U. (1993). Friends, acquaintances, strangers: Gender differences in the structure of
+entrepreneurial networks. Journal of Small Business & Entrepreneurship, 11(1), 73–82.
+23
+
+Strohmeier, M., Olive, X., L¨ubbe, J., Sch¨afer, M., & Lenders, V. (2021). Crowdsourced
+air traffic data from the opensky network 2019–2020. Earth System Science Data, 13(2),
+357–366.
+Tifferet, S. (2019). Gender differences in privacy tendencies on social network sites: A
+meta-analysis. Computers in Human Behavior, 93, 1–12.
+VanderWeele, T. J. (2011). Sensitivity analysis for contagion effects in social networks.
+Sociological Methods & Research, 40(2), 240–255.
+WBG.
+(2020a).
+Available online: https://data.worldbank.org/indicator/ny.gdp.mktp.cd.
+CD (accessed on 2 September 2022).
+WBG. (2020b). Available online: https://data.worldbank.org/indicator/sp.pop.totl. CD
+(accessed on 2 September 2022).
+WBG. (2020c). Available online: https://https://data.worldbank.org/indicator/sp.urb.totl.in.zs.
+CD (accessed on 2 September 2022).
+Zeger, S. (1986). The analysis of discrete and continuous longitudinal data. Biometrics, 42,
+121–130.
+24
+
diff --git a/p9E3T4oBgHgl3EQf8Aun/content/tmp_files/load_file.txt b/p9E3T4oBgHgl3EQf8Aun/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,665 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf,len=664
+page_content='A Generalized Estimating Equation Approach to  Network Regression  Riddhi Pratim Ghosh1, Jukka-Pekka Onnela2, and Ian Barnett3  1Department of Mathematics and Statistics, Bowling Green State University  2Department of Biostatistics, Harvard University  3Department of Biostatistics, University of Pennsylvania  Abstract  Regression models applied to network data where node attributes are the depen-  dent variables poses a methodological challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' As has been well studied, naive re-  gression neither properly accounts for community structure, nor does it account for  the dependent variable acting as both model outcome and covariate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' To address this  methodological gap, we propose a network regression model motivated by the important  observation that controlling for community structure can, when a network is modu-  lar, significantly account for meaningful correlation between observations induced by  network connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' We propose a generalized estimating equation (GEE) approach  to learn model parameters based on clusters defined through any single-membership  community detection algorithm applied to the observed network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' We provide a nec-  essary condition on the network size and edge formation probabilities to establish the  asymptotic normality of the model parameters under the assumption that the graph  structure is a stochastic block model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' We evaluate the performance of our approach  through simulations and apply it to estimate the joint impact of baseline covariates  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='04804v1  [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='ME]  12 Jan 2023  and network effects on COVID-19 incidence rate among countries connected by a net-  work of commercial airline traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' We find that during the beginning of the pandemic  the network effect has some influence, the percentage of urban population has more  influence on the incidence rate compared to the network effect after the travel ban was  in effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Keywords: network regression;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' transportation networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' generalized estimating equations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  COVID-19  2  1  Introduction  Network data provide quantitative information to study and unveil the pattern of interac-  tions among various objects from individuals to countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' The inferential task in scientific  applications requires learning about the influence that an individual has on others, which  is further complicated as the individuals are often nested in communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Two important  methodological challenges of network regression are (1) accounting for correlation induced  by network structure, and (2) allowing for node attributes to appear in the data both as  outcome as well as covariates in the design matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' For example, network regression was  applied in the study of the nature and extent of the person-to-person spread of obesity,  Christakis & Fowler (2007) found that obesity appears to spread through social ties, a phe-  nomenon that may in part be explained by the notion of homophily – that birds of a feather  flock together (Shrum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Igarashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Tifferet, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Sensitivity analyses  suggest that contagion effects for obesity and smoking cessation are reasonably robust to  possible latent homophily or environmental confounding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' those for happiness and loneliness  are somewhat less so (VanderWeele, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' To further investigate the causal relationship,  Shalizi & Thomas (2011) provided three factors underlying such interaction: homophily, or  the formation of social ties due to matching individual traits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' social contagion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' the causal  effect of an individual’s covariates on his or her measurable responses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' and an individual’s  response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' This has led to the development of two types of models: (1) community detection  models that aim to find distinct communities or clusters of similar individuals (Holland et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Newman, 2018), and (2) a more general framework of network regression that  can model such phenomenon using a direct link between observed individual attributes or  covariates and the network interactions (Holland & Leinhardt, 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Hoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Hoff, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Some approaches that aim at merging the above two models exploit distri-  butional assumptions to incorporate covariate information in detecting communities in the  network (Binkiewicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Mu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' These examples highlight the need for  regression techniques that are robust to non-trivial network structure because in all these  3  studies community labels are known which makes the evaluation of covariate effects easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  However, in a more realistic situation, community labels are unknown (unobserved) with  differences in covariates existing through the unobserved communities making the inference  challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  In this article, we aim to develop a network regression model motivated by the important  observation that the differences between the communities in a network can be attributed to  the differences in the influences of covariates responsible for an edge formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Our model  is based on a flexible network effect assumption, and it allows us to perform tests for the  model parameters in addition to the estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Aside from detecting communities solely  based on degree, the degree corrected stochastic block model was one of the first model  that allows within community heterogeniety (Qin & Rohe, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' There has been a surge  of interests among psychologists and social scientists to study such behavior- Aukett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  (1988) shows that gender difference plays an important role in friendship patterns: women  shows a preference for a few, closer, intimate same-sex friendships based on sharing emo-  tions whereas men build up friendship based on the activities they do together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Staber (1993)  studies how women and men form entrepreneurial relationships, concluding that women’s  networks are wider with more strangers and higher proportion of cross-sex ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' In each of  these network regression examples, community labels are known which alleviates the infer-  ential task of finding the effects of covariates and learning community structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' However,  in many other scenarios, community labels are unknown which makes the inferential task  more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' For example, here we will consider the network regression problem of  seeing the impact of air travel network flow between countries on COVID-19 incidence rates,  where network structure is unknown and must be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' The current literature lacks  powerful methods that can address this important issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Our focus is to bridge this gap by  leveraging covariate-assisted information to propose an effective tool which also accounts for  the network modeling through adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Generalized estimating equations (GEE) (Liang & Zeger, 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Zeger, 1986) provide a  4  popular approach that is often used to analyze longitudinal and other types of correlated data  (Burton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Diggle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Mandel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', 2021) We propose first performing  community detection in order to estimate community membership before then using a GEE  regression model to account for the resulting estimated community structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' This approach  is general and agnostic to the community detection algorithm used so long as each node can  belong to at most one community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Given that network regression needs to account for  correlation between attributes from nodes that are connected, a GEE allows for arbitrary  correlation between nodes within the same community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Of note, this approach is best suited  for highly modular networks so that the GEE assumption of independence between different  communities is more accurate, though we explore its performance in less modular networks  with more between-community mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  The rest of this article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' In Section 2 we introduce our network  regression model along with our GEE extension, and we describe the transportation network  we use to model country-specific COVID-19 incidence rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' In Section 3 we present the  theoretical results followed by extensive simulation results and real data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Finally,  Section 4 concludes the article with a discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  2  Methods  In this section, we introduce our network regression model and describe the air travel new-  torks for each month of the first quarter of 2020 (the beginning of the COVID-19 pandemic)  with country-specific information such as COVID-19 incidence rate, GDP, and population  size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Next, we present a generalized estimating equation (GEE) approach to network regres-  sion that accounts for network-induced correlation between observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='1  Model and notation  We consider a directed network of n nodes and an n × n adjacency matrix A = (aij) with  0’s on the diagonal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' no self-loops).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' We denote the feature variable of the ith node by  yi, the l-dimensional vector of covariates by α, and the corresponding design matrix by xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Denoting the coefficient of interest by β, the network regression model for node attribute yi  we consider:  yi = α⊤xi + β ·  �  j̸=i  Ajiyj/(n − 1) + ϵi,  (1)  where ϵi  iid  ∼ N(0, σ2) and α ∈ Rl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  One can note that the above model is reminiscent  of the first-order autoregressive spatial model of Kelejian & Prucha (1998) which frequently  contains a spatial lag of the dependent variable as a covariate that is spatially autoregressive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Using vector and matrix notation, the above model can also be written as  y = X⊤α + β · Ay/(n − 1) + ϵ,  where y is the concatenation of the yis in a vector of length n, and X = [x1 : x2 : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' : xn],  is the matrix of covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Note, model (1) can be trivially adapted to generalized linear  models with non-linear link functions and non-Gaussian data in the scope of GEE models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='2  Data description  With 622 million confirmed cases and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='5 million deaths globally as of October 01, 2022,  the COVID-19 pandemic has had a tremendous impact on the world, shrinking the global  economy by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='2%, the largest recession in the history post World War II (Bank, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' The  travel bans in places worldwide have severely affected the tourism industry, with estimated  losses of 900 billion to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='2 trillion USD and tourism down 58%-78% (Le et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' The  6  airline industry has also suffered heavily, with 43 airlines declaring bankruptcy and 193 of  740 European airlines at risk of closing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Here we focus on the start of the pandemic covering  the transition to travel bans across the world to study the relative effectiveness of travel bans  for controlling and contributing to COVID-19 incidence rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  We use pandemic data from the Johns Hopkins University coronavirus data repository  through April 30, 2020, (COVID, 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Flight data are from the Official Airline Guide (OAG)  (Strohmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Because only data for January and February 2020 are available  from OAG, we used the estimated fight data for other time periods using the OpenSky  Network database (Sch¨afer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Strohmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' This database tracks the  number of fights from one country to another over time, which we use to estimate country-  to-country flight data for other months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' We include as covariates the GDP, total population,  and percentage of the urban population for each country in our network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Constructing a  network based on by the flight data and incorporating the above country-specific attributes  such as GDP, population, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  as covariates through α in (1), we aim to estimate the  effectiveness of travel bans through β in model (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='3  Generalized estimation equation (GEE) approach  Out network contains K communities where community k is defined by  Ek = {i : gi = k},  where index gi represents the community membership of node i, |Ek| = nk (number of nodes  in community k) and � nk = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Let yk and ϵk denote the concatenated vectors of yijs  and ϵijs respectively, and Xk = [x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', xnk] denote the l × nk sub-matrix of covariates  corresponding to cluster k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  To fit our network regression model in the GEE framework, we use communities as  clusters and use the following equation to model the node attributes of members of the kth  7  cluster:  yk = X⊤  k α + βZk + ϵk,  (2)  where Ak is the nk ×nk sub-matrix of A pertaining to the cluster k, and Zk = Akyk/(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  The marginal mean µk of yk has the form:  µk = E(yk|Xk) = (Ink − βAk/(n − 1))−1X⊤  k α,  k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', K,  where Ink is the identity matrix of order nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Adopting a GEE approach (Liang & Zeger, 1986), the resulting estimating equation is  given by  K  �  k=1  D⊤  k V −1  k  (yk − µk) = 0,  k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', K,  (3)  where Dk =  ∂µk  ∂(β,α)⊤ is of dimension nk × (l + 1) and Vk is the nk × nk working covariance  matrix of yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' The explicit form for Dk is  Dk = [− (Ink − (β/(n − 1))Ak)−1Ak(Ink − (β/(n − 1))Ak)−1X⊤α  �  ��  �  nk×1  :  (Ink − (β/(n − 1))Ak)−1X⊤  k  �  ��  �  nk×l  ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  One can note that Dk consists of two partitioned matrices where the first one corresponds  to the network parameter β and the second one is due to the covariate α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' We can solve  for ˆα and ˆβ in equation (3) through iterative reweighted least squares, and use the robust  sandwich covariance estimator to perform inference on α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  8  3  Results  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='1  Theoretical results  In this section, we prove the asymptotic normality of the resulting GEE estimator for β  and α jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Towards this, we assume constant probabilities of edge formation between  and within communities where we denote these quantities by p and q respectively as in a  stochastic block model (Holland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Our proof of asymptotic normality hinges on  Theorem 2 of Liang & Zeger (1986) which establishes the asymptotic normality of the re-  gression parameter in the classical GEE approach under the assumption that the correlation  parameter appropriately scaled by the number of communitites is consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Our primary  distinction from this approach is that we must account for probabilities of edge formation  instead of correlation between observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Therefore, we first establish the consistency of  p and q following Proposition 1 of Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' (2021) and subsequently show asymptotic  normality of (ˆβ, ˆα) from the estimating equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='1  Consistency of ˆp and ˆq  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Consider a network generated from a stochastic block model of K com-  munities of size m so that total number of nodes is n = Km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Assume that mγp → p∗ and  mγq → q∗ as n → ∞, where p∗ and q∗ are positive fixed constants, and γ ∈ [0, 2), then  m1+γ/2(ˆp − p) and m1+γ/2(ˆq − q) are both oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' See the appendix in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Guided by the above proposition, we establish the asymptotic normality of the model  parameters in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' The key difference with the classical formula is the  inclusion of the community size in the covariance coming from the consistency of p and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='2  Asymptotic normality of ˆβ  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Under the conditions of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='1 K1/2m1+γ/2�  (ˆβ, ˆα) − (β, α)  �⊤ is  asymptotically multivariate normal with zero mean and covariance given by  V = lim  K→∞Km2+γ�  K  �  k=1  D⊤  k V −1  k  Dk  �−1�  K  �  k=1  D⊤  k V −1  k  cov(Yk)V −1  k  Dk  ��  K  �  k=1  D⊤  k V −1  k  Dk  �−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' See the appendix in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Remark: The variance formula involves the term m2+γ, where m is the community size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  If m is large, then the covariance will increase at a rate m2+γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' This is reminiscent of the well-  known fact that sandwich estimator ˆV of the covariance of (β, α)⊤ is not stable if m is large  relative to K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' In essence, if m grows at a similar rate to K the sandwich estimator becomes  and unstable estimator of covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' In practice this implies that the GEE approach works  best when the network contains many smaller communities rather than only a few larger  communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='2  Simulations  We simulate a network of n nodes having balanced communities of size m = 10 via a  stochastic block model and vary n in {200, 400} with the number of communities (K) being  20 and 40 for both values of n, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Let p and q denote the within community and  between community probabilities of an edge formation as in a stochastic block model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' In  each setting, we vary (p, q) ∈ {(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='8, 0), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='1), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='2), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='1  Estimation of β and α  The choice of true model parameters and the corresponding data-generating process are  detailed to span networks with varying degrees of modularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' We set β0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='5 and α0 =  10  (1, 1, 1, 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='5, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='5, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='5, 2)⊤ (l = 10 in equation (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' We simulate the l × n matrix  X by a multivariate normal distribution in the following manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' The jth column of X, if  it belongs to community k (k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', K), follows MVN((k/10)1l, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='0001Il), where 1l and  Il are the vector with 1s and identity matrix of dimension l, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' In each setting,  the adjacency matrix A of dimension n is simulated by the stochastic block model which  has K communities of the aforementioned size such that within and between community  edge probabilities are p and q, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Finally, the response variable yi is generated  according to equation (1) with σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  To estimate β and α, we first perform a community detection on the directed graph  obtained from the adjacency matrix A as in Rosvall & Bergstrom (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Next, with the  resulting communities we fit a GEE using the geepack R package (Halekoh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', 2006) and  report the estimates of bias and variance by taking the average of over B = 1000 replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  The squared bias and variance of the estimated β increase as the networks become less  modular (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' q increases) for both our GEE approach as well as naive least squares (see  Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' The naive least squares method does not assume any community structure and  reports the parameter estimates from assuming independence between observations using  equation (1) directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  From Figure 1 we see that less modular networks are more difficult to fit in general based  on the increase in bias for both methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Despite this our GEE approach uniformly is less  biased that naive least squares, which demonstrates that controlling for correlation within  communities is effectively done by GEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' This also exposes the weakness of our GEE ap-  proach: if the network is not modular with high degrees of mixing between communities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  large q) the GEE framework cannot accommodate this due its assumption of independence  between communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' However, even when q is large we still observe that GEE somewhat  mitigates the impact of correlation induced by the network at least partially, which ex-  plains why its bias is smaller than that of naive least squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Essentially, even when the  network structure suggests the GEE assumptions are incorrect, one is still generally better  11  off partially accounting for network structure using our GEE approach rather than ignore  community structure altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  The smaller standard errors demonstrated by least squares in Figure 1 reflect the inaccu-  racy of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' By ignoring community structure and assuming independence between  observations in network regression settings, we expect to see anti-conservative hypothesis  tests, too-narrow confidence intervals, and general overconfidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' This overconfidence is  reflected in the too-small standard errors for naive least squares as well as in the anti-  conservative Type I errors we demonstrate next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Figure 1: Bias squared and standard error of estimates of β with varying degrees  of network modularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' In order to estimate β B = 1000 replications were performed for  n = 200, 400 with the average degree 7 and 16 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' The values of (p, q) are varied  over {(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='8, 0), (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='7, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='1), (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='2), (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='2  Hypothesis testing of β  We consider an approach to testingthe hypothesis of H0 : β = 0 against the alternative  HA : β ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' We perform a simulation study to obtain Type I error in a variety of network  12  SquaredBiasandStandardErrorofβ  Squared bias of β  o(β)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='003  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='25  Method  Method  Squared bias of β  GEE  GEE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='002  LS  LS  a  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='00  n  n  200  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='001  400  400  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content="000-  0'0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='3  q  bstructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' We consider two working correlation structures for our GEE model: indepen-  dence and exchangeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' First, we simulate our data as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='2 with β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' We  obtain an empirical null distribution by replicating this procedure B = 1000 times to ob-  tain ˆβ(1), ˆβ(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', ˆβ(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' We estimate the P-value by �B  b=1 I(|ˆβ(b)| < ˆβ)/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' We summarize  our Type I error results at the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='05 significance level in Table 1 which demonstrates that  as network modularity decreases (q gets larger), the hypothesis test for H0 becomes anti-  conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' This is true for both least squares as well as GEE, although least squares is  more adversely impacted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' In addition, while GEE excels in the highly modular setting, as  expected, LS is still anti-conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' This demonstrates the efficacy of GEE as a better  inferential tool over naive least squares in network regression settings, even when GEE as-  sumptions of between-community independence do not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' It is also instructive to note  that although the P-values for independent correlation structure are generally less than those  for exchangeable structure, the differences are not overwhelming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Table 1:  Type I error for the test of H0 : β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Comparison of Type-I error at the  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='05 significance level for a network of n nodes with K balanced communities for different  choices of within community edge probability (p) and between community edge probability  (q) among GEE with independent and exchangeable correlation structure and naive least  squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  (n, K)  (p, q)  GEE (independent)  GEE (exchangeable)  LS  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
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+page_content='096  13  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='3  Real data analysis  Air travel networks were constructed from flight data retrieved from the Official Airline  Guide (OAG), with node attribute outcomes being the country-specific COVID-19 incidence  rates by month available from the Johns Hopkins University coronavirus data repository  through April 30, 2020, (COVID, 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' In addition we included country-specific GDP (WBG,  2020a),total population (WBG, 2020b), and percentage of the urban population (WBG,  2020c)of the countries from the website of the World Bank as covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' We study the effec-  tiveness of travel bans on the incidence rate of the COVID-19 using our network regression  model by performing a month-by-month analysis from January-April, 2020 which span the  period of before the pandemic till one month after the travel restriction was in effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' In  addition, we study the importance of baseline covariates effects such as GDP, percentage of  urban population, and population size of the countries vs the network effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' The GDP and  the population of the most populated countries are in the order of 1012 and 106 respectively,  so we scale these variables by 1012, 106, and 102, respectively, in order to stablize coefficient  estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  With these data retrieved across different continents to model the network effect through  our porposed model in (1) we assume a directed stochastic block model framework, where  nodes correspond to countries and each block contains countries having a large number of  commercial flights traveling among them compared to the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Edge formation is deter-  mined by thresholding the population-normalized count of flights arriving at the destination  country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  We let the incidence rate yi of the ith country to be the number of cases per 1000  populations, and the covariates xin×4 = (1n×1 : xi2n×1 : xi3n×1 : xi4n×1) to be a matrix  whose first column has 1s as the intercept and the second, third and fourth columns have  the population size, GDP, percentage of the urban population of the ith country scaled by  106, 1012, and 102 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' For constructing the adjacency matrix A, we consider two  scenarios: unweighted and weighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' For unweighted adjacency matrices, from the directed  14  Figure 2: Changes in air travel networks and the impact of incidence rates over  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Monthly average number of flights per million population over the countries and  monthly estimate of β for Jan-Apr 2020 demonstrate a downward and an upward trend,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  graph dictated by the number of flights in a particular month, we first construct a count  matrix C that counts the number of flights from one country to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Then we construct  an unweighted adjacency matrix A with 0s and 1s where the entry of A is 1 if it exceeds  the third quartile of the elements of C, and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' For weighted adjacency matrices  we divide the elements of C by the total population of the destination country per million.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  The main rationale behind this scaling is that one would expect more flights traveling to  a populated country compared to a less populated country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Therefore, dividing it by the  population of the country will result in the elements of the weighted adjacency matrix being  on the same scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  With the two adjacency matrices constructed in this way, we fit model (3) and summarize  the results in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' The estimates of β increase from February to April both for the  weighted and unweighed cases suggesting that there is an increasing association between the  travel and spread of the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' To further investigate this behavior, we plot the average  number of flights per million population in Figure 2 which shows that although the average  15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='03  150  Averageflights permillion population  Estimateof  β  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='02  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='01  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='00  Jan  Feb  Mar  Apr  Jan  Feb  Mar  Apr  Month  Monthnumber of flights is decreasing from January to April, the estimates of β are increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  This reflects the fact that while travel bans led to a decrease in the total number of flights in  March and April, the increasing β in these months implies that each flight had an increased  likelihood of transmitting COVID-19 to the destination country, increasing the correlation  between the incidence rates in the two countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Table 2 also demonstrates that while the overall population of a country has a negligible  effect on the incidence rate leading to smaller values of α2, the percentage of the urban  population plays a crucial role especially when the travel ban has already been in effect, for  example, the value of α4 is 62 (55 for weighted case) in April compared to 0 in February.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Moreover, the corresponding values are consistent for weighted and unweighted networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Table 2 also demonstrates the utility of including baseline covariates alongside network  effect to study the effectiveness of travel bans in mitigating the spread of COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' By  comparing the values of β and α4, one can note that for the initial months of the pandemic,  the network effect is more compared to the effect of urban population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' However, when the  travel ban has already taken place during March (for most of the countries), the effect of  urban population supersedes the network effect as its value increase to 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='33 drastically  from 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='88 suggesting that urbanity is the next important factor to consider for controlling  the spread of the pandemic after the travel bans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  We also compare our results with naive least squares in Table 3 which shows coefficient  estimates for both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Of note, the April estimate of β is dramatically larger in the naive  least squares model, which may be a manifestation of the increased bias we expect to see for  naive least squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Standard errors for the coefficient estimates are uniformly smaller for  least squares as we expect, likely a representation of the overconfidence of the least squares  model that comes from ignoring network induced correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' This case demonstrates that  using least squares would incorrectly dramatically increase both the magnitude and degree  of significance of the network effects on incidence rates, particularly in the months post-  lockdown when travel bans were in place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' In contrast, the GEE model provides a more  16  realistic view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Table 2: Estimates of the parameters for unweighted and (weighted) networks  under the GEE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' β, α1, α2, α3 and α4 correspond to the coefficients of the adjacency  matrix (network effect), intercept, population, GDP, and percentage of the urban population,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Month of 2020  β  α1  α2  α3  α4  Jan  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
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+page_content='9767)  Table 3: Comparison of the naive linear regression with GEE for the weighted  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' The numbers reported in the parenthesis correspond to our GEE method while  the others correspond to least squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' β, α1, α2, α3 and α4 correspond to the coefficients of  the adjacency matrix, intercept, population, GDP, and percentage of the urban population  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Month of 2020  β  α1  α2  α3  α4  Jan  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
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+page_content='5900 (54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='9767)  4  Discussion  We have proposed a generalized estimating equation (GEE) approach to network regression  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Assuming independent community structure and using the simultaneous estimation  of memberships, the GEE approach allows for community dependent covariate coefficient  estimation, and thereby provides a flexible and efficient solution to the network regression  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Moreover, this allows us to do a hypothesis testing of the network regression pa-  rameter β which helps us to decide the importance of including such term in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  We provided a relevant real data example of COVID-19 cases along with baseline covari-  ates such as GDP, population size, and percentage of urban population of countries across  17  different continents, and the number of commercial flights traveling between them to study  the importance of travel bans to mitigate the spread of the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' We have  constructed the adjacency matrix from the count of flights rendered in a network of countries  via a stochastic block model where each block contains countries having a similar number  of flights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Our proposed model has helped us to understand the importance of the baseline  covariates vs network effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Since we have dealt with longitudinal data, it is also instructive  to note that our proposed model offers us the flexibility of clustering both in the network  space and also over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  One limitation of our results is the balanced community assumption made in Proposition  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='1 which may not reflect some unbalanced networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' This assumption can be relaxed some-  what with small departures from the balanced design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Further, under the stochastic block  model, extremely unbalanced networks still allow for consistent estimation of p and q, how-  ever for a more flexible network model that allows for community-specific edge probabilities,  consistent estimation of edge probabilities requires each community to grow asymptotically  with n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  5  Appendix  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='1  Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='1  From the construction of the adjacency matrices, one can note that the entries Aijs are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Bernoulli random variables with  E(Aij) =  �  �  �  �  �  �  �  p,  if i, j belong to the same community  q,  otherwise  18  and  V ar(Aij) =  �  �  �  �  �  �  �  p(1 − p),  if i, j belong to the same community  q(1 − q),  otherwise  Denote by S the set {i, j : 1 ≤ i < j ≤ n, i, j belongs to the same community}, and its  complement by S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Therefore, for a directed graph, one can write  E  � �  i,j∈S  Aij  �  = 2K  �m  2  �  p,  V ar  � �  i,j∈S  Aij  �  = 2K  �m  2  �  p(1 − p) = s2  mp,  E  � �  i,j∈S′  Aij  �  = K(K − 1)m2q, and  V ar  � �  i,j∈S′  Aij  �  = K(K − 1)m2q(1 − q) = s2  mq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Next, one can verify Lindeberg condition to establish the central limit theorem:  �  i,j∈S Aij − 2K  �m  2  �  p  �  2K  �m  2  �  p(1 − p)  and  �  i,j∈S′ Aij − K(K − 1)m2q  �  K(K − 1)m2q(1 − q)  d→ N(0, 1)  (4)  as Km2p(1 − p) and Km2q(1 − q) → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  The Lindeberg’s condition requires us to verify  1  s2  mp  �  i,j∈S  E  �  B2  ijI{|Bij|≥ϵsmp}  �  → 0  and  1  s2  mq  �  i,j∈S′  E  �  B2  ijI{|Bij|≥ϵsmq }  �  → 0,  19  where Bij = Aij −E(Aij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Since |Bij| ≤ 1, the above condition is satisfied when Km2p(1−p)  and Km2q(1 − q) tend to ∞ under which ϵsmp and ϵsmq are both greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Dividing the numerator and denominator of (4) by 2K  �m  2  �  and K(K −1)m2 respectively,  one obtains both  mγ/2�  2K  �m  2  �  �  i,j∈S Aij/  �  2K(m  2)  �  −p  √  mγp(1−p)  and  mγ/2�  K(K − 1)m2  �  i,j∈S′ Aij/  �  K(K−1)m2  �  −q  √  mγq(1−q)  both converge to N(0, 1) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Since ˆp = �  i,j∈S Aij/  �  K  �m  2  ��  , and ˆq = �  i,j∈S′ Aij/  �  K(K−  1)m2�  , both ˆp and ˆq are m1+γ/2 consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='2  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='2  Here we present the sketch of the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' One can write (3) as  K  �  k=1  Uk(α, β, p, q) =  K  �  k=1  D⊤  k V −1  k  Sk = 0,  (5)  where Sk = yk − µk, and U is a function of the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Let b = (β, α)⊤ denote the vector of model parameters, and π = (p, q)⊤ denote the  vector of model parameters, and vector of within and between edge probabilities respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Letting b fixed, the Taylor expansion yields  �K  k=1 Uk(b, π∗)  K1/2m1+γ/2  =  �K  k=1 Uk(b, π)  K1/2m1+γ/2  +  �K  k=1  ∂Uk(b,π)  ∂π  K1/2m1+γ/2 m1+γ/2(π∗ − π) + oP(1)  (6)  = ˜A + ˜B ˜C + oP(1),  One can note that ˜B = oP(1) as ∂Uk(b, π)/∂π are linear functions of Sk’s defined in (5)  whose means are zero, and ˜C = OP(1) thanks to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Therefore,  �K  k=1 Uk(b,π∗)  K1/2m1+γ/2  is asymptotically equivalent to  �K  k=1 Uk(b,π)  K1/2m1+γ/2  whose asymptotic distribution is multivariate  20  normal with zero mean and covariance is equal to V as defined in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' The proof  is thus complete following Liang & Zeger (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  6  Acknowledgement  RG would like to thank Anupam Kundu, postdoctoral associate at Yale School of Public  Health and Thien Le, postdoctoral fellow at Harvard for many useful discussions regarding  real data processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' JP acknowledges support from R01AI138901, and IB acknowledges  support from R01MH116884.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  References  Aukett, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', Ritchie, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', & Mill, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Gender differences in friendship patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Sex  roles, 19(1), 57–66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Bank, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Covid-19 to plunge global economy into worst recession since World War  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' World Bank Group Washington, DC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Binkiewicz, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', Vogelstein, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', & Rohe, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Covariate-assisted spectral clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Biometrika, 104(2), 361–377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Burton, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', Gurrin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', & Sly, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Extending the simple linear regression model to  account for correlated responses: an introduction to generalized estimating equations and  multi-level mixed modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Statistics in Medicine, 17(11), 1261–1291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Chen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', Kato, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', & Leng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Analysis of networks via the sparse β-model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Journal  of the Royal Statistical Society: Series B (Statistical Methodology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Christakis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', & Fowler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' The spread of obesity in a large social network  over 32 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' New England Journal of Medicine, 357(4), 370–379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  21  COVID, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  database (2021) retrieved from https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' com/cssegisanddata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Diggle, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', Diggle, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', Heagerty, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', Liang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', Zeger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Analysis of  longitudinal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Oxford University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Halekoh, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', Højsgaard, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', & Yan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  The r package geepack for generalized  estimating equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Journal of Statistical Software, 15/2, 1–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Hoff, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Additive and multiplicative effects network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Statistical Science,  36(1), 34–50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Hoff, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', Raftery, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', & Handcock, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Latent space approaches to social  network analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Journal of the American Statistical Association, 97(460), 1090–1098.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Holland, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', Laskey, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', & Leinhardt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Stochastic blockmodels: First steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Social Networks, 5(2), 109–137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Holland, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', & Leinhardt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' An exponential family of probability distributions  for directed graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Journal of the American Statistical Association, 76(373), 33–50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Igarashi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', Takai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', & Yoshida, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Gender differences in social network devel-  opment via mobile phone text messages: A longitudinal study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Journal of Social and  Personal Relationships, 22(5), 691–713.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Kelejian, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', & Prucha, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  A generalized spatial two-stage least squares  procedure for estimating a spatial autoregressive model with autoregressive disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  The Journal of Real Estate Finance and Economics, 17(1), 99–121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Le, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', Raynal, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', Talbot, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', Hambridge, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', Drovandi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', Mira, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
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+page_content='  22  Liang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', & Zeger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Longitudinal data analysis using generalized linear  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
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+page_content='  Mandel, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', Ghosh, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', & Barnett, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
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+page_content='  Mu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', Mele, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', Hao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', Cape, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', Athreya, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', & Priebe, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
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+page_content='  Newman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Network structure from rich but noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
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+page_content='  Qin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', & Rohe, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Regularized spectral clustering under the degree-corrected  stochastic blockmodel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Advances in Neural Information Processing Systems, 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Rosvall, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', & Bergstrom, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
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+page_content=' arXiv preprint physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='soc-ph/0707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='0609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Sch¨afer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', Strohmeier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', Lenders, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', Martinovic, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', & Wilhelm, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Bringing  up opensky: A large-scale ads-b sensor network for research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
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+page_content='  83–94).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  Shalizi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', & Thomas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
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+page_content=', Cheek Jr, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', & MacD, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
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+page_content='  Staber, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
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+page_content=', Sch¨afer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=', & Lenders, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
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+page_content='  Zeger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' The analysis of discrete and continuous longitudinal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content=' Biometrics, 42,  121–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
+page_content='  24' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQf8Aun/content/2301.04804v1.pdf'}
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+arXiv:2301.04367v1  [math.PR]  11 Jan 2023
+AN MCMC SAMPLING OF DENSEST k-SUB-GRAPHS OF REGULAR
+GRAPHS WITH CONNECTED COMPLEMENT GRAPH
+JENS WALTER FISCHER♠♣
+Abstract. We present an exclusion process based approach for sampling densest k-sub-
+graphs from regular graphs L with connected complement. By interpreting an exclusion
+process as a Markov chain on a corresponding Token Graph Lk, we make use of classical
+Markov chain theory to obtain quantitative bounds on the convergence speed depending on
+the geometry of L, which are sharp in the sens that variations in the valency lead to an
+equality, which is 1, for the geometric bounds in the case of a complete graph. We propose
+an algorithm which makes use of this particle view to avoid excessive memory use due to
+the state space size and discuss the regularity and connectivity condition on L and Lc in
+the Outlook.
+♠ Universit´e de Toulouse
+♣ University of Potsdam
+Key words : Densest k Sub-graphs, Exclusion Processes, Token Graphs, MCMC Sampling
+MSC 2020 : 82C22, 60K35, 05C60, 65C05, 82M31
+1. Introduction
+In this work we use generalized exclusion processes to approach the task of finding k-
+densest sub-graphs in a ¯d-regular graph L of arbitrary size. This is, in particular, linked to
+the community detection problem which has received wide attention in modern research, for
+example in [10] and [9]. The density of any vertex induced sub-graph Lv = (v, Ev) of some
+graph L = (V, E) on the subset v ⊆ V is defined as ρ(v) = |Ev|
+|v| , see for example [2]. The
+problem of finding the densest sub-graph becomes, then, the maximization problem which
+aims at finding the subset v∗ ⊂ V such that ρ(v∗) = maxv∈P(V ), |v|≥1 ρ(v) where P(V ) is
+the power set of V . In [2] the author shows that this problem can be solved in polynomial
+time. Fixing the size of the sub-graph renders the problem vastly more complicated as being
+discussed in, among many others, [3], [6] and [11]. In particular, the work in [11] and [6],
+the authors conclude, that finding densest sub-graphs of fixed size is NP-hard and finding
+such graphs is, therefore, unfeasible. Hence, the research focus is on approximations of such
+objects using combinatorial selection methods or related families of problems which, reliably,
+give results which are close in density to the desired sub-graph.
+We contribute to this discussion a stochastic approach based on generalized exclusion
+processes and Monte Carlo simulation. The process is constructed on the complement graph
+Lc in such a way, that it exhibits in the long time limit most likely to a densest sub-graph in L
+and, even, gives monotonously sub-graphs of lesser density, i.e., a second densest sub-graph is
+obtained with second largest probability, a third densest with third largest probability and so
+1
+
+2
+J. W. FISCHER
+on. We obtain a quantitative bound, which is tight with respect to variations in the average
+degree ¯d, on the convergence speed based on a binary geometric condition on L.
+2. An appropriate GEP
+To identify densest sub-graphs we want to construct a particle system based on discrete
+time generalized exclusion process (GEP) ηk using the following Definition 2.1.
+Definition 2.1. Let L = (V, E) be a simple connected graph with |V | = ¯n ∈ N and k ∈
+{1, . . . , ¯n − 1}. Denote by (P (η))η∈{0,1}V , |η|=k a family of stochastic matrices. A GEP in
+discrete time ηk := (ηk;t)t∈N of k particles on L is a Markov chain on the set of configurations
+Sk = {η ∈ {0, 1}V | |η| = k}
+defined by the transition matrix Q = (qη,µ)η,µ∈{0,1}V given for η, µ ∈ {0, 1}V by
+qη,µ =
+
+
+
+
+
+P (η)(v, w)1η(v)=1=µ(w),η(w)=0=µ(v)
+η(u)=µ(v)∀u̸∈{v,w}
+,
+η ̸= µ
+1 − �
+µ̸=η qη,µ,
+η = µ.
+(2.1)
+We are going to exploit a link between a system of moving particles ηk on a finite graph
+and a Markov chain in a higher dimensional space, as developed in [8], to have access to
+classical results on the long time behavior of the process. In particular, we consider discrete
+time generalized exclusion processes, choosing the transition probabilities according to the
+application at hand. This gives us access to a process which respects at any time step the fixed
+size of the subgraphs in which we are interested. We translate GEPs as defined hereinabove
+to Markov chains on subsets of fixed size of the vertex set of the underlying graph L. The
+arising state space corresponds to what is know as Token Graphs or Symmetric Powers of
+Graphs, as defined in [1] and [5].
+Definition 2.2 ([5]). Consider a finite simple graph L = (V, E). Define Vk = {v ⊂ v| |v| =
+k} and Ek = {⟨v, w⟩|v△w = {v, w}; ⟨v, w⟩ ∈ E}. The graph Lk = (Vk, Ek) is called the k
+Token Graph of L and we denote by degk(v) the degree of v ∈ Vk in Lk.
+In choosing an adequate family of transition matrices (P θ)θ∈{0,1}|V | we obtain a GEP
+on some graph L, based on which we can propose and quantify a Markov chain Monte
+Carlo approach to sampling densest sub-graphs. This GEP ηk has a discrete time Markov
+chain representation SMC
+k
+on Lk as developed in [8]. Consider as an example the graph
+illustrated in Figure 1b. Indeed, there are various ways to approach GEPs, depending on
+the distributions which govern the transitions of individual particles. One way is by defining
+a Markov chain based on a series of dynamic graphs, i.e., by interpreting ηk as a Markov
+chain in a random environment given by the family (P η)η∈{0,1}V , |η|=k. To this end, define for
+t ∈ N the set Nk;t = {v ∈ V |ηk;t(v) = 1} and the directed bipartite graph B = (Bt)t∈N with
+Bt = ((Nk;t, V \Nk;t, Σk,t), where Σk,t represent all possible transitions of particles at time t,
+with potential loops on vertices in Nk;t. The random graph Bt might be disconnected but
+serves as a graph theoretical representation of transitions of the GEP ηk, i.e., the positive
+transition probabilities of P η conditioned on η.
+The central part is the change of Σk,t when going from time t to t + 1. It captures the
+possible particle movements induced by the distribution used in Definition 2.1. To illustrate
+
+MCMC SAMPLING OF DENSEST k-SUBGRAPHS
+3
+(a) The underlying 3-regular graph on
+6 vertices which we are going to use as
+example for all constructions of exclusion
+processes.
+(b) Potential particle displacements; Uniform distribu-
+tion for drawing a directed edge. States of the vertices
+connected by said edge are interchanged. When drawing
+a loop, everything remains the same.
+the construction, we consider a 3-regular graph on 6 vertices depicted in Figure 1a. It will
+be the reference for missing edges which will illustrate the distinctive parts of each exclusion
+process. To any η ∈ {0, 1}V we can associate a vη ⊂ V with vη = {v ∈ V |η(v) = 1} and in
+this sense we can write with a little abuse of notation η = vη. For our central construction,
+assume that for some v ⊂ V with |v| = k at time t ∈ N the process satisfies ηk;t = v. Set
+vc := V \v. Consider the possibly disconnected bipartite sub-graph Lv,vc = ((v, vc), Ev,vc)
+of L and add a loop ev,v to any v ∈ v. We call the resulting graph B′
+t = (V, Σ′
+k,t) with
+Σ′
+k,t = Ev,vc ⊔ {ev,v|v ∈ v}. Indeed, for a ¯d-regular graph L we obtain |Σk,t| = degk(v) + k,
+where degk(v) denotes the degree of v interpreted as vertex in the token graph Lk associated
+to L, see [1] and [5]for a deeper discussion of Token Graphs. In Figure 1b we illustrate a
+possible example derived from the graph L as in Figure 1a and some configuration v. In
+each time step, the exclusion process makes a step by drawing one edge uniformly from Σ′
+k,t
+and exchanging the state of the endpoints. Note, that in changing the number of present
+edges also the transition probabilities change due to the uniform distribution over Σ′
+k,t. This
+renders the process, in particular, in-homogeneous because the transition probabilities now
+depend on the current configuration v. A configuration v can be seen as a representative
+of the sub-graph class of vertex induced sub-graphs which are spanned by a subset of the
+original vertex set and all edges included therein.
+Definition 2.3. Let L = (V, E) be any simple graph and v ⊆ V . Then the graph Lv = (v, Ev)
+with ⟨v, w⟩ ∈ Ev if and only if v, w ∈ v and ⟨v, w⟩ ∈ E is called the vertex induced sub-graph
+of L on v.
+In what follows, in order to obtain meaningful results we work under the assumption that
+both L and Lc are connected graphs, as explained in the Introduction. A further discussion
+of the limitations is postponed to the Outlook. If this assumption is removed, the additional
+difficulty of initial condition dependencies on disconnected graphs for particle systems has to
+be considered. We leave this open to further research and remain with the assumption on L
+and Lc, giving some pointers in the Outlook.
+
+4
+J. W. FISCHER
+3. Convergence Results
+We consider the generalized exclusion process ηk as constructed above. Then, by [8], there
+is an associated Markov chain SMC
+k
+on Lk such that ηk and SMC
+k
+are equal in law. We make,
+in what follows, exclusively claims about the associated Markov chain SMC
+k
+on Lk.
+Theorem 3.1. Let k ∈ {1, . . . , ¯n−1} and consider the Markov chain SMC
+k
+on Lk = (Vk, Ek).
+Then it is aperiodic, irreducible and, hence, ergodic. Furthermore, it is a reversible chain
+and the stationary distribution πMC
+k
+is given in terms of v ∈ Vk by
+πMC
+k
+(v) = degk(v) + k
+2|Ek| + k
+�¯n
+k
+�
+(3.1)
+Based on the transition matrix given by Proposition 5.1 the stationary distribution πMC
+k
+of SMC
+k
+can be derived directly and reversibility follows as well. Indeed, one can identify
+SMC
+k
+with a random walk on Lk where any vertex in Lk has additionally to its incident edges
+k loops. Under a dichotomy of geometric conditions on the underlying graph L, we find the
+following convergence speed of SMC
+k
+to πMC
+k
+.
+Theorem 3.2. Let L be a ¯d-regular graph on ¯n vertices and k ∈ {1, . . . , ¯n − 1}. Consider
+the Markov chain SMC
+k
+on Lk and ε > 0. Additionally, define
+ρ(¯n, ¯d, k) := 4(¯n − 1)2(¯n − k)2
+¯d4
+ξ(¯n, ¯d, k) := log
+��¯n
+k
+��
++ log
+�k(¯n − k)
+¯n − 1
++ k
+¯d
+�
+.
+If L is such that minv∈Vk avg degk(Lv) < ¯d − 1 then for v, w ∈ Vk and
+t ≥ 1 + ρ(¯n, ¯d, k)
+�
+log
+�4
+ε
+�
++ ξ(¯n, ¯d, k)
+�
+(3.2)
+the Markov chain SMC
+k
+satisfies
+�������
+�
+pMC
+k;v,w
+�(t)
+− πMC
+k
+(w)
+πMC
+k
+(w)
+�������
+≤ ε.
+On the other hand, if L is such that minv∈Vk avg degk(Lv) ≥ ¯d − 1 then there is a constant
+C > 0 such that its ε mixing time w.r.t. the total variation distance is bounded by
+τmix(ε) ≤ C log2(ε−1)
+�
+ξ(¯n, ¯d, k) − (¯n − 1)2
+¯d4
+�
+.
+(3.3)
+Note that the constants we used in Theorem 3.2, namely ρ(¯n, ¯d, k) and ξ(¯n, ¯d, k), grow
+at most like a polynomial as ¯n → ∞ since log
+��¯n
+k
+��
+grows at most like ¯n by Stirling’s
+approximation and the remaining terms are polynomial in ¯n.
+Consequently, SMC
+k
+mixes
+rapidly relative to the size of its actual state space, employing this term as used in [15]. Its
+mixing is, therefore, polynomial fast in terms of the size of the underlying graph L, which
+makes it a suitable process for sampling sub-graphs of L.
+
+MCMC SAMPLING OF DENSEST k-SUBGRAPHS
+5
+In the following section, we propose an algorithm and the idea behind sampling densest
+sub-graphs of L with high probability using the Markov chain SMC
+k
+introduced in this section
+and exploiting its properties.
+4. Particle based MCMC algorithm for sampling densest k-sub-graphs
+By the structure of the stationary distribution we obtain an ordering based on the degree
+of the vertices, where the probability of drawing a densest sub-graph as t → ∞ is smallest
+since degk(v) = k ¯d−2|Ev|. On the other hand, this also implies that it is most probable under
+πMC
+k
+to draw a least dense sub-graph of L. Since the densest k sub-graph in L corresponds
+to the least dense k sub-graph in Lc and vice versa, we can exploit the fact that both are
+connected to inverse the roles under SMC
+k
+with respect to the weights given by its stationary
+distribution. Using this, we propose the following MCMC approach to finding densest k sub-
+graphs in a ¯d-regular graph with high probability. To this end consider a ¯d-regular simple
+connected graph L on ¯n vertices and assume that its graph complement Lc is also connected.
+Then Lc is a ¯n − 1 − ¯d regular simple connected graph. We denote for any k ∈ {1, . . . , ¯n − 1}
+by L(c)
+k
+= (Vk, E(c)
+k ).
+Let k ∈ {1, . . . , ¯n − 1} and define SMC
+k
+as the Markov chain associated to the exclusion
+process on Lc. Then, by Theorem 3.1 the Markov chain SMC
+k
+converges in distribution to
+πMC
+k
+and with maximal probability we obtain a v∗ ∈ Vk which induces a least dense k-sub-
+graph Lc
+v∗ of Lc and, in turn, the set v∗ induces a densest k-sub-graph in L. We propose the
+following algorithm which only uses the local information of L and the current configuration
+v of SMC
+k
+. For the simulation, it is not necessary to construct the whole state space Lk with
+�¯n
+k
+�
+vertices and the size of the edge set given by Proposition B.2. Algorithm 1 only depends
+on the neighborhood of all v ∈ v after having fixed a v ∈ Vk. Exploiting the fact that L is
+assumed to be ¯d-regular, we can bound the number of states we have to access in each turn
+by ¯d·k which is also a very crude upper bound for degk(v). We use multi-sets to describe the
+algorithm and when we set X = ∅ as a multi-set, we impose that (X∪{v})∪{v} = X∪{v, v}.
+The multi-set X can be replaced in real code by any mutable list-type object which allows
+multiple times the same entry. Algorithm 1 mirrors the dynamics of the exclusion process,
+constructed earlier. It, therefore, samples rapidly, in the sense of [15], densest sub-graphs with
+high probability. We want to emphasize the importance of a detailed understanding of the
+underlying state space Lk from the perspective of vertex induced sub-graphs. In particular,
+the geometric results as well as knowledge about the structure of the vertex set presented
+in the appendix allowed for the bounds in Theorem 3.2. Finally, the proposed algorithm
+benefits from the particle perspective in that it only needs to consider at most k ¯d vertices in
+each step which is a considerable reduction from considering the whole vertex set Vk.
+5. Proofs of central results
+We consider the generalized exclusion process ηk as constructed above. Then, by [8], there
+is an associated Markov chain SMC
+k
+on Lk such that the marginal distributions of ηk and
+SMC
+k
+are identical. To start, we characterize the associated Markov chain SMC
+k
+on Lk by
+giving its transition matrix explicitly.
+
+6
+J. W. FISCHER
+Algorithm 1 Particle based algorithm to simulate SMC
+k
+.
+Require: Terminal time for simulation, e.g. tmix, number of trials m, graph L
+Initialize t = 0
+Define X = ∅ as multi-set
+Run following burn in step
+Draw initial state v ∈ Vk
+Set SMC
+k,0 = v
+while t ≤ tmix do
+Construct bipartite graph B′(v) as in Figure 1b
+Draw uniformly an edge ⟨v, w⟩ from B′(v)
+Update SMC
+k,t+1 = (v\{v}) ∪ {w}; t = t + 1
+end while
+Run sampling after burn in step
+while i ≤ m do
+Construct bipartite graph B′(v) as in Figure 1b
+Draw uniformly an edge ⟨v, w⟩ from B′(v)
+Update SMC
+k,t+1 = (v\{v}) ∪ {w}; t = t + 1
+Update i = i + 1;
+Update X = X ∪ Sk,t
+end while
+return X, statistic of m final states after tmix simulation steps.
+Proposition 5.1. Let k ∈ {1, . . . , ¯n − 1} and consider the Markov chain SMC
+k
+on Lk =
+(Vk, Ek). We call its transition matrix P MC
+k
+. Then, it has the form
+pMC
+k;v,w =
+
+
+
+
+
+
+
+
+
+
+
+1
+degk(v) + k ,
+∃⟨x, y⟩ ∈ E s.t. v△w = {x, y},
+k
+degk(v) + k ,
+v = w,
+0,
+otherwise.
+The form of the transition matrix follows directly from the construction of B′
+t in Figure
+1b and the uniform distribution over all edges in B′
+t as discussed above.
+The stationary
+distribution πMC
+k
+of SMC
+k
+can consequently be derived directly and reversibility follows as
+well. Based on the transition matrix given by Proposition 5.1 and the stationary distribution
+given by Theorem 3.1 one can identify SMC
+k
+with a random walk on Lk where any vertex in
+Lk has additionally to its incident edges k loops. Indeed, for almost all cases of k, this is not
+sufficient to render SMC
+k
+a lazy random walk. Indeed, we can quantify this transition.
+Lemma 5.2. The random walk SMC
+k
+is lazy if and only if
+min
+v∈Vk
+avg degk(Lv) ≥ ¯d − 1.
+
+MCMC SAMPLING OF DENSEST k-SUBGRAPHS
+7
+Proof. Recall that the Markov chain is called lazy if and only if
+min
+v,v pMC
+k;v,v ≥ 1
+2.
+(5.1)
+Therefore, using the expression derived in Proposition 5.1 we obtain that SMC
+k
+is lazy if and
+only if for all v ∈ Vk we have
+degk(v)
+k
++ 1 ≤ 2 ⇔ ¯d − avg degk(Lv) ≤ 1
+which proves the claim.
+□
+An obvious property is minv∈Vk avg degk(Lv) ≤ ¯d since Lv is a vertex induced sub-graph
+and, therefore, the degree of any vertex in Lv is bounded by ¯d which implies the same for
+the average. Geometrically, the condition given by Lemma 5.2 leaves, consequently, only
+little room for the parameter triple (¯n, ¯d, k). Due to the intermediate position SMC
+k
+takes
+between the simple random walk on Lk and the lazy random walk on this state space, we
+can in general only use that
+min
+v∈Vk
+pMC
+k;v,v ≥ γ > 0
+for some γ ∈
+�
+0, 1
+2
+�
+with a phase transition, if for a parameter triple (¯n, ¯d, k) we have
+minv∈Vk avg degk(Lv) ≥ ¯d − 1 which is a geometric condition on L. We turn now to the
+proof of Theorem 3.2.
+Proof of Theorem 3.2. The goal of this proof is to obtain bounds on geometric properties
+of vertex induced subgraphs of L, which allow for a meaningful application of known re-
+sults from Markov chain theory, as for example presented in [14].
+To begin, let γ :=
+� ¯d + 1 − minv∈Vk avg degk(Lv)
+�−1 and consider the following two terms which we call the
+additive and the multiplicative constant, respectively, given as, firstly,
+4(1 − γ)2
+γ2
+log
+�
+2|Ek| + k|Vk|
+4(min{degk(v), degk(w)} + k)
+�
+�
+ι(Lk)
+2|Ek|+k|Vk|
+maxv∈Vk degk(v)+k
+�2
+and, secondly,
+4(1 − γ)2
+γ2
+�
+ι(Lk)
+2|Ek| + k|Vk|
+maxv∈Vk degk(v) + k
+�−2
+.
+By [13] we can estimate the isoperimetric constant ι(Lk) from below by a constant times the
+vertex connectivity κ(Lk) of Lk in the form
+ι(Lk) ≥
+2
+|Vk|κ(Lk)
+and we have from Theorem B.1 the value of κ(Lk) as κ(Lk) = minv∈Vk degk(v). This gives
+us, consequently, in the first case the lower bound for
+ι(Lk)
+2|Ek| + k|Vk|
+maxv∈Vk degk(v) + k ≥ 2 min
+v∈Vk
+degk(v)
+avg deg(Lk) + k
+maxv∈Vk degk(v) + k
+≥ min
+v∈Vk
+degk(v)
+avg deg(Lk)
+maxv∈Vk degk(v)
+
+8
+J. W. FISCHER
+using for the last inequality that minv∈Vk avg degk(Lv) < ¯d−1. This in addition to maxv∈Vk degk(v) ≤
+k(¯n−k) and Proposition B.3 gives the following upper bound for the multiplicative constant
+by
+4(1 − γ)2
+γ2
+� ι(Lk)(2|Ek| + k|Vk|)
+maxv∈Vk degk(v) + k
+�−2
+≤ 4
+k2
+�maxv∈Vk degk(v)
+avg deg(Lk)
+�4 �
+avg deg(Lk)
+minv∈Vk degk(v)
+�2
+≤ 4
+k2
+� ¯n − 1
+¯d
+�4 �
+avg deg(Lk)
+minv∈Vk degk(v)
+�2
+.
+Employing the lower bound minv∈Vk degk(v) ≥ ¯d, which can be derived easily by using
+k ≤ ¯n − 1 and the fact that there is at least one empty vertex in L with degree ¯d, one can
+obtain the following upper bound, using Proposition B.3, independent of the geometry of L
+given by
+4
+k2
+� ¯n − 1
+¯d
+�4 �
+avg deg(Lk)
+minv∈Vk degk(v)
+�2
+≤ 4(¯n − 1)2(¯n − k)2
+¯d4
+.
+Since the additive constant equals the multiplicative constant times a logarithmic term, we
+are only going to focus on the logarithmic term and we note
+log
+�
+2|Ek| + k|Vk|
+4(min{degk(v), degk(w)} + k)
+�
+≤ log
+�
+2|Ek| + k|Vk|
+minv∈Vk degk(v) + k
+�
+= log
+��¯n
+k
+��
++ log
+�
+avg deg(Lk) + k
+minv∈Vk degk(v) + k
+�
+.
+For the second summand, we can use the bound which we used already above to obtain
+log
+�
+avg deg(Lk) + k
+minv∈Vk degk(v) + k
+�
+≤ log
+�k(¯n − k)
+¯n − 1
++ k
+¯d
+�
+.
+Therefore, we arrive at the upper bound on the additive constant
+4(¯n − 1)2(¯n − k)2
+¯d4
+�
+log
+��¯n
+k
+��
++ log
+�k(¯n − k)
+¯n − 1
++ k
+.
+¯d
+��
+We now employ some results based on evolving sets presented in [14]. To this end, we will
+focus on lower bounds for Φk(u) := inf
+�
+∂(S,Sc)
+πMC
+k
+(S)
+�� πMC
+k
+(S) ≤ u
+�
+for u ∈
+�
+minv∈Vk πMC
+k
+(v), 1
+2
+�
+as defined in [14]. In particular, we use that Φ(u) ≥ Φ
+�1
+2
+�
+. The first claim then follows
+by Theorem 5 of [14], the second one by [12]. To conclude the proof, note that for ¯S ∈
+�
+S ⊂ Vk
+�� |S| ≤ |Vk|
+2
+�
+∩
+�
+S ⊂ Vk
+�� πMC
+k
+(S) ≤ 1
+2
+�
+we have
+∂( ¯S, ¯Sc)
+πMC
+k
+( ¯S) ≥ ∂( ¯S, ¯Sc)
+| ¯S|
+2|Ek| + k|Vk|
+maxv∈Vk degk(v) + k ≥ ι(Lk)
+2|Ek| + k|Vk|
+maxv∈Vk degk(v) + k
+where we used ¯S ∈
+�
+S ⊂ Vk| |S| ≤ |Vk|
+2
+�
+for the last estimate and the definition of the
+isoperimetric constant of a graph. Since, additionally, ¯S ∈
+�
+S ⊂ Vk
+�� πMC
+k
+(S) ≤ 1
+2
+�
+, taking
+the infimum over all S ∈
+�
+S ⊂ Vk
+�� πMC
+k
+(S) ≤ 1
+2
+�
+we arrive at
+Φk
+�1
+2
+�
+≥ ι(Lk)
+2|Ek| + k|Vk|
+maxv∈Vk degk(v) + k
+
+MCMC SAMPLING OF DENSEST k-SUBGRAPHS
+9
+and, therefore, for all u ∈
+�
+minv∈Vk πMC
+k
+(v), 1
+2
+�
+we obtain
+Φk(u) ≥ ι(Lk)
+2|Ek| + k|Vk|
+maxv∈Vk degk(v) + k.
+The second step consists in finding a lower bound for minv∈Vk pMC
+k;v,v. We have already found
+in the proof of Lemma 5.2 that
+min
+v∈Vk
+pMC
+k;v,v ≥
+1
+¯d + 1 − minv∈Vk avg degk(Lv) = γ.
+Having found the necessary bounds, we can apply Theorem 5 of [14] and obtain after inte-
+gration
+� 4ε−1
+4(πMC
+k
+(v)∧πMC
+k
+(w))
+4 du
+uΦk(u)2 ≤
+�
+log
+�4
+ε
+�
+− log
+�
+4(min{degk(v),degk(w)}+k)
+2|Ek|+k|Vk|
+��
+�
+ι(Lk)
+2|Ek|+k|Vk|
+maxv∈Vk degk(v)+k
+�2
+.
+(5.2)
+We identify the terms in the right hand side of equation 5.2 with the additive constant and the
+multiplicative constant for which we have found meaningful upper bounds at the beginning
+of this proof, such that we obtain the first claim. The second claim follows by the same
+estimate for Φk(u) and Theorem 2.2 of [12] as well as equation (5) of [14] using the bound of
+the additive term by ξ(¯n, ¯d, k) as defined in the Theorem.
+□
+This concludes our section on the proof of the main Theorem 3.2.
+6. Outlook
+The main goal of this document was the presentation of an MCMC algorithm to sample
+densest k sub-graphs from regular graphs L and the quantification of its convergence speed.
+We needed the additional condition that Lc is connected, as well. This gives us two natural
+aspects from which we could start generalizations of the method.
+Firstly, let us consider the generalization to simple non-regular graphs under the condition
+that Lc is connected and fix some k. We construct the following example L by assuming
+k < ⌊ ¯n
+2 ⌋ and looking at two complete graphs Kk and K¯n−k. We assume that L has ¯n vertices
+and is constructed by gluing together the two complete graphs by adding an edge between
+one vertex in Kk and one vertex in K¯n−k. Therefore, Lc is connected being the complete
+bipartite graph minus one edge. A densest k sub-graph Lv is then given by the complete
+graph Kk, which can be realized by placing either k vertices in Kk or K¯n−k calling them v and
+v′, respectively. Seen as vertices in the Token Graph Lk this shows that the identifiability of
+densest k sub-graphs via their degree in Lk is violated if L is not regular because degk(v) <
+degk(v′) even though they have identical density as defined in the introduction. Nonetheless,
+degk(v) = 1 and is the minimal degree. Therefore, the sampling still gives densest subgraphs
+with high probability but the hit rate will be much lower since not all densest sub-graphs lie
+in the same level set w.r.t. degk(·). Adjustments of the exclusion process construction can be
+helpful here, for another example which respects finer geometric features of the sub-graphs
+see [8].
+Keeping on the other hand the regularity condition and dropping the connectedness of Lc,
+we encounter another interesting problem. The graph Lc is ¯n − ¯d − 1 regular by assumption
+
+10
+J. W. FISCHER
+and assuming that it is disconnected, we find a least dense sub-graph on k vertices by finding
+a redistribution of particles on the connected components. Note that each connected com-
+ponent is by assumption itself a ¯n − ¯d − 1 regular graph. Interesting combinatorial questions
+arise then from monotony questions of the number of particles in each connected components
+w.r.t. to its size. Under the condition on L, that there is a monotonously increasing function
+fL which maps integers l ∈ {1, . . . , ¯n} to the number of particles in k′{1, . . . , k} such that the
+least dense sub-graph can be obtained by putting f(l) particles in the connected component
+of Lc of size l, our result works as well. But the geometric implications on L as well as
+interpretations for f need to be worked out before any meaning can be given to a result in
+this case.
+
+REFERENCES
+11
+References
+[1]
+Yousef Alavi, Don R Lick, and Jiuqiang Liu. “Survey of double vertex graphs”. In:
+Graphs and Combinatorics 18.4 (2002), pp. 709–715.
+[2]
+Moses Charikar. “Greedy Approximation Algorithms for Finding Dense Components
+in a Graph”. In: Approximation Algorithms for Combinatorial Optimization. Ed. by
+Klaus Jansen and Samir Khuller. Berlin, Heidelberg: Springer Berlin Heidelberg, 2000,
+pp. 84–95.
+[3]
+D.G. Corneil and Y. Perl. “Clustering and domination in perfect graphs”. In: Discrete
+Applied Mathematics 9 (1984).
+[4]
+Persi Diaconis and Laurent Saloff-Coste. “Comparison Theorems for Reversible Markov
+Chains”. In: The Annals of Applied Probability 3.3 (1993).
+[5]
+Ruy Fabila-Monroy et al. “Token Graphs”. In: Graphs and Combinatorics 28.3 (2012),
+pp. 365–380.
+[6]
+U. Feige, D. Peleg, and G. Kortsarz. “The Dense k -Subgraph Problem”. In: Algorith-
+mica (2001).
+[7]
+Jens Walter Fischer. On the connectivity and diameter of Token graphs from a vertex
+induced sub-graph perspective. 2022.
+[8]
+Jens Walter Fischer. “Random dynamics in collective behavior - consensus, clustering
+& extinction of populations”. Doctoral Thesis. Universit´e de Toulouse, 2022.
+[9]
+Santo Fortunato. “Community detection in graphs”. In: Physics Reports 486.3 (2010),
+pp. 75–174.
+[10]
+M. Girvan and M. E. J. Newman. “Community structure in social and biological net-
+works”. In: Proceedings of the National Academy of Sciences 99.12 (2002), pp. 7821–
+7826.
+[11]
+Samir Khuller and Barna Saha. “On Finding Dense Subgraphs”. In: ICALP. 2009.
+[12]
+Laszlo Lovasz and Ravi Kannan. “Faster Mixing via Average Conductance”. In: Pro-
+ceedings of the Thirty-First Annual ACM Symposium on Theory of Computing. New
+York, NY, USA: Association for Computing Machinery, 1999, 282–287.
+[13]
+Bojan Mohar. “Isoperimetric numbers of graphs”. In: Journal of Combinatorial Theory,
+Series B 47.3 (1989), pp. 274–291.
+[14]
+B. Morris and Yuval Peres. “Evolving sets, mixing and heat kernel bounds”. In: Prob-
+ability Theory and Related Fields 133.2 (2005), pp. 245–266.
+[15]
+Alistair Sinclair. “Improved Bounds for Mixing Rates of Markov Chains and Multicom-
+modity Flow”. In: Combinatorics, Probability and Computing 1.4 (1992), 351–370.
+
+12
+REFERENCES
+Appendix A. Note on the subtleties of the exclusion process construction
+Exclusion processes are not a new subject and have been analyzed, mostly for continuous
+time instead of discrete time, in statistical mechanics at length.
+This first subsection is
+meant to make the point regarding the construction from the first part of this publication.
+We translate a ”classical exclusion process” discussed in [4] into the framework of dynamic
+random graphs and then to the token graph Lk. To this end define Nk;t = {v ∈ V |ηk;t(v) = 1}
+and the time dependent graph B = (Bt) with Bt = ((Nk;t, V \Nk;t, Σk,t).
+We adjust the step of Σk,t when going from time t to t + 1 and illustrate it using the
+same 3-regular graph on 6 vertices depicted in Figure 1a. Now, the discrete time exclusion
+process discussed in [4] can be understood as follows as a sequence of random graphs. To
+any η ∈ {0, 1}V we associate as before a vη ⊂ V with vη = {v ∈ V |η(v) = 1} and in this
+sense we can write with a little abuse of notation η = vη. Recall, that we assume that for
+some v ⊂ V , |v| = k at time t ∈ N the process satisfies ηk;t = v. Set vc := V \v. Consider the
+possibly disconnected bipartite sub-graph Lv,vc = ((v, vc), Ev,vc) of L. Then, add all edges
+to Lv,vc which are contained in the vertex induced sub-graph Lv = (v, Ev) of L. We call
+the resulting graph Bt = (V, Σk,t) with Σk,t = Ev,vc ⊔ Ev. Indeed, for a ¯d-regular graph L
+we obtain |Σk,t| = ¯dk. Figure 2 illustrates one possible situation based on a 3-regular graph
+on six vertices. Remark that all edges incident to vertices in v are also present in Bt which
+preserves their degree and makes it accessible to calculate the number of edges in Bt. The
+Figure 2. For an underlying 3-regular graph we apply the construction based on
+the classical exclusion process. The graph Bt constructed from the set of occupied
+sites v containing blue particles and its complement in V colored in pale blue. A
+particle displacement happens by drawing uniformly one of the directed edges and
+exchanging the state of the vertices connected by said edge.
+exclusion process now consists of drawing one edge uniformly from Σk,t and exchanging the
+state of the endpoints. Note that this might lead to exchanging two occupied sites which
+renders P η independent of η. As an interpretation of the exclusion process in this case one
+can see the exchange of states of two particles as their collision, both jumping back to the
+state they came from. Note that, as discussed in [4], results in a process whose associated
+Markov chain on Lk is ergodic with a uniform stationary distribution over all states in Lk.
+It, therefore, does not ”see” different sub-structures of the graph L and behaves identically
+on all regular graphs with fixed degree ¯d. It is, consequently, not adapted to the problem we
+want to approach. Further structures of the graphs Bt and B′
+t, presented in hereinabove, can
+
+REFERENCES
+13
+be imagined, accentuating different graphs structures in need of being analyzed via MCMC
+approaches.
+Appendix B. Necessary results on Token Graphs
+Theorem B.1 ([7]). Let L be a connected simple graph on ¯n vertices and let k ∈ {1, . . . , ¯n−
+1}. Then, κ(Lk) = minv∈Vk degk(v) and a corresponding vertex cut is the neighborhood of
+v∗ ∈ Vk with degk(v∗) = minv∈Vk degk(v).
+Proposition B.2 ([7]). Consider the graph Lk = (Vk, Ek) for k ∈ {1, . . . , ¯n − 1}. Then
+|Vk| =
+�¯n
+k
+�
+and
+|Ek| = 1
+2
+�
+¯dk
+�¯n
+k
+�
+− ¯n ¯d
+�¯n − 2
+k − 2
+��
+= k(¯n − k)
+�¯n
+k
+�
+¯d
+2(¯n − 1).
+(B.1)
+Proposition B.3 ([7]). Let L be a simple connected graph on ¯n vertices and k ∈ {1, . . . , ¯n −
+1}. Denote by avg deg(Lk) the average degree in Lk, i.e.,
+avg deg(Lk) :=
+1
+|Vk|
+�
+v∈Vk
+degk(v).
+and by ¯d the average degree in L. Then average degree in Lk satisfies
+avg deg(Lk)
+k(¯n − k)
+=
+¯d
+¯n − 1 ≤ 1.
+(B.2)
+Jens Walter FISCHER,, Institut de Math´ematiques de Toulouse. CNRS UMR 5219., Univer-
+sit´e Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse cedex 09.
+Email address: jens.fischer@math.univ-toulouse.fr
+
diff --git a/p9E3T4oBgHgl3EQfLwm1/content/tmp_files/load_file.txt b/p9E3T4oBgHgl3EQfLwm1/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..2c9748d45e742d28dabd3320b346db3d20aa204e
--- /dev/null
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@@ -0,0 +1,421 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf,len=420
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='04367v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='PR]  11 Jan 2023  AN MCMC SAMPLING OF DENSEST k-SUB-GRAPHS OF REGULAR  GRAPHS WITH CONNECTED COMPLEMENT GRAPH  JENS WALTER FISCHER♠♣  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' We present an exclusion process based approach for sampling densest k-sub-  graphs from regular graphs L with connected complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' By interpreting an exclusion  process as a Markov chain on a corresponding Token Graph Lk, we make use of classical  Markov chain theory to obtain quantitative bounds on the convergence speed depending on  the geometry of L, which are sharp in the sens that variations in the valency lead to an  equality, which is 1, for the geometric bounds in the case of a complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' We propose  an algorithm which makes use of this particle view to avoid excessive memory use due to  the state space size and discuss the regularity and connectivity condition on L and Lc in  the Outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  ♠ Universit´e de Toulouse  ♣ University of Potsdam  Key words : Densest k Sub-graphs, Exclusion Processes, Token Graphs, MCMC Sampling  MSC 2020 : 82C22, 60K35, 05C60, 65C05, 82M31  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Introduction  In this work we use generalized exclusion processes to approach the task of finding k-  densest sub-graphs in a ¯d-regular graph L of arbitrary size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' This is, in particular, linked to  the community detection problem which has received wide attention in modern research, for  example in [10] and [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' The density of any vertex induced sub-graph Lv = (v, Ev) of some  graph L = (V, E) on the subset v ⊆ V is defined as ρ(v) = |Ev|  |v| , see for example [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' The  problem of finding the densest sub-graph becomes, then, the maximization problem which  aims at finding the subset v∗ ⊂ V such that ρ(v∗) = maxv∈P(V ), |v|≥1 ρ(v) where P(V ) is  the power set of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' In [2] the author shows that this problem can be solved in polynomial  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Fixing the size of the sub-graph renders the problem vastly more complicated as being  discussed in, among many others, [3], [6] and [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' In particular, the work in [11] and [6],  the authors conclude, that finding densest sub-graphs of fixed size is NP-hard and finding  such graphs is, therefore, unfeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Hence, the research focus is on approximations of such  objects using combinatorial selection methods or related families of problems which, reliably,  give results which are close in density to the desired sub-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  We contribute to this discussion a stochastic approach based on generalized exclusion  processes and Monte Carlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' The process is constructed on the complement graph  Lc in such a way, that it exhibits in the long time limit most likely to a densest sub-graph in L  and, even, gives monotonously sub-graphs of lesser density, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=', a second densest sub-graph is  obtained with second largest probability, a third densest with third largest probability and so  1  2  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' FISCHER  on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' We obtain a quantitative bound, which is tight with respect to variations in the average  degree ¯d, on the convergence speed based on a binary geometric condition on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' An appropriate GEP  To identify densest sub-graphs we want to construct a particle system based on discrete  time generalized exclusion process (GEP) ηk using the following Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Let L = (V, E) be a simple connected graph with |V | = ¯n ∈ N and k ∈  {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' , ¯n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Denote by (P (η))η∈{0,1}V , |η|=k a family of stochastic matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' A GEP in  discrete time ηk := (ηk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='t)t∈N of k particles on L is a Markov chain on the set of configurations  Sk = {η ∈ {0, 1}V | |η| = k}  defined by the transition matrix Q = (qη,µ)η,µ∈{0,1}V given for η, µ ∈ {0, 1}V by  qη,µ =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  P (η)(v, w)1η(v)=1=µ(w),η(w)=0=µ(v)  η(u)=µ(v)∀u̸∈{v,w}  ,  η ̸= µ  1 − �  µ̸=η qη,µ,  η = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='1)  We are going to exploit a link between a system of moving particles ηk on a finite graph  and a Markov chain in a higher dimensional space, as developed in [8], to have access to  classical results on the long time behavior of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' In particular, we consider discrete  time generalized exclusion processes, choosing the transition probabilities according to the  application at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' This gives us access to a process which respects at any time step the fixed  size of the subgraphs in which we are interested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' We translate GEPs as defined hereinabove  to Markov chains on subsets of fixed size of the vertex set of the underlying graph L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' The  arising state space corresponds to what is know as Token Graphs or Symmetric Powers of  Graphs, as defined in [1] and [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='2 ([5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Consider a finite simple graph L = (V, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Define Vk = {v ⊂ v| |v| =  k} and Ek = {⟨v, w⟩|v△w = {v, w};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' ⟨v, w⟩ ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' The graph Lk = (Vk, Ek) is called the k  Token Graph of L and we denote by degk(v) the degree of v ∈ Vk in Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  In choosing an adequate family of transition matrices (P θ)θ∈{0,1}|V | we obtain a GEP  on some graph L, based on which we can propose and quantify a Markov chain Monte  Carlo approach to sampling densest sub-graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' This GEP ηk has a discrete time Markov  chain representation SMC  k  on Lk as developed in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Consider as an example the graph  illustrated in Figure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Indeed, there are various ways to approach GEPs, depending on  the distributions which govern the transitions of individual particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' One way is by defining  a Markov chain based on a series of dynamic graphs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=', by interpreting ηk as a Markov  chain in a random environment given by the family (P η)η∈{0,1}V , |η|=k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' To this end, define for  t ∈ N the set Nk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='t = {v ∈ V |ηk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='t(v) = 1} and the directed bipartite graph B = (Bt)t∈N with  Bt = ((Nk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='t, V \\Nk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='t, Σk,t), where Σk,t represent all possible transitions of particles at time t,  with potential loops on vertices in Nk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' The random graph Bt might be disconnected but  serves as a graph theoretical representation of transitions of the GEP ηk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=', the positive  transition probabilities of P η conditioned on η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  The central part is the change of Σk,t when going from time t to t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' It captures the  possible particle movements induced by the distribution used in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' To illustrate  MCMC SAMPLING OF DENSEST k-SUBGRAPHS  3  (a) The underlying 3-regular graph on  6 vertices which we are going to use as  example for all constructions of exclusion  processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  (b) Potential particle displacements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Uniform distribu-  tion for drawing a directed edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' States of the vertices  connected by said edge are interchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' When drawing  a loop, everything remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  the construction, we consider a 3-regular graph on 6 vertices depicted in Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' It will  be the reference for missing edges which will illustrate the distinctive parts of each exclusion  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' To any η ∈ {0, 1}V we can associate a vη ⊂ V with vη = {v ∈ V |η(v) = 1} and in  this sense we can write with a little abuse of notation η = vη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' For our central construction,  assume that for some v ⊂ V with |v| = k at time t ∈ N the process satisfies ηk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='t = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Set  vc := V \\v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Consider the possibly disconnected bipartite sub-graph Lv,vc = ((v, vc), Ev,vc)  of L and add a loop ev,v to any v ∈ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' We call the resulting graph B′  t = (V, Σ′  k,t) with  Σ′  k,t = Ev,vc ⊔ {ev,v|v ∈ v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Indeed, for a ¯d-regular graph L we obtain |Σk,t| = degk(v) + k,  where degk(v) denotes the degree of v interpreted as vertex in the token graph Lk associated  to L, see [1] and [5]for a deeper discussion of Token Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' In Figure 1b we illustrate a  possible example derived from the graph L as in Figure 1a and some configuration v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' In  each time step, the exclusion process makes a step by drawing one edge uniformly from Σ′  k,t  and exchanging the state of the endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Note, that in changing the number of present  edges also the transition probabilities change due to the uniform distribution over Σ′  k,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' This  renders the process, in particular, in-homogeneous because the transition probabilities now  depend on the current configuration v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' A configuration v can be seen as a representative  of the sub-graph class of vertex induced sub-graphs which are spanned by a subset of the  original vertex set and all edges included therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Let L = (V, E) be any simple graph and v ⊆ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Then the graph Lv = (v, Ev)  with ⟨v, w⟩ ∈ Ev if and only if v, w ∈ v and ⟨v, w⟩ ∈ E is called the vertex induced sub-graph  of L on v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  In what follows, in order to obtain meaningful results we work under the assumption that  both L and Lc are connected graphs, as explained in the Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' A further discussion  of the limitations is postponed to the Outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' If this assumption is removed, the additional  difficulty of initial condition dependencies on disconnected graphs for particle systems has to  be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' We leave this open to further research and remain with the assumption on L  and Lc, giving some pointers in the Outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  4  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' FISCHER  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Convergence Results  We consider the generalized exclusion process ηk as constructed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Then, by [8], there  is an associated Markov chain SMC  k  on Lk such that ηk and SMC  k  are equal in law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' We make,  in what follows, exclusively claims about the associated Markov chain SMC  k  on Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Let k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' , ¯n−1} and consider the Markov chain SMC  k  on Lk = (Vk, Ek).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  Then it is aperiodic, irreducible and, hence, ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Furthermore, it is a reversible chain  and the stationary distribution πMC  k  is given in terms of v ∈ Vk by  πMC  k  (v) = degk(v) + k  2|Ek| + k  �¯n  k  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='1)  Based on the transition matrix given by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='1 the stationary distribution πMC  k  of SMC  k  can be derived directly and reversibility follows as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Indeed, one can identify  SMC  k  with a random walk on Lk where any vertex in Lk has additionally to its incident edges  k loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Under a dichotomy of geometric conditions on the underlying graph L, we find the  following convergence speed of SMC  k  to πMC  k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Let L be a ¯d-regular graph on ¯n vertices and k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' , ¯n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Consider  the Markov chain SMC  k  on Lk and ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Additionally, define  ρ(¯n, ¯d, k) := 4(¯n − 1)2(¯n − k)2  ¯d4  ξ(¯n, ¯d, k) := log  ��¯n  k  ��  + log  �k(¯n − k)  ¯n − 1  + k  ¯d  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  If L is such that minv∈Vk avg degk(Lv) < ¯d − 1 then for v, w ∈ Vk and  t ≥ 1 + ρ(¯n, ¯d, k)  �  log  �4  ε  �  + ξ(¯n, ¯d, k)  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='2)  the Markov chain SMC  k  satisfies  �������  �  pMC  k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='v,w  �(t)  − πMC  k  (w)  πMC  k  (w)  �������  ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  On the other hand, if L is such that minv∈Vk avg degk(Lv) ≥ ¯d − 1 then there is a constant  C > 0 such that its ε mixing time w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' the total variation distance is bounded by  τmix(ε) ≤ C log2(ε−1)  �  ξ(¯n, ¯d, k) − (¯n − 1)2  ¯d4  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='3)  Note that the constants we used in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='2, namely ρ(¯n, ¯d, k) and ξ(¯n, ¯d, k), grow  at most like a polynomial as ¯n → ∞ since log  ��¯n  k  ��  grows at most like ¯n by Stirling’s  approximation and the remaining terms are polynomial in ¯n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  Consequently, SMC  k  mixes  rapidly relative to the size of its actual state space, employing this term as used in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Its  mixing is, therefore, polynomial fast in terms of the size of the underlying graph L, which  makes it a suitable process for sampling sub-graphs of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  MCMC SAMPLING OF DENSEST k-SUBGRAPHS  5  In the following section, we propose an algorithm and the idea behind sampling densest  sub-graphs of L with high probability using the Markov chain SMC  k  introduced in this section  and exploiting its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Particle based MCMC algorithm for sampling densest k-sub-graphs  By the structure of the stationary distribution we obtain an ordering based on the degree  of the vertices, where the probability of drawing a densest sub-graph as t → ∞ is smallest  since degk(v) = k ¯d−2|Ev|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' On the other hand, this also implies that it is most probable under  πMC  k  to draw a least dense sub-graph of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Since the densest k sub-graph in L corresponds  to the least dense k sub-graph in Lc and vice versa, we can exploit the fact that both are  connected to inverse the roles under SMC  k  with respect to the weights given by its stationary  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Using this, we propose the following MCMC approach to finding densest k sub-  graphs in a ¯d-regular graph with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' To this end consider a ¯d-regular simple  connected graph L on ¯n vertices and assume that its graph complement Lc is also connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  Then Lc is a ¯n − 1 − ¯d regular simple connected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' We denote for any k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' , ¯n − 1}  by L(c)  k  = (Vk, E(c)  k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  Let k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' , ¯n − 1} and define SMC  k  as the Markov chain associated to the exclusion  process on Lc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Then, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='1 the Markov chain SMC  k  converges in distribution to  πMC  k  and with maximal probability we obtain a v∗ ∈ Vk which induces a least dense k-sub-  graph Lc  v∗ of Lc and, in turn, the set v∗ induces a densest k-sub-graph in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' We propose the  following algorithm which only uses the local information of L and the current configuration  v of SMC  k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' For the simulation, it is not necessary to construct the whole state space Lk with  �¯n  k  �  vertices and the size of the edge set given by Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Algorithm 1 only depends  on the neighborhood of all v ∈ v after having fixed a v ∈ Vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Exploiting the fact that L is  assumed to be ¯d-regular, we can bound the number of states we have to access in each turn  by ¯d·k which is also a very crude upper bound for degk(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' We use multi-sets to describe the  algorithm and when we set X = ∅ as a multi-set, we impose that (X∪{v})∪{v} = X∪{v, v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  The multi-set X can be replaced in real code by any mutable list-type object which allows  multiple times the same entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Algorithm 1 mirrors the dynamics of the exclusion process,  constructed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' It, therefore, samples rapidly, in the sense of [15], densest sub-graphs with  high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' We want to emphasize the importance of a detailed understanding of the  underlying state space Lk from the perspective of vertex induced sub-graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' In particular,  the geometric results as well as knowledge about the structure of the vertex set presented  in the appendix allowed for the bounds in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Finally, the proposed algorithm  benefits from the particle perspective in that it only needs to consider at most k ¯d vertices in  each step which is a considerable reduction from considering the whole vertex set Vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Proofs of central results  We consider the generalized exclusion process ηk as constructed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Then, by [8], there  is an associated Markov chain SMC  k  on Lk such that the marginal distributions of ηk and  SMC  k  are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' To start, we characterize the associated Markov chain SMC  k  on Lk by  giving its transition matrix explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  6  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' FISCHER  Algorithm 1 Particle based algorithm to simulate SMC  k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  Require: Terminal time for simulation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' tmix, number of trials m, graph L  Initialize t = 0  Define X = ∅ as multi-set  Run following burn in step  Draw initial state v ∈ Vk  Set SMC  k,0 = v  while t ≤ tmix do  Construct bipartite graph B′(v) as in Figure 1b  Draw uniformly an edge ⟨v, w⟩ from B′(v)  Update SMC  k,t+1 = (v\\{v}) ∪ {w};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' t = t + 1  end while  Run sampling after burn in step  while i ≤ m do  Construct bipartite graph B′(v) as in Figure 1b  Draw uniformly an edge ⟨v, w⟩ from B′(v)  Update SMC  k,t+1 = (v\\{v}) ∪ {w};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' t = t + 1  Update i = i + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  Update X = X ∪ Sk,t  end while  return X, statistic of m final states after tmix simulation steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Let k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' , ¯n − 1} and consider the Markov chain SMC  k  on Lk =  (Vk, Ek).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' We call its transition matrix P MC  k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Then, it has the form  pMC  k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='v,w =  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  1  degk(v) + k ,  ∃⟨x, y⟩ ∈ E s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' v△w = {x, y},  k  degk(v) + k ,  v = w,  0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  The form of the transition matrix follows directly from the construction of B′  t in Figure  1b and the uniform distribution over all edges in B′  t as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  The stationary  distribution πMC  k  of SMC  k  can consequently be derived directly and reversibility follows as  well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Based on the transition matrix given by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='1 and the stationary distribution  given by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='1 one can identify SMC  k  with a random walk on Lk where any vertex in  Lk has additionally to its incident edges k loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Indeed, for almost all cases of k, this is not  sufficient to render SMC  k  a lazy random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Indeed, we can quantify this transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' The random walk SMC  k  is lazy if and only if  min  v∈Vk  avg degk(Lv) ≥ ¯d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  MCMC SAMPLING OF DENSEST k-SUBGRAPHS  7  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Recall that the Markov chain is called lazy if and only if  min  v,v pMC  k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='v,v ≥ 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='1)  Therefore, using the expression derived in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='1 we obtain that SMC  k  is lazy if and  only if for all v ∈ Vk we have  degk(v)  k  + 1 ≤ 2 ⇔ ¯d − avg degk(Lv) ≤ 1  which proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  □  An obvious property is minv∈Vk avg degk(Lv) ≤ ¯d since Lv is a vertex induced sub-graph  and, therefore, the degree of any vertex in Lv is bounded by ¯d which implies the same for  the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Geometrically, the condition given by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='2 leaves, consequently, only  little room for the parameter triple (¯n, ¯d, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Due to the intermediate position SMC  k  takes  between the simple random walk on Lk and the lazy random walk on this state space, we  can in general only use that  min  v∈Vk  pMC  k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='v,v ≥ γ > 0  for some γ ∈  �  0, 1  2  �  with a phase transition, if for a parameter triple (¯n, ¯d, k) we have  minv∈Vk avg degk(Lv) ≥ ¯d − 1 which is a geometric condition on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' We turn now to the  proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' The goal of this proof is to obtain bounds on geometric properties  of vertex induced subgraphs of L, which allow for a meaningful application of known re-  sults from Markov chain theory, as for example presented in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  To begin, let γ :=  � ¯d + 1 − minv∈Vk avg degk(Lv)  �−1 and consider the following two terms which we call the  additive and the multiplicative constant, respectively, given as, firstly,  4(1 − γ)2  γ2  log  �  2|Ek| + k|Vk|  4(min{degk(v), degk(w)} + k)  �  �  ι(Lk)  2|Ek|+k|Vk|  maxv∈Vk degk(v)+k  �2  and, secondly,  4(1 − γ)2  γ2  �  ι(Lk)  2|Ek| + k|Vk|  maxv∈Vk degk(v) + k  �−2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  By [13] we can estimate the isoperimetric constant ι(Lk) from below by a constant times the  vertex connectivity κ(Lk) of Lk in the form  ι(Lk) ≥  2  |Vk|κ(Lk)  and we have from Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='1 the value of κ(Lk) as κ(Lk) = minv∈Vk degk(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' This gives  us, consequently, in the first case the lower bound for  ι(Lk)  2|Ek| + k|Vk|  maxv∈Vk degk(v) + k ≥ 2 min  v∈Vk  degk(v)  avg deg(Lk) + k  maxv∈Vk degk(v) + k  ≥ min  v∈Vk  degk(v)  avg deg(Lk)  maxv∈Vk degk(v)  8  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' FISCHER  using for the last inequality that minv∈Vk avg degk(Lv) < ¯d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' This in addition to maxv∈Vk degk(v) ≤  k(¯n−k) and Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='3 gives the following upper bound for the multiplicative constant  by  4(1 − γ)2  γ2  � ι(Lk)(2|Ek| + k|Vk|)  maxv∈Vk degk(v) + k  �−2  ≤ 4  k2  �maxv∈Vk degk(v)  avg deg(Lk)  �4 �  avg deg(Lk)  minv∈Vk degk(v)  �2  ≤ 4  k2  � ¯n − 1  ¯d  �4 �  avg deg(Lk)  minv∈Vk degk(v)  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  Employing the lower bound minv∈Vk degk(v) ≥ ¯d, which can be derived easily by using  k ≤ ¯n − 1 and the fact that there is at least one empty vertex in L with degree ¯d, one can  obtain the following upper bound, using Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='3, independent of the geometry of L  given by  4  k2  � ¯n − 1  ¯d  �4 �  avg deg(Lk)  minv∈Vk degk(v)  �2  ≤ 4(¯n − 1)2(¯n − k)2  ¯d4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  Since the additive constant equals the multiplicative constant times a logarithmic term, we  are only going to focus on the logarithmic term and we note  log  �  2|Ek| + k|Vk|  4(min{degk(v), degk(w)} + k)  �  ≤ log  �  2|Ek| + k|Vk|  minv∈Vk degk(v) + k  �  = log  ��¯n  k  ��  + log  �  avg deg(Lk) + k  minv∈Vk degk(v) + k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  For the second summand, we can use the bound which we used already above to obtain  log  �  avg deg(Lk) + k  minv∈Vk degk(v) + k  �  ≤ log  �k(¯n − k)  ¯n − 1  + k  ¯d  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  Therefore, we arrive at the upper bound on the additive constant  4(¯n − 1)2(¯n − k)2  ¯d4  �  log  ��¯n  k  ��  + log  �k(¯n − k)  ¯n − 1  + k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  ¯d  ��  We now employ some results based on evolving sets presented in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' To this end, we will  focus on lower bounds for Φk(u) := inf  �  ∂(S,Sc)  πMC  k  (S)  �� πMC  k  (S) ≤ u  �  for u ∈  �  minv∈Vk πMC  k  (v), 1  2  �  as defined in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' In particular, we use that Φ(u) ≥ Φ  �1  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' The first claim then follows  by Theorem 5 of [14], the second one by [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' To conclude the proof, note that for ¯S ∈  �  S ⊂ Vk  �� |S| ≤ |Vk|  2  �  ∩  �  S ⊂ Vk  �� πMC  k  (S) ≤ 1  2  �  we have  ∂( ¯S, ¯Sc)  πMC  k  ( ¯S) ≥ ∂( ¯S, ¯Sc)  | ¯S|  2|Ek| + k|Vk|  maxv∈Vk degk(v) + k ≥ ι(Lk)  2|Ek| + k|Vk|  maxv∈Vk degk(v) + k  where we used ¯S ∈  �  S ⊂ Vk| |S| ≤ |Vk|  2  �  for the last estimate and the definition of the  isoperimetric constant of a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Since, additionally, ¯S ∈  �  S ⊂ Vk  �� πMC  k  (S) ≤ 1  2  �  , taking  the infimum over all S ∈  �  S ⊂ Vk  �� πMC  k  (S) ≤ 1  2  �  we arrive at  Φk  �1  2  �  ≥ ι(Lk)  2|Ek| + k|Vk|  maxv∈Vk degk(v) + k  MCMC SAMPLING OF DENSEST k-SUBGRAPHS  9  and, therefore, for all u ∈  �  minv∈Vk πMC  k  (v), 1  2  �  we obtain  Φk(u) ≥ ι(Lk)  2|Ek| + k|Vk|  maxv∈Vk degk(v) + k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  The second step consists in finding a lower bound for minv∈Vk pMC  k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='v,v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' We have already found  in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='2 that  min  v∈Vk  pMC  k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='v,v ≥  1  ¯d + 1 − minv∈Vk avg degk(Lv) = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  Having found the necessary bounds, we can apply Theorem 5 of [14] and obtain after inte-  gration  � 4ε−1  4(πMC  k  (v)∧πMC  k  (w))  4 du  uΦk(u)2 ≤  �  log  �4  ε  �  − log  �  4(min{degk(v),degk(w)}+k)  2|Ek|+k|Vk|  ��  �  ι(Lk)  2|Ek|+k|Vk|  maxv∈Vk degk(v)+k  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='2)  We identify the terms in the right hand side of equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='2 with the additive constant and the  multiplicative constant for which we have found meaningful upper bounds at the beginning  of this proof, such that we obtain the first claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' The second claim follows by the same  estimate for Φk(u) and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='2 of [12] as well as equation (5) of [14] using the bound of  the additive term by ξ(¯n, ¯d, k) as defined in the Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  □  This concludes our section on the proof of the main Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Outlook  The main goal of this document was the presentation of an MCMC algorithm to sample  densest k sub-graphs from regular graphs L and the quantification of its convergence speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  We needed the additional condition that Lc is connected, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' This gives us two natural  aspects from which we could start generalizations of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  Firstly, let us consider the generalization to simple non-regular graphs under the condition  that Lc is connected and fix some k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' We construct the following example L by assuming  k < ⌊ ¯n  2 ⌋ and looking at two complete graphs Kk and K¯n−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' We assume that L has ¯n vertices  and is constructed by gluing together the two complete graphs by adding an edge between  one vertex in Kk and one vertex in K¯n−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Therefore, Lc is connected being the complete  bipartite graph minus one edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' A densest k sub-graph Lv is then given by the complete  graph Kk, which can be realized by placing either k vertices in Kk or K¯n−k calling them v and  v′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Seen as vertices in the Token Graph Lk this shows that the identifiability of  densest k sub-graphs via their degree in Lk is violated if L is not regular because degk(v) <  degk(v′) even though they have identical density as defined in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Nonetheless,  degk(v) = 1 and is the minimal degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Therefore, the sampling still gives densest subgraphs  with high probability but the hit rate will be much lower since not all densest sub-graphs lie  in the same level set w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' degk(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Adjustments of the exclusion process construction can be  helpful here, for another example which respects finer geometric features of the sub-graphs  see [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  Keeping on the other hand the regularity condition and dropping the connectedness of Lc,  we encounter another interesting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' The graph Lc is ¯n − ¯d − 1 regular by assumption  10  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' FISCHER  and assuming that it is disconnected, we find a least dense sub-graph on k vertices by finding  a redistribution of particles on the connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Note that each connected com-  ponent is by assumption itself a ¯n − ¯d − 1 regular graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Interesting combinatorial questions  arise then from monotony questions of the number of particles in each connected components  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' to its size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Under the condition on L, that there is a monotonously increasing function  fL which maps integers l ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' , ¯n} to the number of particles in k′{1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' , k} such that the  least dense sub-graph can be obtained by putting f(l) particles in the connected component  of Lc of size l, our result works as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' But the geometric implications on L as well as  interpretations for f need to be worked out before any meaning can be given to a result in  this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  REFERENCES  11  References  [1]  Yousef Alavi, Don R Lick, and Jiuqiang Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' “Survey of double vertex graphs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' In:  Graphs and Combinatorics 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='4 (2002), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' 709–715.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  [2]  Moses Charikar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' “Greedy Approximation Algorithms for Finding Dense Components  in a Graph”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' In: Approximation Algorithms for Combinatorial Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' by  Klaus Jansen and Samir Khuller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Berlin, Heidelberg: Springer Berlin Heidelberg, 2000,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' 84–95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  [3]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Corneil and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Perl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' “Clustering and domination in perfect graphs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' In: Discrete  Applied Mathematics 9 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  [4]  Persi Diaconis and Laurent Saloff-Coste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' “Comparison Theorems for Reversible Markov  Chains”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' In: The Annals of Applied Probability 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='3 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  [5]  Ruy Fabila-Monroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' “Token Graphs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' In: Graphs and Combinatorics 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='3 (2012),  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' 365–380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  [6]  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Feige, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Peleg, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Kortsarz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' “The Dense k -Subgraph Problem”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' In: Algorith-  mica (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  [7]  Jens Walter Fischer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' On the connectivity and diameter of Token graphs from a vertex  induced sub-graph perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  [8]  Jens Walter Fischer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' “Random dynamics in collective behavior - consensus, clustering  & extinction of populations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Doctoral Thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Universit´e de Toulouse, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  [9]  Santo Fortunato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' “Community detection in graphs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' In: Physics Reports 486.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='3 (2010),  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' 75–174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  [10]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Girvan and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Newman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' “Community structure in social and biological net-  works”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' In: Proceedings of the National Academy of Sciences 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='12 (2002), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' 7821–  7826.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  [11]  Samir Khuller and Barna Saha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' “On Finding Dense Subgraphs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' In: ICALP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  [12]  Laszlo Lovasz and Ravi Kannan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' “Faster Mixing via Average Conductance”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' In: Pro-  ceedings of the Thirty-First Annual ACM Symposium on Theory of Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' New  York, NY, USA: Association for Computing Machinery, 1999, 282–287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  [13]  Bojan Mohar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' “Isoperimetric numbers of graphs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' In: Journal of Combinatorial Theory,  Series B 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='3 (1989), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' 274–291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  [14]  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Morris and Yuval Peres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' “Evolving sets, mixing and heat kernel bounds”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' In: Prob-  ability Theory and Related Fields 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='2 (2005), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' 245–266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  [15]  Alistair Sinclair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' “Improved Bounds for Mixing Rates of Markov Chains and Multicom-  modity Flow”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' In: Combinatorics, Probability and Computing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='4 (1992), 351–370.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  12  REFERENCES  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Note on the subtleties of the exclusion process construction  Exclusion processes are not a new subject and have been analyzed, mostly for continuous  time instead of discrete time, in statistical mechanics at length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  This first subsection is  meant to make the point regarding the construction from the first part of this publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  We translate a ”classical exclusion process” discussed in [4] into the framework of dynamic  random graphs and then to the token graph Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' To this end define Nk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='t = {v ∈ V |ηk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='t(v) = 1}  and the time dependent graph B = (Bt) with Bt = ((Nk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='t, V \\Nk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='t, Σk,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  We adjust the step of Σk,t when going from time t to t + 1 and illustrate it using the  same 3-regular graph on 6 vertices depicted in Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Now, the discrete time exclusion  process discussed in [4] can be understood as follows as a sequence of random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' To  any η ∈ {0, 1}V we associate as before a vη ⊂ V with vη = {v ∈ V |η(v) = 1} and in this  sense we can write with a little abuse of notation η = vη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Recall, that we assume that for  some v ⊂ V , |v| = k at time t ∈ N the process satisfies ηk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='t = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Set vc := V \\v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Consider the  possibly disconnected bipartite sub-graph Lv,vc = ((v, vc), Ev,vc) of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Then, add all edges  to Lv,vc which are contained in the vertex induced sub-graph Lv = (v, Ev) of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' We call  the resulting graph Bt = (V, Σk,t) with Σk,t = Ev,vc ⊔ Ev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Indeed, for a ¯d-regular graph L  we obtain |Σk,t| = ¯dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Figure 2 illustrates one possible situation based on a 3-regular graph  on six vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Remark that all edges incident to vertices in v are also present in Bt which  preserves their degree and makes it accessible to calculate the number of edges in Bt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' The  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' For an underlying 3-regular graph we apply the construction based on  the classical exclusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' The graph Bt constructed from the set of occupied  sites v containing blue particles and its complement in V colored in pale blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' A  particle displacement happens by drawing uniformly one of the directed edges and  exchanging the state of the vertices connected by said edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  exclusion process now consists of drawing one edge uniformly from Σk,t and exchanging the  state of the endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Note that this might lead to exchanging two occupied sites which  renders P η independent of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' As an interpretation of the exclusion process in this case one  can see the exchange of states of two particles as their collision, both jumping back to the  state they came from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Note that, as discussed in [4], results in a process whose associated  Markov chain on Lk is ergodic with a uniform stationary distribution over all states in Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  It, therefore, does not ”see” different sub-structures of the graph L and behaves identically  on all regular graphs with fixed degree ¯d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' It is, consequently, not adapted to the problem we  want to approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Further structures of the graphs Bt and B′  t, presented in hereinabove, can  REFERENCES  13  be imagined, accentuating different graphs structures in need of being analyzed via MCMC  approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Necessary results on Token Graphs  Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='1 ([7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Let L be a connected simple graph on ¯n vertices and let k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' , ¯n−  1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Then, κ(Lk) = minv∈Vk degk(v) and a corresponding vertex cut is the neighborhood of  v∗ ∈ Vk with degk(v∗) = minv∈Vk degk(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='2 ([7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Consider the graph Lk = (Vk, Ek) for k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' , ¯n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Then  |Vk| =  �¯n  k  �  and  |Ek| = 1  2  �  ¯dk  �¯n  k  �  − ¯n ¯d  �¯n − 2  k − 2  ��  = k(¯n − k)  �¯n  k  �  ¯d  2(¯n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='1)  Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='3 ([7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Let L be a simple connected graph on ¯n vertices and k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' , ¯n −  1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Denote by avg deg(Lk) the average degree in Lk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=',  avg deg(Lk) :=  1  |Vk|  �  v∈Vk  degk(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  and by ¯d the average degree in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' Then average degree in Lk satisfies  avg deg(Lk)  k(¯n − k)  =  ¯d  ¯n − 1 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content='2)  Jens Walter FISCHER,, Institut de Math´ematiques de Toulouse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=' CNRS UMR 5219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
+page_content=', Univer-  sit´e Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse cedex 09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E3T4oBgHgl3EQfLwm1/content/2301.04367v1.pdf'}
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+MNRAS 000, 1–19 (2022)
+Preprint 3 January 2023
+Compiled using MNRAS LATEX style file v3.0
+A New Method to Determine X-ray Luminosity Functions of AGN
+and their Evolution with Redshift
+Ahlam Alqasim,1★ Mat J. Page,1
+1UCL Mullard Space Science Laboratory, Holmbury Hill Road, Dorking, Surrey, RH5 6NT, UK
+Accepted 2022 December 22. Received 2022 December 15; in original form 2022 February 8
+ABSTRACT
+Almost all massive galaxies today are understood to contain supermassive black holes (SMBH)
+at their centers. SMBHs grew by accreting material from their surroundings, emitting X-rays
+as they did so. X-ray Luminosity Functions (XLFs) of Active Galactic Nuclei (AGN) have been
+extensively studied in order to understand the AGN population’s cosmological properties and
+evolution. We present a new fixed rest-frame method to achieve a more accurate study of the
+AGN XLF evolution over cosmic time. Normally, XLFs are constructed in a fixed observer-
+frame energy band, which can be problematic because it probes different rest-frame energies at
+different redshifts. In the new method, we construct XLFs in the fixed rest-frame band instead,
+by varying the observed energy band with redshift. We target a rest-frame 2−8 keV band using
+XMM-Newton and HEAO 1 X-ray data, with 7 observer-frame energy bands that vary with
+redshift for 0 < 𝑧 < 3. We produce the XLFs using two techniques; one to construct a binned
+XLF, and one using a Maximum Likelihood (ML) fit, which makes use of the full unbinned
+source sample. We find that our ML best-fit pure luminosity evolution (PLE) results for both
+methods are consistent with each other, suggesting that performing XLF evolution studies with
+the high-redshift data limited to high-luminosity AGN is not very sensitive to the choice of
+fixed observer-frame or rest-frame energy band, which is consistent with our expectation that
+high-luminosity AGN typically show little absorption. We have demonstrated the viability of
+the new method in measuring the XLF evolution.
+Key words: galaxies: active – quasars: supermassive black holes – X-rays: galaxies – galaxies:
+nuclei – galaxies: luminosity function – methods: data analysis
+1
+INTRODUCTION
+It is now understood that almost all massive galaxies today contain
+Supermassive Black Holes (SMBH) at their centers (Kormendy &
+Ho 2013), with masses ranging between ∼ 106 − 109𝑀⊙. Their
+massive growth is mainly due to accretion of matter from their sur-
+roundings, shining as Quasi-Stellar Objects (QSOs) and emitting
+X-rays as they did so. QSOs belong to a larger population of Ac-
+tive Galactic Nuclei (AGN). SMBHs can also grow due to other
+processes such as black hole mergers and the tidal capture of stars.
+Most of them have been observed to be dormant in present-day
+galaxies, exhibiting luminosities largely below their Eddington lim-
+its, leaving only ∼ 10% of galaxies hosting AGN (Ho 2008). Hence,
+AGN studies are very important in understanding the evolution of
+galaxies over cosmic time, mainly because they are strongly linked
+to the growth of SMBHs and can thus tell us useful information
+about the accretion history of the universe (Brandt & Alexander
+2015). It is now also well-known that the evolution of SMBHs and
+the evolution of their host galaxies are strongly connected, or that
+they co-evolve (Symeonidis et al. 2013). AGN feedback plays a sig-
+★ E-mail: ahlam.alqasim.17@ucl.ac.uk
+nificant role in quenching star formation and stopping the growth
+of their host galaxy after a certain point (Bongiorno et al. 2016), so
+understanding how they evolve with time can provide very useful
+insights into the evolution of galaxies.
+Most of the observed luminosity in AGN is radiated in the
+optical-UV, but the X-ray flux of AGN shows the fastest variability
+on short timescales in all of the wavelength ranges (Netzer 2013).
+This suggests that the X-rays originate from a small region close
+to the central object, now known to be a SMBH (Mushotzky et al.
+1993).
+Studies of Seyfert galaxies showed that one of the most plau-
+sible physical processes driving the X-ray emission was inverse
+Compton radiation (Haardt & Maraschi 1991). In this scenario,
+the intrinsic high energy X-rays seen in AGN originate from a hot
+corona plasma (consisting of relativistic electrons) close to the ac-
+cretion disk (Beckmann & Shrader 2013). The hot corona scatters
+the optical-UV photons coming from the inner regions of the ac-
+cretion disk to X-ray energies. This mechanism drives the shape
+observed in the X-ray spectra of AGN (Netzer 2013). Further out
+from the central engine of the AGN is a thick torus surrounding
+the central SMBH and accretion disk, responsible for obscuring
+the X-ray emission due to photoelectric absorption (Morrison &
+© 2022 The Authors
+arXiv:2301.00223v1  [astro-ph.GA]  31 Dec 2022
+
+2
+A. Alqasim et al.
+McCammon 1983; Antonucci 1993). Seyfert I galaxies give a clear
+view of the active nucleus because the line of sight is unobstructed,
+while Seyfert II galaxies have a line of sight that is obscured by
+the torus, causing them to appear to have less evidence of activity.
+XMM-Newton surveys particularly helped constrain the evolution of
+X-ray absorption, as well as other physical properties of AGN, by
+studying faint X-ray sources (Hasinger et al. 2001).
+The luminosity function (LF) of galaxies is generally defined
+as the number of objects per unit volume (i.e. Mpc3) per unit log-
+arithmic luminosity interval. For AGN, the best model shape that
+describes their X-ray LF (XLF) is a double power-law modified by
+a factor for evolution (e.g. Boyle et al. 1988; Miyaji et al. 2000b).
+XLFs of AGN can be constructed using extensive X-ray surveys in
+order to understand their cosmological properties and study their
+evolution over cosmic time. Many techniques have been published
+to quantify the cosmological evolution of AGN, including using
+the ⟨𝑉/𝑉𝑚𝑎𝑥⟩ method (Schmidt 1968), the 𝑉𝑒/𝑉𝑎 method (Avni
+& Bahcall 1979) for combined samples, Monte Carlo simulations
+(Cristiani & Vio 1990), and the 1/𝑉𝑎 method (Maccacaro et al.
+1991). The ⟨𝑉/𝑉𝑎⟩ (or ⟨𝑉/𝑉𝑚𝑎𝑥⟩) approach is mainly used for de-
+termining whether there is evolution with redshift, while the 1/𝑉𝑎
+(or 1/𝑉𝑚𝑎𝑥) approach is used for calculating a binned luminosity
+function. The 1/𝑉𝑎 is more commonly used because it is simpler,
+incorporating a binned differential luminosity function within a red-
+shift interval. Improved methods to 1/𝑉𝑎 for constructing binned
+XLFs have also been demonstrated (Page & Carrera 2000; Cara &
+Lister 2008).
+Some works have shown that the cosmic evolution of AGN is
+consistent with models in which the luminosity varies with redshift,
+e.g. Pure Luminosity Evolution (PLE) model (Barger et al. 2005). In
+such models, the shape of the XLF evolution remains the same and
+the luminosity evolution causes the model curve to simply shift right
+or left on the luminosity plane as the XLF evolves. Other models try
+to explain the XLF evolution by varying the density with redshift,
+e.g. Pure Density Evolution (PDE) model (Fotopoulou et al. 2016).
+In such models, the XLF shape also remains the same and the density
+evolution causes the model curve to simply shift up or down on the
+density plane as the XLF evolves. Other works have proposed mod-
+els that combine both luminosity and density evolution, e.g. Inde-
+pendent Luminosity Density Evolution (ILDE) model (Yencho et al.
+2009) and Luminosity and Density Evolution (LADE) model (Aird
+et al. 2010). In such models, the XLF shape is kept the same, and the
+combined luminosity and density evolution causes the model curve
+to be shifted in any direction across the luminosity-density plane
+as the XLF evolves. Some studies favor a more complicated model
+in which the shape of the XLF evolution changes while also vary-
+ing the density and luminosity, e.g. Luminosity-Dependent Density
+Evolution (LDDE) model (Hasinger et al. 2005; Ueda et al. 2014).
+In such models, the curves are not only shifted in any direction, but
+their shapes can also change when moving around the luminosity-
+density plane as the XLF evolves. Most of these models include a
+critical redshift 𝑧𝑐 value (also referred to as cutoff redshift), after
+which the XLF evolution either changes or stops (Fotopoulou et al.
+2016). Some papers also use a Bayesian approach to explore the
+AGN evolution in a model-independent way (Georgakakis et al.
+2015; Fotopoulou et al. 2016).
+Previous XLF studies of AGN show that the number of AGN
+per unit volume per unit luminosity has been observed to change
+strongly with redshift (Ebrero et al. 2009). One of the main issues
+to be addressed in XLF studies of AGN (especially with Chandra
+and XMM-Newton surveys) is the impact of absorption, which can
+suppress the observed X-ray flux (especially for redshifts 𝑧 ≤ 1),
+and is even more problematic for Compton-thick sources (Aird et al.
+2015b). Most studies try to correct for this absorption and model
+its evolution with redshift. Normally, luminosity functions for AGN
+are constructed in a fixed observed energy band, and there is still no
+consensus on what the best approach is to model how they evolve
+with redshift when looking at the rest-frame energy bands. How
+an observed energy band, 𝐸𝑜𝑏𝑠(𝑧), relates to the rest-frame energy
+band, 𝐸𝑟 𝑓 , for a given redshift 𝑧 is described as
+𝐸𝑜𝑏𝑠(𝑧) =
+𝐸𝑟 𝑓
+1 + 𝑧 .
+(1)
+The most common X-ray bands normally used to study XLFs
+of AGN are 0.5−2 keV (Miyaji et al. 2000a; Hasinger et al. 2005;
+Ebrero et al. 2009) and 2−10 keV (Aird et al. 2010; Miyaji et al.
+2015; Georgakakis et al. 2015). The 2−8 keV band has also been
+studied in some cases instead of 2−10 keV (Barger et al. 2005;
+Silverman et al. 2008). Some other studies have focused on the
+5−10 keV band (Fotopoulou et al. 2016) to avoid correcting for
+the absorbed part of the AGN spectrum. Since hard X-rays (∼ E
+> 2 keV) are significantly less affected by absorption, the soft X-
+ray band (0.5−2 keV) should mainly sample unabsorbed AGN (and
+the derived XLF should only include the unabsorbed population).
+However, since the rest-frame energy band changes with redshift, the
+population will include absorbed AGN at higher redshifts, which
+affects the binned luminosity function because the K-correction
+doesn’t take that effect into account. As a result, the sample will
+start to gain more absorbed sources as you move up in the rest-
+frame energy band with redshift, even if the study is conducted in
+a harder observed X-ray band (Aird et al. 2015b). This problem
+was mitigated to some degree by Cowie et al. (2003) and Barger
+et al. (2005), who used the 2−8 keV observer-frame to study lower
+redshift sources (𝑧 <∼ 1.5), and used the flux from the 0.5−2 keV
+observer-frame to calculate the 2−8 keV luminosity in the rest-frame
+for the high redshift sources (𝑧 >∼ 1.5).
+In this paper, we present a new method that aims to tackle
+this problem by varying the observed energy band with redshift,
+allowing us to fix the X-ray energy band in the rest-frame. This
+eliminates the need to model the redshift-dependence of X-ray ab-
+sorption from material surrounding the SMBHs. We make use of
+X-ray data from XMM-Newton and HEAO 1 in this work to produce
+X-ray luminosity functions in the fixed observed band (the standard
+method) and the fixed rest-frame band (the new method) for the 2−8
+keV band. The fixed rest-frame band requires the analysis of several
+observer-frame bands that correspond to each redshift bin, which
+will be described in more detail in Section 2.
+This paper is structured as follows: in Section 2, we introduce
+the new method used to produce the fixed rest-frame XLF along
+with the corresponding redshifted observed energy bands. We then
+present the X-ray data used in this sample and its selection criteria in
+Section 3 for both the fixed observed band and the fixed rest-frame
+band. In Section 4, we describe the optical identifications used to
+study the completeness of the X-ray sample and obtain redshifts
+for them. In Section 5, we describe the two techniques used to
+compute our XLFs; one using the method of Page & Carrera (2000)
+to construct a binned XLF, and one using a Maximum Likelihood
+(ML) fit on the full unbinned source sample. Both techniques are
+computed for the standard method (fixed observed band) and the
+new method (fixed rest-frame band) introduced in this paper. We
+present the results of the XLFs in Section 6. In Section 7, we discuss
+the performance of the new method compared with the standard
+MNRAS 000, 1–19 (2022)
+
+Redshifted XLF method for AGN
+3
+Figure 1. AGN model spectrum in the rest-frame, constructed for a wide
+range of column densities 𝑁𝐻. The spikes correspond to photoelectric
+absorption edges, which occur when the X-ray emission passes through a
+highly absorbing material, usually the torus in the case of AGN.
+method, as well as with previous XLF studies. The conclusions of
+this work are then summarized in Section 8.
+The cosmological parameters that were assumed in this paper
+were 𝐻0 = 70 km s−1 Mpc−1, Ω𝑀 = 0.3 and ΩΛ = 0.7.
+2
+THE FIXED REST-FRAME XLF METHOD
+In this section, we describe our new method that allows us to vary
+the observed energy band with redshift, and to fix the X-ray energy
+band in the rest-frame. Fig. 1 shows AGN model spectra in the
+rest-frame, constructed using PyXspec1 (the Python interface for
+Xspec). The model curves are produced assuming a model with a
+cold photoelectric absorber and a powerlaw component with Γ =
+2.0. We explored a wide range of column densities 𝑁𝐻 to account for
+AGN that display different levels of intrinsic absorption. The spikes
+correspond to photoelectric absorption edges, which occur when the
+X-ray emission passes through a highly absorbing material, usually
+the torus in the case of AGN. Looking at Fig. 1, if we study an AGN
+at 0.5−2 keV in a fixed observed band, a source at redshift 𝑧 = 0
+(red shaded region) would be in the correct rest-frame energy band.
+However, if the source is redshifted at 𝑧 = 2 (blue shaded region),
+the part of the spectrum that is observed corresponds to 1.5−6 keV
+in the rest-frame. This means that we are inherently looking at very
+different parts of the AGN rest-frame spectrum at different redshifts.
+As a result, studying AGN in a fixed observed energy band can be
+problematic since it probes very different rest-frame energies at
+different redshifts.
+To construct a fixed rest-frame XLF for a given survey, a fixed
+rest-frame energy band 𝐸𝑟 𝑓 must be chosen to cover a redshift
+range 𝑧𝑖 < 𝑧 < 𝑧 𝑓 . The redshift range determines what redshifted,
+observed energy bands 𝐸𝑜𝑏𝑠(𝑧) will be used to generate images and
+sourcelists for the X-ray data. In this new method, one can choose
+1 https://heasarc.gsfc.nasa.gov/xanadu/xspec/python/
+html/index.html
+0
+1
+2
+10
+100
+1000
+104
+105
+106
+LX (erg s−1)
+z
+Figure 2. Plot of Redshift vs. Luminosity (𝐿𝑋 −𝑧 plane) of the XMS survey
+in the fixed rest-frame 2−8 keV band.
+to do this in as many 𝐸𝑜𝑏𝑠(𝑧) bands as desired, depending on
+how many redshift intervals are chosen. The more 𝐸𝑜𝑏𝑠(𝑧) bands
+there are, the more precise the fixed rest-frame XLF will be, but
+a good middle ground will need to be established to minimize
+computational time spent depending on how large the data sample
+is. Once the 𝐸𝑜𝑏𝑠(𝑧) bands are determined, a sourcelist is produced
+for each given 𝐸𝑜𝑏𝑠(𝑧) band at the end of the data reduction process.
+The X-ray fluxes obtained from the sourcelists are used to convert to
+X-ray luminosity. In this work, the X-ray luminosity is for the 2−8
+keV rest-frame band, and is corrected for Galactic absorption, but
+not for absorption occurring within the AGN and its host galaxy.
+As can be seen in Fig. 1, this energy band will only be sensitive
+to sources with column densities of up to 1023 cm−2. Since the
+degree of attenuation due to photoelectric absorption is strongly
+energy dependent (as shown in Fig. 1), it is important to use a
+fixed rest-frame energy band when producing our X-ray sourcelists.
+The sourcelists are filtered to only contain sources with measured
+redshifts corresponding to the 𝐸𝑜𝑏𝑠(𝑧) band of the data sample. A
+separate flux limit 𝐹𝑙𝑖𝑚 is then derived for each 𝐸𝑜𝑏𝑠(𝑧) band. So
+in essence, the fixed rest-frame XLF uses observed bands that vary
+with redshift, and uses a flux limit that is not constant but is also a
+function of redshift 𝐹𝑙𝑖𝑚(𝑧).
+2.1
+Binned XLF
+When applying the new method to a binned XLF (see Section 5.1
+for more details), redshift bins 𝑧𝑏𝑖𝑛 first need to be determined. A
+plot of the 𝐿𝑋 − 𝑧 plane is made for the targeted 𝐸𝑟 𝑓 , where 𝐿𝑋
+is the X-ray luminosity of the sources, shown in Fig. 2. This figure
+shows the redshift distribution of the data used in this paper and
+at which redshifts most AGN with luminosities lie in at 2−8 keV.
+Looking at the diagram, we have enough sources from our sample
+spread over the luminosity-redshift plane to use bins of ∼ 0.5 in
+redshift.
+Once the redshift bins are determined, the midpoint of each
+redshift bin 𝑧𝑚𝑖𝑑 is then used in equation (1) with 𝑧 = 𝑧𝑚𝑖𝑑 to
+determine what the corresponding 𝐸𝑜𝑏𝑠(𝑧) band will be. The mini-
+mum number of 𝐸𝑜𝑏𝑠(𝑧) bands needed will be set by the number of
+𝑧𝑏𝑖𝑛 chosen for the XLF. In principle, the number of 𝐸𝑜𝑏𝑠(𝑧) bands
+need not be restricted by the number of redshift bins used. Each
+𝑧𝑏𝑖𝑛 can include multiple 𝐸𝑜𝑏𝑠(𝑧) bands that correspond to smaller
+MNRAS 000, 1–19 (2022)
+
+NH = 1019 cm-2
+NH = 1022 cm-2
+z=O
+NH = 1020 cm-2
+z=2
+N = 1023 cm-2
+N = 1021 cm-2
+NH = 1024 cm-2
+10000
+8000
+6000
+4000
+2000
+0
+100
+101
+Energy [keV]4
+A. Alqasim et al.
+Table 1. The 𝐸𝑜𝑏𝑠 (𝑧) bands for which images and sourcelists are produced
+to construct the fixed rest-frame XLF. 𝑧𝑏𝑖𝑛 gives the redshift interval, for
+which the midpoint (𝑧𝑚𝑖𝑑) is used to determine the observer-frame energy
+range (𝐸𝑜𝑏𝑠(𝑧)) that corresponds to 2 − 8 keV in the rest-frame band.
+𝑧𝑏𝑖𝑛
+𝑧𝑚𝑖𝑑
+Eobs(𝑧)
+[eV]
+0.0 − 0.5
+0.25
+1600 − 6400
+0.5 − 1.0
+0.75
+1142 − 4571
+1.0 − 1.5
+1.25
+888 − 3555
+1.5 − 2.0
+1.75
+727 − 2909
+2.0 − 2.5
+2.25
+615 − 2461
+2.5 − 3.0
+2.75
+533 − 2133
+redshift intervals within the same bin. For each 𝑧𝑏𝑖𝑛, an XLF is then
+produced using the data from its corresponding 𝐸𝑜𝑏𝑠(𝑧) band (or
+set of bands). For the binned XLF, 𝐹𝑙𝑖𝑚(𝑧) is a function that maps
+the discrete redshift intervals to discrete flux limit values, and each
+𝐸𝑜𝑏𝑠(𝑧) will have its own flux limit.
+2.2
+Model-fitted XLF
+When applying the new method to produce a model fitted XLF using
+the ML technique (see Section 5.3 for more details), the number of
+𝐸𝑜𝑏𝑠(𝑧) bands used is not determined by any redshift binning. The
+more 𝐸𝑜𝑏𝑠(𝑧) bands used, the more precisely the energy range is
+fixed in the rest-frame for each source for the XLF. As was the case
+in the binned XLF, 𝐹𝑙𝑖𝑚(𝑧) is used to derive a flux limit for each
+𝐸𝑜𝑏𝑠(𝑧) band used for the model-fitted XLF.
+2.3
+Redshifted Energy Bands
+For this work, a fixed rest-frame energy band of 2−8 keV was chosen
+to cover a redshift range of 0 < 𝑧 < 3. We use redshift intervals
+of 0.5 for the binned XLF, with good coverage over the 𝐿𝑋 − 𝑧
+plane (see Fig. 2). To obtain the redshifted, observed energy bands
+in which sourcelists are produced, the midpoint of each redshift
+interval was used, giving a total of 6 redshifted energy bands (see
+Table 1).
+3
+X-RAY DATA
+In the following sections, we describe the targets and observations
+used for the XMM-Newton data in this paper. We then describe the
+data processing methods used to reduce the XMM-Newton data, and
+the source selection criteria used when producing the final X-ray
+sourcelists. Finally, we describe the flux-limited X-ray catalogue
+from Piccinotti et al. (1982), from which the HEAO 1 X-ray data
+was taken.
+For the XMM-Newton data reduction and processing, c-shell
+scripts were used to run tasks using sas-18.0.0, heasoft-6.27,
+wcstools-3.9.5, and SAOImageDS9-8.1. Python scripts were
+run using python-3.7.6.
+3.1
+XMM-Newton Observations
+The data used in this work were taken from the XMM-Newton Satel-
+lite, which carries multiple telescopes to study X-ray sources. XMM-
+Newton is sensitive up to 10 keV with a spatial resolution of ∼5 arc-
+sec (Jansen et al. 2001). The XMM-Newton data used were extracted
+from the three EPIC (European Photon Imaging Camera) cameras
+on XMM-Newton. The EPIC cameras are placed at the focus points
+of the X-ray mirror assemblies. Two of the cameras use EPIC-MOS
+CCDs (Turner et al. 2001), while the third camera uses EPIC-PN
+CCDs (Strüder et al. 2001). Each EPIC instrument is fitted with a
+filter wheel carrying X-ray transparent light blocking filters to block
+out background light outside the desired X-ray band.
+In this work, we use 25 XMM-Newton target fields adopted
+from the the XMM-Newton Medium Sensitivity Survey (XMS). The
+XMS is a survey built using a sample of the AXIS survey (Carrera
+et al. 2007) covering a geometric sky area of 3.33 deg2 (Ebrero et al.
+2009). The luminosity distribution of the entire sample shows that
+the survey contains Seyfert-like AGN as well as QSOs. The XMS is
+sensitive to AGN with intrinsic column densities up to 1023 cm−2
+(Mateos et al. 2005).
+We have used the redshifted energy bands 𝐸𝑜𝑏𝑠(𝑧) listed in
+Table 1 for the methods and analysis performed in this work. For
+each 𝐸𝑜𝑏𝑠(𝑧), the data reduction method from Section 3.2 was
+performed for a total of 29 XMM-Newton observations. A list of the
+XMM-Newton observations used and their general properties can
+be seen in Table 2). Taking into account the excluded areas from
+our masks (see Section 3.2), the total geometric sky area covered
+amounts to 2.61 deg2 for the XMM-Newton X-ray data.
+3.2
+XMM-Newton Data Reduction
+The XMM-Newton mission provides the Science Analysis System
+(SAS) pipeline software (Gabriel et al. 2004) specifically designed
+to reduce and analyze data collected by the XMM-Newton obser-
+vatory. The SAS tasks epproc and emproc are used to produce
+a calibrated event list for each instrument. For the EPIC-PN event
+lists, we excluded the PN readout outer-edge regions at the top and
+bottom of the detector chips (as done in Carrera et al. 2007).
+The data was first reduced in in the 0.5−2 keV energy band, as
+described in Section 3.2.1, for the purpose of correcting the offset
+positions of sources within the attitude file of the observation and
+the event lists obtained from epproc and emproc. This only needs
+to be done once for each XMM-Newton observation. This makes it
+more efficient when reducing the data in the multiple energy bands
+required for this work without needing to correct the source position
+offsets for each energy band at a later stage in the processing. Once
+the position offsets were corrected, the new attitude and event list
+files were used to generate images and sourcelists in the desired
+𝐸𝑜𝑏𝑠(𝑧) bands, as described in Section 3.2.2.
+3.2.1
+Reduction Process to Correct Source Position Offsets
+To filter out intervals of particle background flares from EPIC event
+lists, a high energy light curve (𝐸 > 5 keV) was extracted from
+the event file. A background rate threshold was then determined by
+where the light curve is steady with low background intervals (see
+Table 2 for the rate thresholds used for each XMM-Newton observa-
+tion). This threshold varies with each observation, depending on the
+background. All the work in this section after this point was done
+using a 0.5 − 2 keV energy band.
+We first produce X-ray images separately for each EPIC instru-
+ment using evselect in the targeted energy range. For EPIC-PN,
+2 Obtained using the HEASOFT tool nh.
+MNRAS 000, 1–19 (2022)
+
+Redshifted XLF method for AGN
+5
+Table 2. List of the XMM-Newton fields used in this work, with their general properties. This includes the XMM-Newton observation number, the RA/Dec
+coordinates at the center of the field (RA𝐹𝑖𝑒𝑙𝑑, Dec𝐹𝑖𝑒𝑙𝑑), the Galactic column density in the direction of the field (𝑁𝐻,𝐺𝑎𝑙), the filter used for the EPIC
+cameras, the "clean" exposure time for the M1, M2 and PN cameras (T𝑒𝑥 𝑝,𝑀1, T𝑒𝑥 𝑝,𝑀2, T𝑒𝑥 𝑝,𝑃𝑁 , respectively), and the M1, M2 and PN rate thresholds
+(M1𝑡ℎ, M2𝑡ℎ, PN𝑡ℎ, respectively) used to filter out intervals of flaring particle background rate lightcurves (produced at 𝐸 > 5 keV using a time bin size of 20
+s).
+Observation
+RA
+Dec
+𝑁𝐻 2
+Filter
+T𝑒𝑥 𝑝
+T𝑒𝑥 𝑝
+T𝑒𝑥 𝑝
+M1𝑡ℎ
+M2𝑡ℎ
+PN𝑡ℎ
+𝐹𝑖𝑒𝑙𝑑
+𝐹𝑖𝑒𝑙𝑑
+𝐺𝑎𝑙𝑎𝑐𝑡𝑖𝑐
+𝑀1
+𝑀2
+𝑃𝑁
+[deg]
+[deg]
+[1020 cm−2]
+[ks]
+[ks]
+[ks]
+[count s−1]
+[count s−1]
+[count s−1]
+0012440301
+331.2908325
+-1.9216667
+5.94
+Thin
+29.1
+29.3
+24.3
+2.0
+2.0
+8.0
+0081340901
+342.955833
+-17.8731111
+2.27
+Medium
+22.3
+22.3
+17.9
+1.2
+1.2
+4.6
+0092850201𝑎
+324.438501
+-14.5487222
+4.15
+Medium
+41.3
+41.2
+36.6
+14.0
+14.0
+20.0
+0100240801
+233.095833
+-8.5347222
+8.39
+Medium
+26.5
+26.6
+19.3
+2.0
+2.0
+7.0
+0100440101
+337.1266665
+-5.3152778
+4.98
+Thick
+45.3
+45.5
+36.7
+1.5
+1.5
+5.0
+0102040201
+172.789167
+31.2352777
+1.88
+Thin𝑑
+17.6
+23.2
+11.3
+2.0
+1.2
+7.0
+0102040301
+157.7462505
+31.0488889
+1.67
+Thin𝑑
+25.4
+26.0
+20.6
+1.2
+1.2
+5.0
+0103060101
+322.300833
+-15.6447222
+4.45
+Medium
+20.4
+20.5
+13.7
+1.5
+1.5
+8.0
+0106460101
+145.7500005
+46.9916666
+1.1
+Thin
+47.4
+47.6
+36.7
+2.0
+2.0
+6.5
+0109910101
+210.3945
+-11.127
+4.26
+Thin
+48.6
+48.7
+39.2
+1.2
+1.2
+4.0
+0111000101
+4.6375005
+16.4383333
+3.77
+Medium
+31.3
+31.1
+23.6
+1.2
+1.2
+4.0
+0111220201
+93.894375
+71.0353333
+9.29
+Medium
+48.5
+49.4
+40.4
+2.0
+2.0
+12.5
+0112260201
+44.6041665
+13.3
+9.9
+Thin
+18.1
+18.3
+12.4
+1.2
+1.2
+4.0
+0112260201
+44.6041665
+13.3
+9.9
+Thin
+18.1
+18.3
+12.4
+1.2
+1.2
+4.0
+0112370301
+34.9000005
+-5.0
+2.0
+Thin
+44.5
+44.5
+34.3
+2.2
+2.2
+8.0
+0112371001
+34.5
+-5.0
+2.06
+Thin
+43.6
+43.8
+35.8
+1.2
+1.2
+6.0
+0112620101
+130.35
+70.8947222
+2.81
+Medium
+27.6
+27.9
+23.9
+2.5
+2.5
+14.0
+0112650401
+16.100001
+-6.4
+6.19
+Thin𝑑
+23.6
+23.6
+15.2
+1.1
+1.1
+4.0
+0112650501
+16.0000005
+-6.7
+6.26
+Thin𝑑
+20.2
+22.3
+14.8
+1.3
+1.3
+7.0
+0112880301
+352.958334
+19.9380555
+3.96
+Thick
+14.4
+14.5
+10.8
+1.7
+1.7
+8.5
+0124110101𝑏
+185.433333
+75.3102778
+2.9
+Medium
+33.8
+33.9
+29.8
+–
+–
+–
+0124900101
+187.883334
+64.2391666
+2.52
+Thin
+29.7
+30.1
+24.8
+2.0
+2.0
+8.0
+0112370401 + 𝑐
+0112371501
+34.6999995
+-4.6536111
+2.03
+Thin
+23.1
+23.1
+15.2
+2.0
+2.0
+8.0
+0123100101 + 𝑐
+0123100201
+116.0187495
+74.56375
+3.68
+Thin
+58.8
+57.2
+32.2
+2.5
+2.5
+7.0
+0100240101 + 𝑐
+0100240201
+202.695834
+24.2330556
+0.999
+Medium
+65.2
+65.2
+44.7
+2.5
+2.5
+6.0
+0106660101 + 𝑐
+0106660601
+333.881958
+-17.7349166
+1.85
+Thin
+151.1
+151.4
+126.2
+1.5
+1.5
+6.0
+𝑎 Different exposures were merged within the same XMM-Newton observation (see Table A2).
+𝑏 Different exposures (with different frame modes) within the same XMM-Newton observation were reduced separately and
+the final images were summed (see Table A3).
+𝑐 Different XMM-Newton observations were merged for the same target (see Table A4).
+𝑑 Listed filter corresponds to EMOS1 and EPN. The EMOS2 filter was different (Thick for the first two and Medium for the
+second two).
+“Out-Of-Time" (OOT) images were also produced to take into ac-
+count OOT events that end up being mixed within the read-out
+direction in the CCD frame, which then gets added to the PN back-
+ground map once it’s scaled out, and accounts for the bright streaks
+that tend to be seen in PN images. To combine information from all
+three instruments, the X-ray images were then summed up together
+into a final X-ray image using the ftools3 task farith.
+3 https://heasarc.gsfc.nasa.gov/ftools/
+MNRAS 000, 1–19 (2022)
+
+6
+A. Alqasim et al.
+Along with X-ray images, we produce exposure maps for each
+of the 3 EPIC instruments in the relevant energy band. MOS ex-
+posure maps were multiplied by the ratio of MOS/PN countrates
+assuming a power-law spectrum with a photoelectric absorption
+component. This was done using PyXspec with the Galactic hy-
+drogen column density 𝑁𝐻,𝐺𝑎𝑙 (see Table 2) and a photon index
+Γ = 1.9 (Mateos et al. 2005). Energy channels outside the targeted
+energy range were excluded. The MOS exposure maps were then
+added to the PN exposure map to make a summed exposure map.
+An Energy Conversion Factor (ECF), defined as the ratio of the
+count rate to flux, was then calculated using PyXspec to determine
+how to convert EPIC band count rates to fluxes in a given energy
+band, which is then used for further SAS tasks. Since the MOS1 and
+MOS2 images are added on top of the PN image, the final combined
+EPIC image is in the format of an EPIC-PN image. Hence, the ECF
+is calculated assuming a PN image. The summed exposure map was
+then used to make a mask (using the SAS task emask) that filters
+out sources in areas of the image where there were CCD gaps or
+bad pixels.
+Finally, we produce background maps with our own back-
+ground script for each EPIC instrument. This background script
+uses 2 models, a vignetted background model and a flat background
+model, as well as an OOT component for EPIC-PN. The flat back-
+ground model accounts for the particle background. The script used
+here then performs an ML fit to the background in a similar manner
+to the method described in Loaring et al. (2005). For each EPIC
+instrument, we ran our background script using the list of sources
+from the eboxdetect SAS task, along with the EPIC exposure
+map and X-ray image, producing 3 separate background maps. The
+background maps were then summed up into a final background
+map using farith.
+The output from eboxdetect was also used in the SAS task
+emldetect with an ML threshold of 8 to make an X-ray sourcelist.
+We corrected the astrometry of our event lists by correlating the
+positions of the X-ray sources with optical sources from the SDSS
+Photometric DR12 Catalogue, as well as the Pan-STARRS Survey,
+according to the method described in Traulsen et al. (2020). This
+correction only needed to be done once for each XMM-Newton
+observation, after which the event lists and attitude files were used to
+generate images and sourcelists for our set of 𝐸𝑜𝑏𝑠(𝑧) energy bands
+(see Section 3.2.2). New (position-corrected) images, background
+maps and exposure maps in the 0.5 − 2 keV energy band were
+finally reproduced and subsequently summed up in the same manner
+described in this section.
+Using the position-corrected 0.5 − 2 keV exposure maps, we
+produce 2 masks using emask, which are used when producing X-
+ray sourceslists in the 𝐸𝑜𝑏𝑠(𝑧) bands (see Table 3 for parameter
+values used in all the masks produced). The first initial mask 𝑀𝑑𝑒𝑡
+covers the full field of view (FOV) detector image, constructed us-
+ing the summed exposure map. We then run the mask through the
+ftools program fgauss, which convolves the image with a circular
+Gaussian function to produce a smoothed image. The smoothed im-
+age was then used to produce a modified version of the initial mask,
+which was then used to make the XMM-Newton X-ray sourcelist in
+Section 3.2.2.
+The second and final mask 𝑀 𝑓 𝑖𝑛𝑎𝑙 has excluded regions from
+the image rather than retaining the full FOV. This mask is used
+in Section 3.2.2 to filter out sources from the X-ray sourcelist for
+each of the 𝐸𝑜𝑏𝑠(𝑧) bands. 𝑀 𝑓 𝑖𝑛𝑎𝑙 is the mask constructed by
+multiplying the 𝑀 𝑓 𝑖𝑙𝑡𝑒𝑟 and the 𝑀𝑎𝑥𝑖𝑠 masks together, using the
+task farith. We describe what these are as follows. We applied
+two constraints when producing the mask 𝑀𝑎𝑥𝑖𝑠. We used the PN
+Table 3. Parameter values for the Gaussian sigma and mask thresholds used
+in the XMM-SAS tasks fgauss and emask, respectively. 𝑀𝑑𝑒𝑡 is the full
+FOV detector mask constructed using the summed exposure map; 𝑀 𝑓 𝑖𝑙𝑡𝑒𝑟
+is the mask constructed using the summed exposure map with low exposure
+pixels filtered out (set to a minimum exposure threshold of 10 ks); 𝑀𝑎𝑥𝑖𝑠 is
+the mask constructed using the PN exposure map, where fgauss and emask
+parameters were adjusted to add an extra blur to the detector chip edges and
+exclude pixels falling too close to them.
+Task
+Task
+Parameter
+Task Parameter Values
+𝑀𝑑𝑒𝑡
+𝑀 𝑓 𝑖𝑙𝑡𝑒𝑟
+𝑀𝑎𝑥𝑖𝑠
+fgauss
+sigma
+4.0
+4.0
+6.0
+emask
+threshold1
+threshold2
+0.93
+0.5
+0.93
+0.5
+0.98
+0.5
+exposure map rather than the summed exposure map when making
+the mask. We also adjusted the fgauss and emask parameters to
+add an extra blur to the detector chip edges and exclude pixels
+falling too close to them (see Table 3). AXIS only used PN data
+when processing their XMM-Newton observations, and removed an
+extra 5-7 pixels from the edges of their detector chips (Carrera et al.
+2007). Without limiting the mask to PN and incorporating the extra
+blurring to account for the 5-7 pixels that were excluded by AXIS,
+we ended up with a lot of unidentified sources. This brought our
+completeness statistics down, not necessarily because the sources
+were spurious, but simply because they were not included in the
+X-ray source list in AXIS, and as a result they weren’t part of their
+optical identification campaign. Thus, we chose to include these two
+constraints in 𝑀𝑎𝑥𝑖𝑠 to be consistent with the methods adopted in
+the AXIS survey, given that we are using their optical identification
+campaign. When producing the mask 𝑀 𝑓 𝑖𝑙𝑡𝑒𝑟, we set a minimum
+exposure threshold of 10 ks from the summed exposure map to filter
+out regions with low-exposure pixels. This was done to avoid the
+occurrence of spurious sources (see Section 3.4 for more details).
+After multiplying the 𝑀 𝑓 𝑖𝑙𝑡𝑒𝑟 and the 𝑀𝑎𝑥𝑖𝑠 masks together,
+additional regions were excluded from the resulting 𝑀 𝑓 𝑖𝑛𝑎𝑙 mask,
+which we describe as follows. The XMS survey did a serendipitous
+search rather than a blind search, and previously verified sources
+were taken as a target around which sources were searched for, and
+the target source itself was excluded (or masked out) during this
+search. There were a total of 25 target sources that were excluded in
+the XMS, which we also remove from the final mask. We adopt the
+RA, Dec and target exclusion radius used by AXIS from Table 1 in
+Carrera et al. (2007). One XMM-Newton observation (0112260201)
+is pointed between two cluster targets (A 399 and A 401). AXIS
+provided an exclusion region for one of these cluster targets (A 399).
+For that, we adopted a more conservative approach and extended
+the AXIS exclusion radius to 307′′ to remove extra cluster sources.
+We also added an additional exclusion region to remove the second
+cluster target (A 401) as well. We additionally excluded rectangular
+OOT regions using the widths provided from Table 1 in Carrera
+et al. (2007), and we provide the RA, Dec, angle and height of the
+OOT rectangular region. All details regarding the exclusion areas
+applied to the final mask can be found in Table A1.
+3.2.2
+Reduction Process for 𝐸𝑜𝑏𝑠(𝑧) Bands
+The reduction process in the fixed rest-frame method follows a sim-
+ilar procedure as described in Section 3.2.1. The position-corrected
+attitude files and event lists were used to generate images and
+MNRAS 000, 1–19 (2022)
+
+Redshifted XLF method for AGN
+7
+sourcelists for the XMM-Newton observations in 6 redshifted en-
+ergy band 𝐸𝑜𝑏𝑠(𝑧), as listed in Table 1. Some changes and additions
+were incorporated, which will be described in this section.
+The X-ray images and exposure maps were produced in the
+given 𝐸𝑜𝑏𝑠(𝑧) energy band and summed up together using the
+same process described in Section 3.2.1. To check the dependence
+of the ECF and the MOS/PN ratios on intrinsic AGN absorption,
+we recalculated them for each XMM observation assuming 𝑁𝐻 = 0
+and 𝑁𝐻 = 1022.5 cm−2, and derived the fractional change for each
+parameter. We find that the ECF decreases by 11% between 𝑁𝐻 =
+0 and 𝑁𝐻 = 1022.5 cm−2, so luminosities of absorbed sources
+will be slightly underestimated. The MOS/PN ratio is much less
+affected by absorption, decreasing by just 1.6% between 𝑁𝐻 = 0
+and 𝑁𝐻 = 1022.5 cm−2. Since the MOS/PN ratio and ECF are both
+a function of energy, they were derived separately for each 𝐸𝑜𝑏𝑠(𝑧)
+using the same methods described previously. When making the
+summed exposure map, the MOS/PN ratios derived for the 𝐸𝑜𝑏𝑠(𝑧)
+energy band were used. To make the summed background map in
+the targeted 𝐸𝑜𝑏𝑠(𝑧) (as described in the previous section), we run
+the background script with 4 iterations to get the best background
+map. The multiple iterations allow us to get rid of most of the
+bright sources, and help us produce maps that are very close to the
+background for the final summed background map.
+The final X-ray sourcelist is then produced using the final
+summed images, background maps and exposure maps, with an ML
+threshold of 4 in eboxdetect, an ML threshold of 9 for emlde-
+tect, and the ECF derived for the given 𝐸𝑜𝑏𝑠(𝑧) energy band. The
+mask used when making the sourcelist (in both eboxdetect and
+emldetect) was the initial first mask described in Section 3.2.1.
+This means that the sourcelist includes all sources detected from
+the X-ray image, since the initial mask covers the full FOV of the
+detector. The second and final mask (described in Section 3.2.1)
+was then used to remove sources that lie outside of the mask from
+the X-ray sourcelist. This gives us our final X-ray sourcelist for a
+given 𝐸𝑜𝑏𝑠(𝑧) band.
+3.3
+Combining XMM-Newton Event Lists and Observations
+Given that we’re not going down to fluxes below 10−14 ergs s−1
+cm−2, a 50 ks exposure depth is more than sufficient to have good X-
+ray measurements well below the limit of the optical identifications.
+If a given XMM-Newton observation included more than one science
+exposure for any of the EPIC instruments (i.e. extra S00 or U00
+exposures), we sought to merge the top two event lists with the
+longest exposure times using the SAS task merge to achieve a decent
+exposure time (see Table A2). We placed the criteria that the event
+lists being merged had to have the same filter and the same frame
+mode. This resulted in 3 XMM-Newton observations (0092850201,
+0112370401 and 0123100101) that contained merged event lists
+from multiple exposures within the same observation.
+One XMM-Newton observation (0124110101) contained mul-
+tiple science exposures that were taken in different frame modes,
+and thus could not be merged. Instead, we reduced the science ex-
+posures separately for each EPIC instrument, and summed up their
+X-ray images, exposure maps and background maps at the end of
+the data reduction process. We then used these summed images,
+exposure maps and background maps when running the source de-
+tection chain in Section 3.2.2 and making the final X-ray sourcelist
+(see Table A3).
+If there was more than one XMM-Newton observation targeted
+towards the same field, AXIS had a preference for those that were
+public at an earlier period in the programme or those that were
+Figure 3. Comparison of the measured XMM-Newton fluxes to the published
+AXIS fluxes in the 0.5 − 2 keV band.
+part of the SSC Guaranteed Time Program (Carrera et al. 2007).
+Instead, to make use of all the data, we sought to merge together the
+top two observations with the longest exposure times using the SAS
+task merge to achieve a better exposure time (see Table A4). The
+same criteria was placed on merging different XMM-Newton ob-
+servations, requiring the event lists being merged to have the same
+filter and the same frame mode. If any of these observations had
+more than one science exposure, they were merged internally first
+before being merged with another XMM-Newton observation. For
+these merged observations, we used the exclusion area properties
+from the AXIS-chosen observation for the target field when mod-
+ifying the final mask described in Section 3.2.1. These properties
+are listed in Table A1.
+3.4
+XMM-Newton Source Selection
+In this work, a separate sourcelist was produced for each 𝐸𝑜𝑏𝑠(𝑧)
+band, as well as the 𝐸𝑟 𝑓 band (2−8 keV), per XMM-Newton observa-
+tion. This gives us a total of 7 X-ray sourcelists for each observation.
+We then combined the sourcelists from all the XMM-Newton ob-
+servations, resulting in a comprehensive X-ray soureclist spanning
+the entire dataset for each of the 7 energy bands. Extended sources
+were removed from these sourcelists. To make sure our X-ray fluxes
+produced through the data reduction process were reasonable, we
+compared our 0.5 − 2 keV X-ray fluxes to the published 0.5 − 2 keV
+AXIS fluxes. They were found to be in good agreement (see Fig. 3).
+Spurious sources resulting from data artifacts and systematic
+effects were being detected through the source detection chain,
+which posed some issues. Many of these sources had low detection
+likelihoods but very high fluxes (above the flux limit of the 𝐸𝑜𝑏𝑠(𝑧)
+band, determined in Section 4.4). To reduce the occurrence of these
+types of sources, we filtered out low exposure pixels (using a 10 ks
+exposure threshold) from the 0.5 − 2 keV summed exposure map in
+Section 3.2.1, excluding them from the final mask used to filter out
+the X-ray sourcelists. The ML threshold for emldetect was also
+set to 9 in Section 3.2.2 when making the final sourcelist to avoid
+sources with low detection likelihoods. This reduced the number of
+spurious sources significantly. Fig. 4 displays the X-ray source fluxes
+vs. the detection likelihood of the sources from the final filtered
+sourcelists for each 𝐸𝑜𝑏𝑠(𝑧) band. The black dashed line indicates
+the maximum likelihood threshold of 9 used in emldetect. The
+MNRAS 000, 1–19 (2022)
+
+log(XMM Flux [W/m?])
+10-16
+10-17
+10-17
+10-16
+log(AXIS Flux [W/m2])8
+A. Alqasim et al.
+vertical solid line is the flux limit of the X-ray sample, below which
+sources are not included when making the XLF.
+Despite the measures taken, some spurious sources were still
+found above the flux limit of the 𝐸𝑜𝑏𝑠(𝑧) sourcelists. These sources
+were individually followed-up and investigated. Upon visual inspec-
+tion of the X-ray images, some of them were not real sources and
+had a very low signal-to-noise ratio (SNR, defined by the X-ray flux
+of the source divided by the flux error of the source). These were
+likely due to systematic errors resulting from the data reduction
+pipeline process. To avoid sources like these, we set a condition to
+filter out X-ray sources that had a SNR < 2.5.
+In the XMS survey, there were two sets of XMM-Newton ob-
+servations that partially overlapped over the same region of the sky,
+as listed below:
+• G133-69 Pos_1 and G133-69 Pos_2
+• SDS-1, SDS-2, and SDS-3
+AXIS dealt with this by masking out portions of the overlapping
+regions in the second (and third) fields. Instead of doing this, we
+use equation (2) to take the weighted average ¯𝑥 of the RA and
+Dec positions for any detected sources that were overlapped for a
+given energy band. We consider sources to be overlapped if they
+have an angular separation distance 𝜃𝐷 ≤ 10′′. We search for these
+overlapped sources within the combined sourcelist across all XMM-
+Newton observations for each energy band and filter them out, re-
+placing them with the weighted average,
+¯𝑥 =
+�
+𝑥𝑖
+𝜖2
+𝑖
++ 𝑥 𝑗
+𝜖2
+𝑗
+� �
+1
+𝜖2
+𝑖
++ 1
+𝜖2
+𝑗
+�−1
+(2)
+where ¯𝑥 is the weighted average, 𝑥 is the RA/Dec/flux of the 𝑖𝑡ℎ and
+𝑗𝑡ℎ overlapping sources, and 𝜖 is the RA/Dec/flux error of the 𝑖𝑡ℎ
+and 𝑗𝑡ℎ overlapping sources.
+3.5
+HEAO 1 A-2 X-Ray Sample
+To fill in the gaps in our X-ray data for high luminosity sources at low
+redshifts (𝑧 < 0.2), we also include the flux-limited X-ray sample
+by Piccinotti et al. (1982). This is a complete catalogue of X-ray
+sources at Galactic latitudes produced in the 2−10 keV band using
+data from the HEAO 1 experiment A-2 X-ray survey (Rothschild
+et al. 1979). The survey goes down to a limiting sensitivity of
+3.1 × 10−11 ergs cm−2 s−1 and covers a sky area of 2.7 × 104 deg2.
+The survey reports their measurements in two separate scans (1st
+scan and 2nd scan). Since the 1st scan is deeper, Piccinotti et al.
+(1982) treat it as the primary measurement for deriving their best-fit
+parameters, and the 2nd scan fluxes were only used for independent
+confirmation. Hence, we adopt the 1st scan flux values as the X-ray
+flux measurement when adding the data to our XLF.
+Piccinotti et al. (1982) report their X-ray fluxes in units of
+R15, which is a counting rate derived using the 1.5◦ × 3◦ FWHM
+fields of view of the layers of the X-ray counters in the HEAO 1
+A-2 experiment. They also list conversion factors for each R15 flux
+measurement corresponding to each X-ray source. We multiply each
+conversion factor by the first-scan R15 flux measurement to convert
+the fluxes to units of 10−11 ergs cm−2 s−1.
+Given the slight difference in the 2 − 8 keV and 2 − 10 keV
+energy bands, the 2−10 keV source fluxes from the HEAO 1 sample
+had to be corrected as follows. The flux between two energies (for
+a given energy range 𝐸𝑖 < 𝐸 < 𝐸 𝑓 ) is given by equation (3), which
+is the X-ray spectral form of the majority of Seyfert galaxies.
+𝐹𝐸𝑖−𝐸 𝑓 =
+∫ 𝐸 𝑓
+𝐸𝑖
+𝑘𝐸1−Γ𝑑𝐸
+(3)
+where Γ is the photon index, 𝑘 is a normalization constant, and 𝐸𝑖
+and 𝐸 𝑓 define the energy range.
+When making the model spectrum in PyXspec in Section 3.2.1,
+a photon index of Γ = 1.9 was used. To remain consistent, the same
+photon index was assumed when converting the X-ray fluxes. To
+account for the differences between the 2 − 8 keV and 2 − 10 keV
+fluxes, equation (3) was evaluated for the two energy ranges, and the
+ratio between them 𝑅𝐹 (𝑅𝐹 = 𝐹2−8/𝐹2−10) was taken. This gives
+us 𝑅𝐹 = 0.852, which was then multiplied by the HEAO 1 2 − 10
+keV fluxes to give us the 2 − 8 keV fluxes for our work.
+4
+OPTICAL IDENTIFICATIONS
+To construct an XLF, redshifts are required along with source X-ray
+fluxes. In this section, we describe the optical identifications used
+to obtain redshifts for the X-ray sources in our work. We take the
+optical identifications from the XMS survey for the XMM-Newton
+X-ray sources, and the optical identifications from the Piccinotti
+et al. (1982) catalogue for the HEAO 1 X-ray sources. We restrict
+our redshifts to be from spectroscopically identified optical coun-
+terparts.
+4.1
+XMM-Newton Medium Sensitivity Survey
+The XMS survey is comprised of four overlapping samples in the
+0.5−2 keV, 0.5−4.5 keV, 2.0−10 keV and 4.5−7.5 keV energy bands
+with flux limits well above the sensitivity of the data. XMS covers a
+total of 318 distinct X-ray sources, and counterparts for each source
+were searched for in optical catalogues within 5 arcsec from the
+position of the X-ray source. Redshifts were measured by matching
+emission and absorption features to the sliding wavelengths of these
+features (Barcons et al. 2007). The XMS has a high identification
+completeness, giving us 255 AGN sources with counterparts that
+are positively identified via optical spectroscopy.
+To identify the X-ray sources in each of the 7 filtered
+sourcelists, the RA/Dec positions of our sources were matched with
+the XMS optical counterpart source positions, taken from Table 5
+in Barcons et al. (2007). For this work, only sources that were iden-
+tified via optical spectroscopy within the XMS sample were consid-
+ered for our identifications. This criteria was also applied for 𝑁𝑖𝑑
+in equation (4) when calculating the completeness fraction. Since
+XMM-Newton has a ∼ 5′′ spatial resolution (Jansen et al. 2001),
+we matched sources within an angular distance of 5′′. Matched X-
+ray sources were then assigned corresponding redshifts from their
+matched optical counterparts. These were then used when conduct-
+ing the completeness studies of our sample, described in more detail
+in Section 4.4.
+4.2
+HEAO 1 A-2 Survey
+The optical identifications for the X-ray sources from the HEAO 1
+Survey were taken from the Piccinotti et al. (1982) catalogue. We
+included X-ray sources that were classified as either of the following:
+Seyfert-1; Seyfert-2, NELG, N or other active galaxy; BL Lacerate
+object; and QSO. Of these, we only use sources with an ID quality of
+MNRAS 000, 1–19 (2022)
+
+Redshifted XLF method for AGN
+9
+Figure 4. X-ray source flux vs. Detection Likelihood for each of the 𝐸𝑜𝑏𝑠 (𝑧) bands. Red data points are unidentified X-ray sources, and blue data points are
+X-ray sources identified via optical spectroscopy from the AXIS-XMS survey. The black dashed line indicates the maximum likelihood threshold of 9 used in
+emldetect. The vertical solid line is the flux limit of the X-ray sample, below which sources are not included when making the XLF.
+“certain" or “possible", with a redshift measurement 𝑧 < 0.2. This
+gives us a total of 29 spectroscopically identified AGN adopted from
+the Piccinotti et al. (1982) catalogue.
+4.3
+Flux Limits
+To understand what the appropriate flux limits are to use for our
+XMM-Newton X-ray sources, the completeness of the data pool was
+studied for each redshifted energy band (see Section 4.4 for more
+MNRAS 000, 1–19 (2022)
+
+1600ev 6400ev band
+4.0
+Unidentified
+Identified
+3.5
+Flux Limit
+Maximum
+log1o(Likelihood)
+Likelihood
+3.0
+Threshold
+2.5
+2.0
+1.5
+1.0
+-15.0
+-14.5
+-14.0
+-13.5
+-13.0
+-12.5
+log1o(Flux [erg/s/cm?])1142ev 4571ev band
+Unidentified
+4.0
+Identified
+Flux Limit
+3.5
+Maximum
+log1o(Likelihood)
+Likelihood
+Threshold
+3.0
+2.5
+2.0
+1.5
+1.0
+-15.0
+-14.5
+-14.0
+-13.5
+-13.0
+-12.5
+log10(Flux [erg/s/cm?])888ev 3555ev band
+4.5
+Unidentified
+Identified
+4.0
+Flux Limit
+Maximum
+log1o(Likelihood)
+3.5
+Likelihood
+Threshold
+3.0
+2.5
+2.0
+1.5
+1.0
+-15.0
+-14.5
+-14.0
+-13.5
+-13.0
+-12.5
+log10(Flux [erg/s/cm?])727ev 2909ev band
+4.5
+Unidentified
+Identified
+4.0
+Flux Limit
+Maximum
+log1o(Likelihood)
+Likelihood
+3.5
+Threshold
+3.0
+2.5
+2.0
+1.5
+1.0
+-15.0
+-14.5
+-14.0
+-13.5
+-13.0
+-12.5
+-12.0
+log10(Flux [erg/s/cm?])615ev 2461ev band
+Unidentified
+4.5
+Identified
+Flux Limit
+4.0
+Maximum
+log1o(Likelihood)
+Likelihood
+3.5
+Threshold
+3.0
+2.5
+2.0
+1.5
+1.0
+-15.0
+-14.5
+-14.0
+-13.5
+-13.0
+-12.5
+-12.0
+log10(Flux [erg/s/cm?])533ev 2133ev band
+Unidentified
+4.5
+Identified
+Flux Limit
+4.0
+Maximum
+log1o(Likelihood)
+Likelihood
+3.5
+Threshold
+3.0
+2.5
+2.0
+1.5
+1.0
+-15.0
+-14.5
+-14.0
+-13.5
+-13.0
+-12.5
+-12.0
+log10(Flux [erg/s/cm?])10
+A. Alqasim et al.
+details). We list the derived flux limits in Table 5 for each 𝐸𝑜𝑏𝑠(𝑧)
+band.
+The HEAO 1 sample is defined by sources brighter than a
+countrate of 1.25 R15 in the 1st scan, so we use this as the flux limit
+for these X-ray sources. We convert the flux limit to units of 10−11
+ergs cm−2 s−1 using a conversion factor of 2.175, the value most
+used by Piccinotti et al. (1982) for converting their X-ray fluxes.
+Correcting this to the 2 − 8 keV band using 𝑅𝐹 gives us a flux limit
+of 2.315 × 10−11 ergs cm−2 s−1 for the X-ray sources used from the
+HEAO 1 survey.
+4.4
+Completeness Studies of the XMM-Newton Data
+The completeness of the XMM-Newton X-ray data was studied for
+each of the 7 energy bands (6 𝐸𝑜𝑏𝑠(𝑧) bands and 1 𝐸𝑟 𝑓 band).
+This allows us to assign the appropriate flux limits when making
+the XLF. The completeness fraction 𝑓𝑐(𝐹𝑥) as a function of X-ray
+source flux 𝐹𝑥, from brightest to faintest, is given by
+𝑓𝑐(𝐹𝑥) =
+𝑁𝑖𝑑(𝐹 ≥ 𝐹𝑥)
+𝑁𝑡𝑜𝑡𝑎𝑙(𝐹 ≥ 𝐹𝑥) ,
+(4)
+where the numerator term 𝑁𝑖𝑑(𝐹 ≥ 𝐹𝑥) represents the number of
+optically-identified X-ray sources with a flux, 𝐹, greater than or
+equal to 𝐹𝑥, and the denominator term 𝑁𝑡𝑜𝑡𝑎𝑙(𝐹 ≥ 𝐹𝑥) represents
+the total number of X-ray sources with a flux greater than or equal
+to 𝐹𝑥. The 𝑁𝑖𝑑 term in equation (4) thus represents the number
+of X-ray sources identified in the XMS survey. We map out this
+numerator term, 𝑁𝑖𝑑(𝐹 ≥ 𝐹𝑥), as a function of flux 𝐹𝑥 in Fig. 5,
+as well as the denominator term, 𝑁𝑡𝑜𝑡𝑎𝑙(𝐹 ≥ 𝐹𝑥), as a function of
+flux 𝐹𝑥 in Fig. 6.
+The identification criteria for 𝑁𝑖𝑑 required X-ray sources to
+be matched with XMS sources that were positively identified via
+optical spectroscopy, including AGN, clusters, and stars. Additional
+sources were found to have published spectroscopic redshifts that
+were not included in the optical identifications of the XMS survey.
+These were included in 𝑁𝑖𝑑 for the X-ray source identifications in
+this work, listed in Table 4.
+Fig. 5 displays the source flux 𝐹𝑥 vs. 𝑁𝑖𝑑(𝐹 ≥ 𝐹𝑥) for each
+energy band. The figure includes two plots: Fig. 5a displays 𝑁𝑖𝑑
+sources over the entire completeness sample, and Fig. 5b only dis-
+plays 𝑁𝑖𝑑 sources that have redshifts within their corresponding
+𝐸𝑜𝑏𝑠(𝑧) redshift bin 𝑧𝑏𝑖𝑛.
+To study the completeness of our X-ray sourcelists across all
+energy bands, a plot of 𝐹𝑥 vs. 𝑓𝑐 was produced (see Fig. 7). This also
+allowed us to derive flux limits needed to make the redshifted XLF
+for each energy band. To derive the required 𝐹𝑙𝑖𝑚(𝑧) to make the
+fixed rest-frame XLF, a separate 𝐹𝑙𝑖𝑚 is needed for each 𝐸𝑜𝑏𝑠(𝑧)
+band. Each 𝐹𝑙𝑖𝑚(𝑧) was determined based on where the complete-
+ness curve reaches 𝑓𝑐 = 80% for each 𝐸𝑜𝑏𝑠(𝑧) band. A pragmatic
+choice was made in choosing a limit of 80%, as there is a trade off
+between achieving high completeness and having good number of
+sources to produce the XLF. In Fig. 7, we mark this completeness
+flux limit threshold with a horizontal dashed black line, and the set
+of derived 𝐹𝑙𝑖𝑚(𝑧) for the redshifted energy bands are marked by
+the vertical solid lines. Table 5 lists the derived flux limit for each
+energy band. A plot of these flux limits as a function of redshift
+is shown in Fig. 8. On the same figure, we also plot the discreet
+function of 𝐹𝑙𝑖𝑚(𝑧) used when making the fixed rest-frame XLF in
+this work.
+Once the flux limits were derived as a function of redshift,
+a fixed rest-frame XLF could be constructed. Only AGN sources
+with spectroscopic redshifts were used when constructing the XLFs.
+Clusters, stars and AGN with no redshifts or ones that only had pho-
+tometric redshifts were excluded. See Section 5.1 and Section 5.3
+for more details.
+5
+THE X-RAY LUMINOSITY FUNCTION
+In this section, we present two techniques for calculating XLFs of
+AGN in the 𝐸𝑟 𝑓 and 𝐸𝑜𝑏𝑠(𝑧) energy bands. The first technique is
+used to produce a binned XLF over discreet luminosity and redshift
+bins (Section 5.1), while the second technique is used to compute
+the XLF using an ML fit to the full set of sources in the sample
+(Section 5.3). We also describe the analytical function we use to fit
+the XLF (Section 5.2).
+5.1
+Binned XLF
+To construct a binned luminosity function of a sample of objects,
+we divide the Luminosity−Redshift plane into 𝐿 − 𝑧 bins. For this
+paper, the binned XLF was constructed using the method of Page
+& Carrera (2000). Binned luminosity functions are by their nature
+averaged over a luminosity and redshift bin, and hence where 𝜙
+varies significantly with luminosity and/or redshift within the bin
+(as for example, at high 𝐿 where 𝜙 changes rapidly with 𝐿) the
+value expected from a binned estimator may be somewhat different
+to the value of the model at the center of the bin. We have examined
+the magnitude of this difference in our survey by comparing the
+expectation values for the model bin (as defined in Section 5 of
+Page & Carrera (2000)) to the value of the model evaluated at the
+midpoint of the bin. For example, we looked at the difference in
+the 1.0 < 𝑧 < 1.5 range at log 𝐿𝑋 = 44.98 ergs/s, and find that
+the difference between the model and expectation values of log 𝜙 is
+about 0.05.
+We define 𝜙, the differential luminosity function, in terms of
+log 𝐿 rather than in terms of 𝐿, because it is easier to use when
+dealing with a large span of luminosities (Cara & Lister 2008),
+𝜙(log 𝐿, 𝑧) =
+𝑑2𝑁
+𝑑𝑉𝑑 log 𝐿 ,
+(5)
+where 𝑁 is the number of objects, 𝑧 is the redshift, 𝐿 is the luminosity
+and 𝑉 is the comoving volume. Note that 𝜙(𝐿, 𝑧) and 𝜙(log 𝐿, 𝑧)
+are related to each other by a factor of 𝐿 ln(10).
+The binned estimate of the luminosity function using the Page
+& Carrera (2000) method can be obtained for 𝑁 objects found over
+any volume-luminosity region, described by:
+𝜙(log 𝐿, 𝑧) ≈
+𝑁
+∫ log 𝐿𝑚𝑎𝑥
+log 𝐿𝑚𝑖𝑛
+∫ 𝑧𝑚𝑎𝑥(log 𝐿)
+𝑧𝑚𝑖𝑛
+𝑑𝑉
+𝑑𝑧 𝑑𝑧 𝑑 log 𝐿
+,
+(6)
+where ⟨𝑁⟩ is the expectation value of the number of objects, 𝐿 is
+the luminosity, 𝑧𝑚𝑖𝑛 is the minimum redshift in Δ𝑧, and 𝑧𝑚𝑎𝑥(𝐿)
+is the maximum possible redshift for an object of luminosity 𝐿 to
+be detected and remain contained within Δ𝑧. We have adopted a
+uniform bin width in Δ log 𝐿 of 0.3 for our binned XLFs.
+For each redshifted energy band 𝐸𝑜𝑏𝑠(𝑧), an XLF was pro-
+duced within its corresponding redshift bin 𝑧𝑏𝑖𝑛, as listed in Table
+1. For example, the XLF for the 2.5 < 𝑧 < 3.0 bin was done using
+the X-ray sources in the 0.5 − 2.1 keV band. Since we use discreet
+redshift intervals to construct the binned XLF, the equations and
+MNRAS 000, 1–19 (2022)
+
+Redshifted XLF method for AGN
+11
+Table 4. Spectroscopically identified sources from the literature that were not included in the optical identifications of the XMS survey. These were used in 𝑁𝑖𝑑
+for the X-ray source identifications in this work. The table lists the catalogue from which the source was taken, the name of the source, the RA/Dec position of
+the source, and the spectroscopic redshift of the source.
+Source Name
+RA
+Dec
+Redshift
+Redshift Origin
+[deg]
+[deg]
+MS 0737.0+7436
+115.802083
+74.493333
+0.312
+EMSS (Stocke et al. 1991)
+GALEXASC J074202.51+742625.5
+115.512068
+74.440213
+0.599
+XBS (Caccianiga et al. 2008)
+2MASS J21300228-1534131
+322.509363
+-15.570248
+0.562
+2MASS (Caccianiga et al. 2004)
+J133120.3+242304
+202.835000
+24.384472
+0.753
+BUXS (Mateos et al. 2015)
+(a) N𝑖𝑑 ∈ 𝑧𝑎𝑙𝑙
+(b) N𝑖𝑑 ∈ 𝑧𝑏𝑖𝑛
+Figure 5. Plot of flux 𝐹 vs. the number of identified sources 𝑁𝑖𝑑 with a flux greater than or equal to a given X-ray source flux 𝐹𝑥 (𝑁𝑖𝑑 (𝐹 ≥ 𝐹𝑥)) for
+the 7 energy bands. Left: 𝑁𝑖𝑑 includes sources within the entire redshift range 0 < 𝑧 < 3. Right: 𝑁𝑖𝑑 only includes sources that have redshifts within their
+corresponding 𝐸𝑜𝑏𝑠 (𝑧).
+Table 5. Derived flux limits for each energy band in the XMS survey. This
+includes the 𝐸𝑟 𝑓 band (2 − 8 keV) and the 𝐸𝑜𝑏𝑠 (𝑧) bands. The 𝐹𝑙𝑖𝑚(𝑧)
+was derived for each energy band based on where the completeness curve in
+Fig. 7 reaches 𝑓𝑐 = 80% (indicated by the black dashed line). The converted
+2−8 keV flux limit from the HEAO 1 survey is also reported (see Section 4.3
+for more details).
+𝐸 (𝑧)
+𝐹𝑙𝑖𝑚(𝑧)
+[eV]
+[10−14 ergs s−1 cm−2]
+XMS Survey
+2000 − 8000
+4.44186
+1600 − 6400
+2.82524
+1142 − 4571
+2.38016
+888 − 3555
+1.87509
+727 − 2909
+1.68052
+615 − 2461
+1.50592
+533 − 2133
+1.44649
+HEAO 1 Survey
+2000 − 8000
+2315.18
+methods used are the same as Page & Carrera (2000). The key dif-
+ference in the way we construct the binned XLF is that the flux limit
+𝐹𝑙𝑖𝑚(𝑧) is different for each of the redshift shells (see Table 5 and
+Fig. 8). Together, these make up the 2 − 8 keV XLF, fixed in the
+rest-frame.
+The data processed in the 2 − 8 keV band (the 𝐸𝑟 𝑓 band) was
+then used to construct the fixed observer-frame XLF for the entire
+redshift range (0 < 𝑧 < 3) to compare with the fixed rest-frame XLF
+(using the 𝐸𝑜𝑏𝑠(𝑧) bands). The data points from the binned XLFs
+follow the trend of a double power-law, with the break luminosity
+evolving with redshift. This illustrates the expected AGN evolution
+with redshift. Due to the limited number of sources in our higher
+redshift bins, this trend becomes harder to see and analytical models
+are required to understand how AGN evolve.
+5.2
+Analytical Model
+The X-ray luminosity in a pure luminosity evolution (PLE) model
+evolves with redshift, and can be expressed by
+𝜙(log 𝐿, 𝑧) = 𝑑𝜙(𝐿/𝑒(𝑧), 0)
+𝑑 log 𝐿
+.
+(7)
+MNRAS 000, 1–19 (2022)
+
+1600ev 6400ev
+200
+1142ev 4571ev
+888ev 3555ev
+727ev 2909ev
+150
+615ev 2461ev
+533ev 2133ev
+F
+IV
+100
+N
+50
+0
+-14.5
+-14.0
+-13.5
+-13.0
+-12.5
+log10(Flux [erg/s/cm21)60
+1600ev 6400ev
+(0.0 < z < 0.5)
+50 
+1142ev 4571ev
+(0.5 < z < 1.0)
+888ev 3555ev
+40
+(1.0 < z < 1.5)
+727ev 2909ev
+F
+(1.5 < z < 2.0)
+ΛI 30
+615ev 2461ev
+N
+(2.0 < z < 2.5)
+20
+533ev 2133ev
+(2.5 < z < 3.0)
+10
+0
+14.25 -14.00 -13.75 -13.50 -13.25 -13.00 -12.75 -12.50 -12.25
+log1o(Flux [erg/s/cm21)12
+A. Alqasim et al.
+Figure 6. Plot of flux 𝐹 vs. the number of total sources 𝑁𝑡𝑜𝑡𝑎𝑙 with a flux
+greater than or equal to a given X-ray source flux (𝑁𝑡𝑜𝑡𝑎𝑙 (𝐹) ≥ 𝐹𝑥) for
+the 7 energy bands. The purple data points mark the 𝐸𝑟 𝑓 band, and the rest
+of the colors mark the 𝐸𝑜𝑏𝑠 (𝑧) bands.
+The evolution factor 𝑒(𝑧) of the PLE is expressed by
+𝑒(𝑧) =
+����
+����
+(1 + 𝑧) 𝑝1
+if 𝑧 < 𝑧𝑐,
+𝑒(𝑧𝑐)
+� 1 + 𝑧
+1 + 𝑧𝑐
+� 𝑝2
+if 𝑧 ≥ 𝑧𝑐,
+(8)
+where 𝑧𝑐 is the cut-off redshift, 𝑝1 is is the parameter that accounts
+for the evolution below 𝑧𝑐, and 𝑝2 is the parameter that accounts
+for the evolution above 𝑧𝑐 (Miyaji et al. 2000b). The shape of the
+present-day XLF for which we adopt a smoothly connected double
+power law can then be expressed by
+𝜙(log 𝐿, 0) = 𝐴
+�� 𝐿
+𝐿0
+�𝛾1
++
+� 𝐿
+𝐿0
+�𝛾2�−1
+,
+(9)
+where 𝛾1 and 𝛾2 are the slopes, 𝐿0 is the luminosity value where
+the change of slope occurs, and 𝐴 is the normalization constant (e.g.
+Boyle et al. 1988; Miyaji et al. 2000b).
+Equation (9) was then used to plot the PLE analytical model
+on the binned XLF data in the 2−8 keV energy range for 0 < 𝑧 < 3.
+This was done for the fixed rest-frame 2−8 keV band, as well as the
+fixed observed 2−8 keV band for comparison with the new method.
+To be able to compare our results with that of the AXIS survey,
+we first constructed the PLE model curves using parameters taken
+from the PLE fit in the 2 − 10 keV band in Ebrero et al. (2009).
+Given the slight difference in the 2 − 8 keV and 2 − 10 keV energy
+bands, the log10 𝐿0 parameter (43.60 ± 0.13 ℎ−2
+70 erg s−1) had to be
+corrected, as was done for the HEAO 1 X-ray fluxes in Section 3.5.
+As before, we assume a photon index of Γ = 1.9. We use 𝑅𝐹 in log
+space to convert the log10 𝐿0 parameter, giving us a final corrected
+parameter of log10 𝐿0 = 43.53 to be used in the binned XLF.
+The rest of the parameters adopted from Ebrero et al. (2009)
+were 𝛾1 = 0.81 ± 0.06, 𝛾2 = 2.37+0.19
+−0.18, 𝐴 = 17.96+9.97
+−6.09 in units of
+10−6 ℎ3
+70 Mpc−3, 𝑧𝑐 = 1.9 (fixed), and 𝑝1 = 2.04. The parameter
+𝑝2 was fixed to 0, as done in Ebrero et al. (2009) for the 2 − 10
+keV XLF, with the evolution stopping after the cut-off redshift. It is
+important to note that their fitting also took into account the amount
+of absorption in the modeling, which is not done when using the
+new method introduced in this paper. We thus regenerate the PLE
+models for the binned XLFs after performing our own model fitting
+and using our derived best-fit parameters (see Section 5.3 for more
+details).
+5.3
+Maximum Likelihood Fit
+We use the ML method (Crawford et al. 1970) to fit the PLE model
+directly to the sources and obtain the best-fit evolution parameters.
+We perform this technique for XLFs in the fixed rest-frame 2 − 8
+keV band using the new method, as well as the fixed observer-frame
+2 − 8 keV band for comparison. The ML method takes into account
+properties from each individual source, and no information is lost
+as a result of binning the XLF.
+The likelihood function is defined as the product of the prob-
+abilities of the the X-ray sources used in the XLF. This gives us
+the overall probability density for the observed distribution of ob-
+jects. This is normally easier to compute over a logarithmic scale,
+as it allows us to sum over the logarithms of the probabilities. We
+follow the method from Page et al. (2021) to maximize the likeli-
+hood by minimizing the expression 𝐶, when in logarithmic scale,
+as described by,
+𝐶 =2𝑁 ln
+� ∫ log 𝐿𝑚𝑎𝑥
+log 𝐿𝑚𝑖𝑛
+∫ 𝑧𝑚𝑎𝑥 (log 𝐿)
+𝑧𝑚𝑖𝑛
+𝜙(log 𝐿, 𝑧) 𝑑𝑉
+𝑑𝑧 𝑑𝑧 𝑑 log 𝐿
+�
+(10)
+− 2
+𝑁
+∑︁
+𝑖=1
+ln 𝜙(log 𝐿𝑖, 𝑧𝑖).
+We solve this using using the amoeba routine described in
+Press (1997). When applying this method to fitting an XLF, we are
+not just maximizing the values of the luminosity function, but we are
+also turning it into the probability of having observed each source.
+This requires taking into account flux limits when generating the
+probabilities. In the new fixed rest-frame method, the flux limit is a
+function of redshift 𝐹𝑙𝑖𝑚(𝑧). This is incorporated into the ML-fitting
+routine through 𝑧𝑚𝑎𝑥(log 𝐿) (see eq. 10), because the maximum
+redshift that you can see an object of given luminosity depends on
+the flux limit. 𝑧𝑚𝑎𝑥(log 𝐿) is thus the redshift at which the flux limit
+is equal to the flux, which is the maximum redshift to which a source
+could be detected. In previous works which used the ML method to
+model the XLF (e.g. Ebrero et al. 2009; Ueda et al. 2014), the flux
+limit used to determine 𝑧𝑚𝑎𝑥(log 𝐿) was not dependent on redshift,
+whereas in our method the flux limit does depend on redshift.
+The ML method changes the shape of the model distribution
+function to match as best as possible the distribution of the observed
+sources in the sample. The model parameters we fit for are the
+luminosity break log10(𝐿0), the slope before the break 𝛾1, slope
+after the break 𝛾2, and the evolution parameter 𝑝1. To be consistent
+with Ebrero et al. (2009), we fix 𝑝2 to 0, and we also fix the cut-
+off redshift 𝑧𝑐 to 1.9. This means that the evolution law we are
+fitting stops at 𝑧 = 𝑧𝑐, so that equation (8) becomes 𝑒(𝑧) = 𝑒(𝑧𝑐)
+for 𝑧 ≥ 𝑧𝑐. We use these best-fit parameters to generate the PLE
+curves in our binned XLFs for both the fixed rest-frame and fixed
+observer-frame bands.
+6
+RESULTS
+The results of the binned XLFs can be seen in Fig. 9, constructed as
+described in Section 5.1. In the same figure, we display the XLFs
+in both the fixed rest-frame (produced using the combined 𝐸𝑜𝑏𝑠(𝑧)
+MNRAS 000, 1–19 (2022)
+
+1750
+533ev 2133ev
+615ev 2461ev
+1500
+727ev 2909ev
+888ev 3555ev
+1250
+1142ev 4571ev
+1600ev 6400ev
+F
+1000
+2000ev 8000ev
+IV
+N
+750
+500
+250
+0
+-15.0
+-14.5
+-14.0
+-13.5
+-13.0
+-12.5
+-12.0
+log10(Flux [erg/s/cm?])Redshifted XLF method for AGN
+13
+Figure 7. Plot of flux 𝐹 vs. the completeness fraction 𝑓𝑐 for each energy band. The purple data points mark the 𝐸𝑟 𝑓 band, and the rest of the colors mark the
+𝐸𝑜𝑏𝑠 (𝑧) bands. Right: Completeness plot displaying the completeness flux limit threshold at 𝑓𝑐 = 80%, marked by the horizontal black dashed line. The flux
+limits for each 𝐸 (𝑧) band are also plotted as vertical solid lines, with the same color corresponding to their data points. Left: For clarity, we show the same
+plot without the threshold and flux limit lines.
+Figure 8. Plot of redshift 𝑧 vs. the flux limit 𝐹𝑙𝑖𝑚(𝑧) for each energy band,
+based on the values reported in Table 5. The points correspond to the flux
+limits of each energy band used in this work. The purple data point marks
+the 𝐸𝑟 𝑓 band, and the rest of the colors mark the 𝐸𝑜𝑏𝑠 (𝑧) bands. The
+step-function of 𝐹𝑙𝑖𝑚(𝑧) (marked with a gray solid line) is a function that
+maps the discrete redshift intervals to discrete flux limit values.
+band data) and the fixed observer-frame (produced using the 𝐸𝑟 𝑓
+band data). The PLE model curves were also plotted on the binned
+XLFs using our ML best-fit parameters.
+Fig. 10 displays the fixed observer-frame XLFs for each 𝑧 bin in
+a separate plot for clarity, and we additionally plot the model curves
+from the Ebrero et al. (2009) PLE best-fit parameters for comparison
+(marked as magenta dashed lines), as described in Section 5.2. We
+display the same plots for clarity for the fixed rest-frame XLFs in
+Fig. 11.
+Our ML model fitting results are obtained using the methods
+described in Section 5.3. The results of the ML best-fit parameters
+for the fixed rest-frame and fixed observer-frame XLFs are summa-
+rized in Table 6.
+7
+DISCUSSION
+In this section, we compare the performance of our fixed rest-frame
+method with the standard method. We then compare our results with
+the ones obtained by the AXIS survey and some other works. We
+finally describe the future prospects for the fixed rest-frame method.
+7.1
+Comparison between the fixed rest-frame and the fixed
+observer-frame XLFs
+In both the fixed observer-frame and fixed rest-frame XLFs, adding
+the HEAO 1 data allowed us to improve the coverage of the higher
+luminosity sources at low redshifts. This significantly improved the
+constraints on our best-fit parameters. Fig. 9 shows that for both
+methods, the binned data points are reasonably consistent with the
+PLE model curves.
+Our ML best-fit results find that the parameters that define the
+evolution of the XLF (𝑝1), and the parameters that describe the
+shape of the XLF (𝛾1 and 𝛾2, log10(𝐿0)), are in agreement within
+the 1𝜎 confidence intervals for both methods. This means that using
+the new method with a fixed rest-frame band does not appear to have
+a significant effect on the results compared with the standard fixed
+observer-frame method. It is important to note that at high redshifts,
+our data set only spans high-luminosity sources. In comparing the
+two methods to each other in this work, the results suggest that the
+presence of absorbed AGN within the population does not have a
+very significant effect on the observed evolution of the XLF for the
+situation in which at high redshift, only high-luminosity sources are
+sampled.
+We have shown that it is practical to produce luminosity func-
+tions using the new method, and that it has produced results that
+are consistent with expectations in the luminosity-redshift regime
+in which we have tested it. Our finding that both methods are con-
+sistent with each other, and that there appears to be no significant
+difference in using the new and standard methods, is consistent with
+the expected outcome from our data used in this paper. Fig. 11 shows
+that our data are limited to bright AGN at higher redshifts. The PLE
+model assumes that the XLF evolves only with the log10(𝐿0) lu-
+MNRAS 000, 1–19 (2022)
+
+100
+80
+Completeness [%]
+60
+2000ev 8000ev
+1600ev 6400ev
+40
+1142ev 4571ev
+888ev 3555ev
+20
+727ev 2909ev
+615ev 2461ev
+533ev 2133ev
+0
+-15.0
+-14.5
+-14.0
+-13.5
+-13.0
+-12.5
+-12.0
+log10(Flux [erg/s/cm?])100
+80
+Completeness [%]
+60
+2000ev 8000ev
+1600ev 6400ev
+40
+1142ev 4571ev
+888ev 3555ev
+20
+727ev 2909ev
+615ev 2461ev
+533ev 2133ev
+0
+15.0
+-14.5
+-14.0
+-13.5
+-13.0
+-12.5
+-12.0
+log10(Flux [erg/s/cm?])2000ev 8000ev
+1600ev 6400ev
+-13.4
+log1o(Flux Limit [erg/s/cm?])
+1142ev 4571ev
+888ev 3555ev
+727ev 2909ev
+-13.5
+615ev 2461ev
+533ev 2133ev
+-13.6
+-13.7
+-13.8
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Z14
+A. Alqasim et al.
+1
+2
+3
+4
+5
+−8
+−6
+−4
+−2
+log [φ (Mpc−3 ∆ log (L)]
+log [LX (1040 erg/s)]
+ 0.00 < z <  0.50
+ 0.50 < z <  1.00
+ 1.00 < z <  1.50
+ 1.50 < z <  2.00
+ 2.00 < z <  2.50
+ 2.50 < z <  3.00
+ z = 0
+(a) Fixed Observer-frame XLF
+1
+2
+3
+4
+5
+−8
+−6
+−4
+−2
+log [φ (Mpc−3 ∆ log (L)]
+log [LX (1040 erg/s)]
+ 0.00 < z <  0.50
+ 0.50 < z <  1.00
+ 1.00 < z <  1.50
+ 1.50 < z <  2.00
+ 2.00 < z <  2.50
+ 2.50 < z <  3.00
+ z = 0
+(b) Fixed Rest-frame XLF
+Figure 9. X-ray Luminosity Functions in the 2−8 keV band for 0 < 𝑧 < 3. The data points represent the results from the binned XLF and the dashed lines
+are the curves of the analytical PLE model (eq. 9), produced using our own ML best-fit parameters. The orange dashed lines correspond to the PLE model
+evaluated at 𝑧 = 0. Left: Standard XLF in the fixed observed band. Right: New XLF in the fixed rest-frame band.
+Table 6. PLE best-fit parameters for the X-ray luminosity functions in the fixed observer-frame and fixed rest-frame 2−8 keV bands.
+𝐴
+𝛾1
+𝛾2
+log10(𝐿0)
+𝑝1
+[10−6 ℎ3
+70 Mpc−3]
+[ℎ−2
+70 erg s−1]
+Observer-frame
+3.08+0.2
+−0.2
+0.92+0.13
+−0.11
+2.42+0.26
+−0.29
+43.82+0.23
+−0.29
+2.4+0.21
+−0.21
+Rest-frame
+7.06+0.46
+−0.46
+0.77+0.12
+−0.1
+2.45+0.23
+−0.28
+43.63+0.18
+−0.22
+2.31+0.16
+−0.15
+minosity break shifting with redshift. Hence, the fitting of the XLF
+evolution is mainly constrained by the bright end of the XLF, while
+keeping the shape of the XLF fixed. Even at high redshifts, the bright
+X-ray sources are not heavily absorbed (e.g. Ebrero et al. 2009). As
+a consequence, performing XLF evolution studies for only high-
+luminosity AGN sources is less sensitive to the choice of the fixed
+observer-frame or rest-frame luminosity. If our data went to fainter
+fluxes at high redshift, we would be observing sources at or below
+the break in the luminosity function at high redshift. Many of those
+sources are expected to be absorbed (e.g. see Figure 5 in Ebrero
+et al. 2009). Then, the fixed rest-frame and fixed observer-frame
+methods are not expected to give the same results. We expect to
+see the benefits of the new method when lower luminosity AGN are
+included at high redshift, and the absorption distribution becomes
+a key factor in modelling the luminosity function.
+Comparing the fixed rest-frame XLF with the fixed observer-
+frame XLF from Fig.
+9, we also found that the error bars on
+the binned XLF data points are smaller in the fixed rest-frame,
+especially in the high-redshift bins. This effect is due to increased
+number of sources in the sample. At higher redshifts in the fixed rest-
+frame, we are sampling the softer 𝐸𝑜𝑏𝑠(𝑧) energy bands. The softer
+X-ray sources are easier to identify in the optical, and therefore a
+larger sample of them are included at higher redshifts in this study.
+7.2
+Comparison with other XLF studies
+We checked to make sure our fixed observer-frame XLF yields
+results that are in agreement with Ebrero et al. (2009), given that
+we are using a subset of their data sample. We find that our fixed
+observer-frame best-fit results (see Table 6) are consistent with the
+values reported in Table 2 of Ebrero et al. (2009) for the PLE fit
+in the hard band. We also plot in Fig. 12 the binned hard XLF
+data points from AXIS in the 0 < 𝑧 < 0.5 and 0.5 < 𝑧 < 1 bins
+in comparison with our own in the fixed observer-frame band for
+reference. As done before when reproducing their PLE curves, we
+converted the data points from 2−10 keV to 2−8 keV. We did not plot
+the binned XLF data points in the other redshift bins as the AXIS
+XLFs bin their data differently past 𝑧 = 1. This plot, along with the
+plots in Fig. 10, also show that our fixed observer-frame PLE model
+results (which are in agreement with the fixed rest-frame results)
+are consistent with the Ebrero et al. (2009) curves. It is difficult to
+compare our results to more recent relevant works in XLF studies as
+many of them only report best-fit results for LADE or LDDE model
+fits. However, Ueda et al. (2003) construct a hard band (2-10 keV)
+AGN XLF and perform PLE fits, which we can compare with. We
+find that our PLE results are consistent with the values reported in
+their Table 3 for the best-fit PLE parameters.
+MNRAS 000, 1–19 (2022)
+
+Redshifted XLF method for AGN
+15
+1
+2
+3
+4
+5
+−8
+−6
+−4
+−2
+log [φ (Mpc−3 ∆ log (L)]
+log [LX (1040 erg/s)]
+ 0.00 < z <  0.50
+(our best−fit)
+ 0.00 < z <  0.50
+(axis best−fit)
+1
+2
+3
+4
+5
+−8
+−6
+−4
+−2
+log [φ (Mpc−3 ∆ log (L)]
+log [LX (1040 erg/s)]
+ 0.50 < z <  1.00
+(our best−fit)
+ 0.50 < z <  1.00
+(axis best−fit)
+1
+2
+3
+4
+5
+−8
+−6
+−4
+−2
+log [φ (Mpc−3 ∆ log (L)]
+log [LX (1040 erg/s)]
+ 1.00 < z <  1.50
+(our best−fit)
+ 1.00 < z <  1.50
+(axis best−fit)
+1
+2
+3
+4
+5
+−8
+−6
+−4
+−2
+log [φ (Mpc−3 ∆ log (L)]
+log [LX (1040 erg/s)]
+ 1.50 < z <  2.00
+(our best−fit)
+ 1.50 < z <  2.00
+(axis best−fit)
+1
+2
+3
+4
+5
+−8
+−6
+−4
+−2
+log [φ (Mpc−3 ∆ log (L)]
+log [LX (1040 erg/s)]
+ 2.00 < z <  2.50
+(our best−fit)
+ 2.00 < z <  2.50
+(axis best−fit)
+1
+2
+3
+4
+5
+−8
+−6
+−4
+−2
+log [φ (Mpc−3 ∆ log (L)]
+log [LX (1040 erg/s)]
+ 2.50 < z <  3.00
+(our best−fit)
+ 2.50 < z <  3.00
+(axis best−fit)
+Figure 10. X-ray Luminosity Functions in the fixed 2−8 keV observer-frame band for 0 < 𝑧 < 3. The data points represent the results from the binned XLF and
+the dashed lines are the curves of the analytical PLE model (eq. 9). The red dashed lines are the model curves produced using our own ML best-fit parameters.
+The blue dotted lines are the model curves produced using the Ebrero et al. (2009) PLE best-fit parameters, corrected from 2−10 keV to 2−8 keV.
+MNRAS 000, 1–19 (2022)
+
+16
+A. Alqasim et al.
+1
+2
+3
+4
+5
+−8
+−6
+−4
+−2
+log [φ (Mpc−3 ∆ log (L)]
+log [LX (1040 erg/s)]
+ 0.00 < z <  0.50
+(our best−fit)
+ 0.00 < z <  0.50
+(axis best−fit)
+1
+2
+3
+4
+5
+−8
+−6
+−4
+−2
+log [φ (Mpc−3 ∆ log (L)]
+log [LX (1040 erg/s)]
+ 0.50 < z <  1.00
+(our best−fit)
+ 0.50 < z <  1.00
+(axis best−fit)
+1
+2
+3
+4
+5
+−8
+−6
+−4
+−2
+log [φ (Mpc−3 ∆ log (L)]
+log [LX (1040 erg/s)]
+ 1.00 < z <  1.50
+(our best−fit)
+ 1.00 < z <  1.50
+(axis best−fit)
+1
+2
+3
+4
+5
+−8
+−6
+−4
+−2
+log [φ (Mpc−3 ∆ log (L)]
+log [LX (1040 erg/s)]
+ 1.50 < z <  2.00
+(our best−fit)
+ 1.50 < z <  2.00
+(axis best−fit)
+1
+2
+3
+4
+5
+−8
+−6
+−4
+−2
+log [φ (Mpc−3 ∆ log (L)]
+log [LX (1040 erg/s)]
+ 2.00 < z <  2.50
+(our best−fit)
+ 2.00 < z <  2.50
+(axis best−fit)
+1
+2
+3
+4
+5
+−8
+−6
+−4
+−2
+log [φ (Mpc−3 ∆ log (L)]
+log [LX (1040 erg/s)]
+ 2.50 < z <  3.00
+(our best−fit)
+ 2.50 < z <  3.00
+(axis best−fit)
+Figure 11. X-ray Luminosity Functions in the fixed 2−8 keV rest-frame band for 0 < 𝑧 < 3. The data points represent the results from the binned XLF and the
+dashed lines are the curves of the analytical PLE model (eq. 9). The blue dotted lines are the model curves produced using the Ebrero et al. (2009) PLE best-fit
+parameters, corrected from 2−10 keV to 2−8 keV. The red dashed lines are the model curves produced using our own ML best-fit parameters.
+MNRAS 000, 1–19 (2022)
+
+Redshifted XLF method for AGN
+17
+1
+2
+3
+4
+5
+−8
+−6
+−4
+−2
+log [φ (Mpc−3 ∆ log (L)]
+log [LX (1040 erg/s)]
+ 0.00 < z <  0.50
+(our best−fit)
+ 0.00 < z <  0.50
+(axis best−fit)
+1
+2
+3
+4
+5
+−8
+−6
+−4
+−2
+log [φ (Mpc−3 ∆ log (L)]
+log [LX (1040 erg/s)]
+ 0.50 < z <  1.00
+(our best−fit)
+ 0.50 < z <  1.00
+(axis best−fit)
+Figure 12. X-ray Luminosity Functions in the fixed 2−8 keV observer-frame band in the 0 < 𝑧 < 0.5 and 0.5 < 𝑧 < 1 bins. The red data points represent our
+results from the binned XLF and the dashed lines are the curves of the analytical PLE model (eq. 9). The red dashed lines are the model curves produced using
+our own ML best-fit parameters. The blue dotted lines are the model curves produced using the Ebrero et al. (2009) PLE best-fit parameters, converted from
+2−10 keV to 2−8 keV. We also include the data points from their hard XLF in the 0 < 𝑧 < 0.5 and 0.5 < 𝑧 < 1 bins, converted to the 2−8 keV band.
+Another interesting avenue we looked into was the comparison
+between our results and optical QSO surveys. Many of these surveys,
+even to recent times, have used the PLE model to describe the LF
+evolution between 0 < 𝑧 < 3, which aligns well with the work
+done in this paper. Optical QSO surveys rely on the detection of
+UV radiation from the QSOs, and so they are very sensitive to
+dust extinction: dusty QSOs will disappear from their samples, just
+as absorbed AGN will disappear from soft X-ray samples. It is
+reasonable to question whether the evolution measured in optical
+surveys is affected by dust extinction, and just like in X-ray surveys,
+the rest-frame band is shifting with redshift (and so the effects of
+dust extinction affect QSOs at different redshifts differently). We
+have introduced a new method in this paper that disentangles the
+effects of absorption from the measurement of evolution in the XLF.
+Hence, we explore how the evolution we measure with our fixed
+rest-frame method compares to the evolution measured in optical
+surveys, and we do this using the quasar LF studies in Ross et al.
+(2013), Croom et al. (2004) and Croom et al. (2009).
+The Ross et al. (2013) study uses the SDSS/BOSS DR9 data
+in their work, and they describe that their quasar LF is similar to the
+XLF. To compare with our results, we use the PLE best-fit values
+reported in Table 8 in their paper in the 0.3 < 𝑧 < 2.2 range.
+From these best-fit results, they found a decrease in magnitude of
+3.715 when looking at the evolution of the break magnitude in the
+LF for 0 < 𝑧 < 2. Converting our luminosity increase of 𝐿0 to
+the decrease in magnitude between 0 < 𝑧 < 2, we find a decrease
+in magnitude by 2.67. This indicates that the redshift evolution of
+the break luminosity for optical QSO LFs is stronger than in X-ray
+selected AGN LFs.
+The other studies of optical QSO LF also present evidence
+of strong evolution in the break luminosity. The PLE model fitting
+results of the 2dF QSO redshift (2QZ) survey data in Croom et al.
+(2004) show a change in magnitude of 4.35 when looking at the
+evolution of the peak luminosity in the LF for 0 < 𝑧 < 2. Similarly,
+the PLE results of the 2dF-SDSS LRG and QSO (2SLAQ) in Croom
+et al. (2009) indicate a 3.95 mag evolution of the peak luminosity
+for 0 < 𝑧 < 2 (when the sample is limited to brighter QSOs with a
+cut off at −23 mag rather than −21.5 mag).
+For the optical studies, extinction should in principle make the
+measured evolution smaller than the real evolution, because dust
+extinction gets worse further into the UV range, and so becomes
+worse at higher redshift. Hence, the optical QSO LF in these stud-
+ies should suffer from the stronger absorption due to the shorter
+wavelength used for higher redshift samples, which would lead to a
+weaker evolution of the LF. However, we find a stronger evolution
+in the optical QSO LF compared to our results. Since our measure-
+ment is robust to the effects of absorption in the fixed rest-frame
+method, this indicates that the optical QSO LF has an intrinsically
+stronger evolution than the X-ray selected AGN LF in our work.
+7.3
+Future prospects for the fixed rest-frame method
+With the current data used in this work, we are limited in our
+capability of solving the discrepancy on what the best model is to
+describe the XLF evolution. In that regard, this method proves to
+be a useful tool in checking whether this holds true when dealing
+with larger data sets (e.g. Aird et al. 2015a, who include almost
+3000 AGN sources in the hard band). Presently, we have not yet
+analyzed deep X-ray data at the faintest fluxes (e.g. from Chandra
+surveys), which might be useful to fully leverage the capabilities
+of this method at present. For this to work for our new method,
+the X-ray data will need to be treated in an analogous fashion to
+the AXIS data used in our paper. We would need to reduce the
+data from scratch since we are sampling over many energy bands,
+and published studies do not provide X-ray fluxes for the required
+energy bands for each redshift bin, which would be needed to easily
+incorporate more surveys into this work.
+Within the next few years, we expect that eRosita will have
+done the full survey of the X-ray sky. Currently, the largest scale
+of observations of the whole X-ray sky comes from ROSAT, which
+cannot be used for a 2−8 keV band XLF since it only covers the
+0.1−2.0 keV band (hence, only covering the rest-frame 2−8 keV
+band at 𝑧 ≥ 3). Previous XLF studies have made use of wide-
+MNRAS 000, 1–19 (2022)
+
+18
+A. Alqasim et al.
+area Chandra surveys, but these studies use low detection limits
+that carry only a couple of counts, which make observations in
+the 2−8 keV band difficult. On the other hand, the eRosita mission
+will be able to provide the resolution equivalent to XMM-Newton
+data, covering the full energy range from 0.5−8 keV over the entire
+sky. This is very promising in improving our detections of AGN
+and understanding their behavior over a large scale (Kolodzig et al.
+2013). ATHENA will also be able to exceed the capabilities of
+current X-ray observatories with its enhanced performance in X-ray
+spectroscopy and deep wide-field X-ray imaging. For this upcoming
+era of X-ray observatories, our new method could be applied on a
+huge scale. The potential value of our new fixed rest-frame method is
+greater for eRosita and ATHENA, than Chandra and XMM-Newton,
+because it can harness data from all-sky-shallow to very deep (and
+still quite wide, by today’s deep survey scales).
+8
+SUMMARY AND CONCLUSIONS
+In this paper, we have introduced a new fixed rest-frame method
+for constructing X-ray luminosity functions of AGN, which aims to
+give a clearer description of how they evolve over cosmic time. Our
+method fixes the X-ray energy band in the rest-frame by varying the
+observed energy band with redshift. We tested our method against
+two X-ray samples in the hard band: 29 XMM-Newton observations
+following the targets from the XMS survey (Barcons et al. 2007), and
+the 1st scan X-ray sources from the HEAO 1 experiment A2 survey
+(Piccinotti et al. 1982). The XMM-Newton data were used to produce
+images and sourcelists in 6 X-ray energy bands, corresponding to 6
+redshift ranges, to account for a target rest-frame band of 2−8 keV.
+We used the spectroscopic optical identifications from the AXIS
+scheme for the redshift measurements of the XMM-Newton data, as
+well as 4 extra published optical IDs of bright sources that were
+not included in the AXIS optical identifications. We also used the
+optical identifications and redshift measurements from Piccinotti
+et al. (1982) for the HEAO 1 X-ray data.
+We constructed XLFs of AGN using two techniques; one using
+the method of Page & Carrera (2000) to make a binned XLF, and
+one using an ML fit, which makes use of the full unbinned source
+sample (Page et al. 2021). Both techniques were computed for the
+standard method (fixed observer-frame band) and our new method
+(fixed rest-frame band). We then used an analytical model described
+by a pure luminosity evolution (PLE) to fit the XLF data points. The
+model consists of a smoothly connected double power-law with a
+factor accounting for how it evolves with redshift.
+The new method presented here eliminates the need to model
+the effects of intrinsic AGN absorption on the observed XLF be-
+havior. We were able to demonstrate the viability of this method
+in constructing XLFs. We found that for both the fixed rest-frame
+and observer-frame methods, the binned data points are reasonably
+consistent with the PLE model curves. We also find that our PLE
+best-fit results were consistent with Ebrero et al. (2009) and Ueda
+et al. (2003). Furthermore, we found an intrinsically stronger evolu-
+tion in optical QSO LF studies (Ross et al. 2013; Croom et al. 2004,
+2009) compared to our results of the X-ray selected AGN LF in our
+work.
+As was shown by our comparison of the fixed observer-frame
+and fixed rest-frame XLFs in Section 7.1, the ML-fit results for both
+methods were consistent with each other (in the case where only
+high-luminosity sources are sampled at high redshift). Even though
+the two methods produce similar results for the data that we have
+tested them on in our paper, we do not expect that to remain true
+if we were able to reach better coverage of the luminosity-redshift
+plane by going fainter in X-ray flux. The power of the new method
+will be important if we include the fainter sources at high redshift,
+as the evolution of the shape of the XLF at the fainter sources is
+indicated in Ebrero et al. (2009); Aird et al. (2015a). Hence, we
+expect to see the benefits of the new method when lower luminosity
+AGN are included at high redshift, and the absorption distribution
+becomes a key factor in modelling the XLF. Encouraged by the
+success of the pilot study of this paper, this is an obvious direction
+of our future work, which could be applied on a huge scale with the
+upcoming era of X-ray observatories.
+ACKNOWLEDGEMENTS
+AAQ acknowledges the funding bodies, the UAE Ministry of Presi-
+dential Affairs and the UAE Space Agency, for their support through
+the PhD scholarship. MJP acknowledges support from the Sci-
+ence and Technology Facilities Council (STFC) grant numbers
+ST/N000811/1 and ST/S000216/1. We thank the referee for their
+thoughtful and helpful feedback. We thank Daisuke Kawata for his
+valuable comments and input on the paper. We thank Francisco
+Carrera and Silvia Mateos for their help with redshifts for AXIS
+and BUXS sources.
+DATA AVAILABILITY
+The data and reduction scripts used in this work can be made avail-
+able upon request.
+REFERENCES
+Aird J., et al., 2010, MNRAS, 401, 2531
+Aird J., Coil A. L., Georgakakis A., Nandra K., Barro G., Pérez-González
+P. G., 2015a, MNRAS, 451, 1892
+Aird J., et al., 2015b, The Astrophysical Journal, 815, 66
+Antonucci R., 1993, Annual review of astronomy and astrophysics, 31, 473
+Avni Y., Bahcall J., 1979, Technical report, On the simultaneous analysis
+of several complete samples: the V/Vsub (max) and Vsub (e)/Vsub (a)
+variables, with applications to quasars. Weizmann Inst. of Science
+Barcons X., et al., 2007, Astronomy &amp; Astrophysics, 476, 1191
+Barger A. J., Cowie L. L., Mushotzky R. F., Yang Y., Wang W. H., Steffen
+A. T., Capak P., 2005, AJ, 129, 578
+Beckmann V., Shrader C., 2013, Active galactic nuclei. John Wiley &amp;
+Sons
+Bongiorno A., et al., 2016, A&A, 588, A78
+Boyle B. J., Shanks T., Peterson B. A., 1988, MNRAS, 235, 935
+Brandt W., Alexander D., 2015, The Astronomy and Astrophysics Review,
+23, 1
+Caccianiga A., et al., 2004, A&A, 416, 901
+Caccianiga A., et al., 2008, A&A, 477, 735
+Cara M., Lister M. L., 2008, ApJ, 686, 148
+Carrera F. J., et al., 2007, Astronomy &amp; Astrophysics, 469, 27
+Cowie L. L., Barger A. J., Bautz M. W., Brandt W. N., Garmire G. P., 2003,
+ApJ, 584, L57
+Crawford D. F., Jauncey D. L., Murdoch H. S., 1970, ApJ, 162, 405
+Cristiani S., Vio R., 1990, Astronomy and Astrophysics, 227, 385
+Croom S., et al., 2004, in Mújica R., Maiolino R., eds, Multiwavelength
+AGN Surveys. pp 57–62, doi:10.1142/9789812702432_0015
+Croom S. M., et al., 2009, MNRAS, 399, 1755
+Ebrero J., et al., 2009, Astronomy &amp; Astrophysics, 493, 55
+Fotopoulou S., et al., 2016, Astronomy & Astrophysics, 587, A142
+Gabriel C., et al., 2004, in Astronomical Data Analysis Software and Systems
+(ADASS) XIII. p. 759
+MNRAS 000, 1–19 (2022)
+
+Redshifted XLF method for AGN
+19
+Georgakakis A., et al., 2015, MNRAS, 453, 1946
+Haardt F., Maraschi L., 1991, The Astrophysical Journal, 380, L51
+Hasinger G., et al., 2001, Astronomy &amp; Astrophysics, 365, L45
+Hasinger G., Miyaji T., Schmidt M., 2005, Astronomy &amp; Astrophysics,
+441, 417
+Ho L. C., 2008, Annu. Rev. Astron. Astrophys., 46, 475
+Jansen F., et al., 2001, Astronomy &amp; Astrophysics, 365, L1
+Kolodzig A., Gilfanov M., Sunyaev R., Sazonov S., Brusa M., 2013, A&A,
+558, A89
+Kormendy J., Ho L. C., 2013, ARA&A, 51, 511
+Loaring N. S., et al., 2005, MNRAS, 362, 1371
+Maccacaro T., della Ceca R., Gioia I. M., Morris S. L., Stocke J. T., Wolter
+A., 1991, The Astrophysical Journal, 374, 117
+Mateos S., et al., 2005, Astronomy &amp; Astrophysics, 433, 855
+Mateos S., et al., 2015, Monthly Notices of the Royal Astronomical Society,
+449, 1422
+Miyaji T., Hasinger G., Schmidt M., 2000a, Advances in Space Research,
+25, 827
+Miyaji T., Hasinger G., Schmidt M., 2000b, A&A, 353, 25
+Miyaji T., et al., 2015, ApJ, 804, 104
+Morrison R., McCammon D., 1983, ApJ, 270, 119
+Mushotzky R. F., Done C., Pounds K. A., 1993, Annual review of astronomy
+and astrophysics, 31, 717
+Netzer H., 2013, The physics and evolution of active galactic nuclei. Cam-
+bridge University Press
+Page M., Carrera F. J., 2000, Monthly Notices of the Royal Astronomical
+Society, 311, 433
+Page M. J., et al., 2021, MNRAS, 506, 473
+Piccinotti G., Mushotzky R. F., Boldt E. A., Holt S. S., Marshall F. E.,
+Serlemitsos P. J., Shafer R. A., 1982, ApJ, 253, 485
+Press W. H., 1997, The Art Of Scientific Computing 2nd Edition in Numer-
+ical Recipes In C. Cambridge University Press
+Ross N. P., et al., 2013, ApJ, 773, 14
+Rothschild R., et al., 1979, Space Science Instrumentation, 4, 269
+Schmidt M., 1968, The Astrophysical Journal, 151, 393
+Silverman J., et al., 2008, The Astrophysical Journal, 679, 118
+Stocke J. T., Morris S. L., Gioia I. M., Maccacaro T., Schild R., Wolter A.,
+Fleming T. A., Henry J. P., 1991, ApJS, 76, 813
+Strüder L., et al., 2001, Astronomy &amp; Astrophysics, 365, L18
+Symeonidis M., et al., 2013, Monthly Notices of the Royal Astronomical
+Society, 433, 1015
+Traulsen I., et al., 2020, A&A, 641, A137
+Turner M. J., et al., 2001, Astronomy &amp; Astrophysics, 365, L27
+Ueda Y., Akiyama M., Ohta K., Miyaji T., 2003, ApJ, 598, 886
+Ueda Y., Akiyama M., Hasinger G., Miyaji T., Watson M. G., 2014, ApJ,
+786, 104
+Yencho B., Barger A. J., Trouille L., Winter L. M., 2009, ApJ, 698, 380
+APPENDIX A: SUPPLEMENTARY TABLES
+This paper has been typeset from a TEX/LATEX file prepared by the author.
+5 https://ds9.si.edu/
+MNRAS 000, 1–19 (2022)
+
+20
+A. Alqasim et al.
+Table A1. List of the XMM-Newton fields used in this work, with their general properties. This includes the XMM-Newton observation number, the target name,
+the center (RA𝑇 𝑎𝑟𝑔𝑒𝑡,Dec𝑇 𝑎𝑟𝑔𝑒𝑡) and radius (R𝑇 𝑎𝑟𝑔𝑒𝑡) used to exclude the area around the target source with a circular region in ds95, and the center
+(RA𝑂𝑂𝑇 ,Dec𝑂𝑂𝑇 ), angle (OOT𝑎𝑛𝑔𝑙𝑒), width (OOT𝑤𝑖𝑑𝑡ℎ) and height (OOTℎ𝑒𝑖𝑔ℎ𝑡) used to exclude the area around the OOT streaks with a rectangular
+region in ds9.
+Observation
+Target Name
+RA
+Dec
+R
+RA
+Dec
+OOT
+OOT
+OOT
+𝑇 𝑎𝑟𝑔𝑒𝑡
+𝑇 𝑎𝑟𝑔𝑒𝑡
+𝑇 𝑎𝑟𝑔𝑒𝑡
+𝑂𝑂𝑇
+𝑂𝑂𝑇
+𝑎𝑛𝑔𝑙𝑒
+𝑤𝑖𝑑𝑡ℎ
+ℎ𝑒𝑖𝑔ℎ𝑡
+[deg]
+[deg]
+[′′]
+[deg]
+[deg]
+[◦]
+[′′]
+[′′]
+0012440301
+PB5062
+331.292917
+-1.922328
+140
+331.2614
+-1.8289096
+161.2
+40
+765.0
+0081340901
+IRAS22491-18
+342.955833
+-17.873617
+32
+–
+–
+–
+–
+–
+0092850201𝑎
+PKS 2135-147
+324.437917
+-14.548672
+120
+324.40925
+-14.459297
+163.7
+44
+765.0
+0100240801
+UZ LIB
+233.0975
+-8.534811
+140
+233.12562
+-8.4441865
+17.7
+40
+765.0
+0100440101
+PHL 5200
+337.126667
+-5.314756
+16
+–
+–
+–
+–
+–
+0102040201
+B2 1128+31
+172.79
+31.235006
+140
+172.84967
+31.315799
+32.7
+44
+765.0
+0102040301
+B2 1028+31
+157.747083
+31.048911
+140
+157.78327
+31.139418
+19.0
+72
+765.0
+0103060101
+PKS 2126-158
+322.300417
+-15.644567
+120
+322.2726
+-15.554186
+163.5
+40
+765.0
+0106460101
+Cl0939+472
+145.7575
+46.995658
+160
+–
+–
+–
+–
+–
+0109910101
+A 1837
+210.402083
+-11.12865
+440
+–
+–
+–
+–
+–
+0111000101
+CL 0016+16
+4.638333
+16.435547
+148
+–
+–
+–
+–
+–
+0111220201
+Markarian 3
+93.9025
+71.037764
+76
+93.816661
+71.125597
+162.6
+32
+765.0
+0112260201
+A 399
+A 401
+44.472831
+44.704176
+13.110249
+13.489401
+307
+405
+–
+–
+–
+–
+–
+0112370301
+SDS-2
+–
+–
+–
+–
+–
+–
+–
+–
+0112371001
+SDS-1
+–
+–
+–
+–
+–
+–
+–
+–
+0112620101
+S5 0836+716
+130.35125
+70.894739
+160
+130.23089
+70.805634
+24.0
+52
+765.0
+0112650401
+G133-69 Pos_1
+–
+–
+–
+–
+–
+–
+–
+–
+0112650501
+G133-69 Pos_2
+–
+–
+–
+–
+–
+–
+–
+–
+0112880301
+EQ Peg
+352.969583
+19.938461
+160
+352.91519
+20.028585
+151.5
+48
+765.0
+0124110101𝑏
+Mkn205
+185.4325
+75.310856
+140
+185.11684
+75.258375
+56.8
+36
+765.0
+0124900101
+MS1229.2+6430
+187.88
+64.23835
+140
+187.71887
+64.174053
+47.0
+40
+765.0
+0112370401 + 𝑐
+0112371501
+SDS-3
+–
+–
+–
+–
+–
+–
+–
+–
+0123100101 + 𝑐
+0123100201
+MS0737.9+7441
+116.017917
+74.565156
+120
+115.96813
+74.469032
+188.0
+40
+765.0
+0100240101 + 𝑐
+0100240201
+HD 117555
+202.699167
+24.230853
+160
+202.72695
+24.322476
+18.0
+40
+765.0
+0106660101 + 𝑐
+0106660601
+LBQS 2212-1759
+–
+–
+–
+–
+–
+–
+–
+–
+𝑎 Different exposures were merged within the same XMM-Newton observation (see Table A2).
+𝑏 Different exposures (with different frame modes) within the same XMM-Newton observation were reduced separately and
+the final images were summed (see Table A3).
+𝑐 Different XMM-Newton observations were merged for the same target (see Table A4).
+MNRAS 000, 1–19 (2022)
+
+Redshifted XLF method for AGN
+21
+Table A2. Pairs of EPIC exposures (marked as Exposure𝐴 and Exposure𝐵 per instrument) used when merging extra eventlists within the same XMM-Newton
+observation. Instruments that only had one exposure were left blank for Exposure𝐵.
+Observation
+M1 Exposure𝐴
+M1 Exposure𝐵
+M2 Exposure𝐴
+M2 Exposure𝐵
+PN Exposure𝐴
+PN Exposure𝐵
+0092850201
+S001
+U003
+S002
+U003
+S003
+–
+0112370401𝑎
+S002
+U002
+S003
+U003
+U002
+–
+0123100101𝑎
+S002
+–
+S003
+–
+S001
+U014
+𝑎 Merged with another XMM-Newton observation (see Table A4).
+Table A3. Two sets of properties (Set A and Set B) corresponding to different M1, M2 and PN science exposures within the same the XMM-Newton observation
+(0124110101). These sets were reduced separately due to having different frame modes, which prevented us from directly merging the event lists. The final
+images, exposure maps and background maps from Set A and Set B were summed together before making the final sourcelist in the reduction process. The
+properties listed in this table include the M1, M2 and PN rate thresholds used to filter out intervals of flaring particle background rate lightcurves (produced at
+𝐸 > 5 keV using a time bin size of 20 s). The science exposures used for each set are also listed, along with the frame mode for each EPIC instrument.
+Set A
+Set B
+M1 Threshold𝑎
+1.2
+3.7
+M2 Threshold𝑎
+1.2
+3.7
+PN Threshold𝑎
+29.0
+5.7
+M1 Exposure
+S004
+S008
+M2 Exposure
+S005
+S009
+PN Exposure
+S003
+S001
+M1 Frame Mode
+Full Frame
+Large Window
+M2 Frame Mode
+Full Frame
+Large Window
+PN Frame Mode
+Extended Full Frame
+Full Frame
+𝑎 In units of count s−1.
+Table A4. Details regarding the 4 sets of observations that were merged together in this work. The table lists the target name of the XMM-Newton field and the
+two XMM-Newton observations that were merged together for each given target field.
+Merge Set
+Target Name
+Observation
+SET 1
+SDS-3
+0112370401𝑎
+0112371501𝑏
+SET 2
+MS0737.9+7441
+0123100101𝑎
+0123100201𝑏
+SET 3
+HD 117555
+0100240101
+0100240201𝑏
+SET 4
+LBQS 2212-1759
+0106660101𝑏
+0106660601
+𝑎 Different exposures were merged within the same XMM-Newton observation (see Table A2).
+𝑏 Observation chosen by AXIS for the specified target.
+MNRAS 000, 1–19 (2022)
+
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+page_content='MNRAS 000, 1–19 (2022)  Preprint 3 January 2023  Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0  A New Method to Determine X-ray Luminosity Functions of AGN  and their Evolution with Redshift  Ahlam Alqasim,1★ Mat J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Page,1  1UCL Mullard Space Science Laboratory, Holmbury Hill Road, Dorking, Surrey, RH5 6NT, UK  Accepted 2022 December 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Received 2022 December 15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' in original form 2022 February 8  ABSTRACT  Almost all massive galaxies today are understood to contain supermassive black holes (SMBH)  at their centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' SMBHs grew by accreting material from their surroundings, emitting X-rays  as they did so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' X-ray Luminosity Functions (XLFs) of Active Galactic Nuclei (AGN) have been  extensively studied in order to understand the AGN population’s cosmological properties and  evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We present a new fixed rest-frame method to achieve a more accurate study of the  AGN XLF evolution over cosmic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Normally, XLFs are constructed in a fixed observer-  frame energy band, which can be problematic because it probes different rest-frame energies at  different redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' In the new method, we construct XLFs in the fixed rest-frame band instead,  by varying the observed energy band with redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We target a rest-frame 2−8 keV band using  XMM-Newton and HEAO 1 X-ray data, with 7 observer-frame energy bands that vary with  redshift for 0 < 𝑧 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We produce the XLFs using two techniques;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' one to construct a binned  XLF, and one using a Maximum Likelihood (ML) fit, which makes use of the full unbinned  source sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We find that our ML best-fit pure luminosity evolution (PLE) results for both  methods are consistent with each other, suggesting that performing XLF evolution studies with  the high-redshift data limited to high-luminosity AGN is not very sensitive to the choice of  fixed observer-frame or rest-frame energy band, which is consistent with our expectation that  high-luminosity AGN typically show little absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We have demonstrated the viability of  the new method in measuring the XLF evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Key words: galaxies: active – quasars: supermassive black holes – X-rays: galaxies – galaxies:  nuclei – galaxies: luminosity function – methods: data analysis  1  INTRODUCTION  It is now understood that almost all massive galaxies today contain  Supermassive Black Holes (SMBH) at their centers (Kormendy &  Ho 2013), with masses ranging between ∼ 106 − 109𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Their  massive growth is mainly due to accretion of matter from their sur-  roundings, shining as Quasi-Stellar Objects (QSOs) and emitting  X-rays as they did so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' QSOs belong to a larger population of Ac-  tive Galactic Nuclei (AGN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' SMBHs can also grow due to other  processes such as black hole mergers and the tidal capture of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Most of them have been observed to be dormant in present-day  galaxies, exhibiting luminosities largely below their Eddington lim-  its, leaving only ∼ 10% of galaxies hosting AGN (Ho 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Hence,  AGN studies are very important in understanding the evolution of  galaxies over cosmic time, mainly because they are strongly linked  to the growth of SMBHs and can thus tell us useful information  about the accretion history of the universe (Brandt & Alexander  2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' It is now also well-known that the evolution of SMBHs and  the evolution of their host galaxies are strongly connected, or that  they co-evolve (Symeonidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' AGN feedback plays a sig-  ★ E-mail: ahlam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='alqasim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='17@ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='uk  nificant role in quenching star formation and stopping the growth  of their host galaxy after a certain point (Bongiorno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2016), so  understanding how they evolve with time can provide very useful  insights into the evolution of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Most of the observed luminosity in AGN is radiated in the  optical-UV, but the X-ray flux of AGN shows the fastest variability  on short timescales in all of the wavelength ranges (Netzer 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  This suggests that the X-rays originate from a small region close  to the central object, now known to be a SMBH (Mushotzky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Studies of Seyfert galaxies showed that one of the most plau-  sible physical processes driving the X-ray emission was inverse  Compton radiation (Haardt & Maraschi 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' In this scenario,  the intrinsic high energy X-rays seen in AGN originate from a hot  corona plasma (consisting of relativistic electrons) close to the ac-  cretion disk (Beckmann & Shrader 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The hot corona scatters  the optical-UV photons coming from the inner regions of the ac-  cretion disk to X-ray energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This mechanism drives the shape  observed in the X-ray spectra of AGN (Netzer 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Further out  from the central engine of the AGN is a thick torus surrounding  the central SMBH and accretion disk, responsible for obscuring  the X-ray emission due to photoelectric absorption (Morrison &  © 2022 The Authors  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='00223v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='GA]  31 Dec 2022  2  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Alqasim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  McCammon 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Antonucci 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Seyfert I galaxies give a clear  view of the active nucleus because the line of sight is unobstructed,  while Seyfert II galaxies have a line of sight that is obscured by  the torus, causing them to appear to have less evidence of activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  XMM-Newton surveys particularly helped constrain the evolution of  X-ray absorption, as well as other physical properties of AGN, by  studying faint X-ray sources (Hasinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  The luminosity function (LF) of galaxies is generally defined  as the number of objects per unit volume (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Mpc3) per unit log-  arithmic luminosity interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' For AGN, the best model shape that  describes their X-ray LF (XLF) is a double power-law modified by  a factor for evolution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Boyle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Miyaji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2000b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  XLFs of AGN can be constructed using extensive X-ray surveys in  order to understand their cosmological properties and study their  evolution over cosmic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Many techniques have been published  to quantify the cosmological evolution of AGN, including using  the ⟨𝑉/𝑉𝑚𝑎𝑥⟩ method (Schmidt 1968), the 𝑉𝑒/𝑉𝑎 method (Avni  & Bahcall 1979) for combined samples, Monte Carlo simulations  (Cristiani & Vio 1990), and the 1/𝑉𝑎 method (Maccacaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The ⟨𝑉/𝑉𝑎⟩ (or ⟨𝑉/𝑉𝑚𝑎𝑥⟩) approach is mainly used for de-  termining whether there is evolution with redshift, while the 1/𝑉𝑎  (or 1/𝑉𝑚𝑎𝑥) approach is used for calculating a binned luminosity  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The 1/𝑉𝑎 is more commonly used because it is simpler,  incorporating a binned differential luminosity function within a red-  shift interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Improved methods to 1/𝑉𝑎 for constructing binned  XLFs have also been demonstrated (Page & Carrera 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Cara &  Lister 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Some works have shown that the cosmic evolution of AGN is  consistent with models in which the luminosity varies with redshift,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Pure Luminosity Evolution (PLE) model (Barger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' In  such models, the shape of the XLF evolution remains the same and  the luminosity evolution causes the model curve to simply shift right  or left on the luminosity plane as the XLF evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Other models try  to explain the XLF evolution by varying the density with redshift,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Pure Density Evolution (PDE) model (Fotopoulou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  In such models, the XLF shape also remains the same and the density  evolution causes the model curve to simply shift up or down on the  density plane as the XLF evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Other works have proposed mod-  els that combine both luminosity and density evolution, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Inde-  pendent Luminosity Density Evolution (ILDE) model (Yencho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  2009) and Luminosity and Density Evolution (LADE) model (Aird  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' In such models, the XLF shape is kept the same, and the  combined luminosity and density evolution causes the model curve  to be shifted in any direction across the luminosity-density plane  as the XLF evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Some studies favor a more complicated model  in which the shape of the XLF evolution changes while also vary-  ing the density and luminosity, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Luminosity-Dependent Density  Evolution (LDDE) model (Hasinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Ueda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  In such models, the curves are not only shifted in any direction, but  their shapes can also change when moving around the luminosity-  density plane as the XLF evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Most of these models include a  critical redshift 𝑧𝑐 value (also referred to as cutoff redshift), after  which the XLF evolution either changes or stops (Fotopoulou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Some papers also use a Bayesian approach to explore the  AGN evolution in a model-independent way (Georgakakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Fotopoulou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Previous XLF studies of AGN show that the number of AGN  per unit volume per unit luminosity has been observed to change  strongly with redshift (Ebrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' One of the main issues  to be addressed in XLF studies of AGN (especially with Chandra  and XMM-Newton surveys) is the impact of absorption, which can  suppress the observed X-ray flux (especially for redshifts 𝑧 ≤ 1),  and is even more problematic for Compton-thick sources (Aird et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  2015b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Most studies try to correct for this absorption and model  its evolution with redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Normally, luminosity functions for AGN  are constructed in a fixed observed energy band, and there is still no  consensus on what the best approach is to model how they evolve  with redshift when looking at the rest-frame energy bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' How  an observed energy band, 𝐸𝑜𝑏𝑠(𝑧), relates to the rest-frame energy  band, 𝐸𝑟 𝑓 , for a given redshift 𝑧 is described as  𝐸𝑜𝑏𝑠(𝑧) =  𝐸𝑟 𝑓  1 + 𝑧 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  (1)  The most common X-ray bands normally used to study XLFs  of AGN are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5−2 keV (Miyaji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2000a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Hasinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Ebrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2009) and 2−10 keV (Aird et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Miyaji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Georgakakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The 2−8 keV band has also been  studied in some cases instead of 2−10 keV (Barger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Silverman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Some other studies have focused on the  5−10 keV band (Fotopoulou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2016) to avoid correcting for  the absorbed part of the AGN spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Since hard X-rays (∼ E  > 2 keV) are significantly less affected by absorption, the soft X-  ray band (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5−2 keV) should mainly sample unabsorbed AGN (and  the derived XLF should only include the unabsorbed population).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  However, since the rest-frame energy band changes with redshift, the  population will include absorbed AGN at higher redshifts, which  affects the binned luminosity function because the K-correction  doesn’t take that effect into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' As a result, the sample will  start to gain more absorbed sources as you move up in the rest-  frame energy band with redshift, even if the study is conducted in  a harder observed X-ray band (Aird et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2015b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This problem  was mitigated to some degree by Cowie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2003) and Barger  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2005), who used the 2−8 keV observer-frame to study lower  redshift sources (𝑧 <∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5), and used the flux from the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5−2 keV  observer-frame to calculate the 2−8 keV luminosity in the rest-frame  for the high redshift sources (𝑧 >∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  In this paper, we present a new method that aims to tackle  this problem by varying the observed energy band with redshift,  allowing us to fix the X-ray energy band in the rest-frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This  eliminates the need to model the redshift-dependence of X-ray ab-  sorption from material surrounding the SMBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We make use of  X-ray data from XMM-Newton and HEAO 1 in this work to produce  X-ray luminosity functions in the fixed observed band (the standard  method) and the fixed rest-frame band (the new method) for the 2−8  keV band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The fixed rest-frame band requires the analysis of several  observer-frame bands that correspond to each redshift bin, which  will be described in more detail in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  This paper is structured as follows: in Section 2, we introduce  the new method used to produce the fixed rest-frame XLF along  with the corresponding redshifted observed energy bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We then  present the X-ray data used in this sample and its selection criteria in  Section 3 for both the fixed observed band and the fixed rest-frame  band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' In Section 4, we describe the optical identifications used to  study the completeness of the X-ray sample and obtain redshifts  for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' In Section 5, we describe the two techniques used to  compute our XLFs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' one using the method of Page & Carrera (2000)  to construct a binned XLF, and one using a Maximum Likelihood  (ML) fit on the full unbinned source sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Both techniques are  computed for the standard method (fixed observed band) and the  new method (fixed rest-frame band) introduced in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We  present the results of the XLFs in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' In Section 7, we discuss  the performance of the new method compared with the standard  MNRAS 000, 1–19 (2022)  Redshifted XLF method for AGN  3  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' AGN model spectrum in the rest-frame, constructed for a wide  range of column densities 𝑁𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The spikes correspond to photoelectric  absorption edges, which occur when the X-ray emission passes through a  highly absorbing material, usually the torus in the case of AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  method, as well as with previous XLF studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The conclusions of  this work are then summarized in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  The cosmological parameters that were assumed in this paper  were 𝐻0 = 70 km s−1 Mpc−1, Ω𝑀 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='3 and ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  2  THE FIXED REST-FRAME XLF METHOD  In this section, we describe our new method that allows us to vary  the observed energy band with redshift, and to fix the X-ray energy  band in the rest-frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 1 shows AGN model spectra in the  rest-frame, constructed using PyXspec1 (the Python interface for  Xspec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The model curves are produced assuming a model with a  cold photoelectric absorber and a powerlaw component with Γ =  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We explored a wide range of column densities 𝑁𝐻 to account for  AGN that display different levels of intrinsic absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The spikes  correspond to photoelectric absorption edges, which occur when the  X-ray emission passes through a highly absorbing material, usually  the torus in the case of AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Looking at Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 1, if we study an AGN  at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5−2 keV in a fixed observed band, a source at redshift 𝑧 = 0  (red shaded region) would be in the correct rest-frame energy band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  However, if the source is redshifted at 𝑧 = 2 (blue shaded region),  the part of the spectrum that is observed corresponds to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5−6 keV  in the rest-frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This means that we are inherently looking at very  different parts of the AGN rest-frame spectrum at different redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  As a result, studying AGN in a fixed observed energy band can be  problematic since it probes very different rest-frame energies at  different redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  To construct a fixed rest-frame XLF for a given survey, a fixed  rest-frame energy band 𝐸𝑟 𝑓 must be chosen to cover a redshift  range 𝑧𝑖 < 𝑧 < 𝑧 𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The redshift range determines what redshifted,  observed energy bands 𝐸𝑜𝑏𝑠(𝑧) will be used to generate images and  sourcelists for the X-ray data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' In this new method, one can choose  1 https://heasarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='gov/xanadu/xspec/python/  html/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='html  0  1  2  10  100  1000  104  105  106  LX (erg s−1)  z  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Plot of Redshift vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Luminosity (𝐿𝑋 −𝑧 plane) of the XMS survey  in the fixed rest-frame 2−8 keV band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  to do this in as many 𝐸𝑜𝑏𝑠(𝑧) bands as desired, depending on  how many redshift intervals are chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The more 𝐸𝑜𝑏𝑠(𝑧) bands  there are, the more precise the fixed rest-frame XLF will be, but  a good middle ground will need to be established to minimize  computational time spent depending on how large the data sample  is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Once the 𝐸𝑜𝑏𝑠(𝑧) bands are determined, a sourcelist is produced  for each given 𝐸𝑜𝑏𝑠(𝑧) band at the end of the data reduction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  The X-ray fluxes obtained from the sourcelists are used to convert to  X-ray luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' In this work, the X-ray luminosity is for the 2−8  keV rest-frame band, and is corrected for Galactic absorption, but  not for absorption occurring within the AGN and its host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 1, this energy band will only be sensitive  to sources with column densities of up to 1023 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Since the  degree of attenuation due to photoelectric absorption is strongly  energy dependent (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 1), it is important to use a  fixed rest-frame energy band when producing our X-ray sourcelists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  The sourcelists are filtered to only contain sources with measured  redshifts corresponding to the 𝐸𝑜𝑏𝑠(𝑧) band of the data sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' A  separate flux limit 𝐹𝑙𝑖𝑚 is then derived for each 𝐸𝑜𝑏𝑠(𝑧) band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' So  in essence, the fixed rest-frame XLF uses observed bands that vary  with redshift, and uses a flux limit that is not constant but is also a  function of redshift 𝐹𝑙𝑖𝑚(𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='1  Binned XLF  When applying the new method to a binned XLF (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='1  for more details), redshift bins 𝑧𝑏𝑖𝑛 first need to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' A  plot of the 𝐿𝑋 − 𝑧 plane is made for the targeted 𝐸𝑟 𝑓 , where 𝐿𝑋  is the X-ray luminosity of the sources, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This figure  shows the redshift distribution of the data used in this paper and  at which redshifts most AGN with luminosities lie in at 2−8 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Looking at the diagram, we have enough sources from our sample  spread over the luminosity-redshift plane to use bins of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 in  redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Once the redshift bins are determined, the midpoint of each  redshift bin 𝑧𝑚𝑖𝑑 is then used in equation (1) with 𝑧 = 𝑧𝑚𝑖𝑑 to  determine what the corresponding 𝐸𝑜𝑏𝑠(𝑧) band will be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The mini-  mum number of 𝐸𝑜𝑏𝑠(𝑧) bands needed will be set by the number of  𝑧𝑏𝑖𝑛 chosen for the XLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' In principle, the number of 𝐸𝑜𝑏𝑠(𝑧) bands  need not be restricted by the number of redshift bins used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Each  𝑧𝑏𝑖𝑛 can include multiple 𝐸𝑜𝑏𝑠(𝑧) bands that correspond to smaller  MNRAS 000, 1–19 (2022)  NH = 1019 cm-2  NH = 1022 cm-2  z=O  NH = 1020 cm-2  z=2  N = 1023 cm-2  N = 1021 cm-2  NH = 1024 cm-2  10000  8000  6000  4000  2000  0  100  101  Energy [keV]4  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Alqasim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The 𝐸𝑜𝑏𝑠 (𝑧) bands for which images and sourcelists are produced  to construct the fixed rest-frame XLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 𝑧𝑏𝑖𝑛 gives the redshift interval, for  which the midpoint (𝑧𝑚𝑖𝑑) is used to determine the observer-frame energy  range (𝐸𝑜𝑏𝑠(𝑧)) that corresponds to 2 − 8 keV in the rest-frame band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  𝑧𝑏𝑖𝑛  𝑧𝑚𝑖𝑑  Eobs(𝑧)  [eV]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='25  1600 − 6400  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='75  1142 − 4571  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='25  888 − 3555  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='75  727 − 2909  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='25  615 − 2461  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='75  533 − 2133  redshift intervals within the same bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' For each 𝑧𝑏𝑖𝑛, an XLF is then  produced using the data from its corresponding 𝐸𝑜𝑏𝑠(𝑧) band (or  set of bands).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' For the binned XLF, 𝐹𝑙𝑖𝑚(𝑧) is a function that maps  the discrete redshift intervals to discrete flux limit values, and each  𝐸𝑜𝑏𝑠(𝑧) will have its own flux limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2  Model-fitted XLF  When applying the new method to produce a model fitted XLF using  the ML technique (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='3 for more details), the number of  𝐸𝑜𝑏𝑠(𝑧) bands used is not determined by any redshift binning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The  more 𝐸𝑜𝑏𝑠(𝑧) bands used, the more precisely the energy range is  fixed in the rest-frame for each source for the XLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' As was the case  in the binned XLF, 𝐹𝑙𝑖𝑚(𝑧) is used to derive a flux limit for each  𝐸𝑜𝑏𝑠(𝑧) band used for the model-fitted XLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='3  Redshifted Energy Bands  For this work, a fixed rest-frame energy band of 2−8 keV was chosen  to cover a redshift range of 0 < 𝑧 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We use redshift intervals  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 for the binned XLF, with good coverage over the 𝐿𝑋 − 𝑧  plane (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' To obtain the redshifted, observed energy bands  in which sourcelists are produced, the midpoint of each redshift  interval was used, giving a total of 6 redshifted energy bands (see  Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  3  X-RAY DATA  In the following sections, we describe the targets and observations  used for the XMM-Newton data in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We then describe the  data processing methods used to reduce the XMM-Newton data, and  the source selection criteria used when producing the final X-ray  sourcelists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Finally, we describe the flux-limited X-ray catalogue  from Piccinotti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (1982), from which the HEAO 1 X-ray data  was taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  For the XMM-Newton data reduction and processing, c-shell  scripts were used to run tasks using sas-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0, heasoft-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='27,  wcstools-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5, and SAOImageDS9-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Python scripts were  run using python-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='1  XMM-Newton Observations  The data used in this work were taken from the XMM-Newton Satel-  lite, which carries multiple telescopes to study X-ray sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' XMM-  Newton is sensitive up to 10 keV with a spatial resolution of ∼5 arc-  sec (Jansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The XMM-Newton data used were extracted  from the three EPIC (European Photon Imaging Camera) cameras  on XMM-Newton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The EPIC cameras are placed at the focus points  of the X-ray mirror assemblies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Two of the cameras use EPIC-MOS  CCDs (Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2001), while the third camera uses EPIC-PN  CCDs (Strüder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Each EPIC instrument is fitted with a  filter wheel carrying X-ray transparent light blocking filters to block  out background light outside the desired X-ray band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  In this work, we use 25 XMM-Newton target fields adopted  from the the XMM-Newton Medium Sensitivity Survey (XMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The  XMS is a survey built using a sample of the AXIS survey (Carrera  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2007) covering a geometric sky area of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='33 deg2 (Ebrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The luminosity distribution of the entire sample shows that  the survey contains Seyfert-like AGN as well as QSOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The XMS is  sensitive to AGN with intrinsic column densities up to 1023 cm−2  (Mateos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  We have used the redshifted energy bands 𝐸𝑜𝑏𝑠(𝑧) listed in  Table 1 for the methods and analysis performed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' For  each 𝐸𝑜𝑏𝑠(𝑧), the data reduction method from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2 was  performed for a total of 29 XMM-Newton observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' A list of the  XMM-Newton observations used and their general properties can  be seen in Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Taking into account the excluded areas from  our masks (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2), the total geometric sky area covered  amounts to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='61 deg2 for the XMM-Newton X-ray data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2  XMM-Newton Data Reduction  The XMM-Newton mission provides the Science Analysis System  (SAS) pipeline software (Gabriel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2004) specifically designed  to reduce and analyze data collected by the XMM-Newton obser-  vatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The SAS tasks epproc and emproc are used to produce  a calibrated event list for each instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' For the EPIC-PN event  lists, we excluded the PN readout outer-edge regions at the top and  bottom of the detector chips (as done in Carrera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  The data was first reduced in in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5−2 keV energy band, as  described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='1, for the purpose of correcting the offset  positions of sources within the attitude file of the observation and  the event lists obtained from epproc and emproc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This only needs  to be done once for each XMM-Newton observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This makes it  more efficient when reducing the data in the multiple energy bands  required for this work without needing to correct the source position  offsets for each energy band at a later stage in the processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Once  the position offsets were corrected, the new attitude and event list  files were used to generate images and sourcelists in the desired  𝐸𝑜𝑏𝑠(𝑧) bands, as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='1  Reduction Process to Correct Source Position Offsets  To filter out intervals of particle background flares from EPIC event  lists, a high energy light curve (𝐸 > 5 keV) was extracted from  the event file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' A background rate threshold was then determined by  where the light curve is steady with low background intervals (see  Table 2 for the rate thresholds used for each XMM-Newton observa-  tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This threshold varies with each observation, depending on the  background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' All the work in this section after this point was done  using a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 − 2 keV energy band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  We first produce X-ray images separately for each EPIC instru-  ment using evselect in the targeted energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' For EPIC-PN,  2 Obtained using the HEASOFT tool nh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  MNRAS 000, 1–19 (2022)  Redshifted XLF method for AGN  5  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' List of the XMM-Newton fields used in this work, with their general properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This includes the XMM-Newton observation number,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' the RA/Dec  coordinates at the center of the field (RA𝐹𝑖𝑒𝑙𝑑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Dec𝐹𝑖𝑒𝑙𝑑),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' the Galactic column density in the direction of the field (𝑁𝐻,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='𝐺𝑎𝑙),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' the filter used for the EPIC  cameras,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' the "clean" exposure time for the M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' M2 and PN cameras (T𝑒𝑥 𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='𝑀1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' T𝑒𝑥 𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='𝑀2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' T𝑒𝑥 𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='𝑃𝑁 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' respectively),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' and the M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' M2 and PN rate thresholds  (M1𝑡ℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' M2𝑡ℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' PN𝑡ℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' respectively) used to filter out intervals of flaring particle background rate lightcurves (produced at 𝐸 > 5 keV using a time bin size of 20  s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Observation  RA  Dec  𝑁𝐻 2  Filter  T𝑒𝑥 𝑝  T𝑒𝑥 𝑝  T𝑒𝑥 𝑝  M1𝑡ℎ  M2𝑡ℎ  PN𝑡ℎ  𝐹𝑖𝑒𝑙𝑑  𝐹𝑖𝑒𝑙𝑑  𝐺𝑎𝑙𝑎𝑐𝑡𝑖𝑐  𝑀1  𝑀2  𝑃𝑁  [deg]  [deg]  [1020 cm−2]  [ks]  [ks]  [ks]  [count s−1]  [count s−1]  [count s−1]  0012440301  331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
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+page_content='0  0100240101 + 𝑐  0100240201  202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='695834  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2330556  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='999  Medium  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
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+page_content='0  0106660101 + 𝑐  0106660601  333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='881958  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='7349166  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='85  Thin  151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='1  151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='4  126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0  𝑎 Different exposures were merged within the same XMM-Newton observation (see Table A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  𝑏 Different exposures (with different frame modes) within the same XMM-Newton observation were reduced separately and  the final images were summed (see Table A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  𝑐 Different XMM-Newton observations were merged for the same target (see Table A4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  𝑑 Listed filter corresponds to EMOS1 and EPN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The EMOS2 filter was different (Thick for the first two and Medium for the  second two).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  “Out-Of-Time" (OOT) images were also produced to take into ac-  count OOT events that end up being mixed within the read-out  direction in the CCD frame, which then gets added to the PN back-  ground map once it’s scaled out, and accounts for the bright streaks  that tend to be seen in PN images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' To combine information from all  three instruments, the X-ray images were then summed up together  into a final X-ray image using the ftools3 task farith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  3 https://heasarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='gov/ftools/  MNRAS 000, 1–19 (2022)  6  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Alqasim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Along with X-ray images, we produce exposure maps for each  of the 3 EPIC instruments in the relevant energy band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' MOS ex-  posure maps were multiplied by the ratio of MOS/PN countrates  assuming a power-law spectrum with a photoelectric absorption  component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This was done using PyXspec with the Galactic hy-  drogen column density 𝑁𝐻,𝐺𝑎𝑙 (see Table 2) and a photon index  Γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='9 (Mateos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Energy channels outside the targeted  energy range were excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The MOS exposure maps were then  added to the PN exposure map to make a summed exposure map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  An Energy Conversion Factor (ECF), defined as the ratio of the  count rate to flux, was then calculated using PyXspec to determine  how to convert EPIC band count rates to fluxes in a given energy  band, which is then used for further SAS tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Since the MOS1 and  MOS2 images are added on top of the PN image, the final combined  EPIC image is in the format of an EPIC-PN image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Hence, the ECF  is calculated assuming a PN image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The summed exposure map was  then used to make a mask (using the SAS task emask) that filters  out sources in areas of the image where there were CCD gaps or  bad pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Finally, we produce background maps with our own back-  ground script for each EPIC instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This background script  uses 2 models, a vignetted background model and a flat background  model, as well as an OOT component for EPIC-PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The flat back-  ground model accounts for the particle background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The script used  here then performs an ML fit to the background in a similar manner  to the method described in Loaring et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' For each EPIC  instrument, we ran our background script using the list of sources  from the eboxdetect SAS task, along with the EPIC exposure  map and X-ray image, producing 3 separate background maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The  background maps were then summed up into a final background  map using farith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  The output from eboxdetect was also used in the SAS task  emldetect with an ML threshold of 8 to make an X-ray sourcelist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  We corrected the astrometry of our event lists by correlating the  positions of the X-ray sources with optical sources from the SDSS  Photometric DR12 Catalogue, as well as the Pan-STARRS Survey,  according to the method described in Traulsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This  correction only needed to be done once for each XMM-Newton  observation, after which the event lists and attitude files were used to  generate images and sourcelists for our set of 𝐸𝑜𝑏𝑠(𝑧) energy bands  (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' New (position-corrected) images, background  maps and exposure maps in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 − 2 keV energy band were  finally reproduced and subsequently summed up in the same manner  described in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Using the position-corrected 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 − 2 keV exposure maps, we  produce 2 masks using emask, which are used when producing X-  ray sourceslists in the 𝐸𝑜𝑏𝑠(𝑧) bands (see Table 3 for parameter  values used in all the masks produced).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The first initial mask 𝑀𝑑𝑒𝑡  covers the full field of view (FOV) detector image, constructed us-  ing the summed exposure map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We then run the mask through the  ftools program fgauss, which convolves the image with a circular  Gaussian function to produce a smoothed image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The smoothed im-  age was then used to produce a modified version of the initial mask,  which was then used to make the XMM-Newton X-ray sourcelist in  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  The second and final mask 𝑀 𝑓 𝑖𝑛𝑎𝑙 has excluded regions from  the image rather than retaining the full FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This mask is used  in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2 to filter out sources from the X-ray sourcelist for  each of the 𝐸𝑜𝑏𝑠(𝑧) bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 𝑀 𝑓 𝑖𝑛𝑎𝑙 is the mask constructed by  multiplying the 𝑀 𝑓 𝑖𝑙𝑡𝑒𝑟 and the 𝑀𝑎𝑥𝑖𝑠 masks together, using the  task farith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We describe what these are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We applied  two constraints when producing the mask 𝑀𝑎𝑥𝑖𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We used the PN  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Parameter values for the Gaussian sigma and mask thresholds used  in the XMM-SAS tasks fgauss and emask, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 𝑀𝑑𝑒𝑡 is the full  FOV detector mask constructed using the summed exposure map;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 𝑀 𝑓 𝑖𝑙𝑡𝑒𝑟  is the mask constructed using the summed exposure map with low exposure  pixels filtered out (set to a minimum exposure threshold of 10 ks);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 𝑀𝑎𝑥𝑖𝑠 is  the mask constructed using the PN exposure map, where fgauss and emask  parameters were adjusted to add an extra blur to the detector chip edges and  exclude pixels falling too close to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Task  Task  Parameter  Task Parameter Values  𝑀𝑑𝑒𝑡  𝑀 𝑓 𝑖𝑙𝑡𝑒𝑟  𝑀𝑎𝑥𝑖𝑠  fgauss  sigma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0  emask  threshold1  threshold2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='93  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='93  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5  exposure map rather than the summed exposure map when making  the mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We also adjusted the fgauss and emask parameters to  add an extra blur to the detector chip edges and exclude pixels  falling too close to them (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' AXIS only used PN data  when processing their XMM-Newton observations, and removed an  extra 5-7 pixels from the edges of their detector chips (Carrera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Without limiting the mask to PN and incorporating the extra  blurring to account for the 5-7 pixels that were excluded by AXIS,  we ended up with a lot of unidentified sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This brought our  completeness statistics down, not necessarily because the sources  were spurious, but simply because they were not included in the  X-ray source list in AXIS, and as a result they weren’t part of their  optical identification campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Thus, we chose to include these two  constraints in 𝑀𝑎𝑥𝑖𝑠 to be consistent with the methods adopted in  the AXIS survey, given that we are using their optical identification  campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' When producing the mask 𝑀 𝑓 𝑖𝑙𝑡𝑒𝑟, we set a minimum  exposure threshold of 10 ks from the summed exposure map to filter  out regions with low-exposure pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This was done to avoid the  occurrence of spurious sources (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='4 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  After multiplying the 𝑀 𝑓 𝑖𝑙𝑡𝑒𝑟 and the 𝑀𝑎𝑥𝑖𝑠 masks together,  additional regions were excluded from the resulting 𝑀 𝑓 𝑖𝑛𝑎𝑙 mask,  which we describe as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The XMS survey did a serendipitous  search rather than a blind search, and previously verified sources  were taken as a target around which sources were searched for, and  the target source itself was excluded (or masked out) during this  search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' There were a total of 25 target sources that were excluded in  the XMS, which we also remove from the final mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We adopt the  RA, Dec and target exclusion radius used by AXIS from Table 1 in  Carrera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' One XMM-Newton observation (0112260201)  is pointed between two cluster targets (A 399 and A 401).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' AXIS  provided an exclusion region for one of these cluster targets (A 399).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  For that, we adopted a more conservative approach and extended  the AXIS exclusion radius to 307′′ to remove extra cluster sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  We also added an additional exclusion region to remove the second  cluster target (A 401) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We additionally excluded rectangular  OOT regions using the widths provided from Table 1 in Carrera  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2007), and we provide the RA, Dec, angle and height of the  OOT rectangular region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' All details regarding the exclusion areas  applied to the final mask can be found in Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2  Reduction Process for 𝐸𝑜𝑏𝑠(𝑧) Bands  The reduction process in the fixed rest-frame method follows a sim-  ilar procedure as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The position-corrected  attitude files and event lists were used to generate images and  MNRAS 000, 1–19 (2022)  Redshifted XLF method for AGN  7  sourcelists for the XMM-Newton observations in 6 redshifted en-  ergy band 𝐸𝑜𝑏𝑠(𝑧), as listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Some changes and additions  were incorporated, which will be described in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  The X-ray images and exposure maps were produced in the  given 𝐸𝑜𝑏𝑠(𝑧) energy band and summed up together using the  same process described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' To check the dependence  of the ECF and the MOS/PN ratios on intrinsic AGN absorption,  we recalculated them for each XMM observation assuming 𝑁𝐻 = 0  and 𝑁𝐻 = 1022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 cm−2, and derived the fractional change for each  parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We find that the ECF decreases by 11% between 𝑁𝐻 =  0 and 𝑁𝐻 = 1022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 cm−2, so luminosities of absorbed sources  will be slightly underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The MOS/PN ratio is much less  affected by absorption, decreasing by just 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='6% between 𝑁𝐻 = 0  and 𝑁𝐻 = 1022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Since the MOS/PN ratio and ECF are both  a function of energy, they were derived separately for each 𝐸𝑜𝑏𝑠(𝑧)  using the same methods described previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' When making the  summed exposure map, the MOS/PN ratios derived for the 𝐸𝑜𝑏𝑠(𝑧)  energy band were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' To make the summed background map in  the targeted 𝐸𝑜𝑏𝑠(𝑧) (as described in the previous section), we run  the background script with 4 iterations to get the best background  map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The multiple iterations allow us to get rid of most of the  bright sources, and help us produce maps that are very close to the  background for the final summed background map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  The final X-ray sourcelist is then produced using the final  summed images, background maps and exposure maps, with an ML  threshold of 4 in eboxdetect, an ML threshold of 9 for emlde-  tect, and the ECF derived for the given 𝐸𝑜𝑏𝑠(𝑧) energy band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The  mask used when making the sourcelist (in both eboxdetect and  emldetect) was the initial first mask described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  This means that the sourcelist includes all sources detected from  the X-ray image, since the initial mask covers the full FOV of the  detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The second and final mask (described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='1)  was then used to remove sources that lie outside of the mask from  the X-ray sourcelist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This gives us our final X-ray sourcelist for a  given 𝐸𝑜𝑏𝑠(𝑧) band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='3  Combining XMM-Newton Event Lists and Observations  Given that we’re not going down to fluxes below 10−14 ergs s−1  cm−2, a 50 ks exposure depth is more than sufficient to have good X-  ray measurements well below the limit of the optical identifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  If a given XMM-Newton observation included more than one science  exposure for any of the EPIC instruments (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' extra S00 or U00  exposures), we sought to merge the top two event lists with the  longest exposure times using the SAS task merge to achieve a decent  exposure time (see Table A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We placed the criteria that the event  lists being merged had to have the same filter and the same frame  mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This resulted in 3 XMM-Newton observations (0092850201,  0112370401 and 0123100101) that contained merged event lists  from multiple exposures within the same observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  One XMM-Newton observation (0124110101) contained mul-  tiple science exposures that were taken in different frame modes,  and thus could not be merged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Instead, we reduced the science ex-  posures separately for each EPIC instrument, and summed up their  X-ray images, exposure maps and background maps at the end of  the data reduction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We then used these summed images,  exposure maps and background maps when running the source de-  tection chain in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2 and making the final X-ray sourcelist  (see Table A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  If there was more than one XMM-Newton observation targeted  towards the same field, AXIS had a preference for those that were  public at an earlier period in the programme or those that were  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Comparison of the measured XMM-Newton fluxes to the published  AXIS fluxes in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 − 2 keV band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  part of the SSC Guaranteed Time Program (Carrera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Instead, to make use of all the data, we sought to merge together the  top two observations with the longest exposure times using the SAS  task merge to achieve a better exposure time (see Table A4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The  same criteria was placed on merging different XMM-Newton ob-  servations, requiring the event lists being merged to have the same  filter and the same frame mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' If any of these observations had  more than one science exposure, they were merged internally first  before being merged with another XMM-Newton observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' For  these merged observations, we used the exclusion area properties  from the AXIS-chosen observation for the target field when mod-  ifying the final mask described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' These properties  are listed in Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='4  XMM-Newton Source Selection  In this work, a separate sourcelist was produced for each 𝐸𝑜𝑏𝑠(𝑧)  band, as well as the 𝐸𝑟 𝑓 band (2−8 keV), per XMM-Newton observa-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This gives us a total of 7 X-ray sourcelists for each observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  We then combined the sourcelists from all the XMM-Newton ob-  servations, resulting in a comprehensive X-ray soureclist spanning  the entire dataset for each of the 7 energy bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Extended sources  were removed from these sourcelists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' To make sure our X-ray fluxes  produced through the data reduction process were reasonable, we  compared our 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 − 2 keV X-ray fluxes to the published 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 − 2 keV  AXIS fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' They were found to be in good agreement (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Spurious sources resulting from data artifacts and systematic  effects were being detected through the source detection chain,  which posed some issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Many of these sources had low detection  likelihoods but very high fluxes (above the flux limit of the 𝐸𝑜𝑏𝑠(𝑧)  band, determined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' To reduce the occurrence of these  types of sources, we filtered out low exposure pixels (using a 10 ks  exposure threshold) from the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 − 2 keV summed exposure map in  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='1, excluding them from the final mask used to filter out  the X-ray sourcelists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The ML threshold for emldetect was also  set to 9 in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2 when making the final sourcelist to avoid  sources with low detection likelihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This reduced the number of  spurious sources significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 4 displays the X-ray source fluxes  vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' the detection likelihood of the sources from the final filtered  sourcelists for each 𝐸𝑜𝑏𝑠(𝑧) band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The black dashed line indicates  the maximum likelihood threshold of 9 used in emldetect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The  MNRAS 000, 1–19 (2022)  log(XMM Flux [W/m?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='])  10-16  10-17  10-17  10-16  log(AXIS Flux [W/m2])8  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Alqasim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  vertical solid line is the flux limit of the X-ray sample, below which  sources are not included when making the XLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Despite the measures taken, some spurious sources were still  found above the flux limit of the 𝐸𝑜𝑏𝑠(𝑧) sourcelists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' These sources  were individually followed-up and investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Upon visual inspec-  tion of the X-ray images, some of them were not real sources and  had a very low signal-to-noise ratio (SNR, defined by the X-ray flux  of the source divided by the flux error of the source).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' These were  likely due to systematic errors resulting from the data reduction  pipeline process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' To avoid sources like these, we set a condition to  filter out X-ray sources that had a SNR < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  In the XMS survey, there were two sets of XMM-Newton ob-  servations that partially overlapped over the same region of the sky,  as listed below:  G133-69 Pos_1 and G133-69 Pos_2  SDS-1, SDS-2, and SDS-3  AXIS dealt with this by masking out portions of the overlapping  regions in the second (and third) fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Instead of doing this, we  use equation (2) to take the weighted average ¯𝑥 of the RA and  Dec positions for any detected sources that were overlapped for a  given energy band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We consider sources to be overlapped if they  have an angular separation distance 𝜃𝐷 ≤ 10′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We search for these  overlapped sources within the combined sourcelist across all XMM-  Newton observations for each energy band and filter them out, re-  placing them with the weighted average,  ¯𝑥 =  �  𝑥𝑖  𝜖2  𝑖  + 𝑥 𝑗  𝜖2  𝑗  � �  1  𝜖2  𝑖  + 1  𝜖2  𝑗  �−1  (2)  where ¯𝑥 is the weighted average, 𝑥 is the RA/Dec/flux of the 𝑖𝑡ℎ and  𝑗𝑡ℎ overlapping sources, and 𝜖 is the RA/Dec/flux error of the 𝑖𝑡ℎ  and 𝑗𝑡ℎ overlapping sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5  HEAO 1 A-2 X-Ray Sample  To fill in the gaps in our X-ray data for high luminosity sources at low  redshifts (𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2), we also include the flux-limited X-ray sample  by Piccinotti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This is a complete catalogue of X-ray  sources at Galactic latitudes produced in the 2−10 keV band using  data from the HEAO 1 experiment A-2 X-ray survey (Rothschild  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The survey goes down to a limiting sensitivity of  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='1 × 10−11 ergs cm−2 s−1 and covers a sky area of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='7 × 104 deg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  The survey reports their measurements in two separate scans (1st  scan and 2nd scan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Since the 1st scan is deeper, Piccinotti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  (1982) treat it as the primary measurement for deriving their best-fit  parameters, and the 2nd scan fluxes were only used for independent  confirmation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Hence, we adopt the 1st scan flux values as the X-ray  flux measurement when adding the data to our XLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Piccinotti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (1982) report their X-ray fluxes in units of  R15, which is a counting rate derived using the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5◦ × 3◦ FWHM  fields of view of the layers of the X-ray counters in the HEAO 1  A-2 experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' They also list conversion factors for each R15 flux  measurement corresponding to each X-ray source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We multiply each  conversion factor by the first-scan R15 flux measurement to convert  the fluxes to units of 10−11 ergs cm−2 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Given the slight difference in the 2 − 8 keV and 2 − 10 keV  energy bands, the 2−10 keV source fluxes from the HEAO 1 sample  had to be corrected as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The flux between two energies (for  a given energy range 𝐸𝑖 < 𝐸 < 𝐸 𝑓 ) is given by equation (3), which  is the X-ray spectral form of the majority of Seyfert galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  𝐹𝐸𝑖−𝐸 𝑓 =  ∫ 𝐸 𝑓  𝐸𝑖  𝑘𝐸1−Γ𝑑𝐸  (3)  where Γ is the photon index, 𝑘 is a normalization constant, and 𝐸𝑖  and 𝐸 𝑓 define the energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  When making the model spectrum in PyXspec in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='1,  a photon index of Γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='9 was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' To remain consistent, the same  photon index was assumed when converting the X-ray fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' To  account for the differences between the 2 − 8 keV and 2 − 10 keV  fluxes, equation (3) was evaluated for the two energy ranges, and the  ratio between them 𝑅𝐹 (𝑅𝐹 = 𝐹2−8/𝐹2−10) was taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This gives  us 𝑅𝐹 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='852, which was then multiplied by the HEAO 1 2 − 10  keV fluxes to give us the 2 − 8 keV fluxes for our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  4  OPTICAL IDENTIFICATIONS  To construct an XLF, redshifts are required along with source X-ray  fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' In this section, we describe the optical identifications used  to obtain redshifts for the X-ray sources in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We take the  optical identifications from the XMS survey for the XMM-Newton  X-ray sources, and the optical identifications from the Piccinotti  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (1982) catalogue for the HEAO 1 X-ray sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We restrict  our redshifts to be from spectroscopically identified optical coun-  terparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='1  XMM-Newton Medium Sensitivity Survey  The XMS survey is comprised of four overlapping samples in the  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5−2 keV, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 keV, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0−10 keV and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 keV energy bands  with flux limits well above the sensitivity of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' XMS covers a  total of 318 distinct X-ray sources, and counterparts for each source  were searched for in optical catalogues within 5 arcsec from the  position of the X-ray source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Redshifts were measured by matching  emission and absorption features to the sliding wavelengths of these  features (Barcons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The XMS has a high identification  completeness, giving us 255 AGN sources with counterparts that  are positively identified via optical spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  To identify the X-ray sources in each of the 7 filtered  sourcelists, the RA/Dec positions of our sources were matched with  the XMS optical counterpart source positions, taken from Table 5  in Barcons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' For this work, only sources that were iden-  tified via optical spectroscopy within the XMS sample were consid-  ered for our identifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This criteria was also applied for 𝑁𝑖𝑑  in equation (4) when calculating the completeness fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Since  XMM-Newton has a ∼ 5′′ spatial resolution (Jansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2001),  we matched sources within an angular distance of 5′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Matched X-  ray sources were then assigned corresponding redshifts from their  matched optical counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' These were then used when conduct-  ing the completeness studies of our sample, described in more detail  in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2  HEAO 1 A-2 Survey  The optical identifications for the X-ray sources from the HEAO 1  Survey were taken from the Piccinotti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (1982) catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We  included X-ray sources that were classified as either of the following:  Seyfert-1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Seyfert-2, NELG, N or other active galaxy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' BL Lacerate  object;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' and QSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Of these, we only use sources with an ID quality of  MNRAS 000, 1–19 (2022)  Redshifted XLF method for AGN  9  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' X-ray source flux vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Detection Likelihood for each of the 𝐸𝑜𝑏𝑠 (𝑧) bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Red data points are unidentified X-ray sources, and blue data points are  X-ray sources identified via optical spectroscopy from the AXIS-XMS survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The black dashed line indicates the maximum likelihood threshold of 9 used in  emldetect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The vertical solid line is the flux limit of the X-ray sample, below which sources are not included when making the XLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  “certain" or “possible", with a redshift measurement 𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This  gives us a total of 29 spectroscopically identified AGN adopted from  the Piccinotti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (1982) catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='3  Flux Limits  To understand what the appropriate flux limits are to use for our  XMM-Newton X-ray sources, the completeness of the data pool was  studied for each redshifted energy band (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='4 for more  MNRAS 000, 1–19 (2022)  1600ev 6400ev band  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0  Unidentified  Identified  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5  Flux Limit  Maximum  log1o(Likelihood)  Likelihood  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0  Threshold  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
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+page_content=' ])10  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Alqasim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We list the derived flux limits in Table 5 for each 𝐸𝑜𝑏𝑠(𝑧)  band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  The HEAO 1 sample is defined by sources brighter than a  countrate of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='25 R15 in the 1st scan, so we use this as the flux limit  for these X-ray sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We convert the flux limit to units of 10−11  ergs cm−2 s−1 using a conversion factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='175, the value most  used by Piccinotti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (1982) for converting their X-ray fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Correcting this to the 2 − 8 keV band using 𝑅𝐹 gives us a flux limit  of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='315 × 10−11 ergs cm−2 s−1 for the X-ray sources used from the  HEAO 1 survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='4  Completeness Studies of the XMM-Newton Data  The completeness of the XMM-Newton X-ray data was studied for  each of the 7 energy bands (6 𝐸𝑜𝑏𝑠(𝑧) bands and 1 𝐸𝑟 𝑓 band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  This allows us to assign the appropriate flux limits when making  the XLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The completeness fraction 𝑓𝑐(𝐹𝑥) as a function of X-ray  source flux 𝐹𝑥, from brightest to faintest, is given by  𝑓𝑐(𝐹𝑥) =  𝑁𝑖𝑑(𝐹 ≥ 𝐹𝑥)  𝑁𝑡𝑜𝑡𝑎𝑙(𝐹 ≥ 𝐹𝑥) ,  (4)  where the numerator term 𝑁𝑖𝑑(𝐹 ≥ 𝐹𝑥) represents the number of  optically-identified X-ray sources with a flux, 𝐹, greater than or  equal to 𝐹𝑥, and the denominator term 𝑁𝑡𝑜𝑡𝑎𝑙(𝐹 ≥ 𝐹𝑥) represents  the total number of X-ray sources with a flux greater than or equal  to 𝐹𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The 𝑁𝑖𝑑 term in equation (4) thus represents the number  of X-ray sources identified in the XMS survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We map out this  numerator term, 𝑁𝑖𝑑(𝐹 ≥ 𝐹𝑥), as a function of flux 𝐹𝑥 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 5,  as well as the denominator term, 𝑁𝑡𝑜𝑡𝑎𝑙(𝐹 ≥ 𝐹𝑥), as a function of  flux 𝐹𝑥 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  The identification criteria for 𝑁𝑖𝑑 required X-ray sources to  be matched with XMS sources that were positively identified via  optical spectroscopy, including AGN, clusters, and stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Additional  sources were found to have published spectroscopic redshifts that  were not included in the optical identifications of the XMS survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  These were included in 𝑁𝑖𝑑 for the X-ray source identifications in  this work, listed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 5 displays the source flux 𝐹𝑥 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 𝑁𝑖𝑑(𝐹 ≥ 𝐹𝑥) for each  energy band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The figure includes two plots: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 5a displays 𝑁𝑖𝑑  sources over the entire completeness sample, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 5b only dis-  plays 𝑁𝑖𝑑 sources that have redshifts within their corresponding  𝐸𝑜𝑏𝑠(𝑧) redshift bin 𝑧𝑏𝑖𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  To study the completeness of our X-ray sourcelists across all  energy bands, a plot of 𝐹𝑥 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 𝑓𝑐 was produced (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This also  allowed us to derive flux limits needed to make the redshifted XLF  for each energy band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' To derive the required 𝐹𝑙𝑖𝑚(𝑧) to make the  fixed rest-frame XLF, a separate 𝐹𝑙𝑖𝑚 is needed for each 𝐸𝑜𝑏𝑠(𝑧)  band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Each 𝐹𝑙𝑖𝑚(𝑧) was determined based on where the complete-  ness curve reaches 𝑓𝑐 = 80% for each 𝐸𝑜𝑏𝑠(𝑧) band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' A pragmatic  choice was made in choosing a limit of 80%, as there is a trade off  between achieving high completeness and having good number of  sources to produce the XLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 7, we mark this completeness  flux limit threshold with a horizontal dashed black line, and the set  of derived 𝐹𝑙𝑖𝑚(𝑧) for the redshifted energy bands are marked by  the vertical solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Table 5 lists the derived flux limit for each  energy band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' A plot of these flux limits as a function of redshift  is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' On the same figure, we also plot the discreet  function of 𝐹𝑙𝑖𝑚(𝑧) used when making the fixed rest-frame XLF in  this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Once the flux limits were derived as a function of redshift,  a fixed rest-frame XLF could be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Only AGN sources  with spectroscopic redshifts were used when constructing the XLFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Clusters, stars and AGN with no redshifts or ones that only had pho-  tometric redshifts were excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' See Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='1 and Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='3  for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  5  THE X-RAY LUMINOSITY FUNCTION  In this section, we present two techniques for calculating XLFs of  AGN in the 𝐸𝑟 𝑓 and 𝐸𝑜𝑏𝑠(𝑧) energy bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The first technique is  used to produce a binned XLF over discreet luminosity and redshift  bins (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='1), while the second technique is used to compute  the XLF using an ML fit to the full set of sources in the sample  (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We also describe the analytical function we use to fit  the XLF (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='1  Binned XLF  To construct a binned luminosity function of a sample of objects,  we divide the Luminosity−Redshift plane into 𝐿 − 𝑧 bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' For this  paper, the binned XLF was constructed using the method of Page  & Carrera (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Binned luminosity functions are by their nature  averaged over a luminosity and redshift bin, and hence where 𝜙  varies significantly with luminosity and/or redshift within the bin  (as for example, at high 𝐿 where 𝜙 changes rapidly with 𝐿) the  value expected from a binned estimator may be somewhat different  to the value of the model at the center of the bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We have examined  the magnitude of this difference in our survey by comparing the  expectation values for the model bin (as defined in Section 5 of  Page & Carrera (2000)) to the value of the model evaluated at the  midpoint of the bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' For example, we looked at the difference in  the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0 < 𝑧 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 range at log 𝐿𝑋 = 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='98 ergs/s, and find that  the difference between the model and expectation values of log 𝜙 is  about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  We define 𝜙, the differential luminosity function, in terms of  log 𝐿 rather than in terms of 𝐿, because it is easier to use when  dealing with a large span of luminosities (Cara & Lister 2008),  𝜙(log 𝐿, 𝑧) =  𝑑2𝑁  𝑑𝑉𝑑 log 𝐿 ,  (5)  where 𝑁 is the number of objects, 𝑧 is the redshift, 𝐿 is the luminosity  and 𝑉 is the comoving volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Note that 𝜙(𝐿, 𝑧) and 𝜙(log 𝐿, 𝑧)  are related to each other by a factor of 𝐿 ln(10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  The binned estimate of the luminosity function using the Page  & Carrera (2000) method can be obtained for 𝑁 objects found over  any volume-luminosity region,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' described by:  𝜙(log 𝐿,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 𝑧) ≈  𝑁  ∫ log 𝐿𝑚𝑎𝑥  log 𝐿𝑚𝑖𝑛  ∫ 𝑧𝑚𝑎𝑥(log 𝐿)  𝑧𝑚𝑖𝑛  𝑑𝑉  𝑑𝑧 𝑑𝑧 𝑑 log 𝐿  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  (6)  where ⟨𝑁⟩ is the expectation value of the number of objects,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 𝐿 is  the luminosity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 𝑧𝑚𝑖𝑛 is the minimum redshift in Δ𝑧,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' and 𝑧𝑚𝑎𝑥(𝐿)  is the maximum possible redshift for an object of luminosity 𝐿 to  be detected and remain contained within Δ𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We have adopted a  uniform bin width in Δ log 𝐿 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='3 for our binned XLFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  For each redshifted energy band 𝐸𝑜𝑏𝑠(𝑧), an XLF was pro-  duced within its corresponding redshift bin 𝑧𝑏𝑖𝑛, as listed in Table  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' For example, the XLF for the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 < 𝑧 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0 bin was done using  the X-ray sources in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='1 keV band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Since we use discreet  redshift intervals to construct the binned XLF, the equations and  MNRAS 000, 1–19 (2022)  Redshifted XLF method for AGN  11  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Spectroscopically identified sources from the literature that were not included in the optical identifications of the XMS survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' These were used in 𝑁𝑖𝑑  for the X-ray source identifications in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The table lists the catalogue from which the source was taken, the name of the source, the RA/Dec position of  the source, and the spectroscopic redshift of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Source Name  RA  Dec  Redshift  Redshift Origin  [deg]  [deg]  MS 0737.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0+7436  115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='802083  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='493333  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='312  EMSS (Stocke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 1991)  GALEXASC J074202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='51+742625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5  115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='512068  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='440213  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='599  XBS (Caccianiga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2008)  2MASS J21300228-1534131  322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='509363  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='570248  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='562  2MASS (Caccianiga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2004)  J133120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='3+242304  202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='835000  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='384472  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='753  BUXS (Mateos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2015)  (a) N𝑖𝑑 ∈ 𝑧𝑎𝑙𝑙  (b) N𝑖𝑑 ∈ 𝑧𝑏𝑖𝑛  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Plot of flux 𝐹 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' the number of identified sources 𝑁𝑖𝑑 with a flux greater than or equal to a given X-ray source flux 𝐹𝑥 (𝑁𝑖𝑑 (𝐹 ≥ 𝐹𝑥)) for  the 7 energy bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Left: 𝑁𝑖𝑑 includes sources within the entire redshift range 0 < 𝑧 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Right: 𝑁𝑖𝑑 only includes sources that have redshifts within their  corresponding 𝐸𝑜𝑏𝑠 (𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Derived flux limits for each energy band in the XMS survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This  includes the 𝐸𝑟 𝑓 band (2 − 8 keV) and the 𝐸𝑜𝑏𝑠 (𝑧) bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The 𝐹𝑙𝑖𝑚(𝑧)  was derived for each energy band based on where the completeness curve in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 7 reaches 𝑓𝑐 = 80% (indicated by the black dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The converted  2−8 keV flux limit from the HEAO 1 survey is also reported (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='3  for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  𝐸 (𝑧)  𝐹𝑙𝑖𝑚(𝑧)  [eV]  [10−14 ergs s−1 cm−2]  XMS Survey  2000 − 8000  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='44186  1600 − 6400  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='82524  1142 − 4571  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='38016  888 − 3555  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='87509  727 − 2909  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='68052  615 − 2461  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='50592  533 − 2133  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='44649  HEAO 1 Survey  2000 − 8000  2315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='18  methods used are the same as Page & Carrera (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The key dif-  ference in the way we construct the binned XLF is that the flux limit  𝐹𝑙𝑖𝑚(𝑧) is different for each of the redshift shells (see Table 5 and  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Together, these make up the 2 − 8 keV XLF, fixed in the  rest-frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  The data processed in the 2 − 8 keV band (the 𝐸𝑟 𝑓 band) was  then used to construct the fixed observer-frame XLF for the entire  redshift range (0 < 𝑧 < 3) to compare with the fixed rest-frame XLF  (using the 𝐸𝑜𝑏𝑠(𝑧) bands).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The data points from the binned XLFs  follow the trend of a double power-law, with the break luminosity  evolving with redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This illustrates the expected AGN evolution  with redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Due to the limited number of sources in our higher  redshift bins, this trend becomes harder to see and analytical models  are required to understand how AGN evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2  Analytical Model  The X-ray luminosity in a pure luminosity evolution (PLE) model  evolves with redshift, and can be expressed by  𝜙(log 𝐿, 𝑧) = 𝑑𝜙(𝐿/𝑒(𝑧), 0)  𝑑 log 𝐿  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  (7)  MNRAS 000, 1–19 (2022)  1600ev 6400ev  200  1142ev 4571ev  888ev 3555ev  727ev 2909ev  150  615ev 2461ev  533ev 2133ev  F  IV  100  N  50  0  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5  log10(Flux [erg/s/cm21)60  1600ev 6400ev  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5)  50  1142ev 4571ev  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 < z < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0)  888ev 3555ev  40  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0 < z < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5)  727ev 2909ev  F  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 < z < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0)  ΛI 30  615ev 2461ev  N  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0 < z < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5)  20  533ev 2133ev  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 < z < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0)  10  0  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='25 -14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='00 -13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='75 -13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='50 -13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='25 -13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='00 -12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='75 -12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='50 -12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='25  log1o(Flux [erg/s/cm21)12  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Alqasim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Plot of flux 𝐹 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' the number of total sources 𝑁𝑡𝑜𝑡𝑎𝑙 with a flux  greater than or equal to a given X-ray source flux (𝑁𝑡𝑜𝑡𝑎𝑙 (𝐹) ≥ 𝐹𝑥) for  the 7 energy bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The purple data points mark the 𝐸𝑟 𝑓 band, and the rest  of the colors mark the 𝐸𝑜𝑏𝑠 (𝑧) bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  The evolution factor 𝑒(𝑧) of the PLE is expressed by  𝑒(𝑧) =  ����  ����  (1 + 𝑧) 𝑝1  if 𝑧 < 𝑧𝑐,  𝑒(𝑧𝑐)  � 1 + 𝑧  1 + 𝑧𝑐  � 𝑝2  if 𝑧 ≥ 𝑧𝑐,  (8)  where 𝑧𝑐 is the cut-off redshift, 𝑝1 is is the parameter that accounts  for the evolution below 𝑧𝑐, and 𝑝2 is the parameter that accounts  for the evolution above 𝑧𝑐 (Miyaji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2000b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The shape of the  present-day XLF for which we adopt a smoothly connected double  power law can then be expressed by  𝜙(log 𝐿, 0) = 𝐴  �� 𝐿  𝐿0  �𝛾1  +  � 𝐿  𝐿0  �𝛾2�−1  ,  (9)  where 𝛾1 and 𝛾2 are the slopes, 𝐿0 is the luminosity value where  the change of slope occurs, and 𝐴 is the normalization constant (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Boyle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Miyaji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2000b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Equation (9) was then used to plot the PLE analytical model  on the binned XLF data in the 2−8 keV energy range for 0 < 𝑧 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  This was done for the fixed rest-frame 2−8 keV band, as well as the  fixed observed 2−8 keV band for comparison with the new method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  To be able to compare our results with that of the AXIS survey,  we first constructed the PLE model curves using parameters taken  from the PLE fit in the 2 − 10 keV band in Ebrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Given the slight difference in the 2 − 8 keV and 2 − 10 keV energy  bands, the log10 𝐿0 parameter (43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='13 ℎ−2  70 erg s−1) had to be  corrected, as was done for the HEAO 1 X-ray fluxes in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  As before, we assume a photon index of Γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We use 𝑅𝐹 in log  space to convert the log10 𝐿0 parameter, giving us a final corrected  parameter of log10 𝐿0 = 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='53 to be used in the binned XLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  The rest of the parameters adopted from Ebrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2009)  were 𝛾1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='06, 𝛾2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='37+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='19  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='18, 𝐴 = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='96+9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='97  −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='09 in units of  10−6 ℎ3  70 Mpc−3, 𝑧𝑐 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='9 (fixed), and 𝑝1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The parameter  𝑝2 was fixed to 0, as done in Ebrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2009) for the 2 − 10  keV XLF, with the evolution stopping after the cut-off redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' It is  important to note that their fitting also took into account the amount  of absorption in the modeling, which is not done when using the  new method introduced in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We thus regenerate the PLE  models for the binned XLFs after performing our own model fitting  and using our derived best-fit parameters (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='3 for more  details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='3  Maximum Likelihood Fit  We use the ML method (Crawford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 1970) to fit the PLE model  directly to the sources and obtain the best-fit evolution parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  We perform this technique for XLFs in the fixed rest-frame 2 − 8  keV band using the new method, as well as the fixed observer-frame  2 − 8 keV band for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The ML method takes into account  properties from each individual source, and no information is lost  as a result of binning the XLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  The likelihood function is defined as the product of the prob-  abilities of the the X-ray sources used in the XLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This gives us  the overall probability density for the observed distribution of ob-  jects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This is normally easier to compute over a logarithmic scale,  as it allows us to sum over the logarithms of the probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We  follow the method from Page et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2021) to maximize the likeli-  hood by minimizing the expression 𝐶, when in logarithmic scale,  as described by,  𝐶 =2𝑁 ln  � ∫ log 𝐿𝑚𝑎𝑥  log 𝐿𝑚𝑖𝑛  ∫ 𝑧𝑚𝑎𝑥 (log 𝐿)  𝑧𝑚𝑖𝑛  𝜙(log 𝐿, 𝑧) 𝑑𝑉  𝑑𝑧 𝑑𝑧 𝑑 log 𝐿  �  (10)  − 2  𝑁  ∑︁  𝑖=1  ln 𝜙(log 𝐿𝑖, 𝑧𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  We solve this using using the amoeba routine described in  Press (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' When applying this method to fitting an XLF, we are  not just maximizing the values of the luminosity function, but we are  also turning it into the probability of having observed each source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  This requires taking into account flux limits when generating the  probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' In the new fixed rest-frame method, the flux limit is a  function of redshift 𝐹𝑙𝑖𝑚(𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This is incorporated into the ML-fitting  routine through 𝑧𝑚𝑎𝑥(log 𝐿) (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 10), because the maximum  redshift that you can see an object of given luminosity depends on  the flux limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 𝑧𝑚𝑎𝑥(log 𝐿) is thus the redshift at which the flux limit  is equal to the flux, which is the maximum redshift to which a source  could be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' In previous works which used the ML method to  model the XLF (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Ebrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Ueda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2014), the flux  limit used to determine 𝑧𝑚𝑎𝑥(log 𝐿) was not dependent on redshift,  whereas in our method the flux limit does depend on redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  The ML method changes the shape of the model distribution  function to match as best as possible the distribution of the observed  sources in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The model parameters we fit for are the  luminosity break log10(𝐿0), the slope before the break 𝛾1, slope  after the break 𝛾2, and the evolution parameter 𝑝1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' To be consistent  with Ebrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2009), we fix 𝑝2 to 0, and we also fix the cut-  off redshift 𝑧𝑐 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This means that the evolution law we are  fitting stops at 𝑧 = 𝑧𝑐, so that equation (8) becomes 𝑒(𝑧) = 𝑒(𝑧𝑐)  for 𝑧 ≥ 𝑧𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We use these best-fit parameters to generate the PLE  curves in our binned XLFs for both the fixed rest-frame and fixed  observer-frame bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  6  RESULTS  The results of the binned XLFs can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 9, constructed as  described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' In the same figure, we display the XLFs  in both the fixed rest-frame (produced using the combined 𝐸𝑜𝑏𝑠(𝑧)  MNRAS 000, 1–19 (2022)  1750  533ev 2133ev  615ev 2461ev  1500  727ev 2909ev  888ev 3555ev  1250  1142ev 4571ev  1600ev 6400ev  F  1000  2000ev 8000ev  IV  N  750  500  250  0  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0  log10(Flux [erg/s/cm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' ])Redshifted XLF method for AGN  13  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Plot of flux 𝐹 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' the completeness fraction 𝑓𝑐 for each energy band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The purple data points mark the 𝐸𝑟 𝑓 band, and the rest of the colors mark the  𝐸𝑜𝑏𝑠 (𝑧) bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Right: Completeness plot displaying the completeness flux limit threshold at 𝑓𝑐 = 80%, marked by the horizontal black dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The flux  limits for each 𝐸 (𝑧) band are also plotted as vertical solid lines, with the same color corresponding to their data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Left: For clarity, we show the same  plot without the threshold and flux limit lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Plot of redshift 𝑧 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' the flux limit 𝐹𝑙𝑖𝑚(𝑧) for each energy band,  based on the values reported in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The points correspond to the flux  limits of each energy band used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The purple data point marks  the 𝐸𝑟 𝑓 band, and the rest of the colors mark the 𝐸𝑜𝑏𝑠 (𝑧) bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The  step-function of 𝐹𝑙𝑖𝑚(𝑧) (marked with a gray solid line) is a function that  maps the discrete redshift intervals to discrete flux limit values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  band data) and the fixed observer-frame (produced using the 𝐸𝑟 𝑓  band data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The PLE model curves were also plotted on the binned  XLFs using our ML best-fit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 10 displays the fixed observer-frame XLFs for each 𝑧 bin in  a separate plot for clarity, and we additionally plot the model curves  from the Ebrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2009) PLE best-fit parameters for comparison  (marked as magenta dashed lines), as described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We  display the same plots for clarity for the fixed rest-frame XLFs in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Our ML model fitting results are obtained using the methods  described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The results of the ML best-fit parameters  for the fixed rest-frame and fixed observer-frame XLFs are summa-  rized in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  7  DISCUSSION  In this section, we compare the performance of our fixed rest-frame  method with the standard method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We then compare our results with  the ones obtained by the AXIS survey and some other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We  finally describe the future prospects for the fixed rest-frame method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='1  Comparison between the fixed rest-frame and the fixed  observer-frame XLFs  In both the fixed observer-frame and fixed rest-frame XLFs, adding  the HEAO 1 data allowed us to improve the coverage of the higher  luminosity sources at low redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This significantly improved the  constraints on our best-fit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 9 shows that for both  methods, the binned data points are reasonably consistent with the  PLE model curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Our ML best-fit results find that the parameters that define the  evolution of the XLF (𝑝1), and the parameters that describe the  shape of the XLF (𝛾1 and 𝛾2, log10(𝐿0)), are in agreement within  the 1𝜎 confidence intervals for both methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This means that using  the new method with a fixed rest-frame band does not appear to have  a significant effect on the results compared with the standard fixed  observer-frame method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' It is important to note that at high redshifts,  our data set only spans high-luminosity sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' In comparing the  two methods to each other in this work, the results suggest that the  presence of absorbed AGN within the population does not have a  very significant effect on the observed evolution of the XLF for the  situation in which at high redshift, only high-luminosity sources are  sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  We have shown that it is practical to produce luminosity func-  tions using the new method, and that it has produced results that  are consistent with expectations in the luminosity-redshift regime  in which we have tested it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Our finding that both methods are con-  sistent with each other, and that there appears to be no significant  difference in using the new and standard methods, is consistent with  the expected outcome from our data used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 11 shows  that our data are limited to bright AGN at higher redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The PLE  model assumes that the XLF evolves only with the log10(𝐿0) lu-  MNRAS 000, 1–19 (2022)  100  80  Completeness [%]  60  2000ev 8000ev  1600ev 6400ev  40  1142ev 4571ev  888ev 3555ev  20  727ev 2909ev  615ev 2461ev  533ev 2133ev  0  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
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+page_content='0  log10(Flux [erg/s/cm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' ])100  80  Completeness [%]  60  2000ev 8000ev  1600ev 6400ev  40  1142ev 4571ev  888ev 3555ev  20  727ev 2909ev  615ev 2461ev  533ev 2133ev  0  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
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+page_content=' ])2000ev 8000ev  1600ev 6400ev  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='4  log1o(Flux Limit [erg/s/cm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='])  1142ev 4571ev  888ev 3555ev  727ev 2909ev  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
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+page_content=' Alqasim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  1  2  3  4  5  −8  −6  −4  −2  log [φ (Mpc−3 ∆ log (L)]  log [LX (1040 erg/s)]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
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+page_content='50 < z <  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
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+page_content='00  z = 0  (a) Fixed Observer-frame XLF  1  2  3  4  5  −8  −6  −4  −2  log [φ (Mpc−3 ∆ log (L)]  log [LX (1040 erg/s)]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
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+page_content='00 < z <  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
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+page_content='50 < z <  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='00  z = 0  (b) Fixed Rest-frame XLF  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' X-ray Luminosity Functions in the 2−8 keV band for 0 < 𝑧 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The data points represent the results from the binned XLF and the dashed lines  are the curves of the analytical PLE model (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 9), produced using our own ML best-fit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The orange dashed lines correspond to the PLE model  evaluated at 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Left: Standard XLF in the fixed observed band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Right: New XLF in the fixed rest-frame band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' PLE best-fit parameters for the X-ray luminosity functions in the fixed observer-frame and fixed rest-frame 2−8 keV bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  𝐴  𝛾1  𝛾2  log10(𝐿0)  𝑝1  [10−6 ℎ3  70 Mpc−3]  [ℎ−2  70 erg s−1]  Observer-frame  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
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+page_content='21  Rest-frame  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
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+page_content='15  minosity break shifting with redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Hence, the fitting of the XLF  evolution is mainly constrained by the bright end of the XLF, while  keeping the shape of the XLF fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Even at high redshifts, the bright  X-ray sources are not heavily absorbed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Ebrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' As  a consequence, performing XLF evolution studies for only high-  luminosity AGN sources is less sensitive to the choice of the fixed  observer-frame or rest-frame luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' If our data went to fainter  fluxes at high redshift, we would be observing sources at or below  the break in the luminosity function at high redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Many of those  sources are expected to be absorbed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' see Figure 5 in Ebrero  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Then, the fixed rest-frame and fixed observer-frame  methods are not expected to give the same results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We expect to  see the benefits of the new method when lower luminosity AGN are  included at high redshift, and the absorption distribution becomes  a key factor in modelling the luminosity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Comparing the fixed rest-frame XLF with the fixed observer-  frame XLF from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  9, we also found that the error bars on  the binned XLF data points are smaller in the fixed rest-frame,  especially in the high-redshift bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This effect is due to increased  number of sources in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' At higher redshifts in the fixed rest-  frame, we are sampling the softer 𝐸𝑜𝑏𝑠(𝑧) energy bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The softer  X-ray sources are easier to identify in the optical, and therefore a  larger sample of them are included at higher redshifts in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2  Comparison with other XLF studies  We checked to make sure our fixed observer-frame XLF yields  results that are in agreement with Ebrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2009), given that  we are using a subset of their data sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We find that our fixed  observer-frame best-fit results (see Table 6) are consistent with the  values reported in Table 2 of Ebrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2009) for the PLE fit  in the hard band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We also plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 12 the binned hard XLF  data points from AXIS in the 0 < 𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 < 𝑧 < 1 bins  in comparison with our own in the fixed observer-frame band for  reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' As done before when reproducing their PLE curves, we  converted the data points from 2−10 keV to 2−8 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We did not plot  the binned XLF data points in the other redshift bins as the AXIS  XLFs bin their data differently past 𝑧 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This plot, along with the  plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 10, also show that our fixed observer-frame PLE model  results (which are in agreement with the fixed rest-frame results)  are consistent with the Ebrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2009) curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' It is difficult to  compare our results to more recent relevant works in XLF studies as  many of them only report best-fit results for LADE or LDDE model  fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' However, Ueda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2003) construct a hard band (2-10 keV)  AGN XLF and perform PLE fits, which we can compare with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We  find that our PLE results are consistent with the values reported in  their Table 3 for the best-fit PLE parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  MNRAS 000, 1–19 (2022)  Redshifted XLF method for AGN  15  1  2  3  4  5  −8  −6  −4  −2  log [φ (Mpc−3 ∆ log (L)]  log [LX (1040 erg/s)]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='00 < z <  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
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+page_content='00  (axis best−fit)  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' X-ray Luminosity Functions in the fixed 2−8 keV observer-frame band for 0 < 𝑧 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The data points represent the results from the binned XLF and  the dashed lines are the curves of the analytical PLE model (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The red dashed lines are the model curves produced using our own ML best-fit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  The blue dotted lines are the model curves produced using the Ebrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2009) PLE best-fit parameters, corrected from 2−10 keV to 2−8 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  MNRAS 000, 1–19 (2022)  16  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Alqasim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  1  2  3  4  5  −8  −6  −4  −2  log [φ (Mpc−3 ∆ log (L)]  log [LX (1040 erg/s)]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='00 < z <  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='50  (our best−fit)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='00 < z <  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
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+page_content='00  (axis best−fit)  1  2  3  4  5  −8  −6  −4  −2  log [φ (Mpc−3 ∆ log (L)]  log [LX (1040 erg/s)]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
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+page_content='00  (axis best−fit)  1  2  3  4  5  −8  −6  −4  −2  log [φ (Mpc−3 ∆ log (L)]  log [LX (1040 erg/s)]  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='00 < z <  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='50  (our best−fit)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
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+page_content='50  (axis best−fit)  1  2  3  4  5  −8  −6  −4  −2  log [φ (Mpc−3 ∆ log (L)]  log [LX (1040 erg/s)]  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
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+page_content='00  (our best−fit)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='50 < z <  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='00  (axis best−fit)  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' X-ray Luminosity Functions in the fixed 2−8 keV rest-frame band for 0 < 𝑧 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The data points represent the results from the binned XLF and the  dashed lines are the curves of the analytical PLE model (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The blue dotted lines are the model curves produced using the Ebrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2009) PLE best-fit  parameters, corrected from 2−10 keV to 2−8 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The red dashed lines are the model curves produced using our own ML best-fit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  MNRAS 000, 1–19 (2022)  Redshifted XLF method for AGN  17  1  2  3  4  5  −8  −6  −4  −2  log [φ (Mpc−3 ∆ log (L)]  log [LX (1040 erg/s)]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='00 < z <  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='50  (our best−fit)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='00 < z <  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='50  (axis best−fit)  1  2  3  4  5  −8  −6  −4  −2  log [φ (Mpc−3 ∆ log (L)]  log [LX (1040 erg/s)]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='50 < z <  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='00  (our best−fit)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='50 < z <  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='00  (axis best−fit)  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' X-ray Luminosity Functions in the fixed 2−8 keV observer-frame band in the 0 < 𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 < 𝑧 < 1 bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The red data points represent our  results from the binned XLF and the dashed lines are the curves of the analytical PLE model (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The red dashed lines are the model curves produced using  our own ML best-fit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The blue dotted lines are the model curves produced using the Ebrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2009) PLE best-fit parameters, converted from  2−10 keV to 2−8 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We also include the data points from their hard XLF in the 0 < 𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 < 𝑧 < 1 bins, converted to the 2−8 keV band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Another interesting avenue we looked into was the comparison  between our results and optical QSO surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Many of these surveys,  even to recent times, have used the PLE model to describe the LF  evolution between 0 < 𝑧 < 3, which aligns well with the work  done in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Optical QSO surveys rely on the detection of  UV radiation from the QSOs, and so they are very sensitive to  dust extinction: dusty QSOs will disappear from their samples, just  as absorbed AGN will disappear from soft X-ray samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' It is  reasonable to question whether the evolution measured in optical  surveys is affected by dust extinction, and just like in X-ray surveys,  the rest-frame band is shifting with redshift (and so the effects of  dust extinction affect QSOs at different redshifts differently).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We  have introduced a new method in this paper that disentangles the  effects of absorption from the measurement of evolution in the XLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Hence, we explore how the evolution we measure with our fixed  rest-frame method compares to the evolution measured in optical  surveys, and we do this using the quasar LF studies in Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  (2013), Croom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2004) and Croom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  The Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2013) study uses the SDSS/BOSS DR9 data  in their work, and they describe that their quasar LF is similar to the  XLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' To compare with our results, we use the PLE best-fit values  reported in Table 8 in their paper in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='3 < 𝑧 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2 range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  From these best-fit results, they found a decrease in magnitude of  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='715 when looking at the evolution of the break magnitude in the  LF for 0 < 𝑧 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Converting our luminosity increase of 𝐿0 to  the decrease in magnitude between 0 < 𝑧 < 2, we find a decrease  in magnitude by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This indicates that the redshift evolution of  the break luminosity for optical QSO LFs is stronger than in X-ray  selected AGN LFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  The other studies of optical QSO LF also present evidence  of strong evolution in the break luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The PLE model fitting  results of the 2dF QSO redshift (2QZ) survey data in Croom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  (2004) show a change in magnitude of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='35 when looking at the  evolution of the peak luminosity in the LF for 0 < 𝑧 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Similarly,  the PLE results of the 2dF-SDSS LRG and QSO (2SLAQ) in Croom  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2009) indicate a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='95 mag evolution of the peak luminosity  for 0 < 𝑧 < 2 (when the sample is limited to brighter QSOs with a  cut off at −23 mag rather than −21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5 mag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  For the optical studies, extinction should in principle make the  measured evolution smaller than the real evolution, because dust  extinction gets worse further into the UV range, and so becomes  worse at higher redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Hence, the optical QSO LF in these stud-  ies should suffer from the stronger absorption due to the shorter  wavelength used for higher redshift samples, which would lead to a  weaker evolution of the LF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' However, we find a stronger evolution  in the optical QSO LF compared to our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Since our measure-  ment is robust to the effects of absorption in the fixed rest-frame  method, this indicates that the optical QSO LF has an intrinsically  stronger evolution than the X-ray selected AGN LF in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='3  Future prospects for the fixed rest-frame method  With the current data used in this work, we are limited in our  capability of solving the discrepancy on what the best model is to  describe the XLF evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' In that regard, this method proves to  be a useful tool in checking whether this holds true when dealing  with larger data sets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Aird et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2015a, who include almost  3000 AGN sources in the hard band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Presently, we have not yet  analyzed deep X-ray data at the faintest fluxes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' from Chandra  surveys), which might be useful to fully leverage the capabilities  of this method at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' For this to work for our new method,  the X-ray data will need to be treated in an analogous fashion to  the AXIS data used in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We would need to reduce the  data from scratch since we are sampling over many energy bands,  and published studies do not provide X-ray fluxes for the required  energy bands for each redshift bin, which would be needed to easily  incorporate more surveys into this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Within the next few years, we expect that eRosita will have  done the full survey of the X-ray sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Currently, the largest scale  of observations of the whole X-ray sky comes from ROSAT, which  cannot be used for a 2−8 keV band XLF since it only covers the  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='1−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0 keV band (hence, only covering the rest-frame 2−8 keV  band at 𝑧 ≥ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Previous XLF studies have made use of wide-  MNRAS 000, 1–19 (2022)  18  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Alqasim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  area Chandra surveys, but these studies use low detection limits  that carry only a couple of counts, which make observations in  the 2−8 keV band difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' On the other hand, the eRosita mission  will be able to provide the resolution equivalent to XMM-Newton  data, covering the full energy range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='5−8 keV over the entire  sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This is very promising in improving our detections of AGN  and understanding their behavior over a large scale (Kolodzig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' ATHENA will also be able to exceed the capabilities of  current X-ray observatories with its enhanced performance in X-ray  spectroscopy and deep wide-field X-ray imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' For this upcoming  era of X-ray observatories, our new method could be applied on a  huge scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The potential value of our new fixed rest-frame method is  greater for eRosita and ATHENA, than Chandra and XMM-Newton,  because it can harness data from all-sky-shallow to very deep (and  still quite wide, by today’s deep survey scales).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  8  SUMMARY AND CONCLUSIONS  In this paper, we have introduced a new fixed rest-frame method  for constructing X-ray luminosity functions of AGN, which aims to  give a clearer description of how they evolve over cosmic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Our  method fixes the X-ray energy band in the rest-frame by varying the  observed energy band with redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We tested our method against  two X-ray samples in the hard band: 29 XMM-Newton observations  following the targets from the XMS survey (Barcons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2007), and  the 1st scan X-ray sources from the HEAO 1 experiment A2 survey  (Piccinotti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The XMM-Newton data were used to produce  images and sourcelists in 6 X-ray energy bands, corresponding to 6  redshift ranges, to account for a target rest-frame band of 2−8 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  We used the spectroscopic optical identifications from the AXIS  scheme for the redshift measurements of the XMM-Newton data, as  well as 4 extra published optical IDs of bright sources that were  not included in the AXIS optical identifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We also used the  optical identifications and redshift measurements from Piccinotti  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (1982) for the HEAO 1 X-ray data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  We constructed XLFs of AGN using two techniques;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' one using  the method of Page & Carrera (2000) to make a binned XLF, and  one using an ML fit, which makes use of the full unbinned source  sample (Page et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Both techniques were computed for the  standard method (fixed observer-frame band) and our new method  (fixed rest-frame band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We then used an analytical model described  by a pure luminosity evolution (PLE) to fit the XLF data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The  model consists of a smoothly connected double power-law with a  factor accounting for how it evolves with redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  The new method presented here eliminates the need to model  the effects of intrinsic AGN absorption on the observed XLF be-  havior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We were able to demonstrate the viability of this method  in constructing XLFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We found that for both the fixed rest-frame  and observer-frame methods, the binned data points are reasonably  consistent with the PLE model curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We also find that our PLE  best-fit results were consistent with Ebrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2009) and Ueda  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Furthermore, we found an intrinsically stronger evolu-  tion in optical QSO LF studies (Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Croom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 2004,  2009) compared to our results of the X-ray selected AGN LF in our  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  As was shown by our comparison of the fixed observer-frame  and fixed rest-frame XLFs in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='1, the ML-fit results for both  methods were consistent with each other (in the case where only  high-luminosity sources are sampled at high redshift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Even though  the two methods produce similar results for the data that we have  tested them on in our paper, we do not expect that to remain true  if we were able to reach better coverage of the luminosity-redshift  plane by going fainter in X-ray flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The power of the new method  will be important if we include the fainter sources at high redshift,  as the evolution of the shape of the XLF at the fainter sources is  indicated in Ebrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Aird et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' (2015a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Hence, we  expect to see the benefits of the new method when lower luminosity  AGN are included at high redshift, and the absorption distribution  becomes a key factor in modelling the XLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Encouraged by the  success of the pilot study of this paper, this is an obvious direction  of our future work, which could be applied on a huge scale with the  upcoming era of X-ray observatories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  AAQ acknowledges the funding bodies, the UAE Ministry of Presi-  dential Affairs and the UAE Space Agency, for their support through  the PhD scholarship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' MJP acknowledges support from the Sci-  ence and Technology Facilities Council (STFC) grant numbers  ST/N000811/1 and ST/S000216/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We thank the referee for their  thoughtful and helpful feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We thank Daisuke Kawata for his  valuable comments and input on the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' We thank Francisco  Carrera and Silvia Mateos for their help with redshifts for AXIS  and BUXS sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  DATA AVAILABILITY  The data and reduction scripts used in this work can be made avail-  able upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  REFERENCES  Aird J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2010, MNRAS, 401, 2531  Aird J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Coil A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Georgakakis A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Nandra K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Barro G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Pérez-González  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2015a, MNRAS, 451, 1892  Aird J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2015b, The Astrophysical Journal, 815, 66  Antonucci R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 1993, Annual review of astronomy and astrophysics, 31, 473  Avni Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Bahcall J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 1979, Technical report, On the simultaneous analysis  of several complete samples: the V/Vsub (max) and Vsub (e)/Vsub (a)  variables, with applications to quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Weizmann Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' of Science  Barcons X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2007, Astronomy &amp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Astrophysics, 476, 1191  Barger A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Cowie L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Mushotzky R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Yang Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Wang W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Steffen  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Capak P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2005, AJ, 129, 578  Beckmann V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Shrader C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2013, Active galactic nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' John Wiley &amp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Sons  Bongiorno A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2016, A&A, 588, A78  Boyle B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Shanks T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Peterson B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 1988, MNRAS, 235, 935  Brandt W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Alexander D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2015, The Astronomy and Astrophysics Review,  23, 1  Caccianiga A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2004, A&A, 416, 901  Caccianiga A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2008, A&A, 477, 735  Cara M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Lister M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2008, ApJ, 686, 148  Carrera F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2007, Astronomy &amp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Astrophysics, 469, 27  Cowie L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Barger A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Bautz M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Brandt W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Garmire G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2003,  ApJ, 584, L57  Crawford D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Jauncey D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Murdoch H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 1970, ApJ, 162, 405  Cristiani S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Vio R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 1990, Astronomy and Astrophysics, 227, 385  Croom S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2004, in Mújica R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Maiolino R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', eds, Multiwavelength  AGN Surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' pp 57–62, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='1142/9789812702432_0015  Croom S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2009, MNRAS, 399, 1755  Ebrero J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2009, Astronomy &amp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Astrophysics, 493, 55  Fotopoulou S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2016, Astronomy & Astrophysics, 587, A142  Gabriel C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2004, in Astronomical Data Analysis Software and Systems  (ADASS) XIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' 759  MNRAS 000, 1–19 (2022)  Redshifted XLF method for AGN  19  Georgakakis A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2015, MNRAS, 453, 1946  Haardt F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Maraschi L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 1991, The Astrophysical Journal, 380, L51  Hasinger G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2001, Astronomy &amp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Astrophysics, 365, L45  Hasinger G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Miyaji T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Schmidt M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2005, Astronomy &amp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Astrophysics,  441, 417  Ho L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2008, Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 46, 475  Jansen F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2001, Astronomy &amp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Astrophysics, 365, L1  Kolodzig A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Gilfanov M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Sunyaev R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Sazonov S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Brusa M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2013, A&A,  558, A89  Kormendy J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Ho L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2013, ARA&A, 51, 511  Loaring N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2005, MNRAS, 362, 1371  Maccacaro T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', della Ceca R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Gioia I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Morris S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Stocke J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Wolter  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 1991, The Astrophysical Journal, 374, 117  Mateos S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2005, Astronomy &amp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Astrophysics, 433, 855  Mateos S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2015, Monthly Notices of the Royal Astronomical Society,  449, 1422  Miyaji T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Hasinger G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Schmidt M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2000a, Advances in Space Research,  25, 827  Miyaji T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Hasinger G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Schmidt M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2000b, A&A, 353, 25  Miyaji T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2015, ApJ, 804, 104  Morrison R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', McCammon D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 1983, ApJ, 270, 119  Mushotzky R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Done C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Pounds K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 1993, Annual review of astronomy  and astrophysics, 31, 717  Netzer H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2013, The physics and evolution of active galactic nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Cam-  bridge University Press  Page M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Carrera F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2000, Monthly Notices of the Royal Astronomical  Society, 311, 433  Page M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2021, MNRAS, 506, 473  Piccinotti G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Mushotzky R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Boldt E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Holt S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Marshall F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=',  Serlemitsos P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Shafer R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 1982, ApJ, 253, 485  Press W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 1997, The Art Of Scientific Computing 2nd Edition in Numer-  ical Recipes In C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Cambridge University Press  Ross N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2013, ApJ, 773, 14  Rothschild R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 1979, Space Science Instrumentation, 4, 269  Schmidt M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 1968, The Astrophysical Journal, 151, 393  Silverman J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2008, The Astrophysical Journal, 679, 118  Stocke J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Morris S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Gioia I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Maccacaro T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Schild R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Wolter A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=',  Fleming T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Henry J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 1991, ApJS, 76, 813  Strüder L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2001, Astronomy &amp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Astrophysics, 365, L18  Symeonidis M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2013, Monthly Notices of the Royal Astronomical  Society, 433, 1015  Traulsen I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2020, A&A, 641, A137  Turner M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2001, Astronomy &amp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Astrophysics, 365, L27  Ueda Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Akiyama M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Ohta K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Miyaji T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2003, ApJ, 598, 886  Ueda Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Akiyama M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Hasinger G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Miyaji T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Watson M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2014, ApJ,  786, 104  Yencho B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Barger A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Trouille L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', Winter L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=', 2009, ApJ, 698, 380  APPENDIX A: SUPPLEMENTARY TABLES  This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  5 https://ds9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='edu/  MNRAS 000, 1–19 (2022)  20  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Alqasim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' List of the XMM-Newton fields used in this work, with their general properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' This includes the XMM-Newton observation number, the target name,  the center (RA𝑇 𝑎𝑟𝑔𝑒𝑡,Dec𝑇 𝑎𝑟𝑔𝑒𝑡) and radius (R𝑇 𝑎𝑟𝑔𝑒𝑡) used to exclude the area around the target source with a circular region in ds95, and the center  (RA𝑂𝑂𝑇 ,Dec𝑂𝑂𝑇 ), angle (OOT𝑎𝑛𝑔𝑙𝑒), width (OOT𝑤𝑖𝑑𝑡ℎ) and height (OOTℎ𝑒𝑖𝑔ℎ𝑡) used to exclude the area around the OOT streaks with a rectangular  region in ds9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Observation  Target Name  RA  Dec  R  RA  Dec  OOT  OOT  OOT  𝑇 𝑎𝑟𝑔𝑒𝑡  𝑇 𝑎𝑟𝑔𝑒𝑡  𝑇 𝑎𝑟𝑔𝑒𝑡  𝑂𝑂𝑇  𝑂𝑂𝑇  𝑎𝑛𝑔𝑙𝑒  𝑤𝑖𝑑𝑡ℎ  ℎ𝑒𝑖𝑔ℎ𝑡  [deg]  [deg]  [′′]  [deg]  [deg]  [◦]  [′′]  [′′]  0012440301  PB5062  331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
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+page_content='0  0106660101 + 𝑐  0106660601  LBQS 2212-1759  –  –  –  –  –  –  –  –  𝑎 Different exposures were merged within the same XMM-Newton observation (see Table A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  𝑏 Different exposures (with different frame modes) within the same XMM-Newton observation were reduced separately and  the final images were summed (see Table A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  𝑐 Different XMM-Newton observations were merged for the same target (see Table A4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  MNRAS 000, 1–19 (2022)  Redshifted XLF method for AGN  21  Table A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Pairs of EPIC exposures (marked as Exposure𝐴 and Exposure𝐵 per instrument) used when merging extra eventlists within the same XMM-Newton  observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Instruments that only had one exposure were left blank for Exposure𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Observation  M1 Exposure𝐴  M1 Exposure𝐵  M2 Exposure𝐴  M2 Exposure𝐵  PN Exposure𝐴  PN Exposure𝐵  0092850201  S001  U003  S002  U003  S003  –  0112370401𝑎  S002  U002  S003  U003  U002  –  0123100101𝑎  S002  –  S003  –  S001  U014  𝑎 Merged with another XMM-Newton observation (see Table A4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Table A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Two sets of properties (Set A and Set B) corresponding to different M1, M2 and PN science exposures within the same the XMM-Newton observation  (0124110101).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' These sets were reduced separately due to having different frame modes, which prevented us from directly merging the event lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The final  images, exposure maps and background maps from Set A and Set B were summed together before making the final sourcelist in the reduction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The  properties listed in this table include the M1, M2 and PN rate thresholds used to filter out intervals of flaring particle background rate lightcurves (produced at  𝐸 > 5 keV using a time bin size of 20 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The science exposures used for each set are also listed, along with the frame mode for each EPIC instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Set A  Set B  M1 Threshold𝑎  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='7  M2 Threshold𝑎  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='7  PN Threshold𝑎  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='7  M1 Exposure  S004  S008  M2 Exposure  S005  S009  PN Exposure  S003  S001  M1 Frame Mode  Full Frame  Large Window  M2 Frame Mode  Full Frame  Large Window  PN Frame Mode  Extended Full Frame  Full Frame  𝑎 In units of count s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Table A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' Details regarding the 4 sets of observations that were merged together in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content=' The table lists the target name of the XMM-Newton field and the  two XMM-Newton observations that were merged together for each given target field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  Merge Set  Target Name  Observation  SET 1  SDS-3  0112370401𝑎  0112371501𝑏  SET 2  MS0737.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='9+7441  0123100101𝑎  0123100201𝑏  SET 3  HD 117555  0100240101  0100240201𝑏  SET 4  LBQS 2212-1759  0106660101𝑏  0106660601  𝑎 Different exposures were merged within the same XMM-Newton observation (see Table A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  𝑏 Observation chosen by AXIS for the specified target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
+page_content='  MNRAS 000, 1–19 (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfZfc3/content/2301.00223v1.pdf'}
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+Accepted as a conference paper at ICLR 2023
+OUTCOME-DIRECTED
+REINFORCEMENT
+LEARNING
+BY UNCERTAINTY & TEMPORAL DISTANCE-AWARE
+CURRICULUM GOAL GENERATION
+Daesol Cho∗, Seungjae Lee∗, H. Jin Kim
+Seoul National University, AIIS, ASRI
+{dscho1234, ysz0301, hjinkim}@snu.ac.kr
+ABSTRACT
+Current reinforcement learning (RL) often suffers when solving a challenging
+exploration problem where the desired outcomes or high rewards are rarely ob-
+served. Even though curriculum RL, a framework that solves complex tasks by
+proposing a sequence of surrogate tasks, shows reasonable results, most of the
+previous works still have difficulty in proposing curriculum due to the absence
+of a mechanism for obtaining calibrated guidance to the desired outcome state
+without any prior domain knowledge. To alleviate it, we propose an uncertainty
+& temporal distance-aware curriculum goal generation method for the outcome-
+directed RL via solving a bipartite matching problem.1 It could not only provide
+precisely calibrated guidance of the curriculum to the desired outcome states but
+also bring much better sample efficiency and geometry-agnostic curriculum goal
+proposal capability compared to previous curriculum RL methods. We demon-
+strate that our algorithm significantly outperforms these prior methods in a variety
+of challenging navigation tasks and robotic manipulation tasks in a quantitative
+and qualitative way.
+1
+INTRODUCTION
+While reinforcement learning (RL) shows promising results in automated learning of behavioral
+skills, it is still not enough to solve a challenging uninformed search problem where the desired
+behavior and rewards are sparsely observed. Some techniques tackle this problem by utilizing the
+shaped reward (Hartikainen et al., 2019) or combining representation learning for efficient explo-
+ration (Ghosh et al., 2018). But, these not only become prohibitively time-consuming in terms of
+the required human efforts, but also require significant domain knowledge for shaping the reward
+or designing the task-specific representation learning objective. What if we could design the algo-
+rithm that automatically progresses toward the desired behavior without any domain knowledge and
+human efforts, while distilling the experiences into the general purpose policy?
+An effective scheme for designing such an algorithm is one that learns on a tailored sequence of
+curriculum goals, allowing the agent to autonomously practice the intermediate tasks. However, a
+fundamental challenge is that proposing the curriculum goal to the agent is intimately connected to
+the efficient desired outcome-directed exploration and vice versa. If the curriculum generation is
+ineffective for recognizing frontier parts of the explored and feasible areas, an efficient exploration
+toward the desired outcome states cannot be performed. Even though some prior works propose
+to modify the curriculum distribution into a uniform one over the feasible state space (Pong et al.,
+2019; Klink et al., 2022) or generate a curriculum based on the level of difficulty (Florensa et al.,
+2018; Sukhbaatar et al., 2017), most of these methods show slow curriculum progress due to the
+process of skewing the curriculum distribution toward the uniform one rather than the frontier of
+the explored region or the properties that are susceptible to focusing on infeasible goals where the
+agent’s capability stagnates in the intermediate level of difficulty.
+∗Equal contribution.
+1Code is available : https://github.com/jayLEE0301/outpace_official
+1
+arXiv:2301.11741v1  [cs.LG]  27 Jan 2023
+
+Accepted as a conference paper at ICLR 2023
+Figure 1: OUTPACE proposes uncertainty and temporal distance-aware curriculum goals to enable
+the agent to progress toward the desired outcome state automatically. Note that the temporal distance
+estimation is reliable within the explored region where we query the curriculum goals.
+Conversely, without the efficient desired outcome-directed exploration, the curriculum proposal
+could be ineffective when recognizing the frontier parts in terms of progressing toward the de-
+sired outcomes because the curriculum goals, in general, are obtained from the agent’s experiences
+through exploration. Even though some prior works propose the success-example-based approaches
+(Fu et al., 2018; Singh et al., 2019; Li et al., 2021), these are limited to achieving the only given
+example states, which means these cannot be generalized to the arbitrary goal-conditioned agents.
+Other approaches propose to minimize the distance between the curriculum distribution and the de-
+sired outcome distribution (Ren et al., 2019; Klink et al., 2022), but these require an assumption that
+the distance between the samples can be measured by the Euclidean distance metric, which cannot
+be generalized for an arbitrary geometric structure of the environment. Therefore, we argue that
+the development of algorithms that simultaneously address both outcome-directed exploration and
+curriculum generation toward the frontier is crucial to benefit from the outcome-directed curriculum
+RL.
+In this work, we propose Outcome-directed Uncertainty & TemPoral distance-Aware Curriculum
+goal gEneration (OUTPACE) to address such problems, which requires desired outcome examples
+only and not prior domain knowledge nor external reward from the environment. Specifically, the
+key elements of our work consist of two parts. Firstly, our method addresses desired outcome-
+directed exploration via a Bayesian classifier incorporating an uncertainty quantification based on
+the conditional normalized maximum likelihood (Zhou & Levine, 2021; Li et al., 2021), which
+enables our method to propose the curriculum into the unexplored regions and provide directed
+guidance toward the desired outcomes. Secondly, our method utilizes Wasserstein distance with a
+time-step metric (Durugkar et al., 2021b) not only for a temporal distance-aware intrinsic reward but
+also for querying the frontier of the explored region during the curriculum learning. By deploying
+the above two elements, we propose a simple and intuitive curriculum learning objective formalized
+with a bipartite matching to generate a set of calibrated curriculum goals that interpolates between
+the initial state distribution and desired outcome state distribution.
+To sum up, our work makes the following key contributions.
+• We propose an outcome-directed curriculum RL method which only requires desired out-
+come examples and does not require an external reward.
+• To the best of the author’s knowledge, we are the first to propose the uncertainty & tempo-
+ral distance-aware curriculum goal generation method for geometry-agnostic progress by
+leveraging the conditional normalized maximum likelihood and Wasserstein distance.
+• Through several experiments in goal-reaching environments, we show that our method
+outperforms the prior curriculum RL methods, most notably when the environment has a
+geometric structure, and its curriculum proposal shows properly calibrated guidance toward
+the desired outcome states in a quantitative and qualitative way.
+2
+RELATED WORKS
+While a number of works have been proposed to improve the exploration problem in RL, it
+still remains a challenging open problem. For tackling this problem, prior works include state-
+visitation counts (Bellemare et al., 2016; Ostrovski et al., 2017), curiosity/similarity-driven explo-
+ration (Pathak et al., 2017; Warde-Farley et al., 2018), the prediction model’s uncertainty-based
+2
+
+220E
+ : Curriculum goals
+: Agent (@>@→→@)
+F : Desired outcome
+[Uncertainty]
+3
+Explored
+Desired
+Uncertain
+state
+state
+goal
+4
+[Temporal dist. from initial]
+Temporally
+Temporally
+Close
+FarAccepted as a conference paper at ICLR 2023
+exploration (Burda et al., 2018; Pathak et al., 2019), mutual information-based exploration (Ey-
+senbach et al., 2018; Sharma et al., 2019; Zhao et al., 2021; Laskin et al., 2022), maximizing the
+entropy of the visited state distribution (Yarats et al., 2021; Liu & Abbeel, 2021a;b). Unfortunately,
+these techniques are uninformed about the desired outcomes: the trained agent only knows how to
+visit frontier states as diverse as possible. On the contrary, we consider a problem where the de-
+sired outcome can be specified by the given desired outcome examples, allowing for more efficient
+outcome-directed exploration rather than naive frontier-directed exploration.
+Some prior methods that try to accomplish the desired outcome states often utilize the provided suc-
+cess examples (Fu et al., 2018; Singh et al., 2019; Eysenbach et al., 2021; Li et al., 2021). However,
+they do not provide a mechanism for distilling the knowledge obtained from the agent’s experiences
+into general-purpose policies that can be used to achieve new test goals. In this work, we utilize
+the Wasserstein distance not only for an arbitrary goal-conditioned agent but also for querying the
+frontier of the explored region during curriculum learning. Although the Wasserstein distance has
+been adopted by some previous research, they are often limited to imitation learning or skill discov-
+ery (Dadashi et al., 2020; Haldar et al., 2022; Xiao et al., 2019; Durugkar et al., 2021a; Fickinger
+et al., 2021). Another work (Durugkar et al., 2021b) tries to utilize the Wasserstein distance with
+the time-step metric for training the goal-reaching agent, but it requires a stationary goal distribution
+for stable distance estimation. Our work is different from these prior works in that the distribution
+during training is non-stationary for calibrated guidance to the desired outcome states.
+Suggesting a curriculum can also make exploration easier where the agent learns on a tailored se-
+quence of tasks, allowing the agent to autonomously practice the intermediate tasks in a training
+process. However, prior works often require a significant amount of samples to measure the curricu-
+lum’s level of difficulty (Florensa et al., 2018; Sukhbaatar et al., 2017), learning progress (Portelas
+et al., 2020), regret (Jiang et al., 2021). Curricula are often generated by modifying the goal dis-
+tribution into the frontier of the explored region via maximizing a certain surrogate objective, such
+as the entropy of the goal distribution (Pong et al., 2019), disagreement between the value function
+ensembles (Zhang et al., 2020), but these methods do not have convergence mechanism to the de-
+sired outcome distribution. While some algorithms formulate the generation of a curriculum as an
+explicit interpolation between the distribution of target tasks and a surrogate task distribution (Ren
+et al., 2019; Klink et al., 2022), these still depend on the Euclidean distance metric when measuring
+the distance between distributions, which cannot be generalized for an arbitrary geometric structure
+of the environment. In contrast, our method not only provides calibrated guidance to the desired out-
+come distribution in a sample efficient way but also shows geometry-agnostic curriculum progress
+by leveraging the bipartite matching problem with uncertainty & temporal distance-based objective.
+3
+PRELIMINARY
+We consider the Markov decision process (MDP) M = (S, A, G, T, ρ0, γ), where S denotes the
+state space, A the action space, G the goal space, T(s′|s, a) the transition dynamics, ρ0 the initial
+distribution, ρπ the state visitation distribution when the agent follows the policy π, and γ the dis-
+count factor. The MDP in our framework is provided without a reward function and considers an
+environment in which only the desired outcome sample {gk
++}K
+k=1 are given, assuming that gk
++ is
+obtained from the desired outcome distribution G+. Therefore, our work employs a trainable in-
+trinsic reward function r : S × G × A → R, which is detailed in Section 4.1. Also, we represent
+the set of curriculum goals as {gk
+c }N
+k=1 and assume that these are sampled from the curriculum goal
+distribution Gc obtained by our algorithm.
+3.1
+WASSERSTEIN DISTANCE OVER THE TIME-STEP METRIC
+The Wasserstein distance represents how much “work” is required to transport one distribution to
+another distribution following the optimal transport plan (Villani, 2009; Durugkar et al., 2021b).
+In this section, we describe the Wasserstein distance over the time-step metric and how it can be
+obtained with a potential function f. Consider a metric space (X, d), where X is a set and d is a
+metric on X, and two probability measures µ, ν on X. The Wasserstein-p distance for a given metric
+d is defined as follows,
+Wp(µ, ν) :=
+inf
+γ∈Π(µ,ν) E(X,Y )∼γ[d(X, Y )p]1/p if p=1
+=
+sup
+∥f∥L≤1
+[Ey∼ν[f(y)] − Ex∼µ[f(x)]]
+(1)
+3
+
+Accepted as a conference paper at ICLR 2023
+where a joint distribution γ denotes a transport plan, and Π(µ, ν) denotes the set of all possible
+joint distributions γ, and the second equality is held by the Kantorovich-Rubinstein duality with
+1-Lipschitz functions (f : X → R) (Villani, 2009; Arjovsky et al., 2017).
+If we could define the distance metric as dπ(s, sg), which is a time-step metric (quasimetric) based
+on the number of transition steps experienced before reaching the goal sg ∈ G for the first time
+when executing the goal-conditioned policy π(a|s, sg), we could design a temporal-distance aware
+RL by minimizing the Wasserstein distance W1(ρπ, G) that gives an estimate of the work needed to
+transport the state visitation distribution ρπ to the goal distribution G:
+W1(ρπ, G) =
+sup
+∥f∥L≤1
+[Esg∼G[f(sg)] − Es∼ρπ[f(s)]]
+(2)
+Then, the potential function f is approximately increasing along the optimal goal-reaching trajec-
+tory. That is, if ρπ(s) consists of the states optimally reaching toward the goal sg, f(s) increases
+along the trajectory and f(sg) has the maximum value (Durugkar et al., 2021b;a).
+Adopting these prior works, the 1-Lipschitz potential function f with respect to dπ(s, sg) could be
+ensured by enforcing that the difference in values of f on the expected transition from every state is
+bounded by 1 as follows,
+sup
+s∈S
+{Ea∈π(·|s,sg),s′∈T (·|s,a)[|f(s′) − f(s)|]} ≤ 1
+(3)
+More detailed derivations are included in Appendix B.
+3.2
+CONDITIONAL NORMALIZED MAXIMUM LIKELIHOOD (CNML)
+For curriculum learning, our work utilizes conditional normalized maximum likelihood (CNML)
+(Rissanen & Roos; Fogel & Feder, 2018) that can perform an uncertainty-aware classification based
+on previously observed data by minimizing worst-case regret. Let D = {(sp, ep)}n−1
+p=1 be a set
+of data containing pairs of states s1:n−1 and success labels e1:n−1 ∈ {0, 1}, where ‘1’ represents
+the occurrence of the desired event. Given a query point sn, CNML in our framework defines the
+distribution pCNML(en|sn) which predicts the probability that the state sn belongs to the desired
+outcome distribution G+ (e = 1).
+To explain how CNML predicts the label of sn, we suppose Θ is a set of models, where each
+model θ ∈ Θ can represent a conditional distribution of labels, pθ(e1:n|s1:n). CNML considers the
+possibilities that all the possible classes (0,1) will be labeled to the query point sn, and obtains the
+models ˆθ(e1:n|s1:n) ∈ Θ that represent the augmented datasets D ∪ (sn, en) well by solving the
+maximum likelihood estimation (MLE) problem (LHS of Eq. (4)). Then, CNML that minimizes the
+regret over those maximum likelihood estimators can be written as follows (Bibas et al., 2019),
+ˆθi = arg max
+θ∈Θ
+E(s,e)∈D∪(sn,en=i)[log pθ(e|s)],
+pCNML(en = i|sn) =
+pˆθi(e = i|sn)
+�1
+j=0 pˆθj(e = j|sn)
+(4)
+If the query point sn is close to one of the data points in the datasets, CNML will have difficulty in
+assigning a high likelihood to labels that are significantly different from those of nearby data points.
+However, if sn is far from the data points in the dataset, each MLE model ˆθi=0,1 will predict the label
+en as its own class, which leads to a large discrepancy in the predictions between the models and
+provides us with normalized likelihoods closer to uniform (RHS of Eq (4)). Thus, by minimizing the
+regret through labeling all possible classes to the new query data, CNML can provide a reasonable
+uncertainty estimate (Li et al., 2021; Zhou & Levine, 2021) on the queried sn and classify whether
+the queried sn is similar to the previously observed data either the forms of label 0, 1, or out-of-
+distribution data, which is predicted as 0.5.
+However, the classification technique via CNML described above is in most cases computationally
+intractable, as it requires solving separate MLE problems until convergence on every queried data.
+4
+
+Accepted as a conference paper at ICLR 2023
+Figure 2: Visualization of the uncertainty quantification along training progress (left) and trained
+f π
+φ (s) (right) in the Point-N-Maze environment. In the right figure, high reward means temporally
+close to the desired outcome states, and low reward means the opposite.
+Previous methods proposed some ideas to amortize the cost of computing CNML distribution (Zhou
+& Levine, 2021; Li et al., 2021). Following those prior methods, our work adopts MAML (Finn
+et al., 2017) to address the computational intractability of CNML by training one meta-learner net-
+work that can quickly adapt to each model ˆθi rather than training each model separately. As Finn
+et al. (2017) requires samples from G+ and replay buffer B for the inference, the probability should
+be represented as pCNML(e = i|s; G+, B), but we use pCNML(e = i|s) for notational simplicity in
+this work. More details about the meta-learning-based classification are included in Appendix B.
+4
+METHOD
+For a calibrated guidance of the curriculum goals to the desired outcome distribution, we propose to
+progress the curriculum towards the uncertain & temporally distant area before converging to G+,
+as it is not only the most intuitive way for exploration but also enables the agent to progress without
+any prior domain knowledge on the environment such as obstacles. In short, our work tries to obtain
+the distribution of curriculum goals Gc that are considered (a) temporally distant from ρ0 and, (b)
+uncertain and, (c) being progressed toward the desired outcome distribution G+.
+4.1
+TEMPORAL DISTANCE-AWARE RL WITH THE INTRINSIC REWARD
+This section details the intrinsic reward for the RL agent as well as the method of training the
+parameterized potential function f π
+φ , trained with the data collected by the policy π(a|s, sg). We
+consider a 1-Lipschitz potential function f π
+φ whose value increases as the state is far from the initial
+state distribution and getting close to the goals sg ∈ G proposed by curriculum learning. Then, we
+can train an agent that reaches the goals sg in as few steps as possible by minimizing the Wasserstein
+distance W1(ρπ, G).
+Considering that we can obtain the estimate of W1(ρπ, G) by Eq (2), the loss for training the param-
+eterized potential function f π
+φ can be represented as follows (Durugkar et al., 2021b;a):
+Lφ = Es,sg∼B[f π
+φ (s) − f π
+φ (sg)] + λ · Es,s′,sg∼B[max(|f π
+φ (s) − f π
+φ (s′)| − 1, 0))2]
+(5)
+The penalty term with coefficient λ in Eq (5) is from Eq (3) for ensuring the smoothness requirement
+since we consider the Wasserstein distance over the time-step metric. Then, assuming the parameter
+φ is trained by Eq (5) at every training iteration, we could obtain the supremum of Eq (2). Thus, the
+reward can be represented as r = f π
+φ (s) − f π
+φ (sg), which corresponds to −W1(ρπ, G).
+4.2
+CURRICULUM LEARNING
+As CNML can provide a near-uniform prior (prediction of 0.5) for out-of-distribution data given the
+datasets (Section 3.2), we could utilize it by treating the desired outcome states in G+ as (e = 1),
+and data points in the replay buffer B as (e = 0). Then, we could quantify the uncertainty of a state
+s based on CNML as
+ηucert(s, G+) = 1 −
+��pCNML(e = 0|s) − pCNML(e = 1|s)
+��.
+(6)
+which is proportional to the uncertainty of the queried data s. However, ηucert alone cannot provide
+curriculum guidance toward the desired outcome states because it only performs an uninformed
+5
+
+Goal Point
+Initial PointAccepted as a conference paper at ICLR 2023
+search over uncertainties rather than converging to the desired outcome states. Thus, we modify Eq
+(6) with an additional guidance term:
+˜ηucert(Gc, G+) = Es∼Gc[log(ηucert(s, G+) + c · ηguidance(s, G+))].
+(7)
+where ηguidance(s, G+) = (pCNML(e = 1|s) − 0.5) · 1(pCNML(e = 1|s) ≥ 0.5), and c is a
+hyperparameter that adjusts the preference on the desired outcome states. Since the CNML provides
+near-uniform prior for out-of-distribution data, ηucert provides large values in the uncertain areas.
+Also, the guidance term ηguidance(s, G+) reflects the preference for the states considered to be closer
+to the desired outcome state distribution.
+However, in practice, we found the uncertainty quantification itself sometimes has numerical errors,
+and it makes pCNML erroneously predict the states near the initial states or boundaries of the already
+explored regions as uncertain areas. Therefore, we assume that the curriculum should incorporate
+the notion of not only the uncertainty but also the temporally distant states from ρ0 for frontier and
+desired outcome-directed exploration. Thus, we formulate the final curriculum learning objective as
+follows:
+arg max
+Gc
+[˜ηucert(Gc, G+)] + L · W1(ρo, Gc),
+(8)
+where the temporal distance bias term with a coefficient L is represented by the Wasserstein distance
+from initial state distribution ρ0 to the curriculum goal distribution Gc. And, given a parameterized
+1-Lipschitz potential function f π
+φ over the time-step metric dπ(s, sg), we can obtain the estimate of
+W1(ρo, Gc) by RHS of Eq (1). Also, if we assume ˆGc to be a finite set of K particles that is sampled
+from already achieved states in the replay buffer B, the objective function we aim to maximize can
+be represented as follows:
+max
+ˆGc:| ˆGc|=K
+K
+�
+i=1
+[˜ηucert(si, G+) + L · [f π
+φ (si) − f π
+φ (si
+0)]],
+si ∈ ˆGc, si
+0 ∈ ρ0
+(9)
+It enables to propose the curriculum that reflects not only the uncertainty of the states and preference
+on the desired outcomes but also temporally distant states from ρ0, while not requiring prior domain
+knowledge about the environment such as an obstacle.
+4.3
+SAMPLING CURRICULUM GOAL VIA BIPARTITE MATCHING
+Since we assume that desired outcome examples from G+ are given rather than its distribution, we
+could approximate it by the sampled set ˆG+ (| ˆG+| = K). Then, to solve the curriculum learning
+problem of Eq (9), we should address the combinatorial setting that requires assigning ˆGc from the
+entire curriculum goal candidates in the replay buffer B to the ˆG+, which is addressed via bipartite
+matching in this work. With the hyperparameter c = 4, we can rearrange Eq (9) as a minimization
+problem with the costs of cross-entropy loss (CE) and temporal-distance bias (f π
+φ ) term: (Refer to
+the Appendix B for the detailed derivation.)
+min
+ˆGc:| ˆGc|=K
+�
+si∈ ˆ
+Gc,gi
++∈ ˆG+
+w(si, gi
++)
+(10)
+w(si, gi
++) = CE(pCNML(e = 1|si); y = pCNML(e = 1|gi
++)) − L · f π
+φ (si)
+(11)
+Intuitively, before discovering the desired outcome states in G+, the curriculum goal si is proposed
+in a region of the state space considered to be uncertain and temporally distant from ρ0 in order
+to recognize the frontier of the explored regions. And it is kept updated to converge to the desired
+outcome states for minimizing the discrepancy between the predicted labels of ˆ
+G+ and ˆGc.
+Then we can construct a bipartite graph G with the cost of the edges w. Let Va and Vb be the
+sets of nodes representing achieved states in replay buffer and ˆG+ respectively. We define a bipartite
+6
+
+Accepted as a conference paper at ICLR 2023
+(a) OUTPACE(ours)
+(b) CURROT
+(c) HGG
+Figure 3: Visualization of the proposed curriculum goals. First row: Ant Locomotion, Second row:
+Point-N-Maze.
+graph G({Va, Vb}, E) with the weight of the edges E(·, ·) = −w(·, ·) and separated partitions (Va
+and Vb). To solve the bipartite matching problem, we utilize the Minimum Cost Maximum Flow
+algorithm to find K edges with the minimum cost w connecting Va and Vb. (Ahuja et al., 1993;
+Ren et al., 2019). The overall training process is summarized in Algorithm 1 in Appendix B.
+5
+EXPERIMENTS
+We include 6 environments to validate our proposed method.
+Firstly, various maze environ-
+ments (Point U-Maze, N-Maze, Spiral-Maze) are used to validate the geometry-agnostic curricu-
+lum generation capability. Also, we experimented with the Ant-Locomotion and Sawyer-Peg Push,
+Pick&Place environments to evaluate our method in more complex dynamics or other domains rather
+than navigation tasks.
+We compare with other previous curriculum or goal generation methods, where each method has the
+following properties. HGG (Ren et al., 2019): Minimize the distance between the curriculum and
+desired outcome state distributions based on the Euclidean distance metric and value function bias.
+CURROT (Klink et al., 2022): Interpolate between the curriculum and desired outcome state distri-
+bution based on the agent’s current performance via optimal transport. GoalGAN (Florensa et al.,
+2018): Generate curriculum goals that are intermediate level of difficulty by training GAN (Good-
+fellow et al., 2014). PLR (Jiang et al., 2021): Sample increasingly difficult tasks by prioritizing the
+levels of tasks with high TD errors. ALP-GMM (Portelas et al., 2020): Fit a GMM with an absolute
+learning progress score approximated by the absolute reward difference. VDS (Zhang et al., 2020):
+Prioritize goals that maximize the epistemic uncertainty of the value function ensembles. SkewFit
+(Pong et al., 2019): Maximize the entropy of the goal distribution to be uniform on the feasible state
+space via skewing the distribution trained by VAE (Kingma & Welling, 2013).
+5.1
+EXPERIMENTAL RESULTS
+Firstly, to show how each module in our method is trained, we visualized the uncertainty quantifica-
+tion by CNML, and f π
+φ (s) values which are proportional to the required timesteps to reach s from
+ρ0. The uncertainty quantification results (Figure 2) show that the classifier pCNML(·|s) successfully
+discriminates the queried states as already explored region or desired outcome states, otherwise, un-
+certain states. Due to the geometry-agnostic property of the classifier pCNML(·|s), we could propose
+the curriculum in the arbitrary geometric structure of the environments, while most of the previous
+curriculum generation methods do not consider it.
+We also visualized the values of the trained potential function f π
+φ (s) to show how the intrinsic reward
+is shaped (Figure 2). As the potential function f π
+φ (s) is trained to have high values near the desired
+outcome states due to the Wasserstein distance with the time-step metric, the results show gradual
+7
+
+Accepted as a conference paper at ICLR 2023
+(a) U-Maze
+(b) N-Maze
+(c) Spiral-Maze
+(d) Ant Locomotion
+(e) Sawyer Push
+(f) Sawyer Pick&Place
+Figure 4: Average distance from the curriculum goals to the final goals (Lower is better). Our
+method’s increasing tendencies at initial steps in some environments are due to the geometric struc-
+ture of the environments themselves. Shading indicates a standard deviation across 5 seeds.
+(a) U-Maze
+(b) N-Maze
+(c) Spiral-Maze
+(d) Ant Locomotion
+(e) Sawyer Push
+(f) Sawyer Pick&Place
+Figure 5: Episode success rates of the evaluation results. The legends and seeds are the same as Fig
+4. Note that the curves of the baselines in some environments are not visible as they overlap at zero
+success rates.
+increases of f π
+φ (s) values along the trajectory toward the desired outcome states. That is, high values
+of f π
+φ (s) indicate the smaller required timesteps to reach the desired outcome state, and this property
+brings the advantage in identifying the frontier of the explored region.
+To validate whether the curriculum goals are properly interpolated from initial states to desired
+outcome states by combining both objectives for curriculum learning (Eq (8)), we evaluated the
+progress of the curriculum goals in a quantitative and qualitative way. For quantitative evalua-
+tion, we compare with other previous works described above with respect to the distance from the
+proposed curriculum goals to G+. As we can see in Figure 4, our method is the only one that
+consistently interpolates from initial states to G+ as training proceeds, while others have difficulty
+with complex dynamics or geometry of the environments. For qualitative evaluation, we visualized
+the curriculum goals proposed by our method and other baselines that show somewhat comparable
+results as training proceeds (Figure 3). The results show that our method consistently proposes the
+proper curriculum based on the required timesteps and uncertainties regardless of the geometry and
+dynamics of the various environments, while other baselines have difficulty as they utilize the Eu-
+clidean distance metric to interpolate the curriculum distribution to the G+. We also evaluated the
+desired outcome-directed RL performance. As we can see in Figure 5, our method is able to very
+quickly learn how to solve these uninformed exploration problems through calibrated curriculum
+goal proposition.
+8
+
+10
+8
+distance
+6
+4
+2
+0
+0
+200
+400
+600
+800
+1000
+steps (k)16
+14
+12
+distance
+10
+8
+6
+4
+2
+0
+0
+200
+400
+600
+800
+1000
+steps (k)20.0
+17.5
+15.0
+distance
+12.5
+10.0
+7.5
+5.0
+2.5
+0.0
+0
+200
+400
+600
+800
+1000
+steps (k)10
+8
+distance
+6
+4
+2
+0
+500
+1000
+1500
+2000
+2500
+3000
+steps (k)0.8
+0.7
+0.6
+distance
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+500
+1000
+1500
+2000
+2500
+3000
+steps (k)0.7
+0.6
+distance
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+500
+1000
+1500
+2000
+2500
+3000
+steps (k)OUTPACE(ours)
+SkewFit
+CURROT
+PLR
+HGG
+ALP-GMM
+GoalGAN
+VDS1.0
+0.8
+ rate
+0.6
+success_
+0.4
+0.2
+0.0
+0
+200
+400
+600
+800
+1000
+steps (k)1.0
+0.8
+ rate
+0.6
+success.
+0.4
+0.2
+0.0
+0
+200
+400
+600
+800
+1000
+steps (k)1.0
+0.8
+_rate
+0.6
+success_
+0.4
+0.2
+0.0
+0
+200
+400
+600
+800
+1000
+steps (k)1.0
+0.8
+_rate
+0.6
+success
+0.4
+0.2
+0.0
+0
+500
+1000
+1500
+2000
+2500
+3000
+steps (k)1.0
+0.8
+ rate
+0.6
+success_
+0.4
+0.2
+0.0
+0
+500
+1000
+1500
+2000
+2500
+3000
+steps (k)1.0
+0.8
+ rate
+0.6
+success_
+0.4
+0.2
+0.0
+0
+500
+1000
+1500
+2000
+2500
+3000
+steps (k)Accepted as a conference paper at ICLR 2023
+(a) Cost type
+(b) Reward & Goal proposition type
+(c) Effect of c in ηguidance
+Figure 6: Ablation study in terms of the distance from the proposed curriculum goals to the desired
+final goal states (Lower is better). First row: Ant Locomotion. Second row: Sawyer Pick & Place.
+Shading indicates a standard deviation across 5 seeds.
+5.2
+ABLATION STUDY
+Types of curriculum learning cost.
+We first evaluate the importance of each curriculum learning
+objective in Eq (8). Specifically, we experimented only with uncertainty-related objective (only-
+cnml) and timestep-related objective (only-f) when curriculum learning progresses. As we can see
+in Figure 6a, both objectives play complementary roles, which support the requirement of both
+objectives. Without one of them, the agent has difficulty in progressing the curriculum goals toward
+the desired outcome states due to the local optimum of f π
+φ or the numerical error of pCNML, and
+more qualitative/quantitative results and analysis about this and other ablation studies are included
+in Appendix C.
+Reward type & Goal proposition method.
+Secondly, we replace the intrinsic reward with the
+sparse reward, which is typically used in goal-conditioned RL problems, to validate the effects of
+the timestep-proportionally shaped reward. Also, for comparing the curriculum proposition method,
+we replace the Bipartite Matching formulation with a GAN-based generative model, which is similar
+to Florensa et al. (2018), but we label the highly uncertain states as positive labels instead of success
+rates. As we can see in Figure 6b, the timestep-proportionally shaped reward shows consistently
+better results due to the more informed reward signal compared to the sparse one, and the generative
+model has difficulty in sampling the proper curriculum goals because GAN shows training instability
+with the drastic change of the positive labels, while our method is relatively insensitive because
+curriculum candidates are obtained from the experienced states from the buffer B rather than the
+generative model.
+Effect of c in ηguidance.
+Lastly, we experiment with different values of hyperparameter c to validate
+the effect of ηguidance on curriculum learning. When c is smaller than the default value 4, it is still
+possible to explore most of the feasible state space except the area near the desired outcome states
+due to the uncertainty & temporal distance-aware curriculum (Figure 6c). But, we could verify that
+ηguidance’s effect becomes smaller as c decreases and ηguidance helps to guide the curriculum goals
+to the desired outcome states precisely. This is consistent with our analysis that the uncertainty &
+temporal distance themselves can provide curriculum goals in the frontier of the explored region
+while ηguidance can further accelerate the guidance to the desired outcome states.
+6
+CONCLUSIONS
+In this work, we consider an outcome-directed curriculum RL where the agent should progress
+toward the desired outcome automatically without the reward function and prior knowledge about
+the environment. We propose OUTPACE, which performs uncertainty, temporal distance-aware
+curriculum RL with intrinsic reward, based on the classifier by CNML, and Wasserstein distance
+with time-step metric. We show that our method can outperform the previous methods regarding
+9
+
+12
+OUTPACE(ours)
+Ablation_only_f
+10
+Ablation only cnml
+distance
+8
+6
+4
+2
+0
+500
+1000
+1500
+2000
+2500
+3000
+steps (k)12
+OUTPACE(ours)
+Ablation_SparseReward
+10
+Ablation GAN
+distance
+8
+6
+4
+2
+0
+500
+1000
+1500
+2000
+2500
+3000
+steps (k)12
+OUTPACE(ours)
+c=1
+10
+C=2
+C=3
+distance
+8
+6
+4
+2
+0
+500
+1000
+1500
+2000
+2500
+3000
+steps (k)0.9
+OUTPACE(ours)
+0.8
+Ablation_only_f
+Ablation only cnml
+0.7
+distance
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+500
+1000
+1500
+2000
+2500
+3000
+steps (k)0.8
+OUTPACE(ours)
+Ablation SparseReward
+0.7
+Ablation GAN
+0.6
+distance
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+500
+1000
+1500
+2000
+2500
+3000
+steps (k)OUTPACE(ours)
+0.7
+c=1
+C=2
+0.6
+C=3
+distance
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+500
+1000
+1500
+2000
+2500
+3000
+steps (k)Accepted as a conference paper at ICLR 2023
+sample efficiency and curriculum progress quantitatively and qualitatively. Even though our method
+shows promising results, there are some issues with computational complexity due to the innate
+properties of the meta-learning inference procedure itself. Thus, it would be interesting future work
+to find a way to reduce the inference time for less training wall-clock time.
+REFERENCES
+R K Ahuja, T L Magnanti, and J B Orlin. Network Flows: Theory, Algorithms, and Applications.
+Prentice Hall, Englewood Cliffs, NJ, 1st edition, 1993.
+Marcin Andrychowicz, Filip Wolski, Alex Ray, Jonas Schneider, Rachel Fong, Peter Welinder, Bob
+McGrew, Josh Tobin, Pieter Abbeel, and Wojciech Zaremba. Hindsight experience replay. arXiv
+preprint arXiv:1707.01495, 2017.
+Martin Arjovsky, Soumith Chintala, and L´eon Bottou. Wasserstein generative adversarial networks.
+In International conference on machine learning, pp. 214–223. PMLR, 2017.
+Marc Bellemare, Sriram Srinivasan, Georg Ostrovski, Tom Schaul, David Saxton, and Remi Munos.
+Unifying count-based exploration and intrinsic motivation. Advances in neural information pro-
+cessing systems, 29, 2016.
+Koby Bibas, Yaniv Fogel, and Meir Feder. Deep pnml: Predictive normalized maximum likelihood
+for deep neural networks. arXiv preprint arXiv:1904.12286, 2019.
+Yuri Burda, Harrison Edwards, Amos Storkey, and Oleg Klimov. Exploration by random network
+distillation. arXiv preprint arXiv:1810.12894, 2018.
+Robert Dadashi, L´eonard Hussenot, Matthieu Geist, and Olivier Pietquin. Primal wasserstein imita-
+tion learning. arXiv preprint arXiv:2006.04678, 2020.
+Ishan Durugkar, Steven Hansen, Stephen Spencer, and Volodymyr Mnih. Wasserstein distance max-
+imizing intrinsic control. arXiv preprint arXiv:2110.15331, 2021a.
+Ishan Durugkar, Mauricio Tec, Scott Niekum, and Peter Stone. Adversarial intrinsic motivation
+for reinforcement learning. Advances in Neural Information Processing Systems, 34:8622–8636,
+2021b.
+Ben Eysenbach, Sergey Levine, and Russ R Salakhutdinov.
+Replacing rewards with examples:
+Example-based policy search via recursive classification. Advances in Neural Information Pro-
+cessing Systems, 34:11541–11552, 2021.
+Benjamin Eysenbach, Abhishek Gupta, Julian Ibarz, and Sergey Levine. Diversity is all you need:
+Learning skills without a reward function. arXiv preprint arXiv:1802.06070, 2018.
+Arnaud Fickinger, Samuel Cohen, Stuart Russell, and Brandon Amos.
+Cross-domain imitation
+learning via optimal transport. arXiv preprint arXiv:2110.03684, 2021.
+Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model-agnostic meta-learning for fast adaptation
+of deep networks. In International Conference on Machine Learning, pp. 1126–1135. PMLR,
+2017.
+Carlos Florensa, David Held, Xinyang Geng, and Pieter Abbeel. Automatic goal generation for
+reinforcement learning agents. In International conference on machine learning, pp. 1515–1528.
+PMLR, 2018.
+Yaniv Fogel and Meir Feder. Universal supervised learning for individual data. arXiv preprint
+arXiv:1812.09520, 2018.
+Justin Fu, Avi Singh, Dibya Ghosh, Larry Yang, and Sergey Levine. Variational inverse control with
+events: A general framework for data-driven reward definition. Advances in neural information
+processing systems, 31, 2018.
+Dibya Ghosh, Abhishek Gupta, and Sergey Levine. Learning actionable representations with goal-
+conditioned policies. arXiv preprint arXiv:1811.07819, 2018.
+10
+
+Accepted as a conference paper at ICLR 2023
+Ian J Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil
+Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial networks. arXiv preprint
+arXiv:1406.2661, 2014.
+Tuomas Haarnoja, Aurick Zhou, Pieter Abbeel, and Sergey Levine. Soft actor-critic: Off-policy
+maximum entropy deep reinforcement learning with a stochastic actor. In International confer-
+ence on machine learning, pp. 1861–1870. PMLR, 2018.
+Siddhant Haldar, Vaibhav Mathur, Denis Yarats, and Lerrel Pinto. Watch and match: Supercharging
+imitation with regularized optimal transport. arXiv preprint arXiv:2206.15469, 2022.
+Kristian Hartikainen, Xinyang Geng, Tuomas Haarnoja, and Sergey Levine. Dynamical distance
+learning for semi-supervised and unsupervised skill discovery. arXiv preprint arXiv:1907.08225,
+2019.
+Minqi Jiang, Edward Grefenstette, and Tim Rockt¨aschel. Prioritized level replay. In International
+Conference on Machine Learning, pp. 4940–4950. PMLR, 2021.
+Diederik P Kingma and Max Welling.
+Auto-encoding variational bayes.
+arXiv preprint
+arXiv:1312.6114, 2013.
+Pascal Klink, Haoyi Yang, Carlo D’Eramo, Jan Peters, and Joni Pajarinen. Curriculum reinforce-
+ment learning via constrained optimal transport. In International Conference on Machine Learn-
+ing, pp. 11341–11358. PMLR, 2022.
+Michael Laskin, Hao Liu, Xue Bin Peng, Denis Yarats, Aravind Rajeswaran, and Pieter Abbeel. Cic:
+Contrastive intrinsic control for unsupervised skill discovery. arXiv preprint arXiv:2202.00161,
+2022.
+Kevin Li, Abhishek Gupta, Ashwin Reddy, Vitchyr H Pong, Aurick Zhou, Justin Yu, and Sergey
+Levine.
+Mural: Meta-learning uncertainty-aware rewards for outcome-driven reinforcement
+learning. In International Conference on Machine Learning, pp. 6346–6356. PMLR, 2021.
+Hao Liu and Pieter Abbeel.
+Aps: Active pretraining with successor features.
+In International
+Conference on Machine Learning, pp. 6736–6747. PMLR, 2021a.
+Hao Liu and Pieter Abbeel. Behavior from the void: Unsupervised active pre-training. Advances in
+Neural Information Processing Systems, 34:18459–18473, 2021b.
+Georg Ostrovski, Marc G Bellemare, A¨aron Oord, and R´emi Munos. Count-based exploration with
+neural density models. In International conference on machine learning, pp. 2721–2730. PMLR,
+2017.
+Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor
+Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward
+Yang, Zachary DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner,
+Lu Fang, Junjie Bai, and Soumith Chintala. Pytorch: An imperative style, high-performance
+deep learning library.
+In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alch´e-Buc,
+E. Fox, and R. Garnett (eds.), Advances in Neural Information Processing Systems 32, pp.
+8024–8035. Curran Associates, Inc., 2019. URL http://papers.neurips.cc/paper/
+9015-pytorch-an-imperative-style-high-performance-deep-learning-library.
+pdf.
+Deepak Pathak, Pulkit Agrawal, Alexei A Efros, and Trevor Darrell. Curiosity-driven exploration
+by self-supervised prediction. In International conference on machine learning, pp. 2778–2787.
+PMLR, 2017.
+Deepak Pathak, Dhiraj Gandhi, and Abhinav Gupta. Self-supervised exploration via disagreement.
+In International conference on machine learning, pp. 5062–5071. PMLR, 2019.
+Vitchyr H Pong, Murtaza Dalal, Steven Lin, Ashvin Nair, Shikhar Bahl, and Sergey Levine. Skew-
+fit: State-covering self-supervised reinforcement learning.
+arXiv preprint arXiv:1903.03698,
+2019.
+11
+
+Accepted as a conference paper at ICLR 2023
+R´emy Portelas, C´edric Colas, Katja Hofmann, and Pierre-Yves Oudeyer. Teacher algorithms for
+curriculum learning of deep rl in continuously parameterized environments. In Conference on
+Robot Learning, pp. 835–853. PMLR, 2020.
+Zhizhou Ren, Kefan Dong, Yuan Zhou, Qiang Liu, and Jian Peng. Exploration via hindsight goal
+generation. Advances in Neural Information Processing Systems, 32, 2019.
+Jorma Rissanen and Teemu Roos. Conditional nml universal models. In 2007 Information Theory
+and Applications Workshop, pp. 337–341. IEEE.
+John Schulman, Sergey Levine, Pieter Abbeel, Michael Jordan, and Philipp Moritz. Trust region
+policy optimization. In International conference on machine learning, pp. 1889–1897. PMLR,
+2015.
+John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy
+optimization algorithms. arXiv preprint arXiv:1707.06347, 2017.
+Archit Sharma, Shixiang Gu, Sergey Levine, Vikash Kumar, and Karol Hausman. Dynamics-aware
+unsupervised discovery of skills. arXiv preprint arXiv:1907.01657, 2019.
+Archit Sharma, Kelvin Xu, Nikhil Sardana, Abhishek Gupta, Karol Hausman, Sergey Levine, and
+Chelsea Finn. Autonomous reinforcement learning: Benchmarking and formalism. arXiv preprint
+arXiv:2112.09605, 2021.
+Avi Singh, Larry Yang, Kristian Hartikainen, Chelsea Finn, and Sergey Levine. End-to-end robotic
+reinforcement learning without reward engineering. arXiv preprint arXiv:1904.07854, 2019.
+Sainbayar Sukhbaatar, Zeming Lin, Ilya Kostrikov, Gabriel Synnaeve, Arthur Szlam, and Rob
+Fergus. Intrinsic motivation and automatic curricula via asymmetric self-play. arXiv preprint
+arXiv:1703.05407, 2017.
+C´edric Villani. Optimal transport: old and new, volume 338. Springer, 2009.
+David Warde-Farley, Tom Van de Wiele, Tejas Kulkarni, Catalin Ionescu, Steven Hansen, and
+Volodymyr Mnih. Unsupervised control through non-parametric discriminative rewards. arXiv
+preprint arXiv:1811.11359, 2018.
+Huang Xiao, Michael Herman, Joerg Wagner, Sebastian Ziesche, Jalal Etesami, and Thai Hong
+Linh. Wasserstein adversarial imitation learning. arXiv preprint arXiv:1906.08113, 2019.
+Denis Yarats, Rob Fergus, Alessandro Lazaric, and Lerrel Pinto. Reinforcement learning with pro-
+totypical representations. In International Conference on Machine Learning, pp. 11920–11931.
+PMLR, 2021.
+Tianhe Yu, Deirdre Quillen, Zhanpeng He, Ryan Julian, Karol Hausman, Chelsea Finn, and Sergey
+Levine. Meta-world: A benchmark and evaluation for multi-task and meta reinforcement learning.
+In Conference on Robot Learning, pp. 1094–1100. PMLR, 2020.
+Yunzhi Zhang, Pieter Abbeel, and Lerrel Pinto. Automatic curriculum learning through value dis-
+agreement. Advances in Neural Information Processing Systems, 33:7648–7659, 2020.
+Rui Zhao, Yang Gao, Pieter Abbeel, Volker Tresp, and Wei Xu. Mutual information state intrinsic
+control. arXiv preprint arXiv:2103.08107, 2021.
+Aurick Zhou and Sergey Levine. Amortized conditional normalized maximum likelihood: Reliable
+out of distribution uncertainty estimation. In International Conference on Machine Learning, pp.
+12803–12812. PMLR, 2021.
+12
+
+Accepted as a conference paper at ICLR 2023
+A
+TRAINING & EXPERIMENTS DETAILS
+A.1
+TRAINING DETAILS
+Baselines.
+The baseline curriculum RL algorithms are trained as follows,
+• HGG (Ren et al., 2019) : We follow the default setting in the original implementation from
+https://github.com/Stilwell-Git/Hindsight-Goal-Generation.
+• CURROT (Klink et al., 2022): We follow the default setting in the original implementation
+from https://github.com/psclklnk/currot.
+• GoalGAN (Florensa et al., 2018), PLR (Jiang et al., 2021), VDS (Zhang et al., 2020),
+ALP-GMM (Portelas et al., 2020) : We follow the default setting in implementation from
+https://github.com/psclklnk/currot.
+• SkewFit (Pong et al., 2019): We follow the state-based version of SkewFit. The original
+implementation was modified and used since only the image-based version is provided in
+it. (https://github.com/rail-berkeley/rlkit)
+All the baselines are trained by SAC (Haarnoja et al., 2018) with the sparse reward except for the
+SkewFit as it uses a reward based on the conditional entropy. Even though some algorithms’ original
+implementation is based on on-policy algorithms such as TRPO or PPO (Schulman et al., 2015;
+2017), for comparing the sample efficiency, we replace the on-policy algorithm with the off-policy
+algorithm, SAC, following the referred implementation.
+Table 1: Conceptual comparison between our work and the previous curriculum RL algorithms.
+Uncert.
+Timestep
+Target
+Off-
+Curriculum
+Without
+Without non-
+-Aware
+-Aware
+curriculum dist.
+policy
+proposal
+ext. reward
+forgetting mechanism
+HGG
+
+
+G+
+✓
+B
+
+✓
+GoalGAN
+
+
+
+
+GAN
+
+
+CURROT
+
+
+U or G+
+✓
+U
+
+
+PLR
+
+
+
+
+B
+
+
+VDS
+✓
+
+
+✓
+B
+
+✓
+ALP-GMM
+
+
+
+✓
+GMM
+
+
+SkewFit
+✓
+
+
+✓
+VAE
+✓
+
+OUTPACE (ours)
+✓
+✓
+G+
+✓
+B
+✓
+✓
+We included a conceptual comparison between our work and previous other curriculum or goal
+generation methods in Table 1.
+• Uncert.-Aware: whether the curriculum goal proposal process is aware of the uncertainty
+of the candidate goals.
+• Timestep-Aware: whether the curriculum goal proposal process is aware of the temporal
+distance from the initial states or from the desired outcome states.
+• Target curriculum dist: whether there exists a mechanism for the curriculum goals to
+converge to the target distribution. When there is no target distribution (e.g. just exploring
+or expanding the curriculum goal distribution as diverse as possible.), we denoted it as .
+• Off-policy: whether the off-policy RL can be applied. Some baselines need to measure a
+kind of difficulty, which means they require repeated trials and on-policy RL algorithms
+with multi-processing such as TRPO, and PPO (Schulman et al., 2015; 2017).
+• Curriculum proposal: where the curriculum goals are proposed from.
+• Without ext. reward: whether the algorithm requires external environmental reward or
+not.
+• Without non-forgetting mechanism: whether the algorithm requires implicit or explicit
+non-forgetting mechanisms. Some baselines mix the previously practiced curriculum goals
+with a fixed or varying ratio or make the curriculum distribution into uniform over the state
+space to cover all possible test goal states.
+13
+
+Accepted as a conference paper at ICLR 2023
+Training details.
+To train the potential function fφ via Eq. (5), s and sg in the buffer should ideally
+contain all feasible states in the environment. However, until the policy is learned enough to explore
+the map, obtaining such ideal distribution is difficult. To mitigate this issue, following Durugkar
+et al. (2021b), we approximate such a distribution with a small replay buffer Bf containing recent
+trajectories and the relabelling technique (Andrychowicz et al., 2017). While this approximation
+does not provide fφ with the ideal state distribution covering all feasible states, we empirically
+found that this assumption works well since OUTPACE only queries fφ for the explored area when
+it generates curriculum goals.
+Table 2: Hyperparameters for OUTPACE
+Critic hidden dim
+512
+discount factor γ
+0.99
+Critic hidden depth
+3
+fφ update frequency
+1000
+Critic target τ
+0.01
+# of gradient steps for fφ update
+10
+Critic target update frequency
+2
+# of ensemble networks for fφ
+5
+Actor hidden dim
+512
+learning rate for fφ
+1e-4
+Actor hidden depth
+3
+RL optimizer
+adam
+Actor update frequency
+2
+Meta-learner network hidden size
+[2048,2048]
+RL batch size
+512
+Meta-learner train sample size
+512
+Init temperature αinit of SAC
+0.3
+Meta-learner test sample size
+2048
+Replay buffer B size
+3e6
+Meta-learner test batch size
+2048
+Table 3: Env-specific hyperparameters for OUTPACE
+Env name
+Bf size
+λ for fφ
+adam ϵ for SAC
+L
+Point-U-Maze-v0
+10000
+25
+1e-2
+0.02
+Point-N-Maze-v0
+10000
+25
+1e-2
+0.02
+Point-Spiral-Maze-v0
+20000
+50
+1e-8
+0.02
+Ant Locomotion-v0
+50000
+25
+1e-8
+0.02
+Sawyer-Peg-Push
+30000
+25
+1e-2
+2
+Sawyer-Peg-Pick&Place
+30000
+25
+1e-2
+0.02
+A.2
+ENVIRONMENT DETAILS
+• Point-U-Maze : The observation consists of the xy position, angle, velocity, and angular
+velocity of the ‘point’. The action space consists of the velocity and angular velocity of the
+‘point’. The initial state of the agent is [0, 0] and the desired outcome states are obtained
+by adding uniform noise to the default goal point [0, 8]. The size of the map is 12 × 12.
+• Point-N-Maze : It is the same as the Point-U-Maze environment except that the desired
+outcome states are obtained by adding uniform noise to the default goal point [8, 16], and
+the size of the map is 12 × 20.
+• Point-Spiral-Maze : It is the same as the Point-U-Maze environment except that the desired
+outcome states are obtained by adding uniform noise to the default goal point [8, −8], and
+the size of the map is 20 × 20.
+• Ant Locomotion : The observation consists of the xyz position, xyz velocity, joint angle,
+and joint angular velocity of the ‘ant’. The action space consists of the torque applied on
+the rotor of the ‘ant’. The initial state of the agent is [0, 0] and the desired outcome states
+are obtained by adding uniform noise to the default goal point [0, 8]. The size of the map is
+12 × 12.
+• Sawyer-Peg-Push : The observation consists of the xyz position of the end-effector, the
+object, and the gripper’s state. The action space consists of the xyz position of the end-
+effector and gripper open/close control. The initial state of the object is [0.4, 0.8, 0.02] and
+the desired outcome states are obtained by adding uniform noise to the default goal point
+[−0.3, 0.4, 0.02]. We referred to the metaworld (Yu et al., 2020) and EARL (Sharma et al.,
+2021) environments.
+14
+
+Accepted as a conference paper at ICLR 2023
+(a) U-Maze
+(b) N-Maze
+(c) Spiral-Maze
+(d) Ant Locomotion
+(e) Sawyer Manip.
+Figure 7: Environments used for evaluation: (a)-(c) the agent must navigate various kinds of maze
+environments. (d) the quadruped ant must navigate the maze to a particular location. (e) the robot
+has to push or pick&place a peg to the desired location.
+• Sawyer-Peg-Pick&Place : It is the same as the Sawyer-Peg-Push environment except that
+the desired outcome states are obtained by adding uniform noise to the default goal point
+[−0.3, 0.4, 0.2].
+B
+ALGORITHM & DERIVATIONS
+B.1
+ALGORITHM
+Algorithm 1 Overview of OUTPACE algorithm
+1: Input: desired outcome examples ˆG+, total training episodes N, Env, environment horizon H,
+actor network π, critic network Q, potential function network f π
+φ , replay buffer B, Bf
+2: for iteration=1,2,...,N do
+3:
+ˆGc ← sample K curriculum goals that minimize
+�
+si∈B[CE(pCNML(e = 1|si); y = pCNML(e = 1| ˆG+)) − L · f π
+φ (si)] (Section 4.3)
+4:
+for i=1,2,...,K do
+5:
+Env.reset()
+6:
+g ← ˆGc.pop
+7:
+for t=0,1,...,H-1 do
+8:
+if achieved g then
+9:
+g ← random goal (randomly sample a state with high uncertainty measured by pCNML
+in a ball Br(st).)
+10:
+end if
+11:
+at ← π(·|st, g)
+12:
+st+1 ← Env.step(at)
+13:
+end for
+14:
+B ← B ∪ {s0, a0, s1...}, Bf ← Bf ∪ {s0, a0, s1...}
+15:
+end for
+16:
+for i=0,1,...,M do
+17:
+Sample a minibatch b from B and label reward using f π
+φ (st) (Section 4.1)
+18:
+Train π and Q with b via SAC (Haarnoja et al., 2018).
+19:
+Meta-train pCNML with b via meta-NML (Algorithm 2.)
+20:
+Sample a minibatch bf from Bf
+21:
+Train f π
+φ with bf via Eq. (5)
+22:
+end for
+23: end for
+15
+
+★Accepted as a conference paper at ICLR 2023
+Figure 8: The overall diagram of OUTPACE
+Algorithm 2 Meta-NML (Li et al., 2021)
+1: Input: desired outcome examples ˆG+, RL buffer B
+2: Sample ˆG− from RL buffer B
+3: D ← ˆG− ∪ ˆG+ (Let the size of D is n)
+4: Create 2n meta-training tasks by relabelling each data points in D as 0 and 1.
+(D ∪ (xi, y′) where xi ∼ D and y′ ∈ {0, 1})
+5: Meta-learn with 2n meta-training tasks (Finn et al., 2017)
+(Let θ be the parameter of pCNML and L be standard classification loss.
+minθ Exi∼D,y′∼{0,1}[L(D ∪ (xi, y′), θ′(i)
+j=y′)],
+s.t.θ′(i)
+j
+= θ − α∇θL(D ∪ (xi, yi), θ)
+B.2
+DERIVATIONS
+This section contains the definition of time-step metric dπ, derivation of Lipschitz smoothness of
+f, and derivation of Eq. (11). Regarding dπ and Lipschitz smoothness of f, we refer the reader to
+Durugkar et al. (2021b) for more detailed explanation.
+B.2.1
+TIME-STEP METRIC dπ, LIPSCHITZ SMOOTHNESS OF f
+Definition B.1. Given a state space S, action space A, transition dynamics T : S × A → S, and
+agent policy π, the time-step metric dπ is a quasi-metric where the distance from s ∈ S to sg ∈ S is
+the expected number of transitions required with the policy π.
+The time-step metric can be expressed by the expectation of the number of transitions taken under
+the policy π, where T π(·|s, sg) is the probability distribution of the timestep required to go from s
+to sg. dπ can also be written recursively as
+dπ(s, sg) : = Eπ,τ∼T π(·|s,sg)[τ]
+=
+�0
+ifs = sg
+1 + Ea∼π(·|s,sg)Es′∼T (·|s,a)[dπ(s′, sg)]
+otherwise.
+(12)
+16
+
+Sampling Curriculum Goals
+Training Goal Sampling Modules
+Bipartite Matching
+Uncertainty-Aware Module
+Replay
+Buffer
+Augment data
+Calculate
+Calculate
+D U (si,ei E (0,1))
+Uncertainty
+Temp. Dist
+PcNML
+Sample K curriculum
+MAML
+goals that minimize Eq.11
+(Fin et al., 2017)
+Temporal Distance-Aware Module
+Practice Curriculum Goals
+Curriculum Goals
+Train fa
+Off-Policy RL
+by minimizing Lp (Eq.5)
+ Single step adaptation
+Replay Data
+Neural Network ModelAccepted as a conference paper at ICLR 2023
+Lipschitz smoothness of f
+If the difference in values of f on the expected transition from every
+state is bounded by 1, and the policy π can reach the goal sg within a finite number of transitions,
+then f is the 1-Lipschitz function.
+Proof. We can write |f(sg) − f(s0)| via telescopic sum,
+|f(sg) − f(s0)| = Eπ,τ∼T π(·|s0,sg)
+������
+τ−1
+�
+t=0
+(f(st+1) − f(st))
+�����
+�
+≤ Eπ,τ∼T π(·|s0,sg)
+� τ−1
+�
+t=0
+|f(st+1) − f(st)|
+�
+.
+(13)
+Since E[|f(s′) − f(s)|] ≤ 1 by Eq (3), we can write
+Eπ,τ∼T π(·|s0,sg)
+� τ−1
+�
+t=0
+|f(st+1) − f(st)|
+�
+≤ Eπ,τ∼T π(·|s0,sg)
+� τ−1
+�
+t=0
+1
+�
+= Eπ,τ∼T π(·|s0,sg)[τ]
+= dπ(s0, sg)
+(14)
+Thus, f is the 1-Lipschitz function with respect to the time-step metric dπ.
+B.2.2
+DERIVATION OF EQUATION (11)
+By substituting Eq. (7) into Eq (9), and omitting f π
+φ (si
+0) which is not related to ˆGc, we obtain the
+following terms,
+max
+ˆGc:| ˆGc|=K
+K
+�
+i=1
+�
+log
+�
+1 −
+��pCNML(e = 0|si) − pCNML(e = 1|si)
+��
++ c(pCNML(e = 1|si) − 0.5) · 1(pCNML(e = 1|si) > 0.5)
+�
++ L · f π
+φ (si)
+�
+,
+si ∈ ˆGc.
+(15)
+Also, if we use the default value of the hyperparameters c = 4, we can express the above terms as
+follows,
+max
+ˆGc:| ˆGc|=K
+K
+�
+i=1
+�
+log
+�
+1 −
+��1 − 2pCNML(e = 1|si)
+��+
+4(pCNML(e = 1|si) − 0.5) · 1(pCNML(e = 1|si) > 0.5)
+�
++ L · f π
+φ (si)
+�
+,
+si ∈ ˆGc
+(16)
+which can be simplified as
+min
+ˆGc:| ˆGc|=K
+K
+�
+i=1
+[− log(pCNML(e = 1|si)) − L · f π
+φ (si)],
+si ∈ ˆGc
+(17)
+If we could assume that pCNML is well trained to classify the desired outcome examples gi
++ ∈
+ˆG+ from the states s in the already explored region, pCNML(e = 1|gi
++) is approximately equal
+to 1 (pCNML(e = 1|gi
++) ≈ 1). Then, the terms inside the above minimization objective can be
+approximately represented as
+− pCNML(e = 1|gi
++) log(pCNML(e = 1|si)) − L · f π
+φ (si).
+(18)
+17
+
+Accepted as a conference paper at ICLR 2023
+Then, in practical implementation, we can implement the above terms by cross-entropy loss as
+w(si, gi
++) := CE(pCNML(e = 1|si); y = pCNML(e = 1|gi
++)) − L · f π
+φ (si),
+(19)
+which is Eq (11). Even though it is not exactly equivalent to the mathematical definition of the
+cross-entropy, we can just implement it with cross-entropy loss developed in standard deep learning
+framework such as PyTorch (Paszke et al., 2019) because choosing si to maximize log(pCNML(e =
+1|si)) and choosing si close to gi
++ in order to be classified as desired outcome examples by the
+pCNML (predicted labels of si to be close to 1) have the same intuitive meaning.
+B.3
+A DETAILED DESCRIPTION OF META-NML
+Conditional normalized maximum likelihood (CNML) can perform a conservative k-way classi-
+fication based on previously seen data (Li et al., 2021; Zhou & Levine, 2021). Let Dtrain =
+{(xi, yi)}n−1
+i=1 be a set of data containing pairs of inputs x1:n−1 and labels y1:n−1 ∈ {1, · · · , k},
+where k is the number of possible labels, and Θ is a set of models. Given a new input xn, CNML
+defines the distribution pCNML(yn = i|xn)i∈{1,··· ,k} by minimizing the regret R of the worst-case
+label yn as
+pCNML = arg min
+q
+max
+yn R(q, x1:n, y1:n, Θ),
+(20)
+where we define a regret R for label yn of a distribution q and maximum likelihood estimator θ′ as
+R(q, x1:n, y1:n, Θ) := logθ′(y1:n|x1:n)(y1:n|x1:n) − log q(y1:n).
+(21)
+By solving Eq. (20), CNML predicts the distribution of the new label yn as Eq. (22) (Bibas et al.,
+2019).
+pCNML(yn = m, xn) =
+pθ′(n)
+m (y = m|xn)
+�k
+j=1 pθ′(n)
+j
+(y = j|xn)
+,
+(22)
+where a model θ′(n)
+m
+is a model that represents the augmented dataset D = Dtrain ∪ (xn, yn = m)
+well. Thus, the number of total MLE models required is n × k since we should address each data
+by augmenting with the label 1, ..., k, respectively (θ′(α=1:n)
+β=1:k
+). Since our algorithm utilizes 2-way
+classification (k = 2), we define 2n tasks τ −
+i=1:n and τ +
+i=1:n, which are constructed by augmenting
+negative (xi, y = 0) and positive (xi, y = 1) labels respectively for each data point (xi).
+To amortize the training cost of the tasks (tasks τ −,+
+i=1:n) we can apply the meta-learning algorithm
+(Finn et al., 2017; Li et al., 2021) to this setting and train a model θ which can quickly adapt to the
+optimal solution after a single step of gradient update with standard classification loss L as
+min
+θ
+Exi∼D,y′∼{0,1}[L(D ∪ (xi, y′), θ′(i)
+j=y′)],
+(23)
+s.t.θ′(i)
+j
+= θ − α∇θL(D ∪ (xi, yi), θ),
+(24)
+where Eq. (23) and Eq (24) represent the objective of meta-learning and quick adaptation respec-
+tively. Training CNML via meta-learning and leveraging CNML are shown in lines 3 and 19 of the
+algorithm overview (Algorithm 1). Also, we provide the pseudo-code of meta-nml in Algorithm 2.
+C
+MORE EXPERIMENTAL RESULTS
+C.1
+FULL RESULTS OF THE MAIN SCRIPT
+We included the full results of the main script in this section. The uncertainty quantification is
+visualized in Figure 9, the visualization of the trained f π
+φ (s) is in Figure 10, the visualization of
+18
+
+Accepted as a conference paper at ICLR 2023
+the proposed curriculum goals is in Figure 11. We do not include the visualization results of the
+Point-U-Maze, and Sawyer-Peg-Pick&Place results as these environments share the same map with
+the Ant Locomotion, and Sawyer-Peg-Push environments, respectively.
+Figure 9: Visualization of the uncertainty quantification along training progress. First row: U-Maze
+(Point, Ant Locomotion), Second row: N-Maze, Third row: Spiral-Maze, Fourth row: Sawyer Peg
+Push, Pick & Place environments.
+(a) Ant Locomotion
+(b) N-Maze
+(c) Spiral-Maze
+(d) Sawyer Push
+Figure 10: Visualization of the trained f π
+φ (s). High reward means temporally close to the desired
+outcome states (required timesteps are small to reach the goal), and low reward means the opposite
+(required timesteps are large to reach the goal).
+19
+
+Goal Point
+Initial PointInitial Point
+Goal PointInitial Point
+Goal PointGoal Point
+Initial PointAccepted as a conference paper at ICLR 2023
+(a) OUTPACE(ours)
+(b) CURROT
+(c) HGG
+Figure 11: Visualization of the proposed curriculum goals. First row: Ant Locomotion (same
+map size with U-Maze), Second row: Point-N-Maze, Third row: Point-Spiral-Maze, Fourth row:
+Sawyer Peg Push.
+20
+
+Accepted as a conference paper at ICLR 2023
+C.2
+EVALUATION WITH THE GOALS SAMPLED FROM THE UNIFORM DISTRIBUTION
+In some curriculum RL algorithms, they incorporate a mechanism for remembering previously prac-
+ticed curricula either implicitly or explicitly. For example, GoalGAN (Florensa et al., 2018) mixes
+the previously generated goals with currently generated goals in a specified ratio (e.g. 20 % of pre-
+viously used goals), and SkewFit (Pong et al., 2019) set the objective as targeting the uniform goal
+distribution (by maximizing the entropy of the goals H(g)), and CURROT (Klink et al., 2022) also
+utilizes uniform target curriculum distribution in practice, and so do some of the other baselines.
+Due to these algorithmic designs, they require many iterations for explicitly practicing previously
+used curriculum goals, or show slow progress of curriculum to match the uniform target distribution.
+In contrast, our method is based on an intrinsic reward, which is shaped according to the required
+timesteps proportional values f π
+φ (s) as described in our main script. Thus, our method does not
+need to explicitly consider the non-forgetting mechanism when we design the algorithm because the
+reward is already shaped with respect to the timesteps for reaching the arbitrary goal points along
+the trajectory that reaches the desired outcome state. Because of this property, our method does not
+consider uniform target distribution or explicitly mixing the previously practiced curriculum goals,
+and it enables our method to be much faster and show sample-efficient curriculum progress.
+We experimented with the goals sampled from the uniform distribution on the feasible state space for
+validating the previous hypothesis, and the experimental results are shown in Figure 12. Even though
+our method does not explicitly consider the previously practiced curriculum goals, it shows success
+in reaching arbitrary goal points sampled from the uniform distribution. Performance degrades are
+observed in sawyer manipulation environments and these are because we set the uniform distribution
+as areas within the tables in the environment. But the curriculum goals proposed by our method are
+converged before the agent explores the entire state space on the table, thus the agent does not have
+the opportunity to practice the goals from the entire state space.
+(a) U-Maze
+(b) N-Maze
+(c) Spiral-Maze
+(d) Ant Locomotion
+(e) Sawyer Push
+(f) Sawyer Pick&Place
+Figure 12: Episode success rates of the evaluation results with the goals uniformly sampled from
+the feasible state space.
+C.3
+ABLATION STUDY
+We conducted ablation studies described in our main script in all environments. Figure 13 shows
+the average distance from the proposed curriculum goals to the desired final goal states along the
+training steps, and Figure 14 shows the episode success rates along the training steps. As we can
+see in these figures, even though there are not many differences in some environments with simple
+21
+
+OUTPACE(ours)
+SkewFit
+CURROT
+PLR
+HGG
+ALP-GMM
+GoalGAN
+VDS1.0
+0.8
+ rate
+0.6
+success_
+0.4
+0.2
+0.0
+0
+200
+400
+600
+800
+1000
+steps (k)1.0
+0.8
+_rate
+0.6
+success_
+0.4
+0.2
+0.0
+0
+200
+400
+600
+800
+1000
+steps (k)1.0
+0.8
+rate
+0.6
+success_
+0.4
+0.2
+0.0
+0
+200
+400
+600
+800
+1000
+steps (k)1.0
+0.8
+ rate
+0.6
+success_
+0.4
+0.2
+0.0
+0
+500
+1000
+1500
+2000
+2500
+3000
+steps (k)1.0
+0.8
+rate
+0.6
+success_
+0.4
+0.2
+0.0
+0
+500
+1000
+1500
+2000
+2500
+3000
+steps (k)1.0
+0.8
+rate
+0.6
+success
+0.4
+0.2
+0.0
+0
+500
+1000
+1500
+2000
+2500
+3000
+steps (k)Accepted as a conference paper at ICLR 2023
+dynamics, we could obtain consistent analysis with the results in the main script in most of the
+environments.
+We also visualized the curriculum goals obtained by each ablation study as training proceeds in Fig-
+ure 15. As we can see in these figures, the curriculum proposal only based on the uncertainty (only-
+cnml) shows progress toward uncertain areas, but it shows unstable curriculum progress, which is
+depicted by separated curriculum goals despite the same colors or out of order with respect to human
+intuitive optimal curriculum progress. This is because the meta-learning-based inference procedure
+of pCNML has some numerical errors that could lead to the wrong prediction of the states near the
+boundaries of the already explored regions as uncertain areas (For example, in Figure 9, there exist
+some states predicted as uncertain area despite already explored region).
+Also, even though the curriculum proposal based on the temporal distance only obtained by f π
+φ
+(only-f) shows some progress that deviates from the initial states, it still has difficulty in most of
+the environments because f π
+φ is trained to reflect the observed transition data rather than entire state
+space. That is, the temporal distance estimation is reliable within the explored region. Once the
+trained f π
+φ has local optimum before discovering temporally more distant states (Figure 16), the
+proposed curriculum goals can be stuck in some areas that are wrongly predicted to be the most far
+from the initial states in terms of the temporal distance, and it leads to the ineffective exploration of
+the agent and recurrent failures.
+22
+
+Accepted as a conference paper at ICLR 2023
+(a) Cost type
+(b) Reward & Goal proposition type
+(c) Effect of c in ηguidance
+Figure 13: Ablation study in terms of the distance from the proposed curriculum goals to the desired
+final goal states. First row: Point-U-Maze. Second row: Point-N-Maze. Third row: Point-Spiral-
+Maze. Fourth row: Ant Locomotion. Fifth row: Sawyer Push. Sixth row: Sawyer Pick&Place.
+Shading indicates a standard deviation across 5 seeds.
+23
+
+12
+OUTPACE(ours)
+Ablation_only_f
+10
+Ablation only cnml
+distance
+8
+6
+4
+2
+0
+500
+1000
+1500
+2000
+2500
+3000
+steps (k)12
+OUTPACE(ours)
+Ablation_SparseReward
+10
+Ablation GAN
+distance
+8
+6
+4
+2
+0
+500
+1000
+1500
+2000
+2500
+3000
+steps (k)12
+OUTPACE(ours)
+c=1
+10
+C=2
+C=3
+distance
+8
+6
+4
+2
+0
+500
+1000
+1500
+2000
+2500
+3000
+steps (k)0.9
+OUTPACE(ours)
+0.8
+Ablation_only_f
+Ablation only cnml
+0.7
+distance
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+500
+1000
+1500
+2000
+2500
+3000
+steps (k)0.8
+OUTPACE(ours)
+Ablation SparseReward
+0.7
+Ablation GAN
+0.6
+distance
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+500
+1000
+1500
+2000
+2500
+3000
+steps (k)OUTPACE(ours)
+0.7
+c=1
+C=2
+0.6
+C=3
+distance
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+500
+1000
+1500
+2000
+2500
+3000
+steps (k)12
+OUTPACE(ours)
+Ablation only f
+10
+Ablation only cnml
+8
+distance
+6
+4
+2
+0
+200
+400
+600
+800
+1000
+steps (k)12
+OUTPACE(ours)
+Ablation_SparseReward
+10
+Ablation GAN
+distance
+8
+6
+4
+2
+0
+0
+200
+400
+600
+800
+1000
+steps (k)OUTPACE(ours)
+c=1
+8
+C=2
+C=3
+distance
+6
+4
+2
+0
+200
+400
+600
+800
+1000
+steps (k)20.0
+OUTPACE(ours)
+17.5
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+(a) Cost type
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+(c) Effect of c in ηguidance
+Figure 14: Ablation study in terms of the episode success rates. First row: Point-U-Maze. Second
+row: Point-N-Maze. Third row: Point-Spiral-Maze. Fourth row: Ant Locomotion. Fifth row:
+Sawyer Push. Sixth row: Sawyer Pick&Place. Shading indicates a standard deviation across 5
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+Figure 15: Ablation study in terms of curriculum goals visualization. First row: Ant Locomotion,
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+(a) Ant Locomotion
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+Figure 16: Visualization of the not properly trained f π
+φ (s) when ablation study only-f is performed.
+25
+
+Initial PointInitial PointInitial PointInitial Point
\ No newline at end of file
diff --git a/s9FKT4oBgHgl3EQfKS06/content/tmp_files/load_file.txt b/s9FKT4oBgHgl3EQfKS06/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf,len=1063
+page_content='Accepted as a conference paper at ICLR 2023  OUTCOME-DIRECTED  REINFORCEMENT  LEARNING  BY UNCERTAINTY & TEMPORAL DISTANCE-AWARE  CURRICULUM GOAL GENERATION  Daesol Cho∗, Seungjae Lee∗, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Jin Kim  Seoul National University, AIIS, ASRI  {dscho1234, ysz0301, hjinkim}@snu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='kr  ABSTRACT  Current reinforcement learning (RL) often suffers when solving a challenging  exploration problem where the desired outcomes or high rewards are rarely ob-  served.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Even though curriculum RL, a framework that solves complex tasks by  proposing a sequence of surrogate tasks, shows reasonable results, most of the  previous works still have difficulty in proposing curriculum due to the absence  of a mechanism for obtaining calibrated guidance to the desired outcome state  without any prior domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' To alleviate it, we propose an uncertainty  & temporal distance-aware curriculum goal generation method for the outcome-  directed RL via solving a bipartite matching problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='1 It could not only provide  precisely calibrated guidance of the curriculum to the desired outcome states but  also bring much better sample efficiency and geometry-agnostic curriculum goal  proposal capability compared to previous curriculum RL methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' We demon-  strate that our algorithm significantly outperforms these prior methods in a variety  of challenging navigation tasks and robotic manipulation tasks in a quantitative  and qualitative way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  1  INTRODUCTION  While reinforcement learning (RL) shows promising results in automated learning of behavioral  skills, it is still not enough to solve a challenging uninformed search problem where the desired  behavior and rewards are sparsely observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Some techniques tackle this problem by utilizing the  shaped reward (Hartikainen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2019) or combining representation learning for efficient explo-  ration (Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' But, these not only become prohibitively time-consuming in terms of  the required human efforts, but also require significant domain knowledge for shaping the reward  or designing the task-specific representation learning objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' What if we could design the algo-  rithm that automatically progresses toward the desired behavior without any domain knowledge and  human efforts, while distilling the experiences into the general purpose policy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  An effective scheme for designing such an algorithm is one that learns on a tailored sequence of  curriculum goals, allowing the agent to autonomously practice the intermediate tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' However, a  fundamental challenge is that proposing the curriculum goal to the agent is intimately connected to  the efficient desired outcome-directed exploration and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' If the curriculum generation is  ineffective for recognizing frontier parts of the explored and feasible areas, an efficient exploration  toward the desired outcome states cannot be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Even though some prior works propose  to modify the curriculum distribution into a uniform one over the feasible state space (Pong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=',  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Klink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2022) or generate a curriculum based on the level of difficulty (Florensa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=',  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Sukhbaatar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2017), most of these methods show slow curriculum progress due to the  process of skewing the curriculum distribution toward the uniform one rather than the frontier of  the explored region or the properties that are susceptible to focusing on infeasible goals where the  agent’s capability stagnates in the intermediate level of difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  ∗Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  1Code is available : https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='com/jayLEE0301/outpace_official  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='11741v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='LG]  27 Jan 2023  Accepted as a conference paper at ICLR 2023  Figure 1: OUTPACE proposes uncertainty and temporal distance-aware curriculum goals to enable  the agent to progress toward the desired outcome state automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Note that the temporal distance  estimation is reliable within the explored region where we query the curriculum goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Conversely, without the efficient desired outcome-directed exploration, the curriculum proposal  could be ineffective when recognizing the frontier parts in terms of progressing toward the de-  sired outcomes because the curriculum goals, in general, are obtained from the agent’s experiences  through exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Even though some prior works propose the success-example-based approaches  (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2021), these are limited to achieving the only given  example states, which means these cannot be generalized to the arbitrary goal-conditioned agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Other approaches propose to minimize the distance between the curriculum distribution and the de-  sired outcome distribution (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Klink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2022), but these require an assumption that  the distance between the samples can be measured by the Euclidean distance metric, which cannot  be generalized for an arbitrary geometric structure of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Therefore, we argue that  the development of algorithms that simultaneously address both outcome-directed exploration and  curriculum generation toward the frontier is crucial to benefit from the outcome-directed curriculum  RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  In this work, we propose Outcome-directed Uncertainty & TemPoral distance-Aware Curriculum  goal gEneration (OUTPACE) to address such problems, which requires desired outcome examples  only and not prior domain knowledge nor external reward from the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Specifically, the  key elements of our work consist of two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Firstly, our method addresses desired outcome-  directed exploration via a Bayesian classifier incorporating an uncertainty quantification based on  the conditional normalized maximum likelihood (Zhou & Levine, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2021), which  enables our method to propose the curriculum into the unexplored regions and provide directed  guidance toward the desired outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Secondly, our method utilizes Wasserstein distance with a  time-step metric (Durugkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2021b) not only for a temporal distance-aware intrinsic reward but  also for querying the frontier of the explored region during the curriculum learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' By deploying  the above two elements, we propose a simple and intuitive curriculum learning objective formalized  with a bipartite matching to generate a set of calibrated curriculum goals that interpolates between  the initial state distribution and desired outcome state distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  To sum up, our work makes the following key contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  We propose an outcome-directed curriculum RL method which only requires desired out-  come examples and does not require an external reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  To the best of the author’s knowledge, we are the first to propose the uncertainty & tempo-  ral distance-aware curriculum goal generation method for geometry-agnostic progress by  leveraging the conditional normalized maximum likelihood and Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Through several experiments in goal-reaching environments, we show that our method  outperforms the prior curriculum RL methods, most notably when the environment has a  geometric structure, and its curriculum proposal shows properly calibrated guidance toward  the desired outcome states in a quantitative and qualitative way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  2  RELATED WORKS  While a number of works have been proposed to improve the exploration problem in RL, it  still remains a challenging open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' For tackling this problem, prior works include state-  visitation counts (Bellemare et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Ostrovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2017), curiosity/similarity-driven explo-  ration (Pathak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Warde-Farley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2018), the prediction model’s uncertainty-based  2  220E  : Curriculum goals  : Agent (@>@→→@)  F : Desired outcome  [Uncertainty]  3  Explored  Desired  Uncertain  state  state  goal  4  [Temporal dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' from initial]  Temporally  Temporally  Close  FarAccepted as a conference paper at ICLR 2023  exploration (Burda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Pathak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2019), mutual information-based exploration (Ey-  senbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Laskin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2022), maximizing the  entropy of the visited state distribution (Yarats et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Liu & Abbeel, 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Unfortunately,  these techniques are uninformed about the desired outcomes: the trained agent only knows how to  visit frontier states as diverse as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' On the contrary, we consider a problem where the de-  sired outcome can be specified by the given desired outcome examples, allowing for more efficient  outcome-directed exploration rather than naive frontier-directed exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Some prior methods that try to accomplish the desired outcome states often utilize the provided suc-  cess examples (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Eysenbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' However,  they do not provide a mechanism for distilling the knowledge obtained from the agent’s experiences  into general-purpose policies that can be used to achieve new test goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' In this work, we utilize  the Wasserstein distance not only for an arbitrary goal-conditioned agent but also for querying the  frontier of the explored region during curriculum learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Although the Wasserstein distance has  been adopted by some previous research, they are often limited to imitation learning or skill discov-  ery (Dadashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Haldar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Durugkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Fickinger  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Another work (Durugkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2021b) tries to utilize the Wasserstein distance with  the time-step metric for training the goal-reaching agent, but it requires a stationary goal distribution  for stable distance estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Our work is different from these prior works in that the distribution  during training is non-stationary for calibrated guidance to the desired outcome states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Suggesting a curriculum can also make exploration easier where the agent learns on a tailored se-  quence of tasks, allowing the agent to autonomously practice the intermediate tasks in a training  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' However, prior works often require a significant amount of samples to measure the curricu-  lum’s level of difficulty (Florensa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Sukhbaatar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2017), learning progress (Portelas  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2020), regret (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Curricula are often generated by modifying the goal dis-  tribution into the frontier of the explored region via maximizing a certain surrogate objective, such  as the entropy of the goal distribution (Pong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2019), disagreement between the value function  ensembles (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2020), but these methods do not have convergence mechanism to the de-  sired outcome distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' While some algorithms formulate the generation of a curriculum as an  explicit interpolation between the distribution of target tasks and a surrogate task distribution (Ren  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Klink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2022), these still depend on the Euclidean distance metric when measuring  the distance between distributions, which cannot be generalized for an arbitrary geometric structure  of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' In contrast, our method not only provides calibrated guidance to the desired out-  come distribution in a sample efficient way but also shows geometry-agnostic curriculum progress  by leveraging the bipartite matching problem with uncertainty & temporal distance-based objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  3  PRELIMINARY  We consider the Markov decision process (MDP) M = (S, A, G, T, ρ0, γ), where S denotes the  state space, A the action space, G the goal space, T(s′|s, a) the transition dynamics, ρ0 the initial  distribution, ρπ the state visitation distribution when the agent follows the policy π, and γ the dis-  count factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' The MDP in our framework is provided without a reward function and considers an  environment in which only the desired outcome sample {gk  +}K  k=1 are given, assuming that gk  + is  obtained from the desired outcome distribution G+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Therefore, our work employs a trainable in-  trinsic reward function r : S × G × A → R, which is detailed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Also, we represent  the set of curriculum goals as {gk  c }N  k=1 and assume that these are sampled from the curriculum goal  distribution Gc obtained by our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='1  WASSERSTEIN DISTANCE OVER THE TIME-STEP METRIC  The Wasserstein distance represents how much “work” is required to transport one distribution to  another distribution following the optimal transport plan (Villani, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Durugkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  In this section, we describe the Wasserstein distance over the time-step metric and how it can be  obtained with a potential function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Consider a metric space (X, d), where X is a set and d is a  metric on X, and two probability measures µ, ν on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' The Wasserstein-p distance for a given metric  d is defined as follows,  Wp(µ, ν) :=  inf  γ∈Π(µ,ν) E(X,Y )∼γ[d(X, Y )p]1/p if p=1  =  sup  ∥f∥L≤1  [Ey∼ν[f(y)] − Ex∼µ[f(x)]]  (1)  3  Accepted as a conference paper at ICLR 2023  where a joint distribution γ denotes a transport plan, and Π(µ, ν) denotes the set of all possible  joint distributions γ, and the second equality is held by the Kantorovich-Rubinstein duality with  1-Lipschitz functions (f : X → R) (Villani, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Arjovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  If we could define the distance metric as dπ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' sg),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' which is a time-step metric (quasimetric) based  on the number of transition steps experienced before reaching the goal sg ∈ G for the first time  when executing the goal-conditioned policy π(a|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' sg),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' we could design a temporal-distance aware  RL by minimizing the Wasserstein distance W1(ρπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' G) that gives an estimate of the work needed to  transport the state visitation distribution ρπ to the goal distribution G:  W1(ρπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' G) =  sup  ∥f∥L≤1  [Esg∼G[f(sg)] − Es∼ρπ[f(s)]]  (2)  Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' the potential function f is approximately increasing along the optimal goal-reaching trajec-  tory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' That is, if ρπ(s) consists of the states optimally reaching toward the goal sg, f(s) increases  along the trajectory and f(sg) has the maximum value (Durugkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Adopting these prior works, the 1-Lipschitz potential function f with respect to dπ(s, sg) could be  ensured by enforcing that the difference in values of f on the expected transition from every state is  bounded by 1 as follows,  sup  s∈S  {Ea∈π(·|s,sg),s′∈T (·|s,a)[|f(s′) − f(s)|]} ≤ 1  (3)  More detailed derivations are included in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='2  CONDITIONAL NORMALIZED MAXIMUM LIKELIHOOD (CNML)  For curriculum learning, our work utilizes conditional normalized maximum likelihood (CNML)  (Rissanen & Roos;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Fogel & Feder, 2018) that can perform an uncertainty-aware classification based  on previously observed data by minimizing worst-case regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Let D = {(sp, ep)}n−1  p=1 be a set  of data containing pairs of states s1:n−1 and success labels e1:n−1 ∈ {0, 1}, where ‘1’ represents  the occurrence of the desired event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Given a query point sn, CNML in our framework defines the  distribution pCNML(en|sn) which predicts the probability that the state sn belongs to the desired  outcome distribution G+ (e = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  To explain how CNML predicts the label of sn, we suppose Θ is a set of models, where each  model θ ∈ Θ can represent a conditional distribution of labels, pθ(e1:n|s1:n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' CNML considers the  possibilities that all the possible classes (0,1) will be labeled to the query point sn, and obtains the  models ˆθ(e1:n|s1:n) ∈ Θ that represent the augmented datasets D ∪ (sn, en) well by solving the  maximum likelihood estimation (MLE) problem (LHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' (4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Then, CNML that minimizes the  regret over those maximum likelihood estimators can be written as follows (Bibas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2019),  ˆθi = arg max  θ∈Θ  E(s,e)∈D∪(sn,en=i)[log pθ(e|s)],  pCNML(en = i|sn) =  pˆθi(e = i|sn)  �1  j=0 pˆθj(e = j|sn)  (4)  If the query point sn is close to one of the data points in the datasets, CNML will have difficulty in  assigning a high likelihood to labels that are significantly different from those of nearby data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  However, if sn is far from the data points in the dataset, each MLE model ˆθi=0,1 will predict the label  en as its own class, which leads to a large discrepancy in the predictions between the models and  provides us with normalized likelihoods closer to uniform (RHS of Eq (4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Thus, by minimizing the  regret through labeling all possible classes to the new query data, CNML can provide a reasonable  uncertainty estimate (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Zhou & Levine, 2021) on the queried sn and classify whether  the queried sn is similar to the previously observed data either the forms of label 0, 1, or out-of-  distribution data, which is predicted as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  However, the classification technique via CNML described above is in most cases computationally  intractable, as it requires solving separate MLE problems until convergence on every queried data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  4  Accepted as a conference paper at ICLR 2023  Figure 2: Visualization of the uncertainty quantification along training progress (left) and trained  f π  φ (s) (right) in the Point-N-Maze environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' In the right figure, high reward means temporally  close to the desired outcome states, and low reward means the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Previous methods proposed some ideas to amortize the cost of computing CNML distribution (Zhou  & Levine, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Following those prior methods, our work adopts MAML (Finn  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2017) to address the computational intractability of CNML by training one meta-learner net-  work that can quickly adapt to each model ˆθi rather than training each model separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' As Finn  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' (2017) requires samples from G+ and replay buffer B for the inference, the probability should  be represented as pCNML(e = i|s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' G+, B), but we use pCNML(e = i|s) for notational simplicity in  this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' More details about the meta-learning-based classification are included in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  4  METHOD  For a calibrated guidance of the curriculum goals to the desired outcome distribution, we propose to  progress the curriculum towards the uncertain & temporally distant area before converging to G+,  as it is not only the most intuitive way for exploration but also enables the agent to progress without  any prior domain knowledge on the environment such as obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' In short, our work tries to obtain  the distribution of curriculum goals Gc that are considered (a) temporally distant from ρ0 and, (b)  uncertain and, (c) being progressed toward the desired outcome distribution G+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='1  TEMPORAL DISTANCE-AWARE RL WITH THE INTRINSIC REWARD  This section details the intrinsic reward for the RL agent as well as the method of training the  parameterized potential function f π  φ , trained with the data collected by the policy π(a|s, sg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' We  consider a 1-Lipschitz potential function f π  φ whose value increases as the state is far from the initial  state distribution and getting close to the goals sg ∈ G proposed by curriculum learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Then, we  can train an agent that reaches the goals sg in as few steps as possible by minimizing the Wasserstein  distance W1(ρπ, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Considering that we can obtain the estimate of W1(ρπ, G) by Eq (2), the loss for training the param-  eterized potential function f π  φ can be represented as follows (Durugkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='a):  Lφ = Es,sg∼B[f π  φ (s) − f π  φ (sg)] + λ · Es,s′,sg∼B[max(|f π  φ (s) − f π  φ (s′)| − 1, 0))2]  (5)  The penalty term with coefficient λ in Eq (5) is from Eq (3) for ensuring the smoothness requirement  since we consider the Wasserstein distance over the time-step metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Then, assuming the parameter  φ is trained by Eq (5) at every training iteration, we could obtain the supremum of Eq (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Thus, the  reward can be represented as r = f π  φ (s) − f π  φ (sg), which corresponds to −W1(ρπ, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='2  CURRICULUM LEARNING  As CNML can provide a near-uniform prior (prediction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='5) for out-of-distribution data given the  datasets (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='2), we could utilize it by treating the desired outcome states in G+ as (e = 1),  and data points in the replay buffer B as (e = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Then, we could quantify the uncertainty of a state  s based on CNML as  ηucert(s, G+) = 1 −  ��pCNML(e = 0|s) − pCNML(e = 1|s)  ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  (6)  which is proportional to the uncertainty of the queried data s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' However, ηucert alone cannot provide  curriculum guidance toward the desired outcome states because it only performs an uninformed  5  Goal Point  Initial PointAccepted as a conference paper at ICLR 2023  search over uncertainties rather than converging to the desired outcome states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Thus, we modify Eq  (6) with an additional guidance term:  ˜ηucert(Gc, G+) = Es∼Gc[log(ηucert(s, G+) + c · ηguidance(s, G+))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  (7)  where ηguidance(s, G+) = (pCNML(e = 1|s) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='5) · 1(pCNML(e = 1|s) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='5), and c is a  hyperparameter that adjusts the preference on the desired outcome states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Since the CNML provides  near-uniform prior for out-of-distribution data, ηucert provides large values in the uncertain areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Also, the guidance term ηguidance(s, G+) reflects the preference for the states considered to be closer  to the desired outcome state distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  However, in practice, we found the uncertainty quantification itself sometimes has numerical errors,  and it makes pCNML erroneously predict the states near the initial states or boundaries of the already  explored regions as uncertain areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Therefore, we assume that the curriculum should incorporate  the notion of not only the uncertainty but also the temporally distant states from ρ0 for frontier and  desired outcome-directed exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Thus, we formulate the final curriculum learning objective as  follows:  arg max  Gc  [˜ηucert(Gc, G+)] + L · W1(ρo, Gc),  (8)  where the temporal distance bias term with a coefficient L is represented by the Wasserstein distance  from initial state distribution ρ0 to the curriculum goal distribution Gc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' And, given a parameterized  1-Lipschitz potential function f π  φ over the time-step metric dπ(s, sg), we can obtain the estimate of  W1(ρo, Gc) by RHS of Eq (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Also,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' if we assume ˆGc to be a finite set of K particles that is sampled  from already achieved states in the replay buffer B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' the objective function we aim to maximize can  be represented as follows:  max  ˆGc:| ˆGc|=K  K  �  i=1  [˜ηucert(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' G+) + L · [f π  φ (si) − f π  φ (si  0)]],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  si ∈ ˆGc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' si  0 ∈ ρ0  (9)  It enables to propose the curriculum that reflects not only the uncertainty of the states and preference  on the desired outcomes but also temporally distant states from ρ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' while not requiring prior domain  knowledge about the environment such as an obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='3  SAMPLING CURRICULUM GOAL VIA BIPARTITE MATCHING  Since we assume that desired outcome examples from G+ are given rather than its distribution, we  could approximate it by the sampled set ˆG+ (| ˆG+| = K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Then, to solve the curriculum learning  problem of Eq (9), we should address the combinatorial setting that requires assigning ˆGc from the  entire curriculum goal candidates in the replay buffer B to the ˆG+, which is addressed via bipartite  matching in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' With the hyperparameter c = 4, we can rearrange Eq (9) as a minimization  problem with the costs of cross-entropy loss (CE) and temporal-distance bias (f π  φ ) term: (Refer to  the Appendix B for the detailed derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=')  min  ˆGc:| ˆGc|=K  �  si∈ ˆ  Gc,gi  +∈ ˆG+  w(si, gi  +)  (10)  w(si, gi  +) = CE(pCNML(e = 1|si);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' y = pCNML(e = 1|gi  +)) − L · f π  φ (si)  (11)  Intuitively, before discovering the desired outcome states in G+, the curriculum goal si is proposed  in a region of the state space considered to be uncertain and temporally distant from ρ0 in order  to recognize the frontier of the explored regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' And it is kept updated to converge to the desired  outcome states for minimizing the discrepancy between the predicted labels of ˆ  G+ and ˆGc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Then we can construct a bipartite graph G with the cost of the edges w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Let Va and Vb be the  sets of nodes representing achieved states in replay buffer and ˆG+ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' We define a bipartite  6  Accepted as a conference paper at ICLR 2023  (a) OUTPACE(ours)  (b) CURROT  (c) HGG  Figure 3: Visualization of the proposed curriculum goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' First row: Ant Locomotion, Second row:  Point-N-Maze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  graph G({Va, Vb}, E) with the weight of the edges E(·, ·) = −w(·, ·) and separated partitions (Va  and Vb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' To solve the bipartite matching problem, we utilize the Minimum Cost Maximum Flow  algorithm to find K edges with the minimum cost w connecting Va and Vb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' (Ahuja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' The overall training process is summarized in Algorithm 1 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  5  EXPERIMENTS  We include 6 environments to validate our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Firstly, various maze environ-  ments (Point U-Maze, N-Maze, Spiral-Maze) are used to validate the geometry-agnostic curricu-  lum generation capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Also, we experimented with the Ant-Locomotion and Sawyer-Peg Push,  Pick&Place environments to evaluate our method in more complex dynamics or other domains rather  than navigation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  We compare with other previous curriculum or goal generation methods, where each method has the  following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' HGG (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2019): Minimize the distance between the curriculum and  desired outcome state distributions based on the Euclidean distance metric and value function bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  CURROT (Klink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2022): Interpolate between the curriculum and desired outcome state distri-  bution based on the agent’s current performance via optimal transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' GoalGAN (Florensa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=',  2018): Generate curriculum goals that are intermediate level of difficulty by training GAN (Good-  fellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' PLR (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2021): Sample increasingly difficult tasks by prioritizing the  levels of tasks with high TD errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' ALP-GMM (Portelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2020): Fit a GMM with an absolute  learning progress score approximated by the absolute reward difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' VDS (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2020):  Prioritize goals that maximize the epistemic uncertainty of the value function ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' SkewFit  (Pong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2019): Maximize the entropy of the goal distribution to be uniform on the feasible state  space via skewing the distribution trained by VAE (Kingma & Welling, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='1  EXPERIMENTAL RESULTS  Firstly, to show how each module in our method is trained, we visualized the uncertainty quantifica-  tion by CNML, and f π  φ (s) values which are proportional to the required timesteps to reach s from  ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' The uncertainty quantification results (Figure 2) show that the classifier pCNML(·|s) successfully  discriminates the queried states as already explored region or desired outcome states, otherwise, un-  certain states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Due to the geometry-agnostic property of the classifier pCNML(·|s), we could propose  the curriculum in the arbitrary geometric structure of the environments, while most of the previous  curriculum generation methods do not consider it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  We also visualized the values of the trained potential function f π  φ (s) to show how the intrinsic reward  is shaped (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' As the potential function f π  φ (s) is trained to have high values near the desired  outcome states due to the Wasserstein distance with the time-step metric, the results show gradual  7  Accepted as a conference paper at ICLR 2023  (a) U-Maze  (b) N-Maze  (c) Spiral-Maze  (d) Ant Locomotion  (e) Sawyer Push  (f) Sawyer Pick&Place  Figure 4: Average distance from the curriculum goals to the final goals (Lower is better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Our  method’s increasing tendencies at initial steps in some environments are due to the geometric struc-  ture of the environments themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Shading indicates a standard deviation across 5 seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  (a) U-Maze  (b) N-Maze  (c) Spiral-Maze  (d) Ant Locomotion  (e) Sawyer Push  (f) Sawyer Pick&Place  Figure 5: Episode success rates of the evaluation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' The legends and seeds are the same as Fig  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Note that the curves of the baselines in some environments are not visible as they overlap at zero  success rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  increases of f π  φ (s) values along the trajectory toward the desired outcome states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' That is, high values  of f π  φ (s) indicate the smaller required timesteps to reach the desired outcome state, and this property  brings the advantage in identifying the frontier of the explored region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  To validate whether the curriculum goals are properly interpolated from initial states to desired  outcome states by combining both objectives for curriculum learning (Eq (8)), we evaluated the  progress of the curriculum goals in a quantitative and qualitative way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' For quantitative evalua-  tion, we compare with other previous works described above with respect to the distance from the  proposed curriculum goals to G+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' As we can see in Figure 4, our method is the only one that  consistently interpolates from initial states to G+ as training proceeds, while others have difficulty  with complex dynamics or geometry of the environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' For qualitative evaluation, we visualized  the curriculum goals proposed by our method and other baselines that show somewhat comparable  results as training proceeds (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' The results show that our method consistently proposes the  proper curriculum based on the required timesteps and uncertainties regardless of the geometry and  dynamics of the various environments, while other baselines have difficulty as they utilize the Eu-  clidean distance metric to interpolate the curriculum distribution to the G+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' We also evaluated the  desired outcome-directed RL performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' As we can see in Figure 5, our method is able to very  quickly learn how to solve these uninformed exploration problems through calibrated curriculum  goal proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  8  10  8  distance  6  4  2  0  0  200  400  600  800  1000  steps (k)16  14  12  distance  10  8  6  4  2  0  0  200  400  600  800  1000  steps (k)20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
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+page_content='0  0  500  1000  1500  2000  2500  3000  steps (k)Accepted as a conference paper at ICLR 2023  (a) Cost type  (b) Reward & Goal proposition type  (c) Effect of c in ηguidance  Figure 6: Ablation study in terms of the distance from the proposed curriculum goals to the desired  final goal states (Lower is better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' First row: Ant Locomotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Second row: Sawyer Pick & Place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Shading indicates a standard deviation across 5 seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='2  ABLATION STUDY  Types of curriculum learning cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  We first evaluate the importance of each curriculum learning  objective in Eq (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Specifically, we experimented only with uncertainty-related objective (only-  cnml) and timestep-related objective (only-f) when curriculum learning progresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' As we can see  in Figure 6a, both objectives play complementary roles, which support the requirement of both  objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Without one of them, the agent has difficulty in progressing the curriculum goals toward  the desired outcome states due to the local optimum of f π  φ or the numerical error of pCNML, and  more qualitative/quantitative results and analysis about this and other ablation studies are included  in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Reward type & Goal proposition method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Secondly, we replace the intrinsic reward with the  sparse reward, which is typically used in goal-conditioned RL problems, to validate the effects of  the timestep-proportionally shaped reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Also, for comparing the curriculum proposition method,  we replace the Bipartite Matching formulation with a GAN-based generative model, which is similar  to Florensa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' (2018), but we label the highly uncertain states as positive labels instead of success  rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' As we can see in Figure 6b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' the timestep-proportionally shaped reward shows consistently  better results due to the more informed reward signal compared to the sparse one,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' and the generative  model has difficulty in sampling the proper curriculum goals because GAN shows training instability  with the drastic change of the positive labels,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' while our method is relatively insensitive because  curriculum candidates are obtained from the experienced states from the buffer B rather than the  generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Effect of c in ηguidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Lastly, we experiment with different values of hyperparameter c to validate  the effect of ηguidance on curriculum learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' When c is smaller than the default value 4, it is still  possible to explore most of the feasible state space except the area near the desired outcome states  due to the uncertainty & temporal distance-aware curriculum (Figure 6c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' But, we could verify that  ηguidance’s effect becomes smaller as c decreases and ηguidance helps to guide the curriculum goals  to the desired outcome states precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' This is consistent with our analysis that the uncertainty &  temporal distance themselves can provide curriculum goals in the frontier of the explored region  while ηguidance can further accelerate the guidance to the desired outcome states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  6  CONCLUSIONS  In this work, we consider an outcome-directed curriculum RL where the agent should progress  toward the desired outcome automatically without the reward function and prior knowledge about  the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' We propose OUTPACE, which performs uncertainty, temporal distance-aware  curriculum RL with intrinsic reward, based on the classifier by CNML, and Wasserstein distance  with time-step metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' We show that our method can outperform the previous methods regarding  9  12  OUTPACE(ours)  Ablation_only_f  10  Ablation only cnml  distance  8  6  4  2  0  500  1000  1500  2000  2500  3000  steps (k)12  OUTPACE(ours)  Ablation_SparseReward  10  Ablation GAN  distance  8  6  4  2  0  500  1000  1500  2000  2500  3000  steps (k)12  OUTPACE(ours)  c=1  10  C=2  C=3  distance  8  6  4  2  0  500  1000  1500  2000  2500  3000  steps (k)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='9  OUTPACE(ours)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='8  Ablation_only_f  Ablation only cnml  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='7  distance  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='1  0  500  1000  1500  2000  2500  3000  steps (k)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='8  OUTPACE(ours)  Ablation SparseReward  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='7  Ablation GAN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='6  distance  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='1  0  500  1000  1500  2000  2500  3000  steps (k)OUTPACE(ours)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='7  c=1  C=2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='6  C=3  distance  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='1  0  500  1000  1500  2000  2500  3000  steps (k)Accepted as a conference paper at ICLR 2023  sample efficiency and curriculum progress quantitatively and qualitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Even though our method  shows promising results, there are some issues with computational complexity due to the innate  properties of the meta-learning inference procedure itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Thus, it would be interesting future work  to find a way to reduce the inference time for less training wall-clock time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  REFERENCES  R K Ahuja, T L Magnanti, and J B Orlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Network Flows: Theory, Algorithms, and Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Prentice Hall, Englewood Cliffs, NJ, 1st edition, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Marcin Andrychowicz, Filip Wolski, Alex Ray, Jonas Schneider, Rachel Fong, Peter Welinder, Bob  McGrew, Josh Tobin, Pieter Abbeel, and Wojciech Zaremba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Hindsight experience replay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' arXiv  preprint arXiv:1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='01495, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Martin Arjovsky, Soumith Chintala, and L´eon Bottou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Wasserstein generative adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  In International conference on machine learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' 214–223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' PMLR, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Marc Bellemare, Sriram Srinivasan, Georg Ostrovski, Tom Schaul, David Saxton, and Remi Munos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Unifying count-based exploration and intrinsic motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Advances in neural information pro-  cessing systems, 29, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Koby Bibas, Yaniv Fogel, and Meir Feder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Deep pnml: Predictive normalized maximum likelihood  for deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' arXiv preprint arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='12286, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Yuri Burda, Harrison Edwards, Amos Storkey, and Oleg Klimov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Exploration by random network  distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' arXiv preprint arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='12894, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Robert Dadashi, L´eonard Hussenot, Matthieu Geist, and Olivier Pietquin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Primal wasserstein imita-  tion learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' arXiv preprint arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='04678, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Ishan Durugkar, Steven Hansen, Stephen Spencer, and Volodymyr Mnih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Wasserstein distance max-  imizing intrinsic control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' arXiv preprint arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='15331, 2021a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Ishan Durugkar, Mauricio Tec, Scott Niekum, and Peter Stone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Adversarial intrinsic motivation  for reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Advances in Neural Information Processing Systems, 34:8622–8636,  2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Ben Eysenbach, Sergey Levine, and Russ R Salakhutdinov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Replacing rewards with examples:  Example-based policy search via recursive classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Advances in Neural Information Pro-  cessing Systems, 34:11541–11552, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Benjamin Eysenbach, Abhishek Gupta, Julian Ibarz, and Sergey Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Diversity is all you need:  Learning skills without a reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' arXiv preprint arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='06070, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Arnaud Fickinger, Samuel Cohen, Stuart Russell, and Brandon Amos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Cross-domain imitation  learning via optimal transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' arXiv preprint arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='03684, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Chelsea Finn, Pieter Abbeel, and Sergey Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Model-agnostic meta-learning for fast adaptation  of deep networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' In International Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' 1126–1135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' PMLR,  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Carlos Florensa, David Held, Xinyang Geng, and Pieter Abbeel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Automatic goal generation for  reinforcement learning agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' In International conference on machine learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' 1515–1528.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  PMLR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Yaniv Fogel and Meir Feder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Universal supervised learning for individual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' arXiv preprint  arXiv:1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='09520, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Justin Fu, Avi Singh, Dibya Ghosh, Larry Yang, and Sergey Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Variational inverse control with  events: A general framework for data-driven reward definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Advances in neural information  processing systems, 31, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Dibya Ghosh, Abhishek Gupta, and Sergey Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Learning actionable representations with goal-  conditioned policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' arXiv preprint arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='07819, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  10  Accepted as a conference paper at ICLR 2023  Ian J Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil  Ozair, Aaron Courville, and Yoshua Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Generative adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' arXiv preprint  arXiv:1406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='2661, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Tuomas Haarnoja, Aurick Zhou, Pieter Abbeel, and Sergey Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Soft actor-critic: Off-policy  maximum entropy deep reinforcement learning with a stochastic actor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' In International confer-  ence on machine learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' 1861–1870.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' PMLR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Siddhant Haldar, Vaibhav Mathur, Denis Yarats, and Lerrel Pinto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Watch and match: Supercharging  imitation with regularized optimal transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' arXiv preprint arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='15469, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Kristian Hartikainen, Xinyang Geng, Tuomas Haarnoja, and Sergey Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Dynamical distance  learning for semi-supervised and unsupervised skill discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' arXiv preprint arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='08225,  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Minqi Jiang, Edward Grefenstette, and Tim Rockt¨aschel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Prioritized level replay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' In International  Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' 4940–4950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' PMLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Diederik P Kingma and Max Welling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Auto-encoding variational bayes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  arXiv preprint  arXiv:1312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='6114, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Pascal Klink, Haoyi Yang, Carlo D’Eramo, Jan Peters, and Joni Pajarinen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Curriculum reinforce-  ment learning via constrained optimal transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' In International Conference on Machine Learn-  ing, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' 11341–11358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' PMLR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Michael Laskin, Hao Liu, Xue Bin Peng, Denis Yarats, Aravind Rajeswaran, and Pieter Abbeel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Cic:  Contrastive intrinsic control for unsupervised skill discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' arXiv preprint arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='00161,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Kevin Li, Abhishek Gupta, Ashwin Reddy, Vitchyr H Pong, Aurick Zhou, Justin Yu, and Sergey  Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Mural: Meta-learning uncertainty-aware rewards for outcome-driven reinforcement  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' In International Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' 6346–6356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' PMLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Hao Liu and Pieter Abbeel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Aps: Active pretraining with successor features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  In International  Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' 6736–6747.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' PMLR, 2021a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Hao Liu and Pieter Abbeel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Behavior from the void: Unsupervised active pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Advances in  Neural Information Processing Systems, 34:18459–18473, 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Georg Ostrovski, Marc G Bellemare, A¨aron Oord, and R´emi Munos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Count-based exploration with  neural density models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' In International conference on machine learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' 2721–2730.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' PMLR,  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor  Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward  Yang, Zachary DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner,  Lu Fang, Junjie Bai, and Soumith Chintala.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Pytorch: An imperative style, high-performance  deep learning library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  In H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Wallach, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Larochelle, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Beygelzimer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=" d'Alch´e-Buc,  E." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Fox, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Garnett (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' ), Advances in Neural Information Processing Systems 32, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  8024–8035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' URL http://papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='neurips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='cc/paper/  9015-pytorch-an-imperative-style-high-performance-deep-learning-library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Deepak Pathak, Pulkit Agrawal, Alexei A Efros, and Trevor Darrell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Curiosity-driven exploration  by self-supervised prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' In International conference on machine learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' 2778–2787.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  PMLR, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Deepak Pathak, Dhiraj Gandhi, and Abhinav Gupta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Self-supervised exploration via disagreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  In International conference on machine learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' 5062–5071.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' PMLR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Vitchyr H Pong, Murtaza Dalal, Steven Lin, Ashvin Nair, Shikhar Bahl, and Sergey Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Skew-  fit: State-covering self-supervised reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  arXiv preprint arXiv:1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='03698,  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  11  Accepted as a conference paper at ICLR 2023  R´emy Portelas, C´edric Colas, Katja Hofmann, and Pierre-Yves Oudeyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Teacher algorithms for  curriculum learning of deep rl in continuously parameterized environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' In Conference on  Robot Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' 835–853.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Zhizhou Ren, Kefan Dong, Yuan Zhou, Qiang Liu, and Jian Peng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Exploration via hindsight goal  generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Advances in Neural Information Processing Systems, 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Jorma Rissanen and Teemu Roos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Conditional nml universal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' In 2007 Information Theory  and Applications Workshop, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' 337–341.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  John Schulman, Sergey Levine, Pieter Abbeel, Michael Jordan, and Philipp Moritz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Trust region  policy optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' In International conference on machine learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' 1889–1897.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' PMLR,  2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Proximal policy  optimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' arXiv preprint arXiv:1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='06347, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Archit Sharma, Shixiang Gu, Sergey Levine, Vikash Kumar, and Karol Hausman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Dynamics-aware  unsupervised discovery of skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' arXiv preprint arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='01657, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Archit Sharma, Kelvin Xu, Nikhil Sardana, Abhishek Gupta, Karol Hausman, Sergey Levine, and  Chelsea Finn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Autonomous reinforcement learning: Benchmarking and formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' arXiv preprint  arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='09605, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Avi Singh, Larry Yang, Kristian Hartikainen, Chelsea Finn, and Sergey Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' End-to-end robotic  reinforcement learning without reward engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' arXiv preprint arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='07854, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Sainbayar Sukhbaatar, Zeming Lin, Ilya Kostrikov, Gabriel Synnaeve, Arthur Szlam, and Rob  Fergus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Intrinsic motivation and automatic curricula via asymmetric self-play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' arXiv preprint  arXiv:1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='05407, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  C´edric Villani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Optimal transport: old and new, volume 338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Springer, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  David Warde-Farley, Tom Van de Wiele, Tejas Kulkarni, Catalin Ionescu, Steven Hansen, and  Volodymyr Mnih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Unsupervised control through non-parametric discriminative rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' arXiv  preprint arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='11359, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Huang Xiao, Michael Herman, Joerg Wagner, Sebastian Ziesche, Jalal Etesami, and Thai Hong  Linh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Wasserstein adversarial imitation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' arXiv preprint arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='08113, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Denis Yarats, Rob Fergus, Alessandro Lazaric, and Lerrel Pinto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Reinforcement learning with pro-  totypical representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' In International Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' 11920–11931.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  PMLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Tianhe Yu, Deirdre Quillen, Zhanpeng He, Ryan Julian, Karol Hausman, Chelsea Finn, and Sergey  Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Meta-world: A benchmark and evaluation for multi-task and meta reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  In Conference on Robot Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' 1094–1100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Yunzhi Zhang, Pieter Abbeel, and Lerrel Pinto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Automatic curriculum learning through value dis-  agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Advances in Neural Information Processing Systems, 33:7648–7659, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Rui Zhao, Yang Gao, Pieter Abbeel, Volker Tresp, and Wei Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Mutual information state intrinsic  control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' arXiv preprint arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='08107, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Aurick Zhou and Sergey Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Amortized conditional normalized maximum likelihood: Reliable  out of distribution uncertainty estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' In International Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  12803–12812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' PMLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  12  Accepted as a conference paper at ICLR 2023  A  TRAINING & EXPERIMENTS DETAILS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='1  TRAINING DETAILS  Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  The baseline curriculum RL algorithms are trained as follows,  HGG (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2019) : We follow the default setting in the original implementation from  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='com/Stilwell-Git/Hindsight-Goal-Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  CURROT (Klink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2022): We follow the default setting in the original implementation  from https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='com/psclklnk/currot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  GoalGAN (Florensa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2018), PLR (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2021), VDS (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2020),  ALP-GMM (Portelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2020) : We follow the default setting in implementation from  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='com/psclklnk/currot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  SkewFit (Pong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2019): We follow the state-based version of SkewFit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' The original  implementation was modified and used since only the image-based version is provided in  it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='com/rail-berkeley/rlkit)  All the baselines are trained by SAC (Haarnoja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2018) with the sparse reward except for the  SkewFit as it uses a reward based on the conditional entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Even though some algorithms’ original  implementation is based on on-policy algorithms such as TRPO or PPO (Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  2017), for comparing the sample efficiency, we replace the on-policy algorithm with the off-policy  algorithm, SAC, following the referred implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Table 1: Conceptual comparison between our work and the previous curriculum RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Uncert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Timestep  Target  Off-  Curriculum  Without  Without non-  Aware  Aware  curriculum dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  policy  proposal  ext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' reward  forgetting mechanism  HGG  \x17  \x17  G+  ✓  B  \x17  ✓  GoalGAN  \x17  \x17  \x17  \x17  GAN  \x17  \x17  CURROT  \x17  \x17  U or G+  ✓  U  \x17  \x17  PLR  \x17  \x17  \x17  \x17  B  \x17  \x17  VDS  ✓  \x17  \x17  ✓  B  \x17  ✓  ALP-GMM  \x17  \x17  \x17  ✓  GMM  \x17  \x17  SkewFit  ✓  \x17  \x17  ✓  VAE  ✓  \x17  OUTPACE (ours)  ✓  ✓  G+  ✓  B  ✓  ✓  We included a conceptual comparison between our work and previous other curriculum or goal  generation methods in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Uncert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='-Aware: whether the curriculum goal proposal process is aware of the uncertainty  of the candidate goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Timestep-Aware: whether the curriculum goal proposal process is aware of the temporal  distance from the initial states or from the desired outcome states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Target curriculum dist: whether there exists a mechanism for the curriculum goals to  converge to the target distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' When there is no target distribution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' just exploring  or expanding the curriculum goal distribution as diverse as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' ), we denoted it as \x17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Off-policy: whether the off-policy RL can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Some baselines need to measure a  kind of difficulty, which means they require repeated trials and on-policy RL algorithms  with multi-processing such as TRPO, and PPO (Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Curriculum proposal: where the curriculum goals are proposed from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Without ext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' reward: whether the algorithm requires external environmental reward or  not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Without non-forgetting mechanism: whether the algorithm requires implicit or explicit  non-forgetting mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Some baselines mix the previously practiced curriculum goals  with a fixed or varying ratio or make the curriculum distribution into uniform over the state  space to cover all possible test goal states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  13  Accepted as a conference paper at ICLR 2023  Training details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  To train the potential function fφ via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' (5), s and sg in the buffer should ideally  contain all feasible states in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' However, until the policy is learned enough to explore  the map, obtaining such ideal distribution is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' To mitigate this issue, following Durugkar  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' (2021b), we approximate such a distribution with a small replay buffer Bf containing recent  trajectories and the relabelling technique (Andrychowicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' While this approximation  does not provide fφ with the ideal state distribution covering all feasible states, we empirically  found that this assumption works well since OUTPACE only queries fφ for the explored area when  it generates curriculum goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Table 2: Hyperparameters for OUTPACE  Critic hidden dim  512  discount factor γ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='99  Critic hidden depth  3  fφ update frequency  1000  Critic target τ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='01  # of gradient steps for fφ update  10  Critic target update frequency  2  # of ensemble networks for fφ  5  Actor hidden dim  512  learning rate for fφ  1e-4  Actor hidden depth  3  RL optimizer  adam  Actor update frequency  2  Meta-learner network hidden size  [2048,2048]  RL batch size  512  Meta-learner train sample size  512  Init temperature αinit of SAC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='3  Meta-learner test sample size  2048  Replay buffer B size  3e6  Meta-learner test batch size  2048  Table 3: Env-specific hyperparameters for OUTPACE  Env name  Bf size  λ for fφ  adam ϵ for SAC  L  Point-U-Maze-v0  10000  25  1e-2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='02  Point-N-Maze-v0  10000  25  1e-2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='02  Point-Spiral-Maze-v0  20000  50  1e-8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='02  Ant Locomotion-v0  50000  25  1e-8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='02  Sawyer-Peg-Push  30000  25  1e-2  2  Sawyer-Peg-Pick&Place  30000  25  1e-2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='02  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='2  ENVIRONMENT DETAILS  Point-U-Maze : The observation consists of the xy position, angle, velocity, and angular  velocity of the ‘point’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' The action space consists of the velocity and angular velocity of the  ‘point’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' The initial state of the agent is [0, 0] and the desired outcome states are obtained  by adding uniform noise to the default goal point [0, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' The size of the map is 12 × 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Point-N-Maze : It is the same as the Point-U-Maze environment except that the desired  outcome states are obtained by adding uniform noise to the default goal point [8, 16], and  the size of the map is 12 × 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Point-Spiral-Maze : It is the same as the Point-U-Maze environment except that the desired  outcome states are obtained by adding uniform noise to the default goal point [8, −8], and  the size of the map is 20 × 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Ant Locomotion : The observation consists of the xyz position, xyz velocity, joint angle,  and joint angular velocity of the ‘ant’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' The action space consists of the torque applied on  the rotor of the ‘ant’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' The initial state of the agent is [0, 0] and the desired outcome states  are obtained by adding uniform noise to the default goal point [0, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' The size of the map is  12 × 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Sawyer-Peg-Push : The observation consists of the xyz position of the end-effector, the  object, and the gripper’s state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' The action space consists of the xyz position of the end-  effector and gripper open/close control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' The initial state of the object is [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='02] and  the desired outcome states are obtained by adding uniform noise to the default goal point  [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='02].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' We referred to the metaworld (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2020) and EARL (Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=',  2021) environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  14  Accepted as a conference paper at ICLR 2023  (a) U-Maze  (b) N-Maze  (c) Spiral-Maze  (d) Ant Locomotion  (e) Sawyer Manip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Figure 7: Environments used for evaluation: (a)-(c) the agent must navigate various kinds of maze  environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' (d) the quadruped ant must navigate the maze to a particular location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' (e) the robot  has to push or pick&place a peg to the desired location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Sawyer-Peg-Pick&Place : It is the same as the Sawyer-Peg-Push environment except that  the desired outcome states are obtained by adding uniform noise to the default goal point  [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  B  ALGORITHM & DERIVATIONS  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='1  ALGORITHM  Algorithm 1 Overview of OUTPACE algorithm  1: Input: desired outcome examples ˆG+, total training episodes N, Env, environment horizon H,  actor network π, critic network Q, potential function network f π  φ , replay buffer B, Bf  2: for iteration=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=',N do  3:  ˆGc ← sample K curriculum goals that minimize  �  si∈B[CE(pCNML(e = 1|si);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' y = pCNML(e = 1| ˆG+)) − L · f π  φ (si)] (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='3)  4:  for i=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=',K do  5:  Env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='reset()  6:  g ← ˆGc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='pop  7:  for t=0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=',H-1 do  8:  if achieved g then  9:  g ← random goal (randomly sample a state with high uncertainty measured by pCNML  in a ball Br(st).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=')  10:  end if  11:  at ← π(·|st, g)  12:  st+1 ← Env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='step(at)  13:  end for  14:  B ← B ∪ {s0, a0, s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='}, Bf ← Bf ∪ {s0, a0, s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='}  15:  end for  16:  for i=0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=',M do  17:  Sample a minibatch b from B and label reward using f π  φ (st) (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='1)  18:  Train π and Q with b via SAC (Haarnoja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  19:  Meta-train pCNML with b via meta-NML (Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=')  20:  Sample a minibatch bf from Bf  21:  Train f π  φ with bf via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' (5)  22:  end for  23: end for  15  ★Accepted as a conference paper at ICLR 2023  Figure 8: The overall diagram of OUTPACE  Algorithm 2 Meta-NML (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2021)  1: Input: desired outcome examples ˆG+, RL buffer B  2: Sample ˆG− from RL buffer B  3: D ← ˆG− ∪ ˆG+ (Let the size of D is n)  4: Create 2n meta-training tasks by relabelling each data points in D as 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  (D ∪ (xi, y′) where xi ∼ D and y′ ∈ {0, 1})  5: Meta-learn with 2n meta-training tasks (Finn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2017)  (Let θ be the parameter of pCNML and L be standard classification loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  minθ Exi∼D,y′∼{0,1}[L(D ∪ (xi, y′), θ′(i)  j=y′)],  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='θ′(i)  j  = θ − α∇θL(D ∪ (xi, yi), θ)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='2  DERIVATIONS  This section contains the definition of time-step metric dπ, derivation of Lipschitz smoothness of  f, and derivation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Regarding dπ and Lipschitz smoothness of f, we refer the reader to  Durugkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' (2021b) for more detailed explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='1  TIME-STEP METRIC dπ, LIPSCHITZ SMOOTHNESS OF f  Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Given a state space S, action space A, transition dynamics T : S × A → S, and  agent policy π, the time-step metric dπ is a quasi-metric where the distance from s ∈ S to sg ∈ S is  the expected number of transitions required with the policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  The time-step metric can be expressed by the expectation of the number of transitions taken under  the policy π, where T π(·|s, sg) is the probability distribution of the timestep required to go from s  to sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' dπ can also be written recursively as  dπ(s, sg) : = Eπ,τ∼T π(·|s,sg)[τ]  =  �0  ifs = sg  1 + Ea∼π(·|s,sg)Es′∼T (·|s,a)[dπ(s′, sg)]  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  (12)  16  Sampling Curriculum Goals  Training Goal Sampling Modules  Bipartite Matching  Uncertainty-Aware Module  Replay  Buffer  Augment data  Calculate  Calculate  D U (si,ei E (0,1))  Uncertainty  Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Dist  PcNML  Sample K curriculum  MAML  goals that minimize Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='11  (Fin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2017)  Temporal Distance-Aware Module  Practice Curriculum Goals  Curriculum Goals  Train fa  Off-Policy RL  by minimizing Lp (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='5)  Single step adaptation  Replay Data  Neural Network ModelAccepted as a conference paper at ICLR 2023  Lipschitz smoothness of f  If the difference in values of f on the expected transition from every  state is bounded by 1, and the policy π can reach the goal sg within a finite number of transitions,  then f is the 1-Lipschitz function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' We can write |f(sg) − f(s0)| via telescopic sum,  |f(sg) − f(s0)| = Eπ,τ∼T π(·|s0,sg)  ������  τ−1  �  t=0  (f(st+1) − f(st))  �����  �  ≤ Eπ,τ∼T π(·|s0,sg)  � τ−1  �  t=0  |f(st+1) − f(st)|  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  (13)  Since E[|f(s′) − f(s)|] ≤ 1 by Eq (3), we can write  Eπ,τ∼T π(·|s0,sg)  � τ−1  �  t=0  |f(st+1) − f(st)|  �  ≤ Eπ,τ∼T π(·|s0,sg)  � τ−1  �  t=0  1  �  = Eπ,τ∼T π(·|s0,sg)[τ]  = dπ(s0, sg)  (14)  Thus, f is the 1-Lipschitz function with respect to the time-step metric dπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='2  DERIVATION OF EQUATION (11)  By substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' (7) into Eq (9), and omitting f π  φ (si  0) which is not related to ˆGc, we obtain the  following terms,  max  ˆGc:| ˆGc|=K  K  �  i=1  �  log  �  1 −  ��pCNML(e = 0|si) − pCNML(e = 1|si)  ��  + c(pCNML(e = 1|si) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='5) · 1(pCNML(e = 1|si) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='5)  �  + L · f π  φ (si)  �  ,  si ∈ ˆGc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  (15)  Also, if we use the default value of the hyperparameters c = 4, we can express the above terms as  follows,  max  ˆGc:| ˆGc|=K  K  �  i=1  �  log  �  1 −  ��1 − 2pCNML(e = 1|si)  ��+  4(pCNML(e = 1|si) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='5) · 1(pCNML(e = 1|si) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='5)  �  + L · f π  φ (si)  �  ,  si ∈ ˆGc  (16)  which can be simplified as  min  ˆGc:| ˆGc|=K  K  �  i=1  [− log(pCNML(e = 1|si)) − L · f π  φ (si)],  si ∈ ˆGc  (17)  If we could assume that pCNML is well trained to classify the desired outcome examples gi  + ∈  ˆG+ from the states s in the already explored region, pCNML(e = 1|gi  +) is approximately equal  to 1 (pCNML(e = 1|gi  +) ≈ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Then, the terms inside the above minimization objective can be  approximately represented as  − pCNML(e = 1|gi  +) log(pCNML(e = 1|si)) − L · f π  φ (si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  (18)  17  Accepted as a conference paper at ICLR 2023  Then, in practical implementation, we can implement the above terms by cross-entropy loss as  w(si, gi  +) := CE(pCNML(e = 1|si);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' y = pCNML(e = 1|gi  +)) − L · f π  φ (si),  (19)  which is Eq (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Even though it is not exactly equivalent to the mathematical definition of the  cross-entropy, we can just implement it with cross-entropy loss developed in standard deep learning  framework such as PyTorch (Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2019) because choosing si to maximize log(pCNML(e =  1|si)) and choosing si close to gi  + in order to be classified as desired outcome examples by the  pCNML (predicted labels of si to be close to 1) have the same intuitive meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='3  A DETAILED DESCRIPTION OF META-NML  Conditional normalized maximum likelihood (CNML) can perform a conservative k-way classi-  fication based on previously seen data (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Zhou & Levine, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Let Dtrain =  {(xi, yi)}n−1  i=1 be a set of data containing pairs of inputs x1:n−1 and labels y1:n−1 ∈ {1, · · · , k},  where k is the number of possible labels, and Θ is a set of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Given a new input xn, CNML  defines the distribution pCNML(yn = i|xn)i∈{1,··· ,k} by minimizing the regret R of the worst-case  label yn as  pCNML = arg min  q  max  yn R(q, x1:n, y1:n, Θ),  (20)  where we define a regret R for label yn of a distribution q and maximum likelihood estimator θ′ as  R(q, x1:n, y1:n, Θ) := logθ′(y1:n|x1:n)(y1:n|x1:n) − log q(y1:n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  (21)  By solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' (20), CNML predicts the distribution of the new label yn as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' (22) (Bibas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=',  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  pCNML(yn = m, xn) =  pθ′(n)  m (y = m|xn)  �k  j=1 pθ′(n)  j  (y = j|xn)  ,  (22)  where a model θ′(n)  m  is a model that represents the augmented dataset D = Dtrain ∪ (xn, yn = m)  well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Thus, the number of total MLE models required is n × k since we should address each data  by augmenting with the label 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', k, respectively (θ′(α=1:n)  β=1:k  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Since our algorithm utilizes 2-way  classification (k = 2), we define 2n tasks τ −  i=1:n and τ +  i=1:n, which are constructed by augmenting  negative (xi, y = 0) and positive (xi, y = 1) labels respectively for each data point (xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  To amortize the training cost of the tasks (tasks τ −,+  i=1:n) we can apply the meta-learning algorithm  (Finn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2021) to this setting and train a model θ which can quickly adapt to the  optimal solution after a single step of gradient update with standard classification loss L as  min  θ  Exi∼D,y′∼{0,1}[L(D ∪ (xi, y′), θ′(i)  j=y′)],  (23)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='θ′(i)  j  = θ − α∇θL(D ∪ (xi, yi), θ),  (24)  where Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' (23) and Eq (24) represent the objective of meta-learning and quick adaptation respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Training CNML via meta-learning and leveraging CNML are shown in lines 3 and 19 of the  algorithm overview (Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Also, we provide the pseudo-code of meta-nml in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  C  MORE EXPERIMENTAL RESULTS  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='1  FULL RESULTS OF THE MAIN SCRIPT  We included the full results of the main script in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' The uncertainty quantification is  visualized in Figure 9, the visualization of the trained f π  φ (s) is in Figure 10, the visualization of  18  Accepted as a conference paper at ICLR 2023  the proposed curriculum goals is in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' We do not include the visualization results of the  Point-U-Maze, and Sawyer-Peg-Pick&Place results as these environments share the same map with  the Ant Locomotion, and Sawyer-Peg-Push environments, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Figure 9: Visualization of the uncertainty quantification along training progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' First row: U-Maze  (Point, Ant Locomotion), Second row: N-Maze, Third row: Spiral-Maze, Fourth row: Sawyer Peg  Push, Pick & Place environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  (a) Ant Locomotion  (b) N-Maze  (c) Spiral-Maze  (d) Sawyer Push  Figure 10: Visualization of the trained f π  φ (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' High reward means temporally close to the desired  outcome states (required timesteps are small to reach the goal), and low reward means the opposite  (required timesteps are large to reach the goal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  19  Goal Point  Initial PointInitial Point  Goal PointInitial Point  Goal PointGoal Point  Initial PointAccepted as a conference paper at ICLR 2023  (a) OUTPACE(ours)  (b) CURROT  (c) HGG  Figure 11: Visualization of the proposed curriculum goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' First row: Ant Locomotion (same  map size with U-Maze), Second row: Point-N-Maze, Third row: Point-Spiral-Maze, Fourth row:  Sawyer Peg Push.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  20  Accepted as a conference paper at ICLR 2023  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='2  EVALUATION WITH THE GOALS SAMPLED FROM THE UNIFORM DISTRIBUTION  In some curriculum RL algorithms, they incorporate a mechanism for remembering previously prac-  ticed curricula either implicitly or explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' For example, GoalGAN (Florensa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2018) mixes  the previously generated goals with currently generated goals in a specified ratio (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' 20 % of pre-  viously used goals), and SkewFit (Pong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2019) set the objective as targeting the uniform goal  distribution (by maximizing the entropy of the goals H(g)), and CURROT (Klink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=', 2022) also  utilizes uniform target curriculum distribution in practice, and so do some of the other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Due to these algorithmic designs, they require many iterations for explicitly practicing previously  used curriculum goals, or show slow progress of curriculum to match the uniform target distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  In contrast, our method is based on an intrinsic reward, which is shaped according to the required  timesteps proportional values f π  φ (s) as described in our main script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Thus, our method does not  need to explicitly consider the non-forgetting mechanism when we design the algorithm because the  reward is already shaped with respect to the timesteps for reaching the arbitrary goal points along  the trajectory that reaches the desired outcome state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Because of this property, our method does not  consider uniform target distribution or explicitly mixing the previously practiced curriculum goals,  and it enables our method to be much faster and show sample-efficient curriculum progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  We experimented with the goals sampled from the uniform distribution on the feasible state space for  validating the previous hypothesis, and the experimental results are shown in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Even though  our method does not explicitly consider the previously practiced curriculum goals, it shows success  in reaching arbitrary goal points sampled from the uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Performance degrades are  observed in sawyer manipulation environments and these are because we set the uniform distribution  as areas within the tables in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' But the curriculum goals proposed by our method are  converged before the agent explores the entire state space on the table, thus the agent does not have  the opportunity to practice the goals from the entire state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  (a) U-Maze  (b) N-Maze  (c) Spiral-Maze  (d) Ant Locomotion  (e) Sawyer Push  (f) Sawyer Pick&Place  Figure 12: Episode success rates of the evaluation results with the goals uniformly sampled from  the feasible state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='3  ABLATION STUDY  We conducted ablation studies described in our main script in all environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Figure 13 shows  the average distance from the proposed curriculum goals to the desired final goal states along the  training steps, and Figure 14 shows the episode success rates along the training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' As we can  see in these figures, even though there are not many differences in some environments with simple  21  OUTPACE(ours)  SkewFit  CURROT  PLR  HGG  ALP-GMM  GoalGAN  VDS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='8  rate  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='6  success_  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
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+page_content='6  success_  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
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+page_content='0  0  200  400  600  800  1000  steps (k)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
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+page_content='6  success_  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
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+page_content='6  success_  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
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+page_content='0  0  500  1000  1500  2000  2500  3000  steps (k)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
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+page_content='6  success_  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
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+page_content='0  0  500  1000  1500  2000  2500  3000  steps (k)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='8  rate  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='6  success  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='0  0  500  1000  1500  2000  2500  3000  steps (k)Accepted as a conference paper at ICLR 2023  dynamics, we could obtain consistent analysis with the results in the main script in most of the  environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  We also visualized the curriculum goals obtained by each ablation study as training proceeds in Fig-  ure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' As we can see in these figures, the curriculum proposal only based on the uncertainty (only-  cnml) shows progress toward uncertain areas, but it shows unstable curriculum progress, which is  depicted by separated curriculum goals despite the same colors or out of order with respect to human  intuitive optimal curriculum progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' This is because the meta-learning-based inference procedure  of pCNML has some numerical errors that could lead to the wrong prediction of the states near the  boundaries of the already explored regions as uncertain areas (For example, in Figure 9, there exist  some states predicted as uncertain area despite already explored region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Also, even though the curriculum proposal based on the temporal distance only obtained by f π  φ  (only-f) shows some progress that deviates from the initial states, it still has difficulty in most of  the environments because f π  φ is trained to reflect the observed transition data rather than entire state  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' That is, the temporal distance estimation is reliable within the explored region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Once the  trained f π  φ has local optimum before discovering temporally more distant states (Figure 16), the  proposed curriculum goals can be stuck in some areas that are wrongly predicted to be the most far  from the initial states in terms of the temporal distance, and it leads to the ineffective exploration of  the agent and recurrent failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  22  Accepted as a conference paper at ICLR 2023  (a) Cost type  (b) Reward & Goal proposition type  (c) Effect of c in ηguidance  Figure 13: Ablation study in terms of the distance from the proposed curriculum goals to the desired  final goal states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' First row: Point-U-Maze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Second row: Point-N-Maze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Third row: Point-Spiral-  Maze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Fourth row: Ant Locomotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Fifth row: Sawyer Push.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content=' Sixth row: Sawyer Pick&Place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  Shading indicates a standard deviation across 5 seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='  23  12  OUTPACE(ours)  Ablation_only_f  10  Ablation only cnml  distance  8  6  4  2  0  500  1000  1500  2000  2500  3000  steps (k)12  OUTPACE(ours)  Ablation_SparseReward  10  Ablation GAN  distance  8  6  4  2  0  500  1000  1500  2000  2500  3000  steps (k)12  OUTPACE(ours)  c=1  10  C=2  C=3  distance  8  6  4  2  0  500  1000  1500  2000  2500  3000  steps (k)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
+page_content='9  OUTPACE(ours)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FKT4oBgHgl3EQfKS06/content/2301.11741v1.pdf'}
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+Comparisons of five indices for estimating local 
+terrain surface roughness using LiDAR point clouds 
+Lei Fan 
+Department of Civil Engineering, Design School 
+Xi’an Jiaotong-Liverpool University 
+Suzhou, China 
+e-mail: Lei.Fan@xjtlu.edu.cn 
+ 
+ 
+Abstract—Terrain surface roughness is an abstract concept, and 
+its quantitative description is often vague. As such, there are 
+various roughness indices used in the literature, the selection of 
+which is often challenging in applications. This study compared 
+the terrain surface roughness maps quantified by five commonly 
+used roughness indices, and explored their correlations for four 
+terrain surfaces of distinct surface complexities. These surfaces 
+were represented by digital elevation models (DEMs) constructed 
+using airborne LiDAR (Light Detection and Ranging) data. The 
+results of this study reveal the similarity in the global patterns of 
+the local surface roughness maps derived, and the distinctions in 
+their local patterns. The latter suggests the importance of 
+considering multiple indices in the studies where local roughness 
+values are the critical inputs to subsequent analyses. 
+Keywords-roughness; terrain surface; digital elevation model; 
+point cloud; LiDAR; Remote Sensing; 
+I. 
+ INTRODUCTION 
+Terrain surface roughness is an important index in 
+geoscience to describe the complexity or variability of a terrain 
+surface at a particular scale. It is used widely for many 
+applications such as digital terrain analysis, Earth surface 
+process 
+simulations and terrain classifications [1]-[6]. 
+Depending on the application requirements, global or local 
+terrain roughness may be calculated. In the case where point 
+cloud data are the source data, local terrain roughness is often 
+calculated [7]-[11]. This is because such data have fine spatial 
+resolution and record local terrain surface characteristics and 
+the spatial changes in detail. Such data can readily be obtained 
+using the measurement techniques such as Light Detection and 
+Ranging (LiDAR) and Structure from Motion and Multi-View 
+Stereo (SfM-MVS). 
+The definition of terrain surface roughness is often 
+ambiguous. Roughness indices usually rely on a quantitative 
+description of specific terrain characteristics changes, such as 
+the degree of local undulation, the degree of local folds, or the 
+degree of local abrupt changes. The indices commonly used 
+include but are not limited to root mean square height (RMSH) 
+[11], the standard deviation of residual elevation [9], the 
+standard deviation of residual topography [9], the standard 
+deviation of curvature [2], the standard deviation of slope [12]-
+[13], geostatistical analysis [14] and fractal dimension analysis 
+[15]. 
+Due to the wide range of applications of terrain surface 
+roughness and the diversity of user needs, there is not yet a 
+commonly accepted or preferred method for estimating it. In 
+practice and research, people usually consider a commonly 
+used terrain surface index at their preference or consider 
+several indices to compare the results of interest for 
+determining a more suitable index [7]-[11]. In the latter, the 
+comparisons were generally based on a visual inspection [2], 
+[9]. By now, there are few dedicated studies to investigate the 
+quantitative correlations between terrain surface roughness 
+indices, which motivated this study. In the study, five 
+commonly used roughness indices were compared using four 
+LiDAR point cloud datasets representing distinct terrain 
+surfaces of varying surface complexities. 
+II. 
+METHOD 
+A. Study data 
+Fig. 1 shows the study data, which consist of four airborne 
+LIDAR point clouds representing the bare ground surfaces of a 
+similar size of approximately 350 m by 350 m. These four 
+terrain surfaces have different spatial variation characteristics: a 
+hilly rough surface (an average data spacing of 0.67m), a hilly 
+smooth surface (an average data spacing of 0.71 m), a flat 
+rough surface (an average data spacing of 0.71 m), and a flat 
+smooth surface (an average data spacing of 0.63m). These data 
+were used in a previous study [16] and came from a large 
+LiDAR dataset acquired by the National Airborne Laser 
+Mapping Center in the USA at a volcanic field in central 
+Nevada. 
+Each point cloud data has a global trend in elevation. To 
+minimize its effect on the calculation of local surface 
+roughness and to better visualize the spatial variations of the 
+terrain surfaces, the global trend in each point cloud was 
+removed by subtracting a best-fitting plane. This led to a set of 
+residual elevations with a global mean elevation value of zero. 
+B. DEM maps 
+Some surface roughness indices (e.g. RMSH) can be 
+applied to either unstructured scattering point cloud data or 
+structured/gridded data in the form of a digital elevation model 
+(DEM). Meanwhile, there are also indices (e.g. standard 
+deviation of slope) that are typically applicable to only gridded 
+DEM maps. To keep the consistency of the comparisons 
+Accepted version of 2022 29th International Conference on 
+Geoinformatics doi:10.1109/Geoinformatics57846.2022.9963877 
+This research was funded by XJTLU Key Program Special Fund (grant 
+number KSF-E-40) and XJTLU Research Enhancement Funding (grant no. 
+REF-21-01-003). 
+
+between the roughness indices, gridded DEM maps were used 
+to calculate local surface roughness in this study. The 
+triangulation with a liner interpolation method [17] was used to 
+produce those DEM maps of a spatial grid resolution of 1 m. 
+The DEM maps constructed using the detrended point cloud 
+data are shown in Fig. 2. 
+C. Terrain surface roughness indices 
+The surface roughness indices considered in this study are 
+elaborated in the following. 
+1) RMSH: It is one of the most commonly used indices for 
+quantifying local surface roughness for scattered elevation 
+data. The definition of RMSH is given in (1) [2], [11], [18]. 
+ 
+ 
+ 
+(1) 
+ 
+where n represents the number of the data points selected; Zi is 
+the elevation value of the ith data point; 
+is the mean 
+elevation value of all (n) data points selected. 
+2) Standard deviation of locally detrended residual 
+elevations (σLDRE): In this method, the local elevation data 
+selected in a moving window are detrended linearly by fitting 
+a best-fitting plane to obtain the residual elevation values of 
+the data selected. The standard deviation of these residual 
+elevations is calculated to represent the surface roughness. 
+3) Standard deviation of residual topography (σRT): The 
+residual topography is the difference between the original 
+topography (the original DEM) and the smoothed topography 
+(the smoothed DEM) [9]. In this study, the elevation value at a 
+grid location of the smoothed DEM was obtained by averaging 
+the elevation values of its neighboring cells using a 5×5 
+moving window. As the original and the smoothed DEMs 
+have the same spatial resolution, the residual topography is 
+obtained by the arithmetic subtraction of the elevation values 
+of the corresponding cells of the two DEMs. The standard 
+deviation of the residual topography is calculated to represent 
+the surface roughness. 
+(
+)
+2
+1
+RMSE
+1
+n
+i
+i
+Z
+n
+Z
+=
+-
+=
+-
+å
+Z
+ 
+Fig. 1. The four point clouds considered: (a) Hilly Rough, (b) Hilly Smooth, (c) Flat Rough, (d) Flat Smooth. 
+ 
+Fig. 2. The DEM maps of the detrended point cloud data: (a) Hilly Rough, (b) Hilly Smooth, (c) Flat Rough, (d) Flat Smooth. 
+
+2025
+1950
+1980
+1900
+1930
+1850
+1880
+350
+350
+300
+300
+250
+350
+250
+350
+300
+300
+200
+200
+250
+250
+150
+200
+150
+200
+100
+150
+100
+150
+100
+50
+100
+50
+50
+50
+0
+0
+0
+0
+(a) Hilly Rough
+(b) Hilly Smooth
+1757
+1735
+1860
+1710
+1840
+350
+350
+300
+300
+350
+250
+350
+250
+300
+300
+200
+200
+250
+250
+150
+200
+150
+200
+100
+150
+100
+150
+100
+50
+100
+50
+50
+50
+0
+0
+0
+0
+(c) Flat Rough
+(d) FlatSmooth 20
+10
+5
+20 
+300
+300
+300
+300
+4
+10
+10 
+250
+250
+250
+250
+200
+200
+0
+0
+(m
+2
+√ 150
+≥ 150
+-5
+-10 
+ 150
+-10 
+100
+100
+100
+-10
+100
+-20
+0
+50 
+50
+50
+50
+-20
+-15
+-30
+0
+0
+70
+200
+300
+200
+300
+100
+0
+100
+100
+200
+300
+100
+200
+300
+0
+0
+X(m)
+X(m)
+X(m)
+X(m)
+(a) Hilly Rough
+(b) Hilly Smooth
+(c) Flat Rough
+(d) Flat Smooth4) Standard deviation of slope (σslope): To calculate the 
+standard deviation of slope, it is necessary to first calculate the 
+slope values. Slope is defined as the rate of change of terrain 
+elevations, which is expressed in (2). 
+ 
+ 
+ 
+(2) 
+ 
+where dz/dx and dz/dy represent the rate of change in the x and 
+the y directions, respectively, for the cell of interest. 
+Based on a DEM, slope is often calculated using the 3×3 
+moving window shown in Fig. 3. dz/dx and dz/dy for the central 
+cell (Z5) are calculated using (3) and (4), respectively, in which 
+L represents the cell size. When there are cells in the 
+neighborhood (e.g. at the edge of a DEM) that do not contain 
+elevation data, those cells are assumed to have the same 
+elevation value as the central cell. This is useful for the cells at 
+the edge of a DEM raster to ensure that the slope map has the 
+same spatial extent as the DEM map. 
+ 
+dz/dx = [(Z3+2Z6+Z9) – (Z1+2Z4+Z7)]/(8L) 
+(3) 
+ 
+dz/dy = [(Z7+2Z8+Z9) – (Z1+2Z2+Z3)]/(8L) 
+(4) 
+5) Standard deviation of curvature (σcurvature): Curvature is 
+calculated by computing the second derivative of a DEM 
+raster map. The moving window used is the same as that (i.e. 
+Fig. 3) for calculating slope. There are various methods for 
+calculating curvature. In this study, the one proposed by 
+Zevenbergen and Thorne [19]-[20] was adopted, which is 
+described in (5) - (7). Similar to the calculation of slope, the 
+non-value cells in the neighborhood are assumed to have the 
+same elevation value as the central cell. Once the curvature 
+raster map is obtained, the standard deviation of curvature is 
+calculated to represent the surface roughness. 
+ 
+curvature = 2E + 2D 
+(5) 
+where E and D are given in (6) and (7), respectively, using the 
+elevation values within the moving window shown in Fig. 3. 
+ 
+D = [(Z4 + Z6)/2 – Z5]/L2 
+(6) 
+ 
+E = [(Z2 + Z8)/2 – Z5]/L2 
+(7) 
+D. The spatial scale for local terrain surface roughness 
+The local terrain surface roughness is often calculated at a 
+particular scale defined by users, which is implemented using 
+non-overlapping moving windows. Following a survey of the 
+typical scales used in the literature for computing local surface 
+roughness using DEM maps derived from point cloud data, the 
+following set of non-overlapping moving window sizes (3×3, 
+5×5, 7×7, 9×9 and 11×11) were considered in this study to look 
+into effects of the spatial scale on the local surface roughness 
+maps produced by the different roughness indices. 
+III. 
+RESULTS 
+Fig. 4 shows the local terrain surface roughness maps of the 
+five surface roughness indices considered, based on a spatial 
+scale of 5×5 cells as an example. This is also a typical spatial 
+scale that was commonly considered in the literature. Those 
+derived using the other spatial scales are not shown here due to 
+the page limit. In Fig. 4, each individual column of the plots 
+represents a different roughness index while each row 
+represents one of the four terrain surfaces considered. It should 
+be noted that the roughness values shown in Fig. 4 were 
+normalized to range from 0 to 1 for a better visual comparison. 
+It was observed that most of the roughness indices 
+produced local terrain surface roughness maps of similar global 
+patterns, justifying the use of these indices in the literature. To 
+confirm this, the correlation coefficient r for each pair of 
+roughness maps/images compared was calculated using (8) for 
+the Hilly Rough data as an example. The values of correlation 
+coefficient are shown in TABLE I where only those in the 
+upper triangular are shown due to due to symmetry. Apart from 
+RMSH, large correlations were found between two roughness 
+maps derived using the other indices. The largest correlation 
+value (i.e. 0.95) was observed for the pair σRT and σcurvature, 
+which is consistent with the global patterns of the roughness 
+maps for these two indices as shown in Fig. 4.   
+ 
+ (8) 
+ 
+where A and B represent the values of the roughness images 
+compared; the subscript m and n refer to the pixel location in 
+the images; 
+ and 
+represent the mean value.  
+ 
+2
+2
+1
+slope
+tan
+dz
+dz
+dx
+dy
+-
+=
++
+æ
+ö
+æ
+ö
+æ
+ö
+ç
+÷
+ç
+÷
+ç
+÷
+ç
+÷
+è
+ø
+è
+ø
+è
+ø
+(
+)(
+)
+(
+)
+(
+)
+2
+2
+mn
+mn
+m
+n
+mn
+mn
+m
+n
+m
+n
+A
+A
+B
+B
+r
+A
+A
+B
+B
+-
+-
+=
+æ
+öæ
+ö
+-
+-
+ç
+֍
+÷
+è
+øè
+ø
+åå
+åå
+åå
+A
+B
+ 
+Fig. 3.  The 3×3 moving window for calculating slopes using a DEM 
+
+Z1
+Z2
+Z3
+Z4
+Z5
+Z6
+Z7
+Z8
+ZgTABLE I.  THE CORRELATION COEFFICIENT FOR EACH PAIR OF ROUGHNESS 
+MAPS COMPARED, BASED ON THE HILLY ROUGH DATA 
+Roughness 
+Index 
+Roughness Index 
+RMSH 
+ 
+ 
+ 
+ 
+RMSH 
+1 
+0.59 
+0.72 
+0.52 
+0.71 
+ 
+ 
+1 
+0.83 
+0.81 
+0.88 
+ 
+ 
+1 
+0.76 
+0.95 
+ 
+ 
+1 
+0.74 
+ 
+ 
+1 
+ 
+However, the distributions of the roughness values locally 
+vary between those indices. Some (e.g. σRT and σcurvature) exhibit 
+similar local distributions while some others (e.g. RMSH and 
+σslope) are notably different. These variations between the 
+indices suggest that the choice of roughness indices can affect 
+the results of a subsequent analysis because many quantitative 
+studies involving local surface roughness are often based on 
+local roughness values. As such, it would be useful to consider 
+a sensitivity analysis using multiple roughness indices in the 
+quantitative studies where the local roughness values are the 
+critical inputs.  
+Another means of quantitative estimation of the correlation 
+between roughness maps is the coefficient of determination R2, 
+which was calculated using all pixel values of one roughness 
+map against the corresponding pixel values of another 
+roughness map being compared. Two representative datasets 
+(i.e. the most complex surface Hilly Rough and the least 
+complex surface Flat Smooth) were used for the analysis, in 
+which five local scales (i.e. moving windows of 3×3, 5×5, 7×7, 
+9×9 and 11×11 cells) were used. The results are summarized 
+graphically in Fig. 5. The overall correlations between two 
+indices compared appeared to be slightly larger for the rougher 
+surface (i.e. Hilly Rough).  
+The correlation between RMSH and any one of the other 
+four roughness indices was found to be relatively small, as 
+illustrated in Fig. 5. This can also be inferred from the 
+roughness maps in Fig. 4. It is interesting to observe in Fig. 4 
+that the roughness values of RMSH are more spatially coherent 
+than those of the other indices. The likely reason for these is 
+the presence of local elevation trends, which can lead to 
+stronger local spatial autocorrelation in the maps of RMSH. 
+This explanation can partly be justified by the map of σLDRE, 
+which share the same algorithm as RMSH but use locally de-
+trended elevation values.  
+The effects of the spatial scales on the correlations between 
+the roughness indices are shown in Fig. 5. A mixed behavior 
+was observed. For the pairs of roughness indices compared, a 
+larger window size may increase or reduce their correlations. 
+However, based on the results shown in Fig. 5, it appears that 
+the effect of the window size is smaller for the rougher terrain 
+surface (i.e. Hilly Rough). 
+LDRE
+s
+RT
+s
+slope
+s
+curvature
+s
+LDRE
+s
+RT
+s
+slope
+s
+curvature
+s
+ 
+Fig. 4.  The local roughness maps for the four terrain surfaces considered. 
+
+Standard deviation of locally
+Standard deviation of
+RMSH
+detrended residual elevations
+residual topography
+Standard deviation of slope
+Standard deviation of curvature
+300
+300
+300
+300
+300
+0.8
+0.8
+10.8
+0.8
+0.8
+ Rough
+0.6
+0.6
+0.6
+0.6 
+200
+200
+200
+200
+200
+0.6
+Hilly I
+0.4 
+0.4
+0.4 
+0.4 
+0.4
+100
+100
+100
+100
+100
+0.2 
+0.2
+0.2
+0.2
+0.2 
+0
+0
+0
+0
+100
+200300
+0
+100200300
+100200
+300
+200300
+100200300
+0
+0
+300
+300
+300
+300
+300
+0.8 
+0.8
+0.8
+0.8
+0.8
+ Smooth
+200
+0.6 
+0.6 
+200
+0.6
+0.6
+200
+200
+200
+0.6
+Hilly
+0.4 
+0.4 
+0.4 
+0.4 
+0.4 
+100
+100
+100
+100
+100
+0.2
+0.2
+0.2 
+0.2
+0.2
+0
+0
+0
+0
+100
+200
+300
+0
+100
+200
+300
+0
+100
+200
+300
+0
+100 
+200
+300
+0
+100
+200
+300
+300
+300
+300
+300
+300
+0.8
+0.8
+0.8
+0.8
+0.8
+Rough
+0.6
+0.6
+0.6 
+0.6
+200
+200
+200
+200
+200
+0.6
+Flat F
+0.4 
+0.4
+0.4 
+0.4 
+0.4
+100
+100
+100
+100
+100
+0.2 
+0.2
+0.2 
+0.2 
+0.2 
+0
+0
+0
+300
+100
+200
+100
+200300
+200
+300
+100
+300
+100
+200300
+0
+100
+0
+200
+0
+300
+300
+300
+300
+300
+0.8
+0.8
+0.8
+0.8
+0.8
+Smooth
+0.6 
+0.6 
+0.6
+0.6
+200
+200
+200
+200
+200
+0.6
+0.4 
+0.4
+0.4 
+Flat 
+0.4
+0.4
+100
+100
+100
+100
+100
+0.2
+0.2 
+0.2
+0.2 
+0.2 
+0
+0
+0
+0 
+0
+0
+100
+200
+300
+0
+100
+200
+300
+0
+100
+200
+300
+0
+100
+200
+300
+0
+100
+200
+300IV. 
+CONCLUSIONS 
+For the four terrain surfaces of varying complexities, the 
+local roughness maps produced by the roughness indices 
+considered showed similar global patterns, demonstrating their 
+capability of characterizing local terrain surface roughness and 
+their suitability for applications. However, it was also found 
+that the local distributions of the roughness values varied 
+notably between some roughness indices, suggesting the 
+importance of considering multiple roughness indices in the 
+studies where local roughness values are the input for a 
+subsequent analysis. This is particularly the case for RMSH as 
+its correlations with the other four indices considered were 
+found to be small. The effects of local scales (i.e. moving 
+window sizes) on the correlation between two roughness 
+indices compared were briefly investigated, which showed a 
+mixed behavior (i.e. either decreased or increased the 
+correlation). 
+ACKNOWLEDGMENT 
+LiDAR data access is based on [LiDAR, ground] services 
+provided by the OpenTopography Facility with support from 
+the National Science Foundation under NSF Award Numbers 
+1226353 & 1225810. Lidar data acquisition completed by the 
+National Center for Airborne Laser Mapping (NCALM - 
+http://www.ncalm. org). NCALM funding provided by NSF’s 
+Division of Earth Sciences, Instrumentation and Facilities 
+Program. EAR-1043051. https://doi.org/10.5069/G9PR7SX0 
+REFERENCES 
+[1] 
+J. A. Fernando, A. Francisco, A. A. Manuel and C. Fernando, “Effects of 
+terrain morphology, sampling density, and interpolation methods on grid 
+DEM accuracy,” Photogrammetric Engineering & Remote Sensing, vol. 
+71(7), pp. 805-816, 2005. doi:10.14358/pers.71.7.805 
+[2] 
+C. H. Grohmann, M. J. Smith, and C. Riccomini, “Multiscale analysis of 
+topographic surface roughness in the Midland Valley, Scotland,” IEEE 
+Transactions on Geoscience and Remote Sensing, vol. 49(4), pp. 1200-
+1213, 2011. doi:10.1109/tgrs.2010.2053546  
+[3] 
+J. B. Peter, H. S. Vernon, S. Jiro and J. L. Stephen, “Methods for 
+Remote Engineering Geology Terrain Analysis in Boreal Forest Regions 
+of Ontario, Canada,” Environmental and Engineering Geoscience, vol. 
+10(3), pp. 229-241, 2004. doi:10.2113/10.3.229  
+[4] 
+L. Fan and P. M. Atkinson, “Accuracy of digital elevation models 
+derived from terrestrial laser scanning data,” IEEE Geoscience and 
+Remote Sensing Letters, vol. 12(9), pp. 1923-1927, 2015, doi: 
+10.1109/LGRS.2015.2438394 
+[5] 
+L. Fan, Uncertainty in terrestrial laser scanning for measuring surface 
+movements at a local scale. University of Southampton: Southampton, 
+UK, 2014.  
+[6] 
+L. Fan, W. Powrie, J. A. Smethurst, P. M. Atkinson and H. Einstein, 
+“The effect of short ground vegetation on terrestrial laser scans at a local 
+scale,’ ISPRS Journal of Photogrammetry and Remote Sensing, vol.95, 
+pp. 42-52, 2014. doi: 10.1016/j.isprsjprs.2014.06.003 
+[7] 
+K. L. Frankel and J. F. Dolan, “Characterizing arid region alluvial fan 
+surface roughnesswith airborne laser swathmapping digital topographic 
+data,” Journal of Geophysical Research F, vol. 112(2), Article ID 
+F02025, 2007. doi: 10.1029/2006JF000644 
+[8] 
+J. M. Nield and G. F. S. Wiggs, “The application of terrestrial laser 
+scanning to aeolian saltation cloud measurement and its response to 
+changing surface moisture,” Earth Surf. Processes Landforms, vol. 36(2), 
+pp. 273-278, 2011. doi:10.1002/esp.2102 
+[9] 
+M. B. Kristen, L. M. Wayne, J. D. Patrick, A. M. Douglas and W. B. 
+Elizabeth, “The Use of LiDAR Terrain Data in Characterizing Surface 
+Roughness and Microtopography,” Applied and Environmental Soil 
+Science, vol. 2013, pp. 1–13, 2013. doi:10.1155/2013/891534  
+[10] M. Milenković, N. Pfeifer and P. Glira, “Applying Terrestrial Laser 
+Scanning for Soil Surface Roughness Assessment,” Remote Sensing, vol. 
+7, pp. 2007-2045, 2015. doi: 10.3390/rs70202007  
+[11] L. Fan, “A comparison between structure-from-motion and terrestrial 
+laser scanning for deriving surface roughness: A case study on a sandy 
+terrain surface,” The International Archives of Photogrammetry, Remote 
+Sensing and Spatial Information Sciences, vol. 42, pp. 1225-1229, 2020. 
+doi: 10.5194/isprs-archives-XLII-3-W10-1225-2020 
+ 
+Fig. 5. The coefficient of determination for paired roughness indices under different window sizes: (a) Hilly Rough and (b) Flat Smooth. 
+
+0.9
+0.9
+0.8
+0.8
+0.7
+0.7
+-△ - RMSH vs. 0.
+slope
+元
+90 %
+ 0.6
+values
+values
+0.4
+0.44
+0.3
+0.3
+0.2 
+0.2
+0.1
+0.1
+3x3
+5x5
+7×7
+9x9
+11x11
+3x3
+5x5
+7x7
+9x9
+11x11
+Moving Window Size
+Moving Window Size
+(a) Hilly Rough
+(b) Flat Smooth[12] M. N. Joanna and F. S. W. Giles, “The application of terrestrial laser 
+scanning to aeolian saltation cloud measurement and its response to 
+changing surface moisture,” Earth Surface Processes and Landforms, vol. 
+36(2), pp. 273-278, 2010. doi:10.1002/esp.2102  
+[13] L. F. Kurt and F. D. James, “Characterizing arid region alluvial fan 
+surface roughness with airborne laser swath mapping digital topographic 
+data,” Journal of Geophysical Research, vol. 112(F2), pp. 1-14, 2007. 
+doi:10.1029/2006jf000644  
+[14] C. H. Huang and J. M. Bradford, “Applications of a Laser Scanner to 
+Quantify Soil Microtopography,” Soil Science Society of America 
+Journal, 
+vol. 
+56(1), 
+14, 
+1992. 
+doi: 
+10.2136/sssaj1992.03615995005600010002x  
+[15] R. Andrle and A. D. Abrahams, “Fractal techniques and the surface 
+roughness of talus slopes,” Earth Surface Processes and Landforms, vol. 
+14(3), pp. 197–209, 1989. doi:10.1002/esp.3290140303  
+[16] L. Fan and P. M. Atkinson, “An Iterative Coarse-to-Fine Sub-Sampling 
+Method for Density Reduction of Terrain Point Clouds,” Remote 
+Sensing, vol. 11(8), pp. 947, 2019. doi: 10.3390/rs11080947 
+[17] L. Fan, J. A. Smethurst, P. M. Atkinson and W. Powrie, “Propagation of 
+vertical and horizontal source data errors into a TIN with linear 
+interpolation,” International Journal of Geographical Information 
+Science, 
+vol. 
+28(7), 
+pp. 
+1378-1400, 
+2014. 
+doi: 
+10.1080/13658816.2014.889299 
+[18] L. Fan and P. M. Atkinson, “A new multi-resolution based method for 
+estimating local surface roughness from point clouds,” ISPRS Journal of 
+Photogrammetry and Remote Sensing, vol. 144, pp. 369–378, 2018. 
+doi:10.1016/j.isprsjprs.2018.08.003 
+[19] L. W. Zevenbergen and C. R. Thorne, “Quantitative analysis of land 
+surface topography,” Earth Surface Processes and Landforms, vol. 12(1), 
+pp. 47–56, 1987. doi:10.1002/esp.3290120107  
+[20] I. D. Moore, R. B. Grayson and A. R. Ladson, A. R., “Digital terrain 
+modelling: A review of hydrological, geomorphological, and biological 
+applications,” Hydrological Processes, vol. 5(1), pp. 3–30, 1991. doi: 
+10.1002/hyp.3360050103  
+ 
+ 
+
diff --git a/utE0T4oBgHgl3EQfbgCl/content/tmp_files/load_file.txt b/utE0T4oBgHgl3EQfbgCl/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf,len=633
+page_content='Comparisons of five indices for estimating local  terrain surface roughness using LiDAR point clouds  Lei Fan  Department of Civil Engineering, Design School  Xi’an Jiaotong-Liverpool University  Suzhou, China  e-mail: Lei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='Fan@xjtlu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='cn  Abstract—Terrain surface roughness is an abstract concept, and  its quantitative description is often vague.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' As such, there are  various roughness indices used in the literature, the selection of  which is often challenging in applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' This study compared  the terrain surface roughness maps quantified by five commonly  used roughness indices, and explored their correlations for four  terrain surfaces of distinct surface complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' These surfaces  were represented by digital elevation models (DEMs) constructed  using airborne LiDAR (Light Detection and Ranging) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' The  results of this study reveal the similarity in the global patterns of  the local surface roughness maps derived, and the distinctions in  their local patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' The latter suggests the importance of  considering multiple indices in the studies where local roughness  values are the critical inputs to subsequent analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  Keywords-roughness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' terrain surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' digital elevation model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  point cloud;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' LiDAR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Remote Sensing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  INTRODUCTION  Terrain surface roughness is an important index in  geoscience to describe the complexity or variability of a terrain  surface at a particular scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' It is used widely for many  applications such as digital terrain analysis, Earth surface  process  simulations and terrain classifications [1]-[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  Depending on the application requirements, global or local  terrain roughness may be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' In the case where point  cloud data are the source data, local terrain roughness is often  calculated [7]-[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' This is because such data have fine spatial  resolution and record local terrain surface characteristics and  the spatial changes in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Such data can readily be obtained  using the measurement techniques such as Light Detection and  Ranging (LiDAR) and Structure from Motion and Multi-View  Stereo (SfM-MVS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  The definition of terrain surface roughness is often  ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Roughness indices usually rely on a quantitative  description of specific terrain characteristics changes, such as  the degree of local undulation, the degree of local folds, or the  degree of local abrupt changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' The indices commonly used  include but are not limited to root mean square height (RMSH)  [11], the standard deviation of residual elevation [9], the  standard deviation of residual topography [9], the standard  deviation of curvature [2], the standard deviation of slope [12]-  [13], geostatistical analysis [14] and fractal dimension analysis  [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  Due to the wide range of applications of terrain surface  roughness and the diversity of user needs, there is not yet a  commonly accepted or preferred method for estimating it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' In  practice and research, people usually consider a commonly  used terrain surface index at their preference or consider  several indices to compare the results of interest for  determining a more suitable index [7]-[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' In the latter, the  comparisons were generally based on a visual inspection [2],  [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' By now, there are few dedicated studies to investigate the  quantitative correlations between terrain surface roughness  indices, which motivated this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' In the study, five  commonly used roughness indices were compared using four  LiDAR point cloud datasets representing distinct terrain  surfaces of varying surface complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  METHOD  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Study data  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 1 shows the study data, which consist of four airborne  LIDAR point clouds representing the bare ground surfaces of a  similar size of approximately 350 m by 350 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' These four  terrain surfaces have different spatial variation characteristics: a  hilly rough surface (an average data spacing of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='67m), a hilly  smooth surface (an average data spacing of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='71 m), a flat  rough surface (an average data spacing of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='71 m), and a flat  smooth surface (an average data spacing of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='63m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' These data  were used in a previous study [16] and came from a large  LiDAR dataset acquired by the National Airborne Laser  Mapping Center in the USA at a volcanic field in central  Nevada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  Each point cloud data has a global trend in elevation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' To  minimize its effect on the calculation of local surface  roughness and to better visualize the spatial variations of the  terrain surfaces, the global trend in each point cloud was  removed by subtracting a best-fitting plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' This led to a set of  residual elevations with a global mean elevation value of zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' DEM maps  Some surface roughness indices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' RMSH) can be  applied to either unstructured scattering point cloud data or  structured/gridded data in the form of a digital elevation model  (DEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Meanwhile, there are also indices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' standard  deviation of slope) that are typically applicable to only gridded  DEM maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' To keep the consistency of the comparisons  Accepted version of 2022 29th International Conference on  Geoinformatics doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='1109/Geoinformatics57846.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='9963877  This research was funded by XJTLU Key Program Special Fund (grant  number KSF-E-40) and XJTLU Research Enhancement Funding (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  REF-21-01-003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  between the roughness indices, gridded DEM maps were used  to calculate local surface roughness in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' The  triangulation with a liner interpolation method [17] was used to  produce those DEM maps of a spatial grid resolution of 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  The DEM maps constructed using the detrended point cloud  data are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Terrain surface roughness indices  The surface roughness indices considered in this study are  elaborated in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  1) RMSH: It is one of the most commonly used indices for  quantifying local surface roughness for scattered elevation  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' The definition of RMSH is given in (1) [2], [11], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  (1)  where n represents the number of the data points selected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Zi is  the elevation value of the ith data point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  is the mean  elevation value of all (n) data points selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  2) Standard deviation of locally detrended residual  elevations (σLDRE): In this method, the local elevation data  selected in a moving window are detrended linearly by fitting  a best-fitting plane to obtain the residual elevation values of  the data selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' The standard deviation of these residual  elevations is calculated to represent the surface roughness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  3) Standard deviation of residual topography (σRT): The  residual topography is the difference between the original  topography (the original DEM) and the smoothed topography  (the smoothed DEM) [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' In this study, the elevation value at a  grid location of the smoothed DEM was obtained by averaging  the elevation values of its neighboring cells using a 5×5  moving window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' As the original and the smoothed DEMs  have the same spatial resolution, the residual topography is  obtained by the arithmetic subtraction of the elevation values  of the corresponding cells of the two DEMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' The standard  deviation of the residual topography is calculated to represent  the surface roughness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  (  )  2  1  RMSE  1  n  i  i  Z  n  Z  = = å  Z  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' The four point clouds considered: (a) Hilly Rough, (b) Hilly Smooth, (c) Flat Rough, (d) Flat Smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' The DEM maps of the detrended point cloud data: (a) Hilly Rough, (b) Hilly Smooth, (c) Flat Rough, (d) Flat Smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='2025  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='1950  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='1980  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
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+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='X(m)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='X(m)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='X(m)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='X(m)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='(a) Hilly Rough  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='(b) Hilly Smooth  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='(c) Flat Rough  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='(d) Flat Smooth4) Standard deviation of slope (σslope): To calculate the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='standard deviation of slope,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' it is necessary to first calculate the  slope values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Slope is defined as the rate of change of terrain  elevations, which is expressed in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  (2)  where dz/dx and dz/dy represent the rate of change in the x and  the y directions, respectively, for the cell of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  Based on a DEM, slope is often calculated using the 3×3  moving window shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' dz/dx and dz/dy for the central  cell (Z5) are calculated using (3) and (4), respectively, in which  L represents the cell size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' When there are cells in the  neighborhood (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' at the edge of a DEM) that do not contain  elevation data, those cells are assumed to have the same  elevation value as the central cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' This is useful for the cells at  the edge of a DEM raster to ensure that the slope map has the  same spatial extent as the DEM map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  dz/dx = [(Z3+2Z6+Z9) – (Z1+2Z4+Z7)]/(8L)  (3)  dz/dy = [(Z7+2Z8+Z9) – (Z1+2Z2+Z3)]/(8L)  (4)  5) Standard deviation of curvature (σcurvature): Curvature is  calculated by computing the second derivative of a DEM  raster map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' The moving window used is the same as that (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 3) for calculating slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' There are various methods for  calculating curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' In this study, the one proposed by  Zevenbergen and Thorne [19]-[20] was adopted, which is  described in (5) - (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Similar to the calculation of slope, the  non-value cells in the neighborhood are assumed to have the  same elevation value as the central cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Once the curvature  raster map is obtained, the standard deviation of curvature is  calculated to represent the surface roughness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  curvature = 2E + 2D  (5)  where E and D are given in (6) and (7), respectively, using the  elevation values within the moving window shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  D = [(Z4 + Z6)/2 – Z5]/L2  (6)  E = [(Z2 + Z8)/2 – Z5]/L2  (7)  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' The spatial scale for local terrain surface roughness  The local terrain surface roughness is often calculated at a  particular scale defined by users, which is implemented using  non-overlapping moving windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Following a survey of the  typical scales used in the literature for computing local surface  roughness using DEM maps derived from point cloud data, the  following set of non-overlapping moving window sizes (3×3,  5×5, 7×7, 9×9 and 11×11) were considered in this study to look  into effects of the spatial scale on the local surface roughness  maps produced by the different roughness indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  RESULTS  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 4 shows the local terrain surface roughness maps of the  five surface roughness indices considered, based on a spatial  scale of 5×5 cells as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' This is also a typical spatial  scale that was commonly considered in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Those  derived using the other spatial scales are not shown here due to  the page limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 4, each individual column of the plots  represents a different roughness index while each row  represents one of the four terrain surfaces considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' It should  be noted that the roughness values shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 4 were  normalized to range from 0 to 1 for a better visual comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  It was observed that most of the roughness indices  produced local terrain surface roughness maps of similar global  patterns, justifying the use of these indices in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' To  confirm this, the correlation coefficient r for each pair of  roughness maps/images compared was calculated using (8) for  the Hilly Rough data as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' The values of correlation  coefficient are shown in TABLE I where only those in the  upper triangular are shown due to due to symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Apart from  RMSH, large correlations were found between two roughness  maps derived using the other indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' The largest correlation  value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='95) was observed for the pair σRT and σcurvature,  which is consistent with the global patterns of the roughness  maps for these two indices as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  (8)  where A and B represent the values of the roughness images  compared;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' the subscript m and n refer to the pixel location in  the images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  and  represent the mean value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  2  2  1  slope  tan  dz  dz  dx  dy =  +  æ  ö  æ  ö  æ  ö  ç  ÷  ç  ÷  ç  ÷  ç  ÷  è  ø  è  ø  è  ø  (  )(  )  (  )  (  )  2  2  mn  mn  m  n  mn  mn  m  n  m  n  A  A  B  B  r  A  A  B  B =  æ  öæ  ö ç  ÷ç  ÷  è  øè  ø  åå  åå  åå  A  B  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' The 3×3 moving window for calculating slopes using a DEM  Z1  Z2  Z3  Z4  Z5  Z6  Z7  Z8  ZgTABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' THE CORRELATION COEFFICIENT FOR EACH PAIR OF ROUGHNESS  MAPS COMPARED, BASED ON THE HILLY ROUGH DATA  Roughness  Index  Roughness Index  RMSH  RMSH  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='59  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='72  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='71  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='83  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='81  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='88  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='76  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='95  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='74  1  However, the distributions of the roughness values locally  vary between those indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Some (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' σRT and σcurvature) exhibit  similar local distributions while some others (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' RMSH and  σslope) are notably different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' These variations between the  indices suggest that the choice of roughness indices can affect  the results of a subsequent analysis because many quantitative  studies involving local surface roughness are often based on  local roughness values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' As such, it would be useful to consider  a sensitivity analysis using multiple roughness indices in the  quantitative studies where the local roughness values are the  critical inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  Another means of quantitative estimation of the correlation  between roughness maps is the coefficient of determination R2,  which was calculated using all pixel values of one roughness  map against the corresponding pixel values of another  roughness map being compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Two representative datasets  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' the most complex surface Hilly Rough and the least  complex surface Flat Smooth) were used for the analysis, in  which five local scales (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' moving windows of 3×3, 5×5, 7×7,  9×9 and 11×11 cells) were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' The results are summarized  graphically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' The overall correlations between two  indices compared appeared to be slightly larger for the rougher  surface (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Hilly Rough).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  The correlation between RMSH and any one of the other  four roughness indices was found to be relatively small, as  illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' This can also be inferred from the  roughness maps in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' It is interesting to observe in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 4  that the roughness values of RMSH are more spatially coherent  than those of the other indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' The likely reason for these is  the presence of local elevation trends, which can lead to  stronger local spatial autocorrelation in the maps of RMSH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  This explanation can partly be justified by the map of σLDRE,  which share the same algorithm as RMSH but use locally de-  trended elevation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  The effects of the spatial scales on the correlations between  the roughness indices are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' A mixed behavior  was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' For the pairs of roughness indices compared, a  larger window size may increase or reduce their correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  However, based on the results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
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+page_content='  CONCLUSIONS  For the four terrain surfaces of varying complexities, the  local roughness maps produced by the roughness indices  considered showed similar global patterns, demonstrating their  capability of characterizing local terrain surface roughness and  their suitability for applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' However, it was also found  that the local distributions of the roughness values varied  notably between some roughness indices, suggesting the  importance of considering multiple roughness indices in the  studies where local roughness values are the input for a  subsequent analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' This is particularly the case for RMSH as  its correlations with the other four indices considered were  found to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' The effects of local scales (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' moving  window sizes) on the correlation between two roughness  indices compared were briefly investigated, which showed a  mixed behavior (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' either decreased or increased the  correlation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  ACKNOWLEDGMENT  LiDAR data access is based on [LiDAR, ground] services  provided by the OpenTopography Facility with support from  the National Science Foundation under NSF Award Numbers  1226353 & 1225810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Lidar data acquisition completed by the  National Center for Airborne Laser Mapping (NCALM -  http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='ncalm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' NCALM funding provided by NSF’s  Division of Earth Sciences, Instrumentation and Facilities  Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' EAR-1043051.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='5069/G9PR7SX0  REFERENCES  [1]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Fernando, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Francisco, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Manuel and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Fernando, “Effects of  terrain morphology, sampling density, and interpolation methods on grid  DEM accuracy,” Photogrammetric Engineering & Remote Sensing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  71(7), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 805-816, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
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+page_content='805  [2]  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Grohmann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Smith, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Riccomini, “Multiscale analysis of  topographic surface roughness in the Midland Valley, Scotland,” IEEE  Transactions on Geoscience and Remote Sensing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 49(4), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 1200-  1213, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='1109/tgrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='2053546  [3]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Peter, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Vernon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Jiro and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Stephen, “Methods for  Remote Engineering Geology Terrain Analysis in Boreal Forest Regions  of Ontario, Canada,” Environmental and Engineering Geoscience, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  10(3), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 229-241, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='2113/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='229  [4]  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Fan and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Atkinson, “Accuracy of digital elevation models  derived from terrestrial laser scanning data,” IEEE Geoscience and  Remote Sensing Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 12(9), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 1923-1927, 2015, doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='1109/LGRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='2438394  [5]  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Fan, Uncertainty in terrestrial laser scanning for measuring surface  movements at a local scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' University of Southampton: Southampton,  UK, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  [6]  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Fan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Powrie, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Smethurst, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Atkinson and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Einstein,  “The effect of short ground vegetation on terrestrial laser scans at a local  scale,’ ISPRS Journal of Photogrammetry and Remote Sensing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='95,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 42-52, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='isprsjprs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='003  [7]  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Frankel and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Dolan, “Characterizing arid region alluvial fan  surface roughnesswith airborne laser swathmapping digital topographic  data,” Journal of Geophysical Research F, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 112(2), Article ID  F02025, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='1029/2006JF000644  [8]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Nield and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Wiggs, “The application of terrestrial laser  scanning to aeolian saltation cloud measurement and its response to  changing surface moisture,” Earth Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Processes Landforms, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 36(2),  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 273-278, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='1002/esp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='2102  [9]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Kristen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Wayne, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Patrick, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Douglas and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  Elizabeth, “The Use of LiDAR Terrain Data in Characterizing Surface  Roughness and Microtopography,” Applied and Environmental Soil  Science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 1–13, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='1155/2013/891534  [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Milenković, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Pfeifer and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Glira, “Applying Terrestrial Laser  Scanning for Soil Surface Roughness Assessment,” Remote Sensing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 2007-2045, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='3390/rs70202007  [11] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Fan, “A comparison between structure-from-motion and terrestrial  laser scanning for deriving surface roughness: A case study on a sandy  terrain surface,” The International Archives of Photogrammetry, Remote  Sensing and Spatial Information Sciences, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 42, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 1225-1229, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='5194/isprs-archives-XLII-3-W10-1225-2020  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' The coefficient of determination for paired roughness indices under different window sizes: (a) Hilly Rough and (b) Flat Smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Joanna and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Giles, “The application of terrestrial laser  scanning to aeolian saltation cloud measurement and its response to  changing surface moisture,” Earth Surface Processes and Landforms, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  36(2), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 273-278, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='1002/esp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='2102  [13] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Kurt and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' James, “Characterizing arid region alluvial fan  surface roughness with airborne laser swath mapping digital topographic  data,” Journal of Geophysical Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 112(F2), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 1-14, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='1029/2006jf000644  [14] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Huang and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Bradford, “Applications of a Laser Scanner to  Quantify Soil Microtopography,” Soil Science Society of America  Journal,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  56(1),  14,  1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='2136/sssaj1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='03615995005600010002x  [15] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Andrle and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Abrahams, “Fractal techniques and the surface  roughness of talus slopes,” Earth Surface Processes and Landforms, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  14(3), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 197–209, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='1002/esp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='3290140303  [16] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Fan and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Atkinson, “An Iterative Coarse-to-Fine Sub-Sampling  Method for Density Reduction of Terrain Point Clouds,” Remote  Sensing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 11(8), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 947, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='3390/rs11080947  [17] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Fan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Smethurst, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Atkinson and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Powrie, “Propagation of  vertical and horizontal source data errors into a TIN with linear  interpolation,” International Journal of Geographical Information  Science,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  28(7),  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  1378-1400,  2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='1080/13658816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='889299  [18] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Fan and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Atkinson, “A new multi-resolution based method for  estimating local surface roughness from point clouds,” ISPRS Journal of  Photogrammetry and Remote Sensing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 144, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 369–378, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='isprsjprs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='003  [19] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Zevenbergen and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Thorne, “Quantitative analysis of land  surface topography,” Earth Surface Processes and Landforms, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 12(1),  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 47–56, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='1002/esp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='3290120107  [20] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Moore, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Grayson and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' Ladson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=', “Digital terrain  modelling: A review of hydrological, geomorphological, and biological  applications,” Hydrological Processes, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 5(1), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' 3–30, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content=' doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='1002/hyp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
+page_content='3360050103' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQfbgCl/content/2301.02350v1.pdf'}
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf,len=2834
+page_content='MNRAS 000, 1–34 (2022)  Preprint 31 January 2023  Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  The Origin of Stars in the Inner 500 Parsecs in TNG50 Galaxies  Alina Boecker1,2★, Nadine Neumayer1, Annalisa Pillepich1, Neige Frankel1,3,4, Rahul Ramesh5,  Ryan Leaman6, Lars Hernquist7  1Max-Planck Institut für Astronomie, Königstuhl 17, D-69117 Heidelberg, Germany  2Instituto de Astrofísica de Canarias, C/ Vía Láctea s/n, E-38205 La Laguna, Spain  3Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' George Street, Toronto, ON M5S 3H8, Canada  4Department of Astronomy and Astrophysics, University of Toronto, 50 St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' George Street, Toronto, ON M5S 3H4, Canada  5Universität Heidelberg, Zentrum für Astronomie, Institut für theoretische Astrophysik, Albert-Ueberle-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2, D-69120 Heidelberg, Germany  6Department of Astrophysics, University of Vienna, Türkenschanzstrasse 17, 1180 Wien, Austria  7Harvard–Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA  Accepted 2022 December 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Received 2022 November 25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' in original form 2022 April 7  ABSTRACT  We investigate the origin of stars in the innermost 500 pc of galaxies spanning stellar masses of 5 × 108−12 M⊙ at z = 0 using the  cosmological magnetohydrodynamical TNG50 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Three different origins of stars comprise galactic centers: 1) in-situ  (born in the center), 2) migrated (born elsewhere in the galaxy and ultimately moved to the center), 3) ex-situ (accreted from other  galaxies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In-situ and migrated stars dominate the central stellar mass budget on average with 73% and 23% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The  ex-situ fraction rises above 1% for galaxies ≳ 1011 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Yet, only 9% of all galaxies exhibit no ex-situ stars in their centers and  the scatter of ex-situ mass is significant (4−6 dex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Migrated stars predominantly originate closely from the center (1−2 kpc), but  if they travelled together in clumps distances reach ∼ 10 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Central and satellite galaxies possess similar amounts and origins  of central stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Star forming galaxies (≳ 1010 M⊙) have on average more ex-situ mass in their centers than quenched ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  We predict readily observable stellar population and dynamical properties: 1) migrated stars are distinctly young (∼ 2 Gyr) and  rotationally supported, especially for Milky Way mass galaxies, 2) in-situ stars are most metal-rich and older than migrated stars,  3) ex-situ stars are on random motion dominated orbits and typically the oldest, most metal-poor and 𝛼-enhanced population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We  demonstrate that the interaction history with other galaxies leads to diverse pathways of building up galaxy centers in a ΛCDM  universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Our work highlights the necessity for cosmological context in formation scenarios of central galactic components and  the potential to use galaxy centers as tracers of overall galaxy assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Key words: methods: numerical – galaxies: formation – galaxies: evolution – galaxies: stellar content – galaxies: structure –  galaxies: nuclei – galaxies: bulges  1 INTRODUCTION  The center of a galaxy depicts its brightest and densest region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Thus  observations of galaxy centers provide us with the highest data qual-  ity, which should enable us to make the most precise predictions  about their formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' On the other hand, being also the deepest  point of the potential well, the center witnessed the galaxy’s overall  stellar assembly from the earliest cosmic times onward, as understood  from the inside-out formation scenario of galaxies within a ΛCDM  (Lambda-Cold-Dark-Matter) Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Therefore, many transforma-  tive processes of galaxy evolution influence a galaxy’s center until  the present day, which need to be taken into account to uniquely  interpret even the highest quality observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  As a consequence, a variety of central stellar structures are found  in galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Decreasing in size from the order of one kpc to sub  parsec scales, these range from bars and (pseudo)bulges (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Laurikainen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Kormendy & Kennicutt 2004, for a sum-  mary), which can include other structures such as nuclear rings and  ★ E-mail: aboecker@iac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='de  disks, to nuclear star clusters (NSCs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Neumayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2020,  for a summary) and supermassive black holes (SMBHs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Ko-  rmendy & Ho 2013, for a summary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Some galaxies may exhibit  more than one of these components or none at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Many of these  components possess scaling relations of their structural parameters,  such as the Sérsic (1968) index and effective radius of bulges (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Gadotti 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Fisher & Drory 2010) and the luminosity/mass-size  relation of NSCs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Böker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Côté et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Georgiev  & Böker 2014), as well as scaling relations with each other, such as  the bulge-SMBH-mass (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Häring & Rix 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Sani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Läsker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2016) and NSC-SMBH-mass relations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Ferrarese  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Georgiev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2016), which also scale with the stellar  mass of their underlying host galaxy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Scott & Graham 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Reines & Volonteri 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Sánchez-Janssen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Some of  these scaling relations can differ for early-type and late-type galax-  ies, or depend on the bulge type or the presence of a bar (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Gadotti  & Kauffmann 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Georgiev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Sahu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  As diverse as the structural properties of central components are, so  are the formation scenarios trying to explain them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Broadly speaking,  © 2022 The Authors  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='11942v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='GA]  27 Jan 2023  2  Boecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  all of these formation scenarios can be divided into internal and  external processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For example, bulges are thought to form from  merger events (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2009, 2010), from rapid early-on  star formation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Guedes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Okamoto 2013), from secular  evolution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Kormendy & Kennicutt 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Athanassoula 2005) or  from the migration of clumps formed in the disk at high redshift (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Elmegreen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Dekel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' bars form through disk  instabilities either in isolation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Bottema 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Athanassoula et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2013) or in a cosmological context (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Romano-Díaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Kraljic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Peschken & Łokas 2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' nuclear star clusters  are thought to form through either star formation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Maciejewski  2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Aharon & Perets 2015) or through the migration and successive  merging of globular clusters in the center (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Hartmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Agarwal & Milosavljević 2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' SMBHs can grow by accreting gas  and by merging with other SMBHs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Croton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Malbon  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Fanidakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Lapiner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In many  cases, the formation of any one component will also influence the  others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For example, once a bar is formed it can re-arrange the orbits  of stars causing radial migration, or it can efficiently funnel gas to  the center, which can trigger star formation in the center and also  feed the SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In turn, the AGN (active galactic nucleus) feedback  caused by the SMBH will then influence the gas supply and hence  truncate the formation of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Thus, it is important to also understand  the interplay between the presence and formation of several central  components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Observationally, we can only indirectly deduce constraints on any  of these formation scenarios from the stellar population and dynam-  ical properties of a galaxy’s central structure(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For external galax-  ies, such necessary measurements are only possible with integral  field units (IFUs) that provide spatially resolved stellar population  and kinematical maps (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Gadotti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Bittner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  While major progress has been made in producing these maps with  increasing quality, it is still difficult to disentangle stars from centrally  overlapping galaxy components due to the line-of-sight integration  let alone identify stars of different origins within a given central  component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This is possibly further complicated by the fact that  stars with properties characteristic of one formation scenario might  be subdominant in luminosity or mass compared to the bulk stellar  population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Even in the Milky Way, it has only become evident fairly recently  that all major central components contain metal-poor subpopula-  tions of stars that also exhibit different kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For the Galactic  bulge (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Barbuy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2018, for a summary) there is a smooth  transition from rotation to dispersion dominated kinematics for stars  decreasing from (super-)solar metallicity all the way to the lowest  metallicities ([Fe/H] < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0 dex) (Ness et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Zoccali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Arentsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' To a lesser extent this decrease is also  seen for the nuclear stellar disk (Schultheis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2021) with addi-  tional evidence of recent star formation activity (< 1 Gyr) on top of  the overall old bulk population (> 8 Gyr) (Nogueras-Lara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2020,  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The nuclear star cluster, which hosts the most metal-rich stars  in the Milky Way, also has a subpopulation of sub-solar metallicity  stars, which show an asymmetric spatial distribution and a higher  degree of rotation (Feldmeier-Krause et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Do et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Generally, signs of young, metal-rich and kinematically cold stars  in these central structures such as bulges and NSCs, are associated  with being formed in-situ from gas infall, while old, metal-poor  and dispersion dominated systems are thought to originate from  merger processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' However, stars formed in-situ at the beginning  of a galaxy’s lifetime are also metal-poor and might as well become  dispersion dominated over time through various processes, such as  resonances created by the bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Therefore, even though observed prop-  erties of stars in the centers of galaxies act as a fossil record of their  origin, we need simulations to disentangle which (combinations of)  formation scenarios are able to predict those observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Cosmological, hydrodynamical galaxy simulations (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Somerville & Davé 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Vogelsberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2020, for a summary)  are ideal to study the complex formation pathways of galaxy centers  as they encompass the most complete conglomeration of galaxy for-  mation processes in a ΛCDM framework, thus capturing internal and  external formation processes alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The most recent simulations are  able to produce a realistic, diverse population of galaxies (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Vogelsberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2014b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2019a, and references therein  for Illustris/TNG specifically).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Typically, large simulation boxes are  used to study global galaxy properties across an array of different  galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Illustris: Genel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Vogelsberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2014a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  EAGLE: Schaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Crain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Horizon-AGN: Dubois  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2014, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Magneticum: Hirschmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Teklu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Bocquet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' IllustrisTNG: Weinberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Pillepich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' SIMBA: Davé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2019), while zoom-in (re-  )simulations focus on internal galaxy structures and dynamics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  ERIS: Guedes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' NIHAO: Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Latte: Wetzel  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Auriga: Grand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' FIRE-2: Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  NIHAO-UHD: Buck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' To understand the mass build-up of  galaxy centers we need the advantages of both: a big enough box to  probe many different assembly histories and thus galaxy demograph-  ics, and a zoom-in like resolution to focus on the center of galaxies  and capture internal dynamical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  We therefore focus our analysis on the origin of stars in the central  few hundred parsecs of galaxies in TNG50 (Pillepich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Nel-  son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2019b) from the IllustrisTNG simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='73 cMpc3  volume captures two 1014 M⊙ halos and hundreds of Milky Way like  galaxies, whereas the spatial resolution provides hundreds to tens of  thousands stellar particles inside the central 500 pc for a four dex  range in galaxy stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Importantly, TNG50 starts to capture  the diversity of central components, such as low and high Sérsic  index bulges in Milky Way like galaxies (Gargiulo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2021), and  performs well in a statistical comparison of simulated and observed  bar properties (Rosas-Guevara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Frankel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' both  which were previously not possible with zoom-in simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Hence,  TNG50 offers the unique opportunity to study the contribution of  stars with different (internal or external) origins to the formation of  the galaxy center across diverse galaxy formation pathways and de-  mographics, while predicting the observable imprint that the different  formation scenarios impose on the stars in a galaxy’s center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The goal of this study is to appeal to different scientific commu-  nities that focus on various central stellar structures of the Milky  Way and external galaxies to provide an understanding where the  most central stars of galaxies originate across a wide range of galaxy  masses inside the TNG modelling framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Specifically, we also  study, for the first time, stars that have migrated towards the center  to address formation scenarios of central structures that include the  necessity for these processes such as NSC formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Even though  NSCs are not explicitly resolved in TNG50, we hope to offer new in-  centives for simulations (Antonini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Perets & Mastrobuono-  Battisti 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Guillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2016) and (semi-)analytical models  (Antonini 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Antonini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Leaman & van de Ven 2021)  that are tailored towards NSC formation channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Lastly, we aim  to demonstrate that there are possibilities to use the bright centers  of galaxies as a tracer of the galaxy’s overall assembly history with  readily available observables from current surveys such as SDSS  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Gallazzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In Section 2 we briefly describe  the TNG50 simulation and the definition of properties of galaxies  MNRAS 000, 1–34 (2022)  Centers of TNG50 Galaxies  3  and stars (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' stellar particles) that we will analyze at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We  also provide a detailed description and verification of selecting stars  belonging to a galaxy’s center and our galaxy sample selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In  Section 3 we present the three different possible origins for stars re-  siding in a galaxy’s center and discuss their birth locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In Section  4 we show the results of the different contributions of central stars  of different origins across different galaxy population demographics  and their observable stellar population and dynamical properties at  z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In Section 5, we discuss our findings and implications from  TNG50 on the central mass assembly of galaxies in a cosmological  context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We also provide outlooks in the context of the formation of  central galaxy components as well as the assembly of the overall host  galaxy tailored towards measurements of extragalactic observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Finally, we conclude our study in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2 TOOLS AND METHODS  We briefly introduce the TNG50 simulation below as well as the  properties of TNG50 galaxies and their stars (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We then  describe in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='4 how we define stellar particles that belong to  a galaxy’s center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1 The TNG50 simulation  In this work we primarily study galaxies in TNG50 (Pillepich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2019b), which is the highest resolution installment  of the IllustrisTNG (Illustris The Next Generation) (Pillepich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Springel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Naiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Marinacci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2018) suite of cosmological, magnetohydrodynam-  ical simulations1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' It provides unprecedented zoom-in like resolution  within a representative cosmological volume with a box of 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='7 cMpc  on each side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The simulation was performed with the Arepo code (Springel  2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Pakmor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Pakmor & Springel 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Pakmor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2016), which employs a finite-volume method on a moving-mesh  to solve the equations of magnetohydrodynamics coupled with a  tree-particle-mesh method for self-gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' TNG50(-1) has a mass  resolution of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='6 × 105 M⊙ for dark matter and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='4 × 104 M⊙ for  baryonic particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The softening length is 288 cpc for collisionless  particles for z ≤ 1 and 576 cpc for z > 1, whereas the softening length  of the gas particles is adaptive depending on the local cell size of the  moving mesh with a floor value of 74 cpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' TNG50 is accompanied  by three additional simulation runs (-2,-3,-4) that decrease the spatial  resolution each time by half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The initial conditions are set according  to cosmological parameters measured by Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Additionally, the TNG simulations implement a list of physical  subgrid models, which describe galaxy formation and evolution,  such as stellar formation and feedback, chemical enrichment, galactic  winds, supermassive black hole growth and feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Details can be  found in Weinberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Pillepich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' (2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Importantly, the TNG framework successfully reproduces key ob-  servational results such as the galaxy stellar mass function up until  z < 4 (Pillepich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2018b), bi-modality in galaxy color distribution  (Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2018), the fraction of quiescent galaxies (Donnari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2019, 2021b), scaling relations, such as the galaxy mass-size relation  (Genel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2018), the gas-phase mass-metallicity relation (Torrey  1 IllustrisTNG also encompasses two larger volume runs, namely TNG100  and TNG300 with subsequently coarser resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2019) and certain element abundances (Naiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2018),  as well as the clustering of galaxies (Springel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2018) and mag-  netic fields of massive halos (Marinacci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Specifically, the  resolution of TNG50 allows for the study of internal dynamics and  structures of galaxies (Pillepich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2019) as well as the influence  of stellar and black-hole driven outflows on galaxy evolution (Nelson  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Results from the TNG simulation are output in 100 snapshots  ranging from z = 20 until today with an approximate time step  of 150 Myr since z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For each snapshot dark matter halos are  identified by the friends-of-friends (FoF) algorithm (Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  1985) with a linking length of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2, with baryonic particles being  attached to the same FoF group based on their nearest dark matter  particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Substructures within these halos, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' subhalos, are found  through the Subfind algorithm (Springel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2001), which is run on  both dark matter and baryonic particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' To track the mass assembly of  subhalos/galaxies through cosmic time, merger trees are constructed  based on the Sublink algorithm (Rodriguez-Gomez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The  merger trees were constructed twice, once based on dark matter and  once based on baryonic matter alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The entire simulations’ particle information for the 100 snapshots,  the halo and subhalo catalogues, merger trees as well as many more  additional supplementary data catalogues are made publicly available  on the TNG website2 (see also Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2019a, for the public  data release).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2 General note on calculations  Unless otherwise stated we employ the following definitions in our  subsequent calculations and plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' To center the coordinate system  on a galaxy of interest we choose the position of the particle (of any  type) with the minimum gravitational potential energy as the galaxy’s  center, as given by SubhaloPos in the subhalo catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For the  systemic velocity of a galaxy we use the median velocity of the  5% most bound stellar particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For face-on or edge-on projections,  galaxies are oriented such that the z-axis is aligned with the total  angular momentum of stellar particles within twice the stellar half  mass radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' To track back galaxies in time we exclusively use the  merger trees based on following baryonic particles (‘Sublink_gal’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Plots that display summary statistics of galaxy populations use a  running median with a bin size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='25-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='3 dex, which is adapted,  if necessary, to ensure a minimum number of ten galaxies per bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Furthermore, all displayed quantities are in physical units and all  provided SubfindIDs refer to galaxies at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Throughout this study the terms in-situ, migrated and ex-situ al-  ways refer to stars within the central 500 pc of galaxies unless other-  wise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='3 Galaxy characteristics and properties of their stars  Throughout this study we are interested in two sets of demographics:  1) How does the central mass assembly of galaxies change as a  function of a galaxy’s overall bulk properties?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2) How do the intrinsic  properties of stars in the center of galaxies differ for different origins?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  To address the first question we do not only study the central 500 pc  of galaxies as a function of the galaxy’s total stellar (dynamical)  mass, but we also divide our galaxy sample into different types of  galaxies characterized at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' To address the second question  we study individual properties of stars (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' stellar particles) in the  2 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='tng-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='org  MNRAS 000, 1–34 (2022)  4  Boecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Property  Short description  Detailed description  Results  Overall galaxy  Mass  total stellar or dynamical (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' stars+gas+dark) mass  Appendix A1  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1  Environment  central or satellite  Star formation activity  star forming or quenched  Morphology  (kinematically) disk or bulge dominated  Bar-like feature  present or not, based on Fourier decomposition  AGN feedback  above or below average AGN feedback based on mass of the SMBH  Physical Size  compact or extended with respect to the mass-size relation  Individual stellar particle  Age [Gyr]  the lookback time when the star was born  Appendix A2  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2  Metallicity [log10 𝑍/𝑍⊙]  the total amount of metals  [Mg/Fe] [dex]  the abundance of magnesium as a proxy for 𝛼-elements  Circularity 𝜖  indicates the type of orbit the star is on  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Properties of galaxies and their individual stars (stellar particles) in TNG50 at z = 0 investigated in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' A detailed description on the exact  calculation of the properties as well as the results with respect to the centers of galaxies are found in the indicated sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  center of galaxies at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' These investigated characteristics are  briefly summarized in Table 1, whereas a detailed description on  their calculations can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='4 Defining stars belonging to a galaxy’s center  The most straightforward way to define a galaxy’s center at z = 0  is to select all stellar particles within a 3D spherical aperture with a  given radius 𝑟cut around its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This simple selection will give us  knowledge about stellar particles that have an instantaneous radius  smaller than the selected aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' However, as we are interested in  the mass assembly of the center of galaxies, we want to make sure  that selected particles roughly stay inside the spherical aperture over  their orbital time at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This ensures that we track particles that  changed their orbit, should they have migrated to the center, and not  particles that are just on more eccentric orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  To estimate, whether the particles are on such orbits confined to  the center at z = 0, we calculate the specific energy 𝐸cut a particle  on a circular orbit with guiding radius 𝑟cut would have, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' :  𝐸cut = 𝑣circ(𝑟cut)2  2  + Φ(𝑟cut).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  (1)  The circular velocity 𝑣circ is calculated from the spherically enclosed  mass (stellar, gas and dark matter particles) 𝑣circ(𝑟)2 = 𝐺𝑀 (< 𝑟)  𝑟  ,  whereas the gravitational potential energy Φ is given by the simu-  lation and interpolated to 𝑟cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Stellar particles with total energies  𝐸 = |v|2  2  + Φ(x) less than 𝐸cut should roughly be confined on orbits  that are within the spherical volume with radius 𝑟cut, whereas parti-  cles with higher energies are able to move to larger radii and hence  spend less time in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  We additionally enforce that the specific angular momentum in the  z-direction 𝐿z of stellar particles in the center lies between 𝐿cut =  ±𝑣circ𝑟cut, as we noticed that some lower mass galaxies with stellar  masses ≲ 1010 M⊙ had very large 𝐿z and hence large radii with 𝐸 <  𝐸cut, which probably stems from the fact that they are undergoing  tidal stripping at present time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' If particles with large orbital radii  (> 2 kpc) still persisted after this cut, we disregard them as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  These additional steps do not significantly affect the amount of central  particles selected for galaxies with stellar masses ≳ 1010 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  In general, the selection based on Equation 1 is a simplification as  it assumes a spherical mass distribution, but it gives a good enough  estimate of which particles are truly confined to the center without  actually integrating their orbits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' see Appendix B1 for validation of  this with two galaxies contained in the subbox with higher time  cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We visualize the difference between a simple selection in  radius and the one in energy using Equation 1 in Figure B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1 Choice of the central region  The last step in selecting stellar particles belonging to a galaxy’s  center is to set a value for 𝑟cut, with which we can in turn calculate  𝐸cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  We choose a fixed value of 500 pc3 for 𝑟cut across all galaxies  to avoid running too close into the numerical softening length (see  Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='4 for further elaboration on this).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We explicitly do not  choose to adopt a mass-dependent size, as already with a 10% scaling  of the mass-size relation of TNG50 galaxies, we are at the softening  length of 1010 M⊙ in stellar mass galaxies, while for the highest mass  galaxies we approach 5 kpc, which we do not deem to be central  anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We also refer the reader to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='4 and Appendix D  for a more detailed discussion and investigation about numerical  resolution effects and the choices of 𝑟cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2 Galaxy sample selection  Due to the choice of a fixed central aperture of 500 pc we have to make  some selection for our galaxy sample considered in this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Generally, sizes of TNG50 galaxies are numerically well con-  verged above a stellar mass of ∼ 3 × 108 M⊙ (see Pillepich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2019) at z = 0, but we employ a slightly higher lower mass cut  of 5 × 108 M⊙ ensuring that the galaxies have a sufficient number  of stellar particles for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We also only consider subha-  los/galaxies that are of cosmological origin (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' SubhaloFlag is  true).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Any scaling relations used in this analysis such as the galaxy  mass-size or the stellar-mass-black-hole-mass relation (to determine  for example if galaxies lie above or below the median at fixed stellar  mass) are always computed with respect to this galaxy sample, which  contains 4344 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Furthermore, for our main analysis of the centers of TNG galaxies,  3 Due to the selection of stellar particles belonging to the 500 pc center  based on their energies, some particles will have instantaneous radii larger  than 500 pc at z = 0, but typically not larger than 1 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  MNRAS 000, 1–34 (2022)  Centers of TNG50 Galaxies  5  1  10  5  20  Percentage of Stars in Center  109  1010  1011  1012  Total Stellar Mass [M ]  1  10  2  50  3D Stellar Half Mass Radius [kpc]  56%  23%  7%  0%  2531/4344 Galaxies  TNG50  z = 0  10  100  2  4  R1/2 in Factors of rcut  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Sample selection of TNG50 galaxies at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Stellar mass-size  relation of TNG50 galaxies at z = 0 considered in this analysis colored coded  according to their percentage of stars in the center relative to their total number  of stellar particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We employ a lower total stellar mass cut of 5 × 108 M⊙  leaving 4344 Galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We additionally impose a minimum 3D stellar half  mass radius R1/2 of 2 kpc (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', R1/2/rcut ≥ 4 with rcut = 500 pc) resulting in  a sample size of 2531 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Excluded galaxies are shown with grey points  and percentages show their number fractions in bins of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The median  stellar mass-size relation of all TNG50 galaxies is shown as the black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  we enforce that the ratio of the 3D stellar half mass radius R1/2 and the  central aperture (rcut = 500 pc) is greater than four, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' R1/2 ≥ 2 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Otherwise, galaxies are too compact for our selected central aperture  and about half of the entire galaxy will be classified as “central”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Additionally, we make sure that at least a hundred stellar particles  are within the central 500 pc according to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='4, otherwise the  galaxy is disregarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Our final galaxy sample selection yields 2531 TNG50 galaxies  and their masses and sizes are visualized in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The data points  are color-coded by the percentage of stars inside the central 500 pc  compared to the total amounts of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The color trend is neither  uniform in the direction of increasing stellar mass nor size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This  hints at different density profiles for galaxies across their stellar  mass-size plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We note that our subsequent results do not show any  strong differential trends, even though our constrain of R1/2 ≥ 2 kpc  disregards half the galaxies with stellar masses < 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 M⊙ (see  Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='4 for a further discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  In Figure 2 we show the cumulative number of stellar particles as a  function of their instantaneous radius at z = 0 for our galaxy sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The average number of stellar particles in the center, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' within  500 pc, is around 103 for galaxy stellar masses between 5 × 108 M⊙  and 5 × 109 M⊙ and increases towards 105 for the highest mass  bin4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Hence, our choice for 𝑟cut ensures that we have enough stellar  4 A synonymous measure can be achieved with the StellarHsml field,  which gives an approximation for the spatial extent a single stellar particle  samples from the underlying stellar density field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The spherical radius of  stars within 500 pc ranges between 10 − 100 pc for the highest to lowest mass  galaxies respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  5 × 108 5 × 109  1010  5 × 1010 5 × 1012  Total Stellar Mass [M ]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1  1  10  Radius [kpc]  1  102  104  106  108  Cumulative Stellar Particle Number  TNG50  z = 0  2531 Galaxies  rcut  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Number of stellar particles in the central 500 pc of our TNG50  galaxy sample at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Cumulative number of stellar particles as a function  of radius per individual galaxy are shown as thin gray lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The thick colored  lines show the average per galaxy stellar mass bin as depicted by the colorbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The dashed black line shows our adopted rcut value of 500 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The average  number of stellar particles within rcut lies between 103 and 105 for the lowest  and highest galaxy mass bins respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' No individual galaxy in our sample  has less than 100 stellar particles in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  particles in the center to reliably study their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We can also  observe a turn-over in the stellar particle number profile at radii  around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5−1 kpc confirming that we are indeed probing the densest  (central) region of TNG50 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  3 THE DIFFERENT ORIGINS OF STARS IN THE CENTER  OF TNG50 GALAXIES  After selecting stars in the center of TNG50 galaxies at z = 0, we  investigate their different origins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We find three general populations  of stars in the central region of galaxies, which we describe in Section  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1 in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We also present the distribution of their birth origin for  stacks in galaxy stellar mass in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1 Definition of different origins  We define the following different origins of stars in the center of  TNG50 galaxies at z = 0:  in-situ: stars were born inside the host galaxy’s center and are  still found there at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  migrated: stars were born gravitationally bound inside the host  galaxy but outside its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' At z = 0 they reside in the host galaxy’s  center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  ex-situ: stars were born inside other galaxies, which merged with  the host and are ultimately found inside the host’s center at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  MNRAS 000, 1–34 (2022)  6  Boecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1 Born inside or outside the host galaxy  To determine whether a star is born inside a galaxy or was brought  in through merger events, we use the stellar assembly catalogue5  produced by methods of Rodriguez-Gomez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' (2016) for TNG50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  This classifies stellar particles that formed along the main progenitor  branch of a galaxy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' the galaxy with the most massive history  behind it, as in-situ (InSitu = 1) and otherwise as ex-situ (InSitu =  0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The ex-situ stars generally have two possible origins: they either  came from galaxies that completely merged with the main galaxy,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' they are present in the merger tree of the host, or were stripped  from galaxies that do not belong to the host’s merger tree, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' flybys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Additionally, we treat subhalos/satellites that directly merged onto  the main progenitor branch of a galaxy but are flagged as not be-  ing of cosmological origin (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' SubhaloFlag = 0 in the subhalo  catalogue) differently in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' These subhalos are often formed  within another galaxy as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' a fragment of the baryonic disk, contain  little dark matter and hence are not thought of as galaxies (see also  Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2019a, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Because the construction of the  stellar assembly catalogue involves the use of merger trees, which  only track stellar particles and star-forming gas cells of subhalos,  these spurious galaxies are counted as of ex-situ origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Here, we  change their labelling back to in-situ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' their InSitu flag in the  stellar assembly catalogue becomes true again) for now (see Section  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='3 for the implications of this), because we only consider ex-situ  particles coming from true external galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We verify with Figure  C1 in Appendix C that this change does not alter the overall total  ex-situ stellar mass fraction of TNG50 galaxies significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We note  that spurious galaxies brought to the main progenitor branch of the  host galaxy through prior merging with a real galaxy are continued  to be counted as ex-situ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2 Born in-situ or migrated to the center  To address whether a stellar particle is born inside the center of the  host galaxy or migrated to the center from elsewhere inside the host  galaxy, we need to determine its birth radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' A stellar particle with  a birth radius smaller than rcut = 500 pc is then consequently born  in-situ and otherwise counts as migrated6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  In TNG, two new fields (BirthPos and BirthVel) for their stellar  particles were added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' These represent the spatial position and velocity  of the star-forming gas cell that parented the stellar particle at its  exact time of birth (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' GFM_StellarFormationTime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In theory,  this provides us with knowledge of the exact birth condition of a  stellar particle at the original time step resolution of the simulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  and not only at the output time steps of the snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Because these quantities are provided in the reference frame of  the simulation box, we need to center them on the reference frame  of the galaxy of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This however becomes an impossible task  to do to the precision needed for our analysis, as we only know the  center position of subhalos at the one hundred output snapshots, but  the information of its trajectory in-between is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We find that even  interpolating the subhalos’ position with a higher order spline to the  exact birth times of stars can lead to centering offsets of several kpc,  especially when there is a merger in process or a pericenter passage  around another galaxy (see Figure B2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' As we are interested in typical  scales of one kpc or less in this study, this problem is severe and will  5 This particular catalogue has not been publicly released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  6 We here apply a simple cut in the birth radius instead of calculating 𝐸cut  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' following Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='4), as the potential is not recorded for every snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  result in a strong bias towards stars being classified as migrated even  though they where formed inside our selected spherical aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  We therefore define the birth position of stellar particles as the  position they have in the snapshot they first appear in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Practically this  is done by matching particles at z = 0 to their birth snapshot through  their unique ParticleIDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The caveat of this approach is that the  stellar particles have already moved since their exact formation time,  which can also lead to a wrong classification of migrated and in-situ  stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' However, the error created by this approach is much smaller  than the incorrect centering described above (see Figure B3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  We verify this approach by looking at two subhalos that reside  in the subboxes of TNG50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The subbox has 3600 snapshot outputs,  which makes it possible to track the center position of galaxies across  a much finer time resolution of a few Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The reader is referred to  Appendix B2 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='3 Clumpy or smoothly migrated  Because we have changed the InSitu flag from the stellar assembly  catalogue for spurious galaxies, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1, we now find two  types of migrated stellar particles in the center of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Stars  either travelled individually (‘smoothly’ migrated) or together in  clumps (‘clumpy’ migrated) to their galaxy’s center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Smoothly mi-  grated stars are genuinely born on the main progenitor branch of the  subhalo/galaxy in question and the clumpy migrated stars originate  from these spurious galaxies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' stellar clumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Generally, these clumps are ubiquitous in TNG50 galaxies, about  36% of all galaxies considered in this work have at least one through-  out their life time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In stellar or gas surface mass density maps they  look like massive star cluster like objects (see Figure E1 for an exam-  ple) that form within spiral arms or gaseous (disk) fragments during  galaxy interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' However, we want to be extremely cautious here,  as it is unclear, if their formation is physical or due to some numerical  artifact, even though measures against artificial fragmentation are in  place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In fact, their sizes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 3D stellar half mass radii) lie mostly  below the gravitational softening length of TNG50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Once these clumps are formed, however, their dynamical evolution  within the host galaxy is determined by gravity, which we believe is  well captured in TNG50 (modulo the softening).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Hence, depending  on their density and the exerted tidal forces on the clumps, they are  either completely disrupted or travel to the center of their host galaxy  due to dynamical friction and deposit their stellar particles there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Their typical stellar masses are ∼ 108 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We point the interested  reader to Appendix E for more statistics on the clumps and their  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We provide an extensive discussion on the existence and  formation of stellar clumps in simulations and observation in Section  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  For the rest of the paper, we sometimes make the distinction be-  tween migrated particles coming from the ‘smooth’ or ‘clumpy’  migration, if it is explicitly stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Otherwise, all general references  to migrated properties always include both types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2 Birth locations of the central stars  The distributions of birth radii of in-situ, migrated and ex-situ central  stars are illustrated in Figure 3 in stacks of galaxy stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The in-situ stars are born (by definition) in the center of the host  galaxy at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The peak of the birth radii distribution is around  200 − 300 pc for galaxies larger than 1010 M⊙ and shifts slightly  towards larger radii for the lower mass galaxy bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We also see that  higher mass galaxies birth more in-situ stars at all radii and hence  are more centrally concentrated (see also Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  MNRAS 000, 1–34 (2022)  Centers of TNG50 Galaxies  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  Birth Radius [kpc]  106  108  1010  Particle Mass [M ]  TNG50  z = 0  Insitu  1  10  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content="5  Birth Radius [kpc]  TNG50  z = 0  Migrated  'Smooth'  1  10  100  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content="5  Birth Radius [kpc]  106  108  1010  Particle Mass [M ]  TNG50  z = 0  Migrated  'Clumpy'  5 × 1010 M  < M , tot  5 × 1012 M  1010 M  < M , tot  5 × 1010 M  5 × 109 M  < M , tot  1010 M  5 × 108 M  < M , tot  5 × 109 M  1  102  104  106  # Particles  1  102  104  106  # Particles  106  108  1010  Particle Mass [M ]  TNG50  z = 0  Exsitu  106  108  1010  Particle Mass [M ]  TNG50  z = 0  Exsitu  1  102  104  106  # Particles  1  102  104  106  # Particles  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1  1  10  100  Birth Radius [kpc]  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='t Host at Birth  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1  1  10  100  Radius [kpc]  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='t Primary at Stripping  TNG50  z = 0  Exsitu  20%  50%  90%  99%  1  102  104  # Particles  106  108  Particle Mass [M ]  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Distribution of birth radii of in-situ, migrated and ex-situ stellar populations in the central 500 pc of TNG50 galaxies at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Left: Histograms  of birth radii of the in-situ, ‘smooth’ migrated and ‘clumpy’ migrated stars colored according to stacks of galaxy stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The left hand side of the y-axis  shows stacked particle mass, whereas the right hand side shows the number of stellar particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The migrated stars can originate from large radii (> 10 kpc)  and the smoothly and clumpy migrated stars show different distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Right: 2D histogram of the birth radius with respect to the host galaxy at birth and  the radius with respect to the primary galaxy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' final z = 0 host) at the time of stripping for all central ex-situ stars in our galaxy sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The contours (from  thicker to thinner lines) include 20%, 50%, 90% and 99% of all ex-situ particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The respective 1D histograms for stacks in galaxy stellar mass are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Most ex-situ stars are born in the central ∼ 1 kpc of their birth galaxy and stay together until they are deposited also within the central ∼ 1 kpc of their z = 0 host  galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1 Individually migrated stars originate close to the galaxy’s  center  Most of the smoothly migrated stellar particles were also born close  to the center with radii between 500 pc and 1 kpc, which is partly  due to how we have defined them (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' purely based on their birth  radius) and partly a consequence of the typical density profile of  galaxies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' more stars reside in the center of galaxies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For galaxies  below 1010 M⊙ the distribution of birth radii declines exponentially  reaching the highest values of about 10 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The lowest mass galaxies  in our sample (≤ 5 × 109 M⊙) have 11% of smoothly migrated stars,  which are born in the range of 1 − 10 kpc, whereas this increases  slightly to 17% for the next higher mass bin (≤ 1010 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  For galaxies above 1010 M⊙ we observe a plateau for the distribu-  tion of birth radii starting at ∼ 10 kpc, which stops at around 30 kpc  and 60 kpc for galaxies with ≤ 5 × 109 M⊙ and > 5 × 109 M⊙ re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The migrated stars originating from these large distances  likely come from gas that was stripped during a merger, but which  was already attributed to be gravitationally bound to the primary  galaxy according to the Subfind algorithm and hence was counted  as being born in-situ to the primary host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The percentage of smoothly  migrated stars with birth radii larger than 1 kpc is around 20% and  14% for the two highest stellar mass bins respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2 Stars migrated in clumps originate from the outskirts of  galaxies  The clumpy migrated stars show a distinctively different distribution  than the smoothly migrated ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For galaxies below 1010 M⊙ their  contribution is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For galaxies between 1010 M⊙ and 5 ×  1010 M⊙ the clumpy migrated stars are only 3% of the total migrated  stars, whereas for galaxies above 5 × 1010 M⊙ the contribution rises  to almost 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Therefore, clump migration is only important for high  mass galaxies, where it becomes the dominant driver for contributing  migrated stars in the centers (see also Section E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Furthermore, the peak of the birth radii distribution of clumpy  migrated stars is above 10 kpc for the high mass galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This is in  agreement with the fact that the gaseous disk of galaxies is much  more extended than the stellar one (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Stars  travelling in stellar clumps are therefore able to migrate to the center  of galaxies from much farther distances compared to when they travel  individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='3 Central ex-situ stars originate from the nuclei of their birth  galaxies  Regarding the ex-situ stars, we investigate two different locations: 1)  their birth place with respect to their birth host galaxy and 2) the  location they were deposited inside their z = 0 host (primary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The  latter is defined as the radius the stellar particles have with respect to  the primary at stripping time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' the time they last switched galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  We show the distribution of these two quantities also in Figure 3 in  the same stacks of galaxy stellar mass, as well as the 2D distribution  of all ex-situ stars for these two radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' About half of all ex-situ stars  that reside in the center of galaxies at z = 0 exhibit values between  100 pc and 1 kpc for both radii respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This means that the ex-  situ stars are also born in the center of their respective birth galaxies  MNRAS 000, 1–34 (2022)  8  Boecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  as well as remain in said center until they are deposited right in the  center of the primary galaxy during the merger process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Hence, the  central, most bound cores of galaxies are more likely to stay together  during accretion events until they arrive close to the center of the  primary galaxy and ultimately deposit a large quantity of stars there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  This is a consequence of mergers preserving the rank order of the  particles’ binding energy (Barnes 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  We also find two other cases of ex-situ stars, albeit much lower  in number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Firstly, TNG predicts a slight excess of ex-situ stars that  are born at larger radii (1 − 100 kpc), but are still deposited close  to the primary galaxy at stripping time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' within ∼ 1 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' These  stars represent a second generation of ‘migrated’ stars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' or likely in  the case of ex-situ stars with birth radii of ≥ 10 kpc, stars that were  formed from stripped gas during secondary mergers, which only  appear for the most massive z = 0 hosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Consequently, these ex-situ  stars were born at large radii in their respective host galaxies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  which will become the secondary galaxy during the merger process  onto the z = 0 host), then migrated to the center of said galaxy in  order to be deposited close to the center of the primary host during  accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We confirm this by explicitly checking that their radii are  indeed central (≲ 1 kpc) with respect to the merging host galaxy one  snapshot before the merger coalesces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The second case represents ex-situ stars that were deposited at  larger radii from the primary (> 1 kpc), but born within the central  1 kpc of their birth galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Despite being stripped outside the center  of the z = 0 host galaxy, these stars still were able to migrate such that  they are found in the center of their respective galaxy at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' There  is a possibility that these stars were stripped earlier, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' before the  merger coalesces, but their dynamics were still following the orbit of  the galaxy undergoing the merger and hence they could arrive at the  center of the final host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  4 THE CENTRAL IN-SITU, MIGRATED AND EX-SITU  POPULATIONS ACROSS TNG50 GALAXIES  In this section we present our results of in-situ, migrated and ex-situ  populations within the central ∼ 500 pc of TNG50 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  We study their contributions across different galaxy properties  (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1) and examine differences in their stellar population and  dynamical properties (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1 Galaxy population demographics  Below we depict the contribution of the central stellar mass of the  different origins as an overall trend with galaxy mass (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1),  in correlation to each other (Sec 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2) and for different galaxy types  (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1 Galaxy mass trends  In Figure 4 we give an overview of the absolute and relative contri-  bution of the ex-situ, migrated and in-situ population across galaxy  masses (both stellar and dynamical) in TNG50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  For all three populations the central stellar mass increases with  increasing galaxy mass with the in-situ population dominating at all  galaxy masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Whereas the relation for the in-situ and the smoothly  migrated stars have the same shape, the slope for the ex-situ popu-  lation is steeper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The latter also shows a larger overall scatter due to  the stochasticity of merger events contributing stars to the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Even though the fractional mass of the ex-situ population in the  center is negligible for galaxy stellar masses below 1011 M⊙, there  are only 227 (9%) galaxies in our total sample that have no central  ex-situ mass, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' they do not possess a single stellar particle of ex-  situ origin in their central 500 pc, or in other words they possess less  than ∼ 8 × 104 M⊙, the mass of a stellar particle, in ex-situ stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Above 1011 M⊙ the ex-situ mass becomes of the same order as the  in-situ and migrated population, which is a consequence of mergers  contributing a significant amount stellar mass to build up of these  galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The ex-situ mass reaches about 10% of the total central  stellar mass at the highest galaxy masses, albeit with a large scatter  of up to 60%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Around galaxy stellar masses of about 5 × 1010 M⊙, the relation  flattens for the in-situ and smoothly migrated stars, with the in-situ  population reaching about 4% of the total galaxy stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Al-  though we have low number statistics of galaxies in this regime within  the TNG50 volume (there are 18 galaxies with stellar masses above  5×1011 M⊙), it is reasonable that the in-situ mass goes down, because  the ex-situ mass increases in addition to galaxies being quenched  by AGN feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The consequential increased stochasticity is also  seen by the larger scatter in the in-situ and migrated population at  the highest galaxy stellar masses7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The contribution of clumpy migrated stars to the overall central  migrated population only starts to significantly affect galaxies with  stellar masses higher than 5 × 1010 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For galaxies higher than  2 × 1011 M⊙ the clumps are responsible for roughly quadrupling  the mass of migrated stars, or, in fractional terms, increasing the  contribution of migrated stars to the total central mass from below  10% to slightly above 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Hence, the clumps are important driver  to bring in stars from the outskirts of galaxies in TNG50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Taking  into  account  the  entire  migrated  population  (‘smooth+clumpy’), we find a contribution of around 20% to  the total stellar mass in the center across all TNG50 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Interestingly, the total central migrated fraction around galaxy stellar  masses of ∼ 1010 M⊙ slightly increases, with the 84th percentile  reaching almost 40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We explicitly confirm that this is not due to  mixing galaxies with different sizes and hence different total central  stellar masses (see also Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The statements made so far also apply when correlating the central  stellar masses of the three populations with the total dynamical mass  of TNG50 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The larger scatter in all three relations is due to  the scatter in the stellar-to-halo mass relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2 The diversity of central stellar mass at fixed galaxy mass  In Figure 5 three correlations between the central ex-situ, migrated  and in-situ stellar mass as 2D Gaussian kernel estimates in bins of  total galaxy stellar masses are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The bins were specifically  chosen based on the change of the average migrated fraction as a  function of galaxy stellar mass and, in case of the highest mass bin,  to ensure enough galaxies per bin to reliably perform the kernel  density estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  In-situ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' migrated mass (Figure 5, first panel): At all stellar  masses, the mass of migrated stars correlates strongly with the in-  situ stellar mass with three times as much in-situ than migrated mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  This reflects our previous statements that the shape of the median  7 Another possibility for the large scatter at high galaxy stellar masses for the  in-situ and migrated central stellar mass could be stars formed from accreted  gas, which was brought in by gas-rich mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We do not quantify this further  as it is beyond the scope of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content="  MNRAS 000, 1–34 (2022)  Centers of TNG50 Galaxies  9  104  105  106  107  108  109  1010  Central Stellar Mass [M ]  TNG50  z = 0  2531 Galaxies  Insitu  Migrated 'Smooth+Clumpy'  Migrated 'Smooth'  Exsitu  TNG50  z = 0  2531 Galaxies  109  1010  1011  1012  Total Stellar Mass [M ]  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  Central Stellar Mass Fraction  TNG50  z = 0  1011  1012  1013  Total Dynamical Mass [M ]  TNG50  z = 0  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Central (500 pc) stellar mass of the in-situ, migrated and ex-situ populations of TNG50 galaxies at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Median trends of central stellar mass  (top panels) and central stellar mass fraction (bottom panels) as a function of the galaxies’ total stellar mass (left) and total dynamical mass (right) divided in  the three origins: in-situ (pink), migrated (orange) and ex-situ (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The migrated population is shown for both ‘smooth+clumpy’ (solid line) and just ‘smooth’  (dashed line) migration (see 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='3 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Shaded areas show the 16th and 84th percentiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Overall the in-situ population dominates on average across  all galaxy masses with the migrated population contributing around 20% to the total central stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Only above galaxy masses of 1011 M⊙ the ex-situ  population starts to significantly contribute to the central mass build-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  107  108  109  1010  Migrated Mass in Center [M ]  107  108  109  1010  Insitu Mass in Center [M ]  TNG50  z = 0  5 × 108 M  < M ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' tot  5 × 109 M  5 × 109 M  < M ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' tot  1010 M  1010 M  < M ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' tot  5 × 1010 M  5 × 1010 M  < M ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' tot  5 × 1012 M  1%  20%  50%  90%  107  108  109  1010  Insitu Mass in Center [M ]  104  105  106  107  108  109  1010  1011  Exsitu Mass in Center [M ]  TNG50  z = 0  107  108  109  1010  Migrated Mass in Center [M ]  TNG50  z = 0  107  108  109  1010 1011 1012  Total Exsitu Mass [M ]  TNG50  z = 0  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2D distributions of the central (500 pc) in-situ, migrated, ex-situ as well as the total ex-situ stellar mass of TNG50 galaxies at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Gaussian  kernel density estimates for different combinations of in-situ, migrated and ex-situ mass in the center as well as total ex-situ mass color-coded according to four  different total stellar mass bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The contours show different percentiles encompassing 1%, 20%, 50% and 90% of all data points (thickest to thinnest line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The black dashed line shows the one-to-one relation in all panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The correlation between the central mass of the in-situ and migrated population follow the  one-to-one relation closely with a fixed offset, whereas the central ex-situ population exhibits a large scatter across the other central populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  MNRAS 000, 1–34 (2022)  10  Boecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  relation of the central in-situ and migrated mass versus the total  stellar galaxy mass is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Nevertheless, for some galaxies the migrated mass is larger than  the in-situ mass in the center as seen by the 90% contours for the three  highest galaxy mass bins in the range of 5×109−5×1012 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Galax-  ies in this regime are dominated by clumpy migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Hence, the  mass contributed to the center by clumps can be significant enough to  break the otherwise tight one-to-one relation of migrated and in-situ  mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Lastly, we find for the 90% contour in the highest mass bin for  galaxies above 5 × 1010 M⊙ that there is a larger tail of galaxies with  lower in-situ and migrated mass in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Most galaxies situated  in this space have a high ex-situ central mass fraction of 40% or  higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Ex-situ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' in-situ and migrated mass (Figure 5, second and third  panels): At roughly fixed central in-situ or migrated masses there is a  large variety of ex-situ mass that is deposited in the center of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The scatter of the ex-situ mass in the center increases roughly from  four to six dex the from smallest to largest galaxy stellar mass bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Compared to that the scatter in the in-situ and migrated mass direction  is rather small, being roughly one dex across all galaxy stellar mass  bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' However, some galaxies in mass bins below 5 × 1010 M⊙ have  lower in-situ masses of up to one dex below the majority of the other  galaxies in their respective bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Galaxies lying in this region have  above average migrated fractions of 40% or more with some reaching  extreme values of above 80%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The spread in migrated mass compared to the in-situ mass is larger  for the highest mass galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This is mainly due to the increased  stochasticity in the total central stellar mass for the 18 galaxies above  5 × 1011 M⊙ in galaxy stellar mass, which almost spans one dex  as opposed to only a quarter dex for galaxies between 5 × 1010 and  5×1011 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' These 18 galaxies lie between the 50% and 90% contour  and have ex-situ masses spanning from 107 to 1011 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Their in-  situ masses are exclusively below the 50% contour, whereas their  respective migrated masses can lie towards lower or higher values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The peak of the central ex-situ mass distributions begins to rise  for galaxies above 1010 M⊙ in total stellar mass, going from about  three dex below the one-to-one relation to one dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This break  point roughly translates to 4 × 108 M⊙ in central in-situ mass and  1 − 2 × 108 M⊙ in central migrated mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The former roughly  coincides with the critical mass needed for the SMBH to be in the  kinetic feedback mode (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Zinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2020, Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The total  ex-situ mass also begins to rise for galaxies with total stellar masses  above a few 1010 M⊙ (see Figure C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Central ex-situ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' total ex-situ mass (Figure 5, fourth panel):  Lastly, we also show the correlation between the central ex-situ mass  and the total ex-situ mass, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' all stars that were ever accreted onto the  z = 0 host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For fixed total galaxy stellar mass the slope of the  contours depict that a higher total ex-situ mass generally also implies  a higher central ex-situ mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The slope of this correlation is rather  steep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' While the total ex-situ mass spans approximately two dex per  galaxy stellar mass bin, the ex-situ mass in the center spans four to six  dex from the lowest to highest galaxy stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Consequently,  it is quite stochastic which merging satellite galaxies deposit stellar  mass in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Furthermore, the central density contours shift closer to the one-  to-one relation with increasing galaxy stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This means that  more galaxies in the highest mass bin have mergers that are more  effective in bringing a larger fraction of their total ex-situ mass into  their center as opposed to lower mass galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Nevertheless, the  90% contours for galaxy stellar masses above 5 × 109 M⊙ extend  right up to the one-to-one relation, meaning that there some galaxies  that have almost all their ex-situ mass in the central 500 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='3 Trends for different galaxy types  Galaxies with different present-day properties are thought to have  undergone different formation pathways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Is this reflected in different  contributions of in-situ, migrated and ex-situ stars building up the  center of these galaxies?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  In Figure 6 we show the running median of the central stellar mass  of the three origins as a function of total galaxy stellar mass split  into six different galaxy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The definitions of the different  galaxy properties are summarized in Table 1 and described in detail  in Appendix A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  All in all, the most significant differences are seen in the central  ex-situ population across various galaxy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This significance  manifests in separation between the median relations including  the scatter (which we do not show, however, in favour of clar-  ity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Small differences for the in-situ and migrated populations are  not significant with respect to the scatter around the median relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Centrals vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Satellites (Figure 6, top left panel): On average,  centrals and satellites contain the same amount of central in-situ,  migrated and ex-situ mass, showing that their central 500 pc  is unaffected by their environment at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This is sensible  considering that galaxy centers likely assemble before the galaxy  becomes a satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Additionally, most environmental effects  should first take effect in the outskirts of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Similarly, we  find no significant difference in the central mass of the three pop-  ulations, when the galaxies are divided by the mass of their host halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Quenched vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Star Forming (Figure 6, top middle panel):  Quenched galaxies between 5 × 109 and 5 × 1010 M⊙ have slightly  higher central in-situ and migrated mass than for star forming ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  This difference primarily arises because the star forming galaxies  tend to have lower central densities on average than quenched ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  A larger difference is seen in the ex-situ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For galaxy stellar masses  above 1010 M⊙ the average ex-situ mass starts to rise more rapidly  for star forming galaxies than for quenched ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For galaxies around  5×1010 M⊙ this difference becomes largest, with the median central  ex-situ mass of star forming galaxies being higher by more than one  dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  While this trend may seemingly be counter-intuitive for the  current consensus of galaxy evolution, the difference is also true  when considering the total ex-situ stellar mass in TNG50 (and also  TNG100) as seen in Figure C1 in Appendix B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This could be an  indication that today’s star forming galaxies had more or larger  mass ratio mergers with galaxies with high gas content at later  cosmic times (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1 for a further discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We obtain a  consistent picture when galaxies are divided according to their 𝑔 − 𝑖  colour or total gas mass at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Bulgey vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Disky (Figure 6, top right panel): The in-situ and  migrated central mass for disky and bulgey galaxies show a similar  trend as for the star forming and quenched population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' However, the  trend for the central ex-situ mass is distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Bulgey galaxies below  1010 M⊙ in stellar mass have higher ex-situ masses (by roughly half  a dex) in their centers than their disky counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This difference  disappears for galaxy stellar masses above 1010 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  We have checked the median relation for the total ex-situ mass and  find that bulgey galaxies have a constant higher offset of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='25  MNRAS 000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 1–34 (2022)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='Centers of TNG50 Galaxies  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='109  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1010  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1011  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1012  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='Total Stellar Mass [M ]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='105  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='106  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='107  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='108  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='109  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1010  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='Central Stellar Mass [M ]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='TNG50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='z = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='Insitu  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content="Migrated 'Smooth+Clumpy'  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='Exsitu  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='Centrals (1678)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='Satellites (853)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='109  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1010  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1011  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1012  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='Total Stellar Mass [M ]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='TNG50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='z = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='Quenched (615)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='Star Forming (1916)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='109  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1010  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1011  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1012  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='Total Stellar Mass [M ]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='TNG50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='z = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='Bulgey (1150)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='Disky (1381)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='109  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1010  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1011  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1012  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='Total Stellar Mass [M ]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='105  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='106  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='107  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='108  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='109  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1010  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='Central Stellar Mass [M ]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='TNG50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='z = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='Insitu  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content="Migrated 'Smooth+Clumpy'  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='Exsitu  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content="'Barred' (1610)  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content="No 'Bar' (921)  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='109  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1010  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1011  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1012  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='Total Stellar Mass [M ]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='TNG50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='z = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='Overmassive BH (832)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='Undermassive BH (1699)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='109  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1010  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1011  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1012  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='Total Stellar Mass [M ]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='TNG50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='z = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='Extended (1888)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='Compact (643)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Differences in the central (500 pc) in-situ, migrated and ex-situ populations for different galaxy properties of TNG50 galaxies at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Each  panel shows median trends of central stellar mass divided in the three origins: in-situ (pink), migrated (orange) and ex-situ (blue) as a function of the galaxies’  total stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The dashed and solid lines split the TNG50 galaxy population according to different properties, which are (form left to right): central vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  satellite, quenched vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' star forming, bulgey vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' disky, ‘barred’ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' ‘non barred’, overmassive vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' undermassive black holes (BH) and extended vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' compact  galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The bracketed numbers show the total amount of galaxies in each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  dex compared to disk galaxies across the whole galaxy mass range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Hence, disky galaxies below 1010 M⊙ have not only lower absolute  central and total ex-situ masses, but also a lower central-to-total ex-  situ fraction of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='4% as compared to 1% for bulge dominated  galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Thus the relative amount of ex-situ mass that is deposited  in the center might be an important driver for morphological trans-  formation in these lower galaxy mass regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  For galaxies above 1010 M⊙ in stellar mass, the central-to-total  ex-situ fraction decreases strongly as a function of galaxy stellar  mass, with disk galaxies having consequently slightly higher values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  This could be an indication that once a massive rational support  exists in the stellar component it is hard to destroy it through mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Similar relations are found when adopting other definitions for disky  and bulgy galaxies, such as the ratio of the kinetic energy in ordered  motion compared to the total kinetic energy (see Rodriguez-Gomez  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Barred vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' No Bar (Figure 6, bottom left panel): For galaxies  below 1010 M⊙ TNG50 predicts no difference in the central in-situ  and migrated mass, however the galaxies with bar-like features have  higher ex-situ masses than galaxies with no bar-like features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This  trend is similar for the bulgey vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' disky galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We have explicitly  checked that indeed high ex-situ masses in the center of galaxies  within this mass regime mainly occur in bulgey and barred galaxies,  whereas bulgey and unbarred galaxies as well as disky galaxies, both  barred and unbarred, have lower central ex-situ masses by approxi-  mately one dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  For galaxies above 1010 M⊙ in total stellar mass, this relation for  the central ex-situ mass swaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In this regime unbarred galaxies have  higher ex-situ masses in the center regardless whether they are disky  or bulgey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We find that the same statements for barred and unbarred  galaxies across the entire mass range are true when correlating the  total ex-situ mass of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Lastly, the in-situ and migrated mass in the center is higher for  barred galaxies between 1010 M⊙ and 1011 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Hence, barred  galaxies in this mass regime have higher central densities than  unbarred galaxies in TNG50, which is consistent with observations  (see Díaz-García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2016b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Over- vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Undermassive Black Holes (Figure 6, bottom middle  panel): We could expect that AGN feedback has an influence on the  stellar mass growth in the center of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We therefore split our  TNG50 sample according to whether the galaxies have an over- or  undermassive black hole at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Identical relations are found when  the galaxy population is split according to the cumulative energy  injection of each feedback mode or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  MNRAS 000, 1–34 (2022)  12  Boecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  On average, galaxies between 1010 M⊙ and 1011 M⊙ in stellar  mass with an undermassive black hole have a higher central ex-situ  mass by about one dex than galaxies with an overmassive black hole  at the same stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For galaxies with total stellar masses in  the range of 5 × 109 − 5 × 1010 M⊙, the ones with an overmassive  black hole have in-situ and migrated masses in the center that are  about half a dex higher than for galaxies with an undermassive black  hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Consequently, galaxies with overmassive black holes in this  mass regime have higher central densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  We find that mainly all of these differences in the in-situ, migrated  and ex-situ mass for galaxies with over- and undermassive black  holes emerge because galaxies at fixed stellar mass with overmassive  black holes tend to be more compact in TNG50 and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Therefore, a similar behaviour of the central stellar mass in the  three populations with total galaxy stellar mass is found when  the galaxy population is split into compact and extended galaxies  (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This connection between black hole masses, central  densities and sizes of galaxies at fixed galaxy stellar mass is also  found in observations (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Extended vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Compact (Figure 6, bottom right panel): Extended  galaxies tend to have on average more ex-situ mass in the center than  compact galaxies at the same total stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The difference is  around one dex for galaxies above 1010 M⊙ in stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  When we correlate with the total ex-situ mass, we find an opposite  behaviour in TNG50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Galaxies ≲ 5 × 1010 M⊙ and with higher total  ex-situ fractions are on average more extended (see Figure C2 in  Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Compact galaxies between 5 × 109 M⊙ and 5 × 1010 M⊙ have  more in-situ and migrated mass in the center, and therefore higher  central densities (and black hole masses, see above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' As a matter of  fact, this difference is also seen for quenched vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' star forming and  bulgey vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' disky galaxies, even though to a lesser extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This stems  from the fact that generally star forming galaxies tend to be more  disky and hence more extended and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2 Stellar population and dynamical properties  Are there distinguishable features in the stellar population and dy-  namical properties of the in-situ, migrated and ex-situ stars?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The  short answer is yes, especially for galaxies below ≲ 1011 M⊙, where  the majority of ex-situ stars originate from lower mass satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1 Average age, metallicity and [Mg/Fe] of central stars  Metallicities, ages and magnesium-to-iron abundances [Mg/Fe] of  stars encode information about their birth places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Figure 7 illustrates  average quantities of stellar populations belonging to the in-situ,  migrated and ex-situ origin as a function of their galaxy’s stellar  mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We also show separate relations for migrated stars that have  birth radii larger than 1 kpc to exclude the majority of migrated stars  that were born close to the center, which dominate the average stellar  population properties (see smoothly migrated stars in Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Metallicity (Figure 7, left panel): Stars in the central 500 pc follow  a mass-metallicity relation, where galaxies at the lowest mass end  (5 × 108 M⊙) have on average solar metallicity and galaxies at the  highest mass end (∼ 1012 M⊙) have metallicities of around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The total mass-metallicity relation of all the stars in the center is very  close to the one for the in-situ population only, as they dominate the  mass in the center of galaxies on average (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Furthermore, the average metallicity for central stars is consistently  offset by about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='3 dex towards higher metallicities across the whole  galaxy mass range compared to the mass-metallicity relation which  takes into account all the stars belonging to a given galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This  emphasizes the self-similarity of galactic chemical enrichment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  On top of that, the total central mass-metallicity relation is tight  having a scatter of around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1 dex, which also holds when only in-situ  or only migrated stars are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Hence, there is little galaxy-to-  galaxy variation at fixed stellar mass regarding in-situ star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The average metallicity of the in-situ population is the highest,  followed by the migrated stars, which is less than a quarter dex  lower across the whole galaxy mass range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This small difference is  expected as most of the migrated stars are born very close to the  center (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 − 1 kpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' When including only migrated stars with large  birth radii (> 1 kpc), the difference becomes larger to about half a  dex due to internal metallicity gradients present in galaxies, which is  in turn caused by less efficient star formation in the galactic outskirts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Above galaxy stellar masses of 2 × 1011 M⊙ the average metallicity  of all migrated stars and only those with birth radii larger than 1 kpc  becomes similar again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This is because migrated stars from clumps  are dominating at these galaxy masses, which originate from larger  distances (median distance is 30 kpc) and have high metallicities  (median metallicity is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2 dex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The mass-metallicity relation for the central ex-situ stars follows a  steeper slope than the one for the in-situ and migrated stars, because  we are showing the mass of the z = 0 host galaxy and not of the galaxy  they were born in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The average metallicity of ex-situ stars is around  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 dex lower at the lowest galaxy masses compared to the metallicity  for the in-situ stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' At the highest mass end the average metallicity  of the ex-situ stars becomes close to the one for the migrated stars,  which is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='25 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This steeper slope emphasizes that ex-situ  stars in the center of low mass galaxies originate from galaxies of  even lower mass, while most of the central ex-situ stars in high mass  galaxies originate from galaxies of more similar stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Lastly, the galaxy-to-galaxy variation at fixed galaxy stellar mass  for the average metallicity of ex-situ stars is much larger compared to  the in-situ and migrated population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The scatter varies from around  one dex at the low mass galaxy end to close to a quarter dex for the  highest galaxy masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This emphasizes that at lower host galaxy  stellar mass, a larger variety of satellite galaxies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' with different  stellar masses) can deposit stars in the center of their respective  z = 0 hosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Age (Figure 7, middle panel): The ex-situ stars have a rather con-  stant, old age of around 10 Gyr across the whole galaxy mass range,  albeit with a large scatter of around 2 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This is not surprising as  most mergers happen before the redshift of one, which corresponds to  a lookback time of around 8 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The flat relation for the average age  of the ex-situ stars is not in conflict with their corresponding mass-  metallicity relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Because high mass galaxies are more efficient in  chemical enrichment than low mass galaxies, they will consequently  have higher metallicities at fixed stellar age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The median relations for the average age for the in-situ and mi-  grated stars are again similar to each other with the in-situ stars being  slightly older by around 1 Gyr or less at fixed galaxy stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Overall, in-situ and migrated stars are younger, with average ages  between 3 and 6 Gyr, in the lowest mass galaxies (∼ 109 M⊙), and  become increasingly older with average ages of around 8 − 10 Gyr at  the highest mass end (∼ 1012 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The scatter of the average ages for the in-situ and migrated stars is  much larger than their corresponding variations in metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This  could have multiple reasons, for example: different pathways in star  MNRAS 000, 1–34 (2022)  Centers of TNG50 Galaxies  13  109  1010  1011  1012  Total Stellar Mass [M ]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content="5  Central Metallicity  [log10(Z/Z )]  TNG50  z = 0  Insitu  Migrated 'Smooth+Clumpy'  Migrated Birth Radius  1 kpc  Exsitu  109  1010  1011  1012  Total Stellar Mass [M ]  2  4  6  8  10  12  14  Central Age  [Gyr]  TNG50  z = 0  109  1010  1011  1012  Total Stellar Mass [M ]  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='7  Central [Mg/Fe]  [dex]  TNG50  z = 0  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Average stellar population properties of in-situ, migrated and ex-situ stars in the central 500 pc of TNG50 galaxies at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' From left to right:  The central mass-weighted average metallicity, age and magnesium-to-iron abundance [Mg/Fe] as a function of total galaxy stellar mass for the in-situ (pink),  migrated (orange) and ex-situ (blue) stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The shaded bands depict the 16th and 84th percentile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The dashed orange line only shows the quantities for migrated  stars that have birth radii larger than 1 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Overall, the stellar population properties of in-situ and migrated stars are similar, whereas the ex-situ stars are more  metal-poor, older and have higher [Mg/Fe] across the galaxy mass range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  formation histories (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' star formation rate as a function of time)  can result in the same metallicity but different average ages, or the  metallicity enrichment starts to saturate once a metallicity above  solar is reached and therefore it does not matter, if star formation  continues for another few Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Galaxies above 1011 M⊙ in stellar mass exhibit a larger scatter of  the average age of their migrated population compared to their in-  situ stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This arises because migration to the center in this regime  is dominated by clumps, which have a rather flat formation time  distribution with the majority forming between 4 and 10 Gyr ago  (see Figure E1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Below galaxy stellar mass of 1011 M⊙, the migrated stars, which  were born at distances larger than 1 kpc, have a running median of  averages ages that are around 1 − 2 Gyr older than the total migrated  population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' As these stars need significantly more time to arrive in  the center, their ages are consequently older.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  [Mg/Fe] (Figure 7, right panel): In extragalactic studies magne-  sium is the predominant 𝛼-element present in optical spectra (see  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Martín-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2018a, 2019, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Gallazzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  We therefore show the running median of the mass-weighted average  magnesium-to-iron abundance as a function of galaxy stellar mass  as a proxy for the total 𝛼-to-iron abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This abundance ratio  provides to first hand information about the star formation time scale  before supernovae type Ia significantly enrich the interstellar medium  with iron peak elements8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The average central [Mg/Fe] is almost constant for the in-situ  population across the whole galaxy mass range with a value of about  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='3 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For galaxies between 2 × 109 M⊙ and 6 × 1010 M⊙, the  migrated stars have slightly lower values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The lowest average [Mg/Fe]  of around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='25 dex is reached for galaxies around 2 × 1010 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This  directly maps to the increased difference of the average age between  8 Influences on [𝛼/Fe] due to IMF (initial mass function) changes are not  captured in the simulation, as a Chabrier IMF (Chabrier 2003) is assumed for  every stellar particle  in-situ and migrated stars of around 1 Gyr in the same mass regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Hence, in-situ stars of these galaxies form on average earlier and  more rapidly as opposed to their migrated stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Above 6 × 1010 M⊙, the average [Mg/Fe] for the migrated pop-  ulations rises above the one for the in-situ stars to around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='35 dex  at the highest mass end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This cross-over is not seen in the average  ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' An explanation for this could be that in the high galaxy mass  regime, an increasing number of migrated stars can originate from  larger distances and possibly formed from stripped gas of merging  lower mass systems (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2), which have larger [Mg/Fe]  values due to lesser efficiency in chemical enrichment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  When only including migrated stars originating from distances  farther than 1 kpc away from the center, the average [Mg/Fe] be-  comes larger by around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1 dex across all galaxy stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For  galaxies below ∼ 1011 M⊙ the corresponding ages become older,  which is thus consistent in having formed from true in-situ gas of  their respective host galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The age for migrated stars with birth radii > 1 kpc in galaxies  above 1010 M⊙ does not increase even though their [Mg/Fe] increase  as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This could indeed provide evidence for some migrated stars  having formed from stripped gas for galaxies in this mass regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The ex-situ stars have overall higher average [Mg/Fe] values of  around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='45 dex, which decreases to around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='35 dex for galaxies  above 1011 M⊙ in stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This is consistent with their old ages  and being formed in lower mass satellite galaxies that produced stars  less efficiently than their respective z = 0 hosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The scatter in average [Mg/Fe] for the ex-situ population is signif-  icantly larger than for the in-situ and migrated population across all  galaxy masses, but especially ≲ 1011 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The onset of type Ia su-  pernovae creates probably more stochasticity in lower mass galaxies  as single supernovae events can significantly enrich the interstellar  medium of the entire host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  MNRAS 000, 1–34 (2022)  14  Boecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  109  1010  1011  1012  Total Stellar Mass [M ]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  Central Circularity  TNG50  z = 0  Insitu  Bulgey  Disky  109  1010  1011  1012  Total Stellar Mass [M ]  TNG50  z = 0  Migrated  Bulgey  Disky  109  1010  1011  1012  Total Stellar Mass [M ]  TNG50  z = 0  Exsitu  Bulgey  Disky  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='10  Fraction in  per Stellar Mass Bin  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Average differences in the dynamical properties of in-situ, migrated and ex-situ stars in the central 500 pc of TNG50 galaxies at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' From  left to right: The central circularity distributions in stacks of total galaxy stellar mass for in-situ (pink), migrated (orange) and ex-situ (blue) stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Per galaxy,  stellar particles in the center belonging to either the in-situ, migrated or ex-situ population are binned according to their circularity 𝜖 and normalized to unity  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' They are then stacked together according to the displayed galaxy stellar mass bins and then normalized again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The lines trace the circularity bin  with the maximum fractional mass across galaxy stellar masses divided by bulgey (solid) and disky (dashed-dotted) galaxies respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Clearly the migrated  population has the most rotational support for galaxies around 1010 M⊙ in total stellar mass, regardless of the host being disk or bulge dominated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2 Stacked circularity distributions  Different birth origins also leave imprints on the stars’ dynamics,  which can still be visible until the present-day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We investigate such  imprints by quantifying the instantaneous circularity 𝜖 of stars (see  Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Circularities close to one indicate circular orbits,  values around zero indicate random motion dominated orbits and  negative ones show counter-rotating orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We then compute the  normalized circularity distribution for each galaxy with a bin size of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1 for 𝜖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Circularity distributions are than stacked together according  to the galaxy’s total stellar mass in bins of approximate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 dex and  are re-normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The results for the different in-situ, migrated and ex-situ popula-  tions are displayed in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The lines in Figure 8 trace the peak  of the circularity distributions across galaxy stellar mass, separately  for disky and bulgey galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The circularity distribution of the in-situ population is centered  on random motion dominated orbits for galaxies with stellar masses  smaller than 3 × 109 M⊙ and larger than 1011 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Galaxies with  stellar masses in between have a circularity distribution with a peak  shifted towards slightly higher circularities of around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We see  that this shift is caused by galaxies that are overall disky, as the  bulge dominated galaxies have a circularity peak that stays around  zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Nevertheless, the in-situ stars are in summary on warm to hot  orbits even for disk dominated galaxies, which is not surprising as  the velocity dispersion generally rises towards the center of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  For galaxies below 1010 M⊙ the stacked circularity distributions  for in-situ stars have a sharper peak, whereas galaxies of higher  masses have an overall broader distribution in 𝜖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This could be an  indication of a smaller galaxy-to-galaxy variation of the circular-  ity distribution in the center of the smallest galaxies, regardless of  whether they are disky or bulgey, as in this mass regime the abso-  lute numbers of those two galaxy types are approximately the same in  TNG50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' At the high mass end on the other hand, a broader circularity  distribution could indicate that in-situ stars become redistributed in  their orbits due to the increased influence of mergers and contribution  of ex-situ stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  For the migrated population the circularity distribution is again  centered on random motion orbits for galaxies < 3 × 109 M⊙ and  > 1011 M⊙, although now the distribution is also overall broader for  the low mass galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Migrated stars in intermediate mass galaxies  are on even higher circularity orbits than their corresponding in-  situ stars reaching a peak of around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 for galaxy stellar masses of  ∼ 1010 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This peak is seen in disk and bulge dominated galaxies  alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Hence, migrated stars tend to have the most rotational support  for galaxies in the intermediate mass regime, which could be an  indication for migration being caused by different mechanisms across  the galaxy mass range in TNG50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' However, migrated stars are also on  average younger than the in-situ stars in these galaxies, which might  be the reason why they are still more on circular orbits (see Figure  7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We also point out that some galaxies above ∼ 3 × 1010 M⊙ have  a very double peaked (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' one around zero and around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 or higher)  circularity distribution, which is washed out in Figure 8 due to the  stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' These stars originate from (recently) migrated clumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Ex-situ stars have circularities centered around zero across the en-  tire galaxy mass range in TNG50 and also for both disk and bulge  dominated galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Because they originate from stochastic merger  events, stars are put on average on hot, random motion dominated  orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Nevertheless, we see a a large scatter throughout the circu-  larity distributions for the ex-situ stars in the different galaxy stellar  mass bins indicating a lot of individual galaxy-to-galaxy variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Depending on the exact time the merger occurred and how the orbits  between the host and merging satellite were configured, ex-situ stars  can very well retain some rotational support and often be on counter  rotating orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='3 2D distributions of ages, metallicity and circularities  In Figures 9 and 10 we show the 2D distributions of age and metallic-  ity and age and circularity of the central in-situ, migrated and ex-situ  stars respectively in stacks of galaxy stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For each galaxy  we first compute the mass-weighted and normalized 2D histogram  of the respective quantities with bin sizes of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 Gyr for age, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='25  dex for metallicity and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1 for circularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We then stack those ac-  cording to the total stellar mass bin of the galaxies, normalize again  MNRAS 000, 1–34 (2022)  Centers of TNG50 Galaxies  15  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content="0  Central Metallicity  [log10(Z/Z )]  TNG50  z = 0  145 Galaxies  Insitu  Migrated 'Smooth+Clumpy'  Exsitu  Insitu  Migrated 'Smooth+Clumpy'  Exsitu  TNG50  z = 0  129 Galaxies  TNG50  z = 0  92 Galaxies  TNG50  z = 0  95 Galaxies  Quenched  TNG50  z = 0  145 Galaxies  0  4  8  12  Central Age [Gyr]  2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  Central Metallicity  [log10(Z/Z )]  109 M  TNG50  z = 0  587 Galaxies  0  4  8  12  Central Age [Gyr]  109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 M  TNG50  z = 0  573 Galaxies  0  4  8  12  Central Age [Gyr]  1010 M  TNG50  z = 0  415 Galaxies  0  4  8  12  Central Age [Gyr]  1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 M  TNG50  z = 0  222 Galaxies  0  4  8  12  Central Age [Gyr]  Star Forming  1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 M  TNG50  z = 0  76 Galaxies  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Age-metallicity distributions of central (500 pc) stars of TNG50 galaxies in stacks of stellar mass at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Gaussian kernel density estimates for  in-situ (pink), migrated (orange) and ex-situ (blue) stars encompassing 1%, 20%, 50% and 90% of all central stellar mass (thickest to thinnest) are shown in the  respective galaxy stellar mass bin increasing from left to right as depicted by the colorbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The galaxy mass bins are centered on the indicated stellar mass and  are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 dex wide, except for the last panel, which is approximately one dex wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Prior to stacking the age-metallicity distribution of each galaxy is normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The top row shows quenched and bottom row shows star forming galaxies respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In each panel the number of galaxies in the corresponding stellar mass  bin are indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The galaxy-averaged age-metallicity distribution of the three origins becomes best separated around galaxies with stellar masses of 1010 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  and then compute the Gaussian kernel density estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The galaxy  stellar mass bins are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 dex wide for galaxies between 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='75 M⊙  and 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='75 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We stack all galaxies with stellar masses between  1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='75 M⊙ and 1012 M⊙ together as a finer binning did not reveal  any mass-dependent trends and also became stochastic due to low  number statistics in this mass regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The five galaxies with stel-  lar masses above 1012 M⊙ are not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Additionally, we show  the stacked age-metallicity distributions for quenched and star form-  ing galaxies separately to avoid averaging over too many dissimilar  galaxies in this parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Similarly, we divide between bulge  and disk dominated galaxies for the age-circularity distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  With the 2D distributions we can observe a couple of new trends  that are not necessarily apparent from the average stellar population  properties in Figure 7 and the 1D circularity distributions of Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Age-metallicity (Figure 9): For star forming galaxies (bottom row  of Figure 9), the average distribution of the migrated stars changes  very little in shape and position, apart from shifting towards higher  metallicities, from the lowest mass galaxies until the 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 M⊙  galaxy stellar mass bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' They are centered between 2−4 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The ex-  situ stars behave similarly and are centered around 12 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' However,  the average distribution of the in-situ stars shows an entirely different  mass trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' While the in-situ stars are almost entirely coinciding with  the migrated stars in age-metallicity space for the lowest mass bin,  the peak of the in-situ distribution gradually shifts towards older ages  from 2 Gyr to 8 Gyr for increasing galaxy mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In the process the  in-situ average age-metallicity distribution becomes more elongated  around 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 M⊙ in the age direction when focusing on the 20%  contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  For 1010 M⊙ galaxies the in-situ distribution becomes more cen-  trally concentrated again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Furthermore, the average age-metallicity  distributions of the three origins are maximally separated in this mass  regime, with the migrated stars being the youngest (1 − 6 Gyr for the  20% contour), followed by the in-situ stars (6 − 10 Gyr for the 20%  contour) at similar metallicity and the ex-situ stars populating the  oldest (10 − 13 Gyr for the 20% contour) and most metal-poor tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Thus, there must be a mechanism for these galaxies that halts in-situ  star formation in their centers, while it continues outside of it in order  to be able to produce young migrated stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' It is likely that this is  connected to the (kinetic) AGN feedback implement in TNG, which  quenches galaxies from inside-out (Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2021, see also 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2  and Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Starting at 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 M⊙ the average age-metallicity distribution of  the migrated stars also becomes more elongated towards older ages  and above 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 M⊙ coincides again with the one of the in-situ stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The peak of the ex-situ distribution increases towards metallicities  similar to those of the in-situ and migrated stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For galaxies between  1011 M⊙ and 1012 M⊙ in stellar mass the average distributions for the  in-situ, migrated and ex-situ stars become almost indistinguishable  in age-metallicity space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  For quenched galaxies (top row of Figure 9) the behaviour for the  age-metallicity distributions of the in-situ and migrated stars across  galaxy stellar mass is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' They are not clearly separated in any  galaxy stellar mass bin as was the case for the star forming galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Both the in-situ and migrated average age-metallicity distributions  are more centrally concentrated than for star forming galaxies, their  shapes are very similar to each other and their peaks are both at old  ages (around 8 Gyr) exhibiting little galaxy mass dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The  peak of the age-metallicity distribution for the migrated stars seem to  be slightly younger for galaxies in mass bins between 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 M⊙ and  1010 M⊙ and slightly older otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Interestingly at both the low  and high mass end, the separation in metallicity between the in-situ  and migrated stars is larger for the quenched galaxies as for the star  forming ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For galaxies between 1011 M⊙ and 1012 M⊙ the three  distributions are again indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Age-circularity (Figure 10): Stars with higher circularities are  MNRAS 000, 1–34 (2022)  16  Boecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content="0  Central Circularity  TNG50  z = 0  441 Galaxies  Insitu  Migrated 'Smooth+Clumpy'  Exsitu  Insitu  Migrated 'Smooth+Clumpy'  Exsitu  TNG50  z = 0  297 Galaxies  TNG50  z = 0  129 Galaxies  TNG50  z = 0  96 Galaxies  Bulgey  TNG50  z = 0  145 Galaxies  0  4  8  12  Central Age [Gyr]  1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  Central Circularity  109 M  TNG50  z = 0  291 Galaxies  0  4  8  12  Central Age [Gyr]  109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 M  TNG50  z = 0  405 Galaxies  0  4  8  12  Central Age [Gyr]  1010 M  TNG50  z = 0  378 Galaxies  0  4  8  12  Central Age [Gyr]  1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 M  TNG50  z = 0  221 Galaxies  0  4  8  12  Central Age [Gyr]  Disky  1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 M  TNG50  z = 0  76 Galaxies  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Age-circularity distributions of central (500 pc) stars of TNG50 galaxies in stacks of stellar mass at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Gaussian kernel density estimates  for in-situ (pink), migrated (orange) and ex-situ (blue) stars encompassing 1%, 20%, 50% and 90% of all stellar mass (thickest to thinnest) are shown in the  respective galaxy stellar mass bin increasing from left to right as depicted by the colorbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The galaxy mass bins are centered on the indicated stellar mass and  are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 dex wide, except for the last panel, which is approximately one dex wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Prior to stacking the age-circularity distribution of each galaxy is normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The top row shows bulge dominated and bottom row shows disk dominated galaxies respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In each panel the number of galaxies in the corresponding  stellar mass bin are indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The galaxy-averaged age-circularity distribution of the three origins becomes best separated around galaxies with stellar masses  of 1010 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  usually younger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The distribution for migrated stars of disky galax-  ies (bottom row of Figure 10) in the lowest galaxy stellar peaks at  around 2 Gyr with high circularity values of around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5, whereas the  distributions for the in-situ stars peaks at older ages (6 Gyr) centered  on circularity values of zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Nevertheless, the 20% contour for the  in-situ stars still has a tails towards younger ages and slightly above  zero circularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In the next higher galaxy stellar mass bin the 20%  contour of the distribution for the migrated stars looses its tail of  older ages (4−8 Gyr) and zero circularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The 20% contour for the  in-situ stars in now centered on even older ages (8 Gyr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Beginning  around galaxy stellar masses of 1010 M⊙ the 20% contour for the  migrated stars elongates towards older ages spanning now 1 − 8 Gyr,  while roughly maintaining the high circularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The distribution for  the in-situ stars becomes broader and slightly shifts towards above  zero circularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In the 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 M⊙ galaxy stellar mass bin the peak of  the migrated stars shifts from young (2 Gyr) to old (8 Gyr) ages with  just a slight decrease in circularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Above galaxies with 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 M⊙ in  stellar mass the migrated stars switch from a rotationally supported  distribution to random motion dominated one until they coincide  with the age-circularity distributions of the in-situ and ex-situ stars  at the highest galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The peak of the age-circularity distribution for the in-situ stars,  albeit having the same young age as for the migrated stars in the lowest  stellar mass bin, is near zero circularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' With increasing galaxy mass  the age-circularity distribution for in-situ stars shifts towards old ages  and becomes broader in the circularity direction, but stays mostly  centered around zero circularity with perhaps a slight shift towards  higher circularities around the 1010 M⊙ galaxy stellar mass bin as  already observed in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The age-circularity distributions for the  ex-situ stars show practically no galaxy mass dependence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' they are  centered on random motion dominated orbits and the oldest ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The centers of bulge dominated galaxies (top row of Figure 10)  above 109 M⊙ have overall similar age-circularity distributions as  disk dominated galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' However, the absolute values of the mi-  grated distribution do not reach the same high circularities as for the  disky galaxies and its peak transitions quicker to old ages (8 Gyr) be-  tween mass bins of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 and 10 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Below 109 M⊙ both the migrated  and in-situ distrbition are centered on zero circularities and old ages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  distinct to the disky galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  For both bulge and disk dominated galaxies the average age-  circularity distribution of the in-situ, migrated and ex-situ stars are  well separated in mass ranges between 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 M⊙ and 1010 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This  dependence of increasing circularity for younger ages, especially  prominent for the migrated stars, gives an indication that recently  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' young) migrated stars travel to the center of their host galaxies  by loosing their angular momentum (“churning”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Frankel  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2020, for the Milky Way disk) and then, once they have arrived  in the center, become dynamically heated over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  5 DISCUSSION, IMPLICATIONS AND OUTLOOKS  In this section we discuss the implications of the studied mass as-  sembly of the central 500 pc in TNG50 galaxies on the formation  scenarios of central galaxy components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We also discuss the clumps  found in TNG50 as well as the robustness of our results within the  TNG modelling framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In addition, we assess how our results  on the stellar population and dynamical properties can be compared  to observations and used to understand the mass build-up of galaxies  in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  MNRAS 000, 1–34 (2022)  Centers of TNG50 Galaxies  17  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1 The build-up of galaxy centers in a 𝚲CDM cosmology  Throughout this paper, we have unravelled a set of relations between  the properties of the stellar centers of galaxies at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Galaxy  centers are dominated by in-situ stars (see Figure 4) and follow well  established relations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Gallazzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2005) that correlate their  increasing stellar masses with increasing average ages and metallici-  ties (see Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Stars that migrated to the center are second most  abundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' They follow the trends for the in-situ stars, however are  often distinctively younger and on more rotation supported orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Ex-situ stars in the center become only significant (in mass) at high  galaxy stellar masses (> 1011 M⊙) (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The majority of  ex-situ stars originate from the centers of the accreted galaxies (see  also Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Moreover, they are amongst the oldest and most  metal-poor and random motion dominated stars (see Figures 7 and 8  as well as), which is in agreement with El-Badry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2018),  who studied three MW-like galaxies from the Latte project (Wetzel  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  While these trends are consistent with our general understanding  of galaxy formation in a ΛCDM cosmology, we find others that  may be more surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For example, there seems to be no average  difference between the central mass assembly of central and satellite  galaxies (see Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Generally, central galaxies are thought to have  accreted more satellite galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We have checked thisrelation also for  the total accreted mass within TNG50 and also found no significant  difference between centrals and satellites on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Thus, perhaps  TNG50 is not probing enough very high mass central galaxies around  stellar masses of 1012 M⊙, where this trend might become apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Another, rather unexpected result compared to usual assumptions,  is that star forming galaxies above 1010 M⊙ possess on average more  ex-situ mass in their centers compared to quenched ones (see Figure  6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Again, this difference, even though much less significant remains  when considering the total amount of ex-situ mass (see Figure C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  This trend also exists for the larger box of TNG100, thus eliminating  the fact for low number statistics at the higher mass end, and is in  contrast to the original Illustris simulation (see Rodriguez-Gomez  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2016, Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We have checked the median mass growth of  the central ex-situ stars for star forming and quenched galaxies alike  between stellar masses of 1010 − 1011 M⊙ and found that quenched  galaxies stop acquiring ex-situ mass in their centers after z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='7  (lookback time ∼ 10 Gyr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Only if we split quenched galaxies further  into bulgey and disky as well as barred and non barred do we see that  quenched, bulgey and non barred galaxies have a similarly high ex-  situ mass in their centers as their star forming counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Together,  this is an indication that the time of accretion and consequently the  absolute amount of stellar and gas mass of the secondary galaxy (the  former will be higher at later cosmic times and the latter will influence  the amount of newly formed stars during the merger process) will  matter in the build-up of ex-situ mass in the center of the primary  and ultimately dictate what properties it has today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  On top of that, the fraction of in-situ, migrated and ex-situ stars  in the center of galaxies has a significant scatter at fixed galaxy  stellar mass regardless of the galaxy’s bulk properties at z = 0 (see  Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Hence, median trends for different galaxy populations only  reveal half of the picture, as the stochasticity of galaxy mergers and  interactions in a ΛCDM cosmology leads to diverse pathways in the  build-up of stellar mass in the centers of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Thus, characteristic  properties of galaxies at z = 0 are only a limited indicator of the exact  formation history of an individual galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For example, there are  perfectly regular MW-like spiral galaxies in TNG50 z = 0, of which  some have experienced (multiple) major mergers and of which some  had a more quiet assembly (see also Sotillo-Ramos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  This diversity in the central 500 pc of TNG50 galaxies potentially  reflects the variety of central galaxy components seen in observations  (see Section 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Even though nuclear rings, disks and star clusters  are at or below the resolution limit of TNG50, the stellar population  and dynamical properties that we find for central stars of different  origins might be a first indication that this would also manifest in  structurally distinct components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For example, the distinctly high  circularities of migrated stars in 109 − 1010 M⊙ galaxies (see Figure  10) reflect that nuclear disk-like configurations are able to arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Even  more intriguing are their predominantly younger ages of 1 − 2 Gyr  compared to the underlying old (∼ 8 Gyr) in-situ population, which  is in line with observational findings of nuclear disks/rings (Bittner  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Typically, the formation of nuclear rings in disk galaxies  is associated with bars funnelling gas towards the center (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Seo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Tress et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Sormani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2020, for dedicated  simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Even though we did not explicitly investigate the inflow  of gas in this study, we see that the migration of stars to the center is  likely connected to temporarily induced non-axisymmetries during  galaxy interactions (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Hence, this shows that mech-  anisms that are associated with producing distinct nuclear galaxy  components are captured in TNG50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Follow-up zoom-in simulations  of TNG50 galaxies would show if indeed nuclear components such  as disks and rings form from these mechanisms (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2 Mechanisms for the formation and deposit of stars in the  center of galaxies  The cosmological framework of TNG50 produces diverse properties  of galaxies and their centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Consequently, the mechanisms that are  responsible for the formation and deposit of stars in the centers of  galaxies also have to be diverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  To visualize possible mechanisms for the formation and deposit of  stars in the center of galaxies we walk through the central assembly  history of an individual galaxy as seen in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We picked this  particular galaxy, which has a stellar mass of 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='8 M⊙ at z = 0,  as it shows many of the possible mechanisms that can be present  in the formation of galaxy centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This however does not mean  that all galaxies show the same amount of complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Most galaxies  will only exhibit one or two of these mechanisms with varying impact  depending on their individual formation pathway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The galaxy’s center  at z = 0 consists of around 50% in-situ, 30% migrated and 20% ex-  situ stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The main summary of the subsequent sections and Figure 11 is  the following: galaxy mergers and other interactions are probably  the most important driver in central stellar mass assembly, as they  also strongly influence the formation of in-situ stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Due to the  diverse statistics of galaxy interactions, many of the proposed for-  mation scenarios of central galaxy components arise naturally and in  conjunction to each other, when hierarchical galaxy formation is con-  sidered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Thus, TNG50 highlights the necessity to study the central  mass assembly of galaxies in a cosmological context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1 In-situ stars  Galaxy mergers can trigger bursts of star formation as the tidal forces  compress and shock gas efficiently (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g Mihos & Hernquist 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Di  Matteo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Cox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Di Matteo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2008), even in  its nuclear region (Powell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' While the relative enhance-  ment of star formation rates depend on the specific configuration  of the merging galaxies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' merger mass ratio, gas content, orbital  infall parameters, the times of intense star formation coincide with  MNRAS 000, 1–34 (2022)  18  Boecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Central (500 pc) assembly history of an individual galaxy (SubfindID 184937) in TNG50 with a total stellar mass of 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='8 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This galaxy  encompasses many mechanisms that can shape the stellar mass build-up in the center of galaxies in a 𝚲CDM cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Top panel: Points show all  individual stellar particles that belong to that galaxy at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Their distance at the time of birth is shown with respect to their current host in the case of in-situ  formed stars (light gray: all in-situ particles, pink: central in-situ stars only, orange: central migrated stars only).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In the case of the ex-situ formed stars the  distance is shown with respect to their future host galaxy at the time of birth (color-coded according to the colorbar: all ex-situ stars, blue: central ex-situ stars  only).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The distance to individual satellite galaxies (only with maximum stellar masses above 106 M⊙) that will merge with the primary at some point are shown  with thinner solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Their coloring also follows the colorbar, which visualizes the merger mass ratio taken at the time tmax, when the secondary galaxy  reaches maximum stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The thick black solid line shows the radius of the FoF Group the galaxy belongs to at a given lookback time (represented as R200,  where the group’s density is 200 times the critical density of the Universe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The thick gray dashed line shows the distance between the individual galaxy and the  central galaxy of the FoF group it belongs to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Approximately 7 Gyr ago the galaxy fell into another group and became a satellite galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Before that it was the  central of its own FoF group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The vertical black dotted line represents the time the kinetic AGN feedback starts to take effect, which quenches the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This  galaxy has 50% in-situ, 30% migrated (of which only 9% are ’smoothly’ migrated and the rest comes from migrated clumps) and 20% ex-situ stars in its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Bottom panel: Histograms of formation times of in-situ (top), formation (solid) and arrival (dashed-dotted) times at the center for ‘smoothly’ migrated (middle)  as well as ex-situ and ‘clumpy’ migrated (bottom) stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Additionally, in the panel for the in-situ stars, we mark the time of coalescence for the six most massive  mergers of this galaxy with thin blue colored solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' According to the colorbar of the top panel, a darker blue means a higher merger mass ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Pericenter  passages for two mergers are shown by thin dashed lines following the same colorcode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The approximate time of the galaxy falling into its z = 0 FoF group is  shown by the thick black solid line and the onset of the kinetic AGN feedback is shown as the black dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In the panel for the ‘smoothly’ migrated stars  we also show the A2 mode of the stars for a given lookback time (see Appendix A1 for a definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In the panel for the ex-situ and ‘clumpy’ migrated stars, we  show the time of coalescence of the two mergers that deposited ex-situ stars in the galaxy’s center (blue solid lines) as well as the three pericenter passages of  the galaxy around its central galaxy after it became a satellite (gray dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  MNRAS 000, 1–34 (2022)  SubfindID 184937  1000  Ratio at tmax  100=  R200 of FoF Group  10#  Distance to Central  [odx]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1  Clumps  Merger Mass  Birth Radius  of Stellar Particles  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='t Primary  Insitu: 50 %  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1  TT  Migrated: 30 %  Onset of  Kinetic AGN Feedback  Exsitu: 20 %  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='01  14  12  10  6  4  0  Lookback Time [Gyr]  Merger Pericenter  Merger Coalescence  Infall into Group  Kinetic AGN Feedback  Pericenters around Central  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='8  Insitu  #  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='6  At Formation  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content="8  Migrated  #  Smooth'  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='4  At Formation  At Arrival in Center  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2  A2 of Stars  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content="0  Exsitu  Migrated  Normalized  'Clumpy'  2  At Formation  At Arrival in Center  0-  14  12  10  8  6  Lookback Time [Gyr]Centers of TNG50 Galaxies  19  pericenter passages and coalescence (see also Sotillo-Ramos et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Peaks in the formation time of central in-situ stars in Figure 11  coincide with times of pericenters and coalescence of mergers that  this galaxy has experienced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Thus the formation history of the central  in-situ stars is directly connected with the merger history of a galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  However, it has to be further quantified whether also the bulk of  in-situ stars is formed during such events or if that actually happens  in-between galaxy interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Nevertheless, it is clear that a variety  of different mergers are able to produce peaks in the formation of  central in-situ stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  For example, the peak between 10 Gyr and 12 Gyr ago was induced  by a very minor merger9 with a stellar merger mass ratio of around  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' At these high redshifts the primary still had a high gas fraction  (∼ 80%) and thus the minor merger was enough to trigger a peak  of star formation in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Evidently, the formation of in-situ  stars in the center decreased between 10 Gyr and 8 Gyr ago, as the  amount of available gas decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Thus, in order to trigger another  significant peak in in-situ star formation later on, the merger between  8 Gyr and 7 Gyr had to bring in a large amount of gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' While the ratio  of the stellar mass between the secondary and primary was around  one (and therefore a major merger) at the time when the secondary  reached its maximum stellar mass, the secondary still had around 20  times more gas than the primary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Furthermore, this major merger, as well as two smaller ones that  coalesced around 6 Gyr ago, happened while the primary galaxy was  in the process of falling into another FoF Group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' transitioning  from being a central galaxy of its own FoF group to being a satellite  galaxy of another FoF group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This is seen by the two sharp jumps in  R200 between 9 Gyr and 7 Gyr ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Evidently, this process produced  another peak of in-situ star formation at around 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 Gyr ago, which  could stem from the new, higher density environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We have also  seen in other galaxies, that were able to retain enough gas in their  centers after infalling into a group, that in-situ star formation was  triggered during the pericenter passages around the central until the  galaxy became quenched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In such occasions, again tidal forces are  able to compress the gas efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Within TNG50, there are two main processes that can quench the  in-situ star formation in the center of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The first one is the  onset of the kinetic AGN feedback mode implemented in TNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Often  this feedback mode switches on after a merger has been completed,  as is the case for our galaxy in Figure 11 at around 7 Gyr, shortly  after the major merger coalesces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We see that only the central 1 kpc  becomes quenched, while the outskirts of the galaxy continue to form  stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This is because in TNG, AGN driven quenching proceeds from  inside out (see Weinberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2019b, 2021,  for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' After this mode is switched on only occasional gas-rich  mergers or migrated clumps are able to bring in new gas to the center  to cause new in-situ star formation, as seen at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 Gyr in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Lastly, the thermal feedback mode, which is often active prior to the  kinetic mode switches on and also injects relatively more energy,  is not responsible for quenching the centers of galaxies in TNG50  (Zinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The second process that will shut down star formation in the center  of TNG50 galaxies is when the galaxy as a whole becomes quenched,  either through environmental processes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' after a few pericenters  after infall into a group (as is the case for the galaxy in Figure 11  around 3 Gyr ago) or through AGN feedback, which is primarily  9 Weadopt the definition ofRodriguez-Gomez etal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' (2016)for thecalculation  of merger ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  important for the highest mass galaxies (see also Donnari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2 Migrated stars  The formation times of ‘smoothly’ migrated stars in Figure 11 is  closely related to the formation times of the in-situ stars, which is not  the case for the ‘clumpy’ migrated stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This is reasonable, because  the majority of ‘smoothly’ migrated stars are born already close  to the center (≲ 2 kpc), while the ‘clumpy’ migrated stars formed  predominantly in the outer disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The ‘clumpy’ migrated stars make  up 91% of the total mass of migrated stars in the center of this galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  However, the star formation in the central 2 kpc is not a guarantee  to produce a significant amount of ‘smoothly’ migrated stars, as seen  between lookback times of 9 − 11 Gyr and also between 7 − 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 Gyr  in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Thus, specific conditions must be met that transport  stars from around 1 − 2 kpc to the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Non-axisymmetric features, such as spiral arms and bars, are well  known to be able to diffuse the angular momentum of stars and cause  radial migration (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Sellwood & Binney 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Minchev & Famaey  2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' While this effect is mainly studied in the (outer) disk of  galaxies, we show here in Figure 11 that similar non-axisymmetries  are likely responsible for the inward migration of stars to center  of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We see that peaks in the A2 mode (see Appendix A1  for a definition) of the stellar mass distribution occur before peaks  of migrated stars arriving in the galaxy center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We detect similar  enhancements for the Fourier modes of the gas mass distribution  (see also Di Matteo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  These temporary enhancements of non-axisymmetric features are  clearly induced during galaxy interactions and the exerted torques  on the gas and stars can produce these ‘smoothly’ migrated stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  This also indicates that it is possible for a galaxy to have experienced  migration events of stars, even if the galaxy itself does not exhibit  any signs of bar- or spiral-like features today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The ‘clumpy’ migrated stars form after the first pericenter passage  of the galaxy around its central around 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 Gyr ago and arrive at  the center shortly before the second pericenter passage around 2 Gyr  later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Similarly, we have seen qualitatively for other galaxies that  clumps formed rather recently (z < 1) are mainly induced by fly-bys,  as these are still able to destabilize the disk significantly after the  predominant merger phase of the Universe is over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' However, clumps  are also able to form without any significant galaxy interactions and  a follow-up study is needed to characterize this further as well as  establish overall the credibility of the formation of the clumps (see  Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='3 for a further discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='3 Ex-situ stars  The two mergers that are responsible for the majority of the ex-situ  stars in the center of the galaxy in Figure 11, are the 1:1 and 1:10  merger that coalesced around 7 Gyr and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 Gyr ago respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Both mergers brought in a comparable total amount of stellar mass  of around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2 × 1010 M⊙ and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1 × 109 M⊙ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' However,  the major merger deposited around 10 times less stars in the central  500 pc compared to the minor merger, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='6% and 5% of their  respective total stellar mass arrived in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This highlights  that the merger mass ratio cannot be the only parameter determining  the amount of ex-situ stellar mass that is deposited in the center of  galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We expect that the spin-orbit coupling of the primary and  secondary galaxy as well as other orbital parameters play a role in  this, as the exerted tidal forces and the influence of dynamical friction  MNRAS 000, 1–34 (2022)  20  Boecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  differ for different configurations (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Renaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2009, for a  study).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Around 67% and 93% of the ex-situ stars that arrived from the  major and minor merger respectively were formed after both satellite  galaxies entered R200 of the primary’s FoF halo around 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='75 Gyr ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  As also all central ex-situ stars were born in the center (∼ 500 pc) of  their respective birth galaxies, this confirms that significant nuclear  star formation is also triggered in the secondary galaxy after infall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Most of the ex-situ stars in the center arrive there immediately  after the merger coalesces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This is the case if their distance to the  center of the primary was less than 500 pc at the time of stripping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Otherwise it can take up to 2 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Interestingly, the arrival of the  ‘clumpy’ migrated stars at the center around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 Gyr ago induced  a second peak in the arrival of ex-situ stars from the minor merger  into the center, albeit being ten times lower and hence not visible in  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='3 The case of stellar clumps  Disk fragmentation can occur due to gravitational instabilities in a  galaxy’s gas-rich and turbulent disk (Toomre 1964;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Springel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Hopkins 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This fragmentation can form highly star form-  ing clumps, which have been reproduced in several studies using  hydrodynamical galaxy simulations, either isolated or fully cosmo-  logical ones (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Bournaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Genel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Bournaud  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Mandelker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2014, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Buck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The  execution of these simulations was motivated by the discovery of  the clumpy morphology in the rest-frame UV light of high redshift,  star forming galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Elmegreen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Therefore, these simulations are tailored to focus on clump formation  in massive disk galaxies 1010−11 M⊙ at z ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  In observations, clumps have masses between 107 M⊙ and  109 M⊙, as well as sizes of 1 kpc or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Clumps in the simula-  tions are usually identified via regions of enhanced gaseous surface  mass density or from mock stellar light images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In TNG50, the iden-  tification of clumps is (so far) a passive byproduct of the Subfind  algorithm, nevertheless the extracted baryonic mass distribution of  the clumps peaks at 108 M⊙ exhibiting overall high gas fractions  (see Figure E1) are in agreement with the other studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' However, all  clumps in TNG50 have 3D baryonic half mass radii below 300 pc and  therefore seem to be much more compact compared to observations  and some simulation studies (see Figure 9 in Buck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The  latter could be a result of the different treatments of star formation  and feedback in the simulations or the clump identification, as the  numerical resolution of TNG50 is largely comparable to those of  the previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Additionally, in TNG50 clumps seem to form  continuously throughout cosmic time (see Figure E1), which has not  been investigated in other studies of clump formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  These clumps can migrate to the center of their host galaxies due  to dynamical friction, as well as merge with each other while doing  so (Bournaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Dekel & Krumholz 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Bournaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Mandelker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Dekel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The migration time  to the center is found to be of the order of a few hundred Myr, which  is similar for clumps in TNG50, where most of the clumps arrive at  their respective galaxy’s center after 1 − 2 snapshots (∼ 200 Myr)  (see Figure E1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This mechanism contributes to the formation of  the bulge (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Bournaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Elmegreen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Dekel  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Consequently, the properties of the clumps need to be  specific, such that they survive their own internal stellar feedback and  the tidal forces on their way to the center with enough stellar mass  to significantly contribute to the formation of the bulge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In TNG50,  the fraction of clumpy migration stars to the total stellar mass in  the center is greater than 40% for about 12% of galaxies with any  clumpy migrated stars in their centers (all of those galaxies have  stellar masses above 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 M⊙;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' see Figure E1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Thus, according  to TNG50, the stellar mass transported by the migration of clumps  is likely not very important for bulge formation for the majority of  high mass galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Nevertheless, the clumps often retain a large  amount of gas until the center, or drag gas along (see clump closest  to the galaxy’s center in the top right of Figure E1), from which  stars might form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We have not checked explicitly if this increases the  contribution of stellar mass in the center significantly, or if such gas  is lost by directly funneling into the SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  In contrast to that, some simulations report clump formation, but  then no migration due to almost immediate dissolution or disruption  (Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2012, 2013a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Mayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Oklopčić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Buck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This is likely due to the different simulation  set ups, as well as the exact implementation of stellar feedback, or  simply because not enough galaxy diversity is probed with isolated  or zoom-in galaxy simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For example, in Figure E1, we see  that galaxies above 1010 M⊙ in stellar mass start to exhibit more  than one clump on average, however significant amounts of clumps  that are able to migrate to the center reside in galaxies with stellar  masses around 1011 M⊙ and above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Hence, an investigation of clump  formation and migration in a fully cosmological box is necessary to  not only capture galaxies of different masses but also different galaxy  assembly histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In TNG50 mergers and other galaxy interactions,  such as flybys, can trigger a significant amount of clump formation  (although not exclusively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' see also Di Matteo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Hopkins  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2013a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Calabrò et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2019, for similar reports).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Still, within TNG50 we want to exercise caution when it comes  to the trustworthiness of clump formation and their exact properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Only follow-up zoom-in simulations with higher resolution for dif-  ferent galaxies, as well as different treatment of star forming gas and  stellar feedback (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2013b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Smith 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2021, for the influence of highly resolved  star formation and different stellar feedback schemes in galaxy sim-  ulations), will allow for a more robust quantification of clumps in  TNG50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Nevertheless, the clump formation in TNG50 is unlikely to  be a numerical artifact in its entirety, as the adaptive mesh refinement  naturally allows for smaller cell sizes in areas of high gas density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='4 The predictive power of TNG50 at small(er) scales  While the TNG modelling framework is extremely successful in re-  producing key observational results of galaxy populations, numerical  resolution and the implementation of sub-grid physics are insupera-  ble limitations of the physical model of galaxy formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Regarding  the former we demonstrate in Figure D1 in Appendix D that the total  stellar mass within the central 500 pc of galaxies in TNG50 is con-  verging (see also Pillepich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' When splitting the central  mass into the contribution of in-situ, migrated and ex-situ stars, the  start of convergence is more difficult to assess due to the fixed size of  rcut (see Appendix D for a more detailed discussion) as well as the  overall influence of resolution on the amount of accreted stellar mass  (which should overall increase with resolution, see also Grand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Thus higher resolution runs are needed to fully determine the  amount of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Higher resolution zoom-in simulations of some TNG50 galaxies  additionally with models variation of stellar feedback and a better  resolved cold gas phase of the star forming gas - are certainly inter-  esting and needed to properly evaluate the convergence of the central  stellar mass, the formation of stellar clumps and observed nuclear  galaxy components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Nevertheless, our study of TNG50 shows that  MNRAS 000, 1–34 (2022)  Centers of TNG50 Galaxies  21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content="0  rcut [kpc]  104  105  106  107  108  109  1010  1011  Central Stellar Mass [M ]  2  37  72  74  17  48  69  36  49  z = 0  Insitu  Migrated 'Smooth+Clumpy'  Exsitu  TNG50-1  TNG50-2  TNG50-3  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  rcut [kpc]  96  54  21  16  100  73  43  22  100  77  52  42  z = 0  1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 M  M , tot  1011 M  TNG50-1  TNG50-2  TNG50-3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  rcut [kpc]  0  1  4  0  1  0  z = 0  TNG50-1  TNG50-2  TNG50-3  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Effects of numerical resolution and aperture size on the central stellar mass for in-situ, migrated and ex-situ stars in TNG50 galaxies with  1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5−11 M⊙ in stellar mass at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Lines show the median central stellar mass for in-situ (pink), migrated (orange) and ex-situ (blue) stars for four choices of  rcut = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 and 1 kpc and three resolution realizations of TNG50 (thicker lines indicate better resolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' TNG50-1 is the highest resolution (flagship),  followed by TNG50-2 and -3, which have 2 and 4 times lower spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The mass resolution is 8 and 64 times lower respectively and indicated by the  dotted horizontal lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The numbers indicate the respective central stellar mass fraction in percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Decreasing size of the center means decreasing stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  However the fraction of migrated mass increases, while the in-situ fraction consequentially decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' At a hypothetical higher resolution (TNG50-0) the latter  effect would be lessened as more stellar mass is formed within a given aperture size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Similar trends are recovered for other galaxy masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  the cosmological context plays a major role in the assembly of galaxy  centers, which is unlikely to become less significant with numerical  resolution and other modelling aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Already at the resolution of  TNG50 it is rare to find a galaxy with no ex-situ stars in its central  500 pc, which is only the case for around 9% of all galaxies in our  sample spanning a range between 5 × 108 − 5 × 1012 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We note  that this percentage remains the same, even if we do not impose a  constraint on the physical size of galaxies included in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  In fact, our selection of galaxies with half-mass radii > 2 kpc does  not impose any strong differential effects on our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We have ex-  plicitly checked Figures 4 and 6, which show the same trends when  galaxies with R1/2 < 2 kpc are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Thus, we do not expect that  any other results of our study are significantly effected by our choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Therefore, our findings highlight that the high density, nuclear  regions of galaxies can survive tidal forces and contribute to the  build-up of the centers of other galaxies, including low-mass galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Additionally, if the two merging galaxies are massive enough to both  host a black hole, the merger of the central regions of galaxies will  contribute to the growth of SMBHs (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Schweizer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Voggel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2021, for a recent observational confirmation of such a  system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Currently there are no large-box, cosmological, hydrodynami-  cal simulations (including TNG50) that can resolve smaller central  galaxy components, such as nuclear star clusters (NSCs), until z = 0  (see Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2018, that investigate NSC formation in a cosmo-  logical set-up until z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' However, by extending the trends in  our study to even smaller scales, it is not impossible to think that  the formation and evolution of NSCs are also governed by galaxy  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Even though the relative fraction of ex-situ stars on tens  of pc scales is likely very small for the majority of galaxies, it is clear  that the in-situ star formation and the migration of stars to the center  is closely connected to the formation pathway of the entire galaxy  (see Figure 11), because galaxy interactions are able to create the  conditions needed to funnel gas and stars to the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Therefore, it  is important to treat NSC formation in the context of the hierarchi-  cal build-up of galaxies in a ΛCDM cosmology (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2018)  and consider the influence of galaxy interactions in (semi-)analytical  models (Leaman & van de Ven 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  We have explicitly checked within TNG50 if we can make predic-  tions at smaller scales than 500 pc, by repeating our entire analysis  for rcut of 250 pc (approximately the softening length of TNG50)  and 100 pc, as well as 1 kpc for a consistency check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In addition  to TNG50-1 (the highest resolution), we repeated this for the two  lower resolution realizations, which are TNG50-2 and TNG50-3 re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The results are shown in Figure 12 for galaxies between  1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 and 1011 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  With decreasing size of the center (rcut) the absolute mass de-  creases for all three origins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' However, the fraction of the in-situ  population decreases with decreasing central size, while the mi-  grated fraction increases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' a consequence of the smaller volume that  is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' At the same time, this behaviour is also affected by the res-  olution, which not only sets the absolute normalization of the mass  fraction, but also the spatial size at which the relative contribution of  the in-situ and migrated fraction swaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We therefore conclude that a  hypothetical higher resolution (TNG50-0) would increase (decrease)  the in-situ (migrated) fraction below 250 pc due to the convergence  behaviour of the absolute stellar mass at fixed aperture size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Simi-  larly, the contribution of ex-situ stars will increase at a given aperture  and also likely reach scales smaller than 500 pc (see Appendix D for  more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  This behaviour emphasizes that the contribution of all three origins  will likely remain relevant on scales of 100 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  MNRAS 000, 1–34 (2022)  22  Boecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 Galaxy centers as tracers of overall galaxy assembly  Unveiling the merger history of galaxies proves difficult to tackle  outside our own Galaxy due to many reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Perhaps the most  severe one is the fact that accreted material is not necessarily visually  apparent in the forms of streams and shells (or any other form of  irregularity), especially when the merger coalesced many Gyr ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Since deep photometry of galaxies (initially stacked for many  galaxies) revealed the need for an additional Sérsic component to  accurately fit their surface brightness profiles beyond tens of kpc  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Zibetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Tal & van Dokkum 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' D’Souza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2014), the focus of quantifying accreted material has primarily been  on the outskirts of galaxies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' their stellar halos (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Monachesi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Merritt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Spavone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Spavone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' It is understood that the excess of light  at large galactic radii should mark the transition from the in-situ to  ex-situ dominated areas of a galaxy, as (significant) stellar mass can  be build-up through minor merging there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  However, the new era of cosmological hydrodynamical simula-  tions suggest that such a transition does not necessarily exist for  every galaxy, as especially high mass galaxies can be dominated by  ex-situ stars at all radii (Tacchella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Pulsoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Furthermore, Remus & Forbes (2021) showed that the transition in  surface brightness profiles traced by two Sérsic fits does not corre-  spond to the true in-situ and ex-situ dominated regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Similarly,  changes in kinematic profiles at large radii, which can, for example,  be obtained with globular clusters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Arnold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2014) or plane-  tary nebulae (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Pulsoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2018), do not, in general, correspond  to transitions between in-situ and ex-situ dominated regions (Schulze  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Pulsoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  While detailed studies of stellar halos are certainly important and  necessary, our study suggests that there lies potential in using the  centers of galaxies to study their accretion history (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Not  only are the centers the brightest region of a galaxy and hence deliver  the highest quality data, but they are also increasingly covered in  numbers by IFU surveys, which provide detailed kinematic and stellar  population information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' SAMI: Bryant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2015, MaNGA:  Bundy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In particular, our results in Figures 9 and 10  show that in-situ and ex-situ stars in the center are (on average) well  separated in age-metallicity-circularity space for galaxies with stellar  masses ≤ 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Newly developed techniques are able to extract  such distributions in ages and metallicity as well as circularities  from IFU measurements, and already have been proven to be able  to estimate the true underlying accreted stellar material much more  realistically (Boecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2020a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Davison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Even though in-situ and ex-situ stars separate better in their stellar  population and dynamical properties for these lower mass galaxies,  the ex-situ stellar mass fraction in the central 500 pc is on average  tenth of percent, thus picking up accreted signatures in the very  centers will still be challenging even with these new techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  However, low redshift IFU observations easily cover 1 − 2 half-light  radii, which extend beyond 500 pc and hence should encompass more  accreted material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' More follow up work will be needed to quantify  the optimal extent needed from a galaxy’s center to reliably pick up  ex-situ fractions in observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  On top of that, the large spread seen in the central ex-situ mass  at fixed total ex-situ mass (see Figure 5) points towards significant  spatial variation of ex-situ stars in the host galaxy, regardless of  whether the galaxy has accreted a lot of stellar material or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' It is  likely that measuring these spatial variations, will inform us about the  types of mergers that have happened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Typical characteristics could be  the merger ratio, for example major mergers will have the ability to  deposit more of their stars in the center of galaxies, but also the gas  content or orbital infall parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We plan to exploit this in future  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='6 Hints for (SDSS-like) observations  TNG50 predicts a diverse mass build-up of galaxy centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' What are  the prospects to learn about a galaxy’s central in-situ, migrated and  ex-situ fraction from more “traditional” observations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For example,  from SDSS DR7 (Abazajian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2009), that provides single 3′′ fiber  spectra for centers of hundred of thousand of galaxies, average ages,  metallicities and [𝛼/Fe] abundances can be determined (Gallazzi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' How much information do such measurements contain  about the contribution of stellar populations of different origins to a  galaxy’s center?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  In Figure 13 we show the mass-weighted average central age and  metallicity for our sample of TNG50 galaxies color-coded by the  mass fraction attributed to each origin (LOESS smoothed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Cappellari  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  If the measured average central age and metallicity of a galaxy  lies on the respective mass-age and mass-metallicity relation, the  galaxy has likely a high fraction of in-situ stars in its center, except  if its larger than 1011 M⊙ in stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' If the measured average  metallicity is below the 16th percentile at fixed stellar mass, it is  more likely that the galaxy’s center is dominated by ex-situ stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Similarly, if the galaxy has an average age below the 16th percentile,  it has likely a high amount of migrated stars in its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' High mass  galaxies above 1011 M⊙ with significant amounts of ex-situ stars  in their centers, also have a slightly younger age (between the 16th  percentile and the median) than the typical average galaxy in that  mass regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Naturally, a proper mocking of observed average stellar population  properties from TNG50 is needed to compare accurately to measure-  ments from Gallazzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' However, Figure 13 seems to  acknowledge that such measurements provide some leverage in de-  termining the fraction of stars with different origins in the centers of  galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  With respect to comparisons to the whole SDSS galaxy sample, it  would be necessary to repeat the analysis of this paper for different  spatial apertures, The fixed 3′′ diameter of the SDSS fibers will  already encompass larger physical sizes than 1 kpc for galaxies with  z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' It would be interesting to understand how the relative  contribution of stars from the different origins change with greater  spatial extent, especially for the ex-situ stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  6 SUMMARY AND CONCLUSIONS  Galaxies growth hierarchically in a ΛCDM universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Their centers  are the regions where usually the highest quality observations are  available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' What information about the hierarchical growth of galaxy  formation is encoded in this observationally favourable region?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' To  answer this, we investigated the central 500 pc mass assembly of  galaxies ranging from 5 × 108 M⊙ to 5 × 1012 M⊙ with half-mass  radii > 2 kpc in the TNG50 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Stars that are found at the center of TNG50 galaxies at z = 0  originate from one of the three possibilities: in-situ (formed inside  the center), migrated (formed inside the host galaxy, but outside the  center), ex-situ (brought in by mergers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Stars can migrate to the  center either as as continuous distribution of individuals (smooth) or  in clumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  MNRAS 000, 1–34 (2022)  Centers of TNG50 Galaxies  23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='6  Central Metallicity  [log10(Z/Z )]  TNG50  z = 0  Insitu  TNG50  z = 0  Migrated  TNG50  z = 0  Exsitu  109  1010  1011  1012  Total Stellar Mass [M ]  0  2  4  6  8  10  12  14  Central Age  [Gyr]  TNG50  z = 0  Insitu  109  1010  1011  1012  Total Stellar Mass [M ]  TNG50  z = 0  Migrated  109  1010  1011  1012  Total Stellar Mass [M ]  TNG50  z = 0  Exsitu  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='8  Central Stellar Mass Fraction  Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Information about the central (500 pc) fractional mass associated with in-situ, migrated and ex-situ stars contained in average age and  metallicity measurements from TNG50 galaxies at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Top row: Mass-weighted average metallicities for all central stars as a function of the galaxy’s total  stellar mass, but color-coded in each panel (from left to right) according to their fraction of in-situ, migrated and ex-situ stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The colors are LOESS (Cappellari  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2013) smoothed to show the average around neighbouring galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The thick dashed line shows the median relation in each panel, and the thin dotted lines  show the 16th and 84th percentile respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Bottom row: The same but for the mass-weighted average age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Galaxies that are more metal-poor than the 16th  percentile for their corresponding stellar mass are more likely to have high ex-situ fractions in their centers, while galaxies with high central migrated fractions  are younger than the 16th percentile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  For each origin we characterized their radius with respect to their  host galaxy at birth to understand the travelling distances for the  migrated stars as well as the spatial environment of ex-situ stars at  the time of birth and deposit into the z = 0 host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  We then investigated the amount of the central stellar mass con-  tained in each of three origins and their relative contribution as well  as their correlation to each other across the entire TNG50 galaxy  mass range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Additionally, we studied differences in central in-situ,  migrated and ex-situ for different galaxy types at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  To address whether the different origins of central stars leave a  discernible imprint on their (observable) features, we characterized  and correlated their ages, metallicities, [Mg/Fe] abundances as well  as dynamical properties with their distributions in circularity as a  function of the galaxy’s total stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We summarize our most  important findings below:  In-situ stars are on average the dominant component in stellar  mass in the central 500 pc of TNG50 galaxies across the entire mass  range of 5 × 108−12 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Migrated stars contribute on average 20%  to the total stellar mass in the center, where below (above) 5 ×  1010 M⊙ in galaxy stellar mass smoothly (clumpy) migrated stars  encompass their majority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The central stellar mass fraction of ex-  situ stars becomes on average non negligible above galaxy masses  of 5 × 1010 M⊙ with a large scatter of up to 80%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' However, it is  the exception to find a galaxy without any ex-situ stellar mass in its  central 500 pc, which is only the case for about 9% of galaxies in our  total sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' (Figure 4)  The majority of smoothly migrated stars originate close to the  center (radii between 500 pc and 1 kpc), whereas ∼ 15% come from  larger distances up until 10 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Compared to that, clumpy migrated  stars possess a distinctively different distribution of birth radii which  peaks around 20−30 kpc for galaxies with stellar masses greater than  5 × 1010 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Most of the ex-situ stars originate in the central 1 kpc  of their birth galaxies, where they remain until they are deposited  inside their z = 0 host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' (Figure 3)  At fixed galaxy stellar mass the amount of central ex-situ stellar  mass exhibits a significant scatter between 4 − 6 dex, reflecting the  stochasticity of the merger history of individual galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In some  cases, close to the entire total amount of ex-situ stellar material ever  deposited inside the host galaxy resides within the central 500 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  (Figure 5)  In TNG at z = 0, star forming galaxies with stellar masses  above 1010 M⊙ have on average larger ex-situ central stellar masses  than their quenched counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Only quenched galaxies that are  additionally bulgey and have no bar signature show a rise of central  ex-situ stellar mass above 1010 M⊙ similar to the star forming galax-  ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Galaxies between 5 × 109 − 5 × 1010 M⊙ with an overmassive  (undermassive) SMBH in the center are more compact (extended)  and show on average a higher (lower) in-situ and migrated central  stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' There is no difference in neither in-situ, migrated nor  ex-situ central stellar masses for central or satellite galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' (Figure  6)  Central ex-situ stars have on average the lowest metallicities, the  oldest ages and the highest [Mg/Fe] abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The slope of their  mass-metallicity relation is slightly steeper than that of the in-situ  and migrated stars, and their mass-age relation is flat compared to the  positive correlation between central age and galaxy stellar mass for  the in-situ and migrated stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Overall, the average stellar populations  MNRAS 000, 1–34 (2022)  24  Boecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  properties of in-situ and migrated stars are very similar, with in-situ  stars being slightly more metal-rich and older.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' (Figure 7)  The majority of central stars for galaxies with stellar masses  below 109 M⊙ and above 1011 M⊙ regardless of their origin are  on random motion dominated orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For galaxies in between those  stellar masses, the peak of the circularity distribution shifts by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='25) towards rotationally supported orbits for migrated (in-situ)  stars for both disk and bulge dominated galaxies, whereas ex-situ  stars remain random motion dominated at all galaxy masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' (Figure  8)  For star forming galaxies around 1010 M⊙ in stellar mass, in-  situ, migrated and ex-situ stars clearly separate in age-metallicity  space, while the distinction becomes less clear for star forming galax-  ies outside that mass range and quenched galaxies in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' (Figure  9)  For  both  disk  and  bulge  dominated  galaxies  between  109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5−10 M⊙ in stellar mass, in-situ, migrated and ex-situ stars clearly  separate in age-circularity space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The migrated stars are the youngest  with the highest amount of rotational support and the ex-situ stars are  the oldest and purely random motion supported, whereas the in-situ  stars are situated in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' (Figure 10)  Furthermore, we have demonstrated the diversity of the central  500 pc of galaxies as governed by the hierarchical mass build-up in a  ΛCDM universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Galaxy interactions are an important driver in not  only contributing ex-situ stars to the center of galaxies, but also in  dictating the formation of in-situ stars and the migration of stars to  the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This leads to an entanglement of different mechanisms  that influence the formation history of stars in the center of galax-  ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In Figure 11 we have qualitatively identified these mechanisms  that are present in TNG50, which includes episodes of in-situ star  formation and stellar migration to the center during times of peri-  center passages and/or coalescence of mergers or flybys, infall into  galaxy groups/clusters as well as depletion of the central gas reservoir  through kinetic AGN feedback and environmental effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  In the future, higher resolution simulations (not only spatially but  also concerning star formation and stellar feedback prescriptions)  will be needed to fully address the formation and migration of stellar  clumps and to study the formation of nuclear galaxy structures, such  as nuclear disks and star clusters, in a fully cosmological context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Bright galaxy centers have the potential to be used in observations  as tracers of the overall galaxy assembly history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' TNG50 predicts  distinct stellar populations and dynamical properties for the stars of  different origins in the center of galaxies, which can be observed  with today’s IFU capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Figure 13 demonstrates that there is  even promise to deduce the fractional contribution of central in-situ,  migrated and ex-situ stars from SDSS-like observations in a galaxy  population averaged sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  In summary, TNG50 is a tremendous advancement in predicting  the stellar build-up of sub-kpc scales in a fully cosmological context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Its predictive power is valuable to consider new pathways in mod-  elling formation scenarios of central stellar components as well as to  push forward novel observational techniques to unveil the formation  history of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  DATA AVAILABILITY  The IllustrisTNG simulations, including TNG50, are publicly avail-  able and accessible at www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='tng-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='org/data (Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Data directly related to this publication and its figures are  available upon reasonable request from the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  AB is grateful to Ignacio Martín-Navarro and Jesús Falcón-Barroso  for their support and scientific discussion throughout this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  AB also likes to thank Glenn van de Ven and Francisco Aros for use-  ful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We thank the anonymous referee for helpful com-  ments on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This work was funded by the Deutsche  Forschungsgemeinschaft (DFG, German Research Foundation) –  Project-ID 138713538 – SFB 881 (“The Milky Way System”, sub-  project B08).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' NF was supported by the Natural Sciences and Engi-  neering Research Council of Canada (NSERC), [funding reference  number CITA 490888-16] through the CITA postdoctoral fellowship  and acknowledges partial support from a Arts & Sciences Postdoc-  toral Fellowship at the University of Toronto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' RR acknowledges fund-  ing from the Deutsche Forschungsgemeinschaft (DFG) through an  Emmy Noether Research Group (grant number NE 2441/1-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The Il-  lustrisTNG simulations were undertaken with compute time awarded  by the Gauss Centre for Supercomputing (GCS) under GCS Large-  Scale Projects GCS-ILLU and GCS-DWAR on the GCS share of the  supercomputer Hazel Hen at the High Performance Computing Cen-  ter Stuttgart (HLRS), as well as on the machines of the Max Planck  Computing and Data Facility (MPCDF) in Garching, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The  computations for this work were performed on the ISAAC cluster  of the Max Planck Institute for Astronomy at the Rechenzentrum in  Garching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  REFERENCES  Abadi M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Navarro J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Steinmetz M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Eke V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2003, ApJ, 597, 21  Abazajian K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2009, ApJS, 182, 543  Agarwal M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Milosavljević M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2011, ApJ, 729, 35  Aharon D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Perets H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2015, ApJ, 799, 185  Antonini F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2013, ApJ, 763, 62  Antonini F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Capuzzo-Dolcetta R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Mastrobuono-Battisti A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Merritt D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2012,  ApJ, 750, 111  Antonini F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Barausse E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Silk J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2015, ApJ, 812, 72  Arentsen A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2020, MNRAS, 491, L11  Arnold J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2014, ApJ, 791, 80  Asplund M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Grevesse N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Sauval A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Scott P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2009, ARA&A, 47, 481  Athanassoula E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2005, MNRAS, 358, 1477  Athanassoula E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Machado R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Rodionov S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2013, MNRAS, 429,  1949  Barbuy B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Chiappini C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Gerhard O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2018, ARA&A, 56, 223  Barnes J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 1988, ApJ, 331, 699  Bittner A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2020, A&A, 643, A65  Bocquet S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Saro A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Dolag K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Mohr J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2016, MNRAS, 456, 2361  Boecker A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Leaman R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', van de Ven G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Norris M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Mackereth J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Crain  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2020a, MNRAS, 491, 823  Boecker A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Alfaro-Cuello M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Neumayer N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Martín-Navarro I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Leaman R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=',  2020b, ApJ, 896, 13  Böker T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Sarzi M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', McLaughlin D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', van der Marel R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Rix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Ho  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Shields J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2004, AJ, 127, 105  Bottema R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2003, MNRAS, 344, 358  Bournaud F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Elmegreen B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Elmegreen D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2007, ApJ, 670, 237  Bournaud F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2014, ApJ, 780, 57  Brown G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Gnedin O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Li H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2018, ApJ, 864, 94  Bryant J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2015, MNRAS, 447, 2857  Buck T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Macciò A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Obreja A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Dutton A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Domínguez-Tenreiro R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=',  Granato G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2017, MNRAS, 468, 3628  Buck T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Obreja A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Macciò A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Minchev I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Dutton A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Ostriker J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=',  2020, MNRAS, 491, 3461  Bundy K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2015, ApJ, 798, 7  Calabrò A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2019, A&A, 632, A98  Cappellari M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2013, MNRAS, 432, 1862  Chabrier G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2003, PASP, 115, 763  MNRAS 000, 1–34 (2022)  Centers of TNG50 Galaxies  25  Chen Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2020, ApJ, 897, 102  Côté P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2006, ApJS, 165, 57  Cox T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Jonsson P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Somerville R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Primack J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Dekel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2008,  MNRAS, 384, 386  Crain R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2015, MNRAS, 450, 1937  Croton D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2006, MNRAS, 365, 11  D’Souza R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Kauffman G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Wang J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Vegetti S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2014, MNRAS, 443, 1433  Davé R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Anglés-Alcázar D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Narayanan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Li Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Rafieferantsoa M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=',  Appleby S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2019, MNRAS, 486, 2827  Davis M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Efstathiou G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Frenk C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', White S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 1985, ApJ, 292, 371  Davis B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Graham A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Cameron E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2019, ApJ, 873, 85  Davison T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Norris M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Pfeffer J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Davies J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Crain R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2020,  MNRAS, 497, 81  Davison T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Norris M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Leaman R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Kuntschner H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Boecker A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', van de  Ven G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2021, MNRAS, 507, 3089  Dekel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Krumholz M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2013, MNRAS, 432, 455  Dekel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Sari R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Ceverino D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2009, ApJ, 703, 785  Dekel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Mandelker N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Bournaud F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Ceverino D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Guo Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', primack J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2021,  arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='13561  Di Matteo P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Combes F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Melchior A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Semelin B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2007, A&A, 468, 61  Di Matteo P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Bournaud F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Martig M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Combes F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Melchior A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Semelin  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2008, A&A, 492, 31  Díaz-García S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Salo H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Laurikainen E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Herrera-Endoqui M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2016a, A&A,  587, A160  Díaz-García S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Salo H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Laurikainen E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2016b, A&A, 596, A84  Do T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', David Martinez G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Kerzendorf W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Feldmeier-Krause A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Arca Sedda  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Neumayer N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Gualandris A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2020, ApJ, 901, L28  Donnari M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2019, MNRAS, 485, 4817  Donnari M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2021a, MNRAS, 500, 4004  Donnari M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Pillepich A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Nelson D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Marinacci F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Vogelsberger M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Hern-  quist L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2021b, MNRAS, 506, 4760  Dubois Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2014, MNRAS, 444, 1453  Dubois Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Peirani S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Pichon C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Devriendt J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Gavazzi R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Welker C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Volon-  teri M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2016, MNRAS, 463, 3948  El-Badry K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2018, MNRAS, 480, 652  Elmegreen D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Elmegreen B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Ravindranath S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Coe D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2007, ApJ,  658, 763  Elmegreen D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Elmegreen B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Marcus M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Shahinyan K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Yau A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=',  Petersen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2009, ApJ, 701, 306  Fanidakis N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Baugh C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Benson A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Bower R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Cole S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Done C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=',  Frenk C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2011, MNRAS, 410, 53  Feldmeier-Krause A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2020, MNRAS, 494, 396  Ferrarese L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2006, ApJ, 644, L21  Fisher D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Drory N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2010, ApJ, 716, 942  Frankel N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Sanders J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Ting Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Rix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2020, ApJ, 896, 15  Frankel N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='08406  Gadotti D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2009, MNRAS, 393, 1531  Gadotti D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Kauffmann G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2009, MNRAS, 399, 621  Gadotti D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2020, A&A, 643, A14  Gallazzi A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Charlot S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Brinchmann J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', White S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Tremonti C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2005,  MNRAS, 362, 41  Gallazzi A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Pasquali A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Zibetti S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Barbera F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2021, MNRAS, 502,  4457  Gao L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Loeb A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Peebles P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', White S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Jenkins A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2004, ApJ,  614, 17  Gargiulo I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2021, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='13712  Genel S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2012, ApJ, 745, 11  Genel S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2014, MNRAS, 445, 175  Genel S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2018, MNRAS, 474, 3976  Georgiev I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Böker T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2014, MNRAS, 441, 3570  Georgiev I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Böker T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Leigh N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Lützgendorf N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Neumayer N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2016,  MNRAS, 457, 2122  Grand R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2017, MNRAS, 467, 179  Grand R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2021, MNRAS, 507, 4953  Guedes J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Callegari S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Madau P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Mayer L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2011, ApJ, 742, 76  Guedes J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Mayer L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Carollo M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Madau P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2013, ApJ, 772, 36  Guillard N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Emsellem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Renaud F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2016, MNRAS, 461, 3620  Guo Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2015, ApJ, 800, 39  Häring N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Rix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2004, ApJ, 604, L89  Hartmann M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Debattista V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Seth A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Cappellari M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Quinn T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2011,  MNRAS, 418, 2697  Hirschmann M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Dolag K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Saro A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Bachmann L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Borgani S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Burkert A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=',  2014, MNRAS, 442, 2304  Hopkins P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2013, MNRAS, 430, 1653  Hopkins P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2009, MNRAS, 397, 802  Hopkins P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2010, ApJ, 715, 202  Hopkins P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Kereš D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Murray N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Quataert E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Hernquist L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2012, MNRAS,  427, 968  Hopkins P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Cox T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Hernquist L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Narayanan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Hayward C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Murray  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2013a, MNRAS, 430, 1901  Hopkins P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Narayanan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Murray N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2013b, MNRAS, 432, 2647  Hopkins P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2018, MNRAS, 480, 800  Huang S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Leauthaud A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Greene J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Bundy K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Lin Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Tanaka M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=',  Miyazaki S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Komiyama Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2018, MNRAS, 475, 3348  Joshi G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Pillepich A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Nelson D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Marinacci F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Springel V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Rodriguez-  Gomez V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Vogelsberger M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Hernquist L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2020, MNRAS, 496, 2673  Kormendy J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Ho L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2013, ARA&A, 51, 511  Kormendy J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Kennicutt Robert C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2004, ARA&A, 42, 603  Kraljic K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Bournaud F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Martig M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2012, ApJ, 757, 60  Lapiner S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Dekel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Dubois Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2021, MNRAS, 505, 172  Läsker R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Greene J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Seth A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', van de Ven G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Braatz J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Henkel C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Lo  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2016, ApJ, 825, 3  Laurikainen E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Peletier R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Gadotti D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', eds, 2016, Galactic Bulges Astro-  physics and Space Science Library Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 418, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1007/978-3-319-  19378-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Leaman R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', van de Ven G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2021, MNRAS,  Maciejewski W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2004, MNRAS, 354, 892  Malbon R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Baugh C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Frenk C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Lacey C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2007, MNRAS, 382,  1394  Mandelker N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Dekel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Ceverino D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Tweed D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Moody C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Primack J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=',  2014, MNRAS, 443, 3675  Mandelker N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Dekel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Ceverino D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', DeGraf C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Guo Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Primack J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2017,  MNRAS, 464, 635  Marinacci F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Pakmor R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Springel V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2014, MNRAS, 437, 1750  Marinacci F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2018, MNRAS, 480, 5113  Martín-Navarro I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Vazdekis A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Falcón-Barroso J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', La Barbera F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Yıldırım  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', van de Ven G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2018a, MNRAS, 475, 3700  Martín-Navarro I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Brodie J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Romanowsky A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Ruiz-Lara T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', van de Ven  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2018b, Nature, 553, 307  Martín-Navarro I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2019, A&A, 626, A124  Martín-Navarro I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2021, A&A, 654, A59  Mayer L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Tamburello V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Lupi A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Keller B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Wadsley J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Madau P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2016,  ApJ, 830, L13  Merritt A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', van Dokkum P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Abraham R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Zhang J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2016, ApJ, 830, 62  Merritt A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Pillepich A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', van Dokkum P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Nelson D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Hernquist L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Marinacci  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Vogelsberger M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2020, MNRAS, 495, 4570  Mihos J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Hernquist L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 1996, ApJ, 464, 641  Minchev I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Famaey B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2010, ApJ, 722, 112  Monachesi A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Bell E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Radburn-Smith D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Bailin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', de Jong R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=',  Holwerda B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Streich D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Silverstein G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2016, MNRAS, 457, 1419  Naiman J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2018, MNRAS, 477, 1206  Nelson E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2012, ApJ, 747, L28  Nelson D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2018, MNRAS, 475, 624  Nelson D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2019a, Computational Astrophysics and Cosmology, 6, 2  Nelson D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2019b, MNRAS, 490, 3234  Nelson E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2021, MNRAS, 508, 219  Ness M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2013, MNRAS, 432, 2092  Neumayer N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Seth A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Böker T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2020, A&ARv, 28, 4  Nogueras-Lara F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2020, Nature Astronomy, 4, 377  Nogueras-Lara F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Schödel R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Neumayer N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2021, ApJ, 920, 97  Okamoto T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2013, MNRAS, 428, 718  Oklopčić A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Hopkins P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Feldmann R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Kereš D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Faucher-Giguère C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=',  Murray N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2017, MNRAS, 465, 952  Pakmor R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Springel V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2013, MNRAS, 432, 176  Pakmor R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Bauer A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Springel V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2011, MNRAS, 418, 1392  MNRAS 000, 1–34 (2022)  26  Boecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Pakmor R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Springel V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Bauer A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Mocz P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Munoz D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Ohlmann S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=',  Schaal K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Zhu C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2016, MNRAS, 455, 1134  Perets H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Mastrobuono-Battisti A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2014, ApJ, 784, L44  Peschken N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Łokas E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2019, MNRAS, 483, 2721  Pillepich A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2018a, MNRAS, 473, 4077  Pillepich A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2018b, MNRAS, 475, 648  Pillepich A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2019, MNRAS, 490, 3196  Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2016, A&A, 594, A13  Powell L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Bournaud F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Chapon D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Teyssier R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2013, MNRAS, 434, 1028  Pulsoni C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2018, A&A, 618, A94  Pulsoni C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Gerhard O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Arnaboldi M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Pillepich A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Rodriguez-Gomez V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=',  Nelson D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Hernquist L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Springel V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2021, A&A, 647, A95  Reines A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Volonteri M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2015, ApJ, 813, 82  Remus R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Forbes D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2021, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='12216  Renaud F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Boily C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Naab T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Theis C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2009, ApJ, 706, 67  Rodriguez-Gomez V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2015, MNRAS, 449, 49  Rodriguez-Gomez V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2016, MNRAS, 458, 2371  Rodriguez-Gomez V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2017, MNRAS, 467, 3083  Romano-Díaz E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Shlosman I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Heller C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Hoffman Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2008, ApJ, 687, L13  Rosas-Guevara Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2020, MNRAS, 491, 2547  Rosas-Guevara Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2021, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='04537  Sahu N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Graham A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Davis B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2019, ApJ, 876, 155  Sánchez-Janssen R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2019, ApJ, 878, 18  Sani E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Marconi A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Hunt L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Risaliti G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2011, MNRAS, 413, 1479  Schaye J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2015, MNRAS, 446, 521  Schultheis M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2021, A&A, 650, A191  Schulze F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Remus R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Dolag K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Bellstedt S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Burkert A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Forbes D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=',  2020, MNRAS, 493, 3778  Schweizer F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Seitzer P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Whitmore B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Kelson D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Villanueva E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=',  2018, ApJ, 853, 54  Scott N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Graham A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2013, ApJ, 763, 76  Sellwood J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Binney J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2002, MNRAS, 336, 785  Seo W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Kim W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Kwak S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Hsieh P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Han C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Hopkins P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2019,  ApJ, 872, 5  Sérsic J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 1968, Atlas de Galaxias Australes  Sheth K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2008, ApJ, 675, 1141  Simmons B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2014, MNRAS, 445, 3466  Smith M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2021, MNRAS, 502, 5417  Smith M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Sijacki D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Shen S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2018, MNRAS, 478, 302  Smith M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Bryan G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Somerville R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Hu C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Teyssier R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Burkhart  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Hernquist L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2021, MNRAS, 506, 3882  Somerville R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Davé R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2015, ARA&A, 53, 51  Sormani M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Tress R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Glover S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Klessen R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Battersby C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=',  Clark P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Hatchfield H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Smith R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2020, MNRAS, 497, 5024  Sotillo-Ramos D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2022, MNRAS, 516, 5404  Spavone M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2017, A&A, 603, A38  Spavone M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2020, A&A, 639, A14  Springel V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2010, MNRAS, 401, 791  Springel V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', White S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Tormen G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Kauffmann G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2001, MNRAS, 328,  726  Springel V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Di Matteo T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Hernquist L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2005, MNRAS, 361, 776  Springel V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2018, MNRAS, 475, 676  Tacchella S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2019, MNRAS, 487, 5416  Tal T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', van Dokkum P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2011, ApJ, 731, 89  Teklu A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Remus R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Dolag K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Beck A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Burkert A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Schmidt A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=',  Schulze F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Steinborn L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2015, ApJ, 812, 29  Toomre A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 1964, ApJ, 139, 1217  Torrey P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2019, MNRAS, 484, 5587  Tress R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Sormani M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Glover S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Klessen R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Battersby C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=',  Clark P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Hatchfield H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Smith R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2020, MNRAS, 499, 4455  Vogelsberger M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2014a, MNRAS, 444, 1518  Vogelsberger M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2014b, Nature, 509, 177  Vogelsberger M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Marinacci F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Torrey P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Puchwein E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2020, Nature Reviews  Physics, 2, 42  Voggel K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2021, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='14854  Wang L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Dutton A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Stinson G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Macciò A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Penzo C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Kang X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=',  Keller B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Wadsley J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2015, MNRAS, 454, 83  Weinberger R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2017, MNRAS, 465, 3291  Wetzel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Hopkins P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Kim J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='-h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Faucher-Giguère C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Kereš D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=',  Quataert E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2016, ApJ, 827, L23  Zhu L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2020, MNRAS, 496, 1579  Zhu L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2021, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='13172  Zibetti S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', White S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Brinkmann J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2004, MNRAS, 347, 556  Zinger E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2020, MNRAS, 499, 768  Zoccali M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2017, A&A, 599, A12  van den Bosch F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', Ogiya G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=', 2018, MNRAS, 475, 4066  APPENDIX A: PROPERTIES OF TNG50 GALAXIES  Below we describe in detail the properties of TNG50 galaxies as well  as their stellar particles that were mentioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' They are  either directly available from the corresponding halo/subhalo, stellar  particle or supplementary data catalogues on the TNG website10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The classification of ‘bar-like’ signatures for galaxies is available  upon reasonable request from the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  A1 Bulk Properties  We here describe how different galaxy bulk properties are defined  and measured in order to define different galaxy populations that are  analyzed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' All properties refer to z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Mass: Generally all galaxy masses are reported to be the total  mass of particles of a given type (or all types in the case of dynam-  ical mass) bound to a specific subhalo as identified by the Subfind  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Environment: We crudely define the environment of a galaxy  by distinguishing between centrals and satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' A central galaxy  is the most massive subhalo in its corresponding FoF halo, all other  galaxies within the same FoF halo are satellite galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Star formation activity: Whether a galaxy is actively forming  stars or not is classified according to Pillepich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' (2019) (see  also Donnari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2019, 2021a,b), who determined the logarith-  mic distance from the star forming main sequence for each galaxy  (Δ log10 SFR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For this, the instantaneous star formation rate (SFR)  of the gas cells as well as galaxy stellar masses were calculated  within twice the stellar half mass radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Star forming galaxies have  Δ log10 SFR ≥ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 and quenched galaxies have Δ log10 SFR ≤ −1,  whereas galaxies in the green valley are in between those two values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Unless otherwise stated we will omit the distinction of green valley  galaxies and also classify them as quenched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Morphology: We quantify disk or bulge dominated galaxies  based on the kinematic classification by Genel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For  each stellar particle the circularity parameter 𝜖 is calculated, which  gives the ratio of the particle’s specific angular momentum in z-  direction and its theoretical maximum angular momentum at that  specific binding energy (Abadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Marinacci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Then the mass fraction of all stellar particles with 𝜖 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='7 and within  ten stellar half mass radii is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' If that fractional mass is above  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='4 we classify that galaxy as disky (see Joshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2020), otherwise  the galaxy is bulge dominated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Bar-like signatures: We also provide a quick estimate of whether  a galaxy has a bar-like structure in its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For this, we calculate  𝐴2, the ratio between the second and zeroth term of the amplitude  of the Fourier expansion, from the face-on stellar surface density of  each galaxy as a function of the 2D radius in ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='04 dex steps, where  each bin is ensured to have at least 100 stellar particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We then  10 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='tng-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='org/data/docs/specifications/  MNRAS 000, 1–34 (2022)  Centers of TNG50 Galaxies  27  identify peaks in the 𝐴2-radius plane with a prominence of at least  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' After that, the value of 𝐴2 for the largest peak within a radius of  10 kpc is recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We impose this radius cut to mitigate the effect of  other 𝐴2 features that may be present at larger radii (see also Frankel  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This is done for all snapshots between z = 0 (SnapNum  99) and z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2 (SnapNum 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Similarly to Rosas-Guevara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  (2020)11, we then define a bar-like structure, when the maximum  𝐴2 value (at a given time step) is above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Bar-like structures at  z = 0 are defined solely based on their instantaneous 𝐴2 value at  that snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' While this method leads to accurate identification of  symmetrically elongated ‘bar-like’ features, we do not check if this  is actually a ‘proper’ bar in the astrophysical sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Nevertheless,  our classification leads to a bar fraction of around 40% (50%) for  disk (all) galaxies above 1010 M⊙ in stellar mass, which is consistent  with observations (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Sheth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Simmons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Díaz-García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2016a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  AGN feedback: We quantify the severity of feedback from su-  permassive black holes by determining whether a specific galaxy  lies below or above the median scaling relation of TNG50 galax-  ies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' similarly to Martín-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2018b, for observations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Typical AGN feedback defining properties could be the black hole  (BH) mass and the cumulative energy injection in the thermal and/or  kinetic feedback modes (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Weinberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Zinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2020, for background).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Such scaling relations are always computed  with respect to the total stellar mass of galaxies as well as for the  total TNG50 galaxy sample above 5 × 108 M⊙ (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Properties of black hole particles per galaxy are computed as the sum  of all black holes particles associated to a given galaxy via Subfind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Physical size: Similarly, we define extended or compact galaxies  depending on whether they are below or above the median stellar  mass-size relation of TNG50 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The size is the 3D stellar half  mass radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  A2 Stellar Particle Properties  Stellar particle properties at z = 0 that are analyzed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2  are briefly described here:  Age: We define the age of a stellar particle as the look-  back time in Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This quantity is calculated from the field  GFM_StellarFormationTime, which provides the exact time of  birth of a star in scale factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Metallicity: We convert the mass fraction in metals 𝑍 as pro-  vided by the field GFM_Metallicity to log10 𝑍/𝑍⊙ with 𝑍⊙ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  This follows conventions adopted in observations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Gallazzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  [Mg/Fe]: We calculate the magnesium-to-iron abundance from  the mass fraction in magnesium 𝑍Mg and iron 𝑍Fe provided by the  simulation (GFM_Metals) via [Mg/Fe] = log10(𝑍Mg/𝑍Mg, ⊙) −  log10(𝑍Fe/𝑍Fe, ⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The adopted solar values are  𝑍Mg, ⊙  =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='00064298 and 𝑍Fe, ⊙ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='001218 respectively from Asplund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Circularity 𝜖: We calculate the instantaneous circularity of each  stellar particle following Genel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For this we first com-  pute the specific angular momentum of each particle in z-direction  by aligning the z-axis of the simulation box with the total angular  momentum of stellar particles within twice the stellar half mass ra-  dius of a given galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The theoretical maximum angular momentum  11 We find that our computed 𝐴2 values are slightly lower than those mea-  sured by methods of Rosas-Guevara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' (2020), hence we adopt their ‘weak  bar’ threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2, whereas their ‘strong bars’ have values of 𝐴2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  each stellar particle can have at its recorded specific binding energy  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 1  2 |v|2 + Φ) is calculated by sliding a maximum filter across the  particle list of specific angular momenta sorted by their total spe-  cific binding energy with a window size of one hundred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Stars with  circularities around zero are on random motion dominated orbits,  whereas values close to one indicate more circular orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Negative  circularities depict counter rotating orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  APPENDIX B: VALIDATION FOR ANALYSIS OF TNG50  GALAXY CENTERS  We briefly validate and justify here the analysis choices we made in  Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1 of the main text using two TNG50 galaxies as  examples where applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Both of these galaxies are also contained  within the subboxes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' smaller regions of the full simulation box,  which offer 3600 snapshots resulting in a time sampling of 2−3 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  At z = 0, SubfindID 537941 is a Milky Way like galaxy, found in  ‘Subbox0’, and SubfindID 35, found in ‘Subbox2’, is a compact  ∼ 1010 M⊙ in stellar mass galaxy, that quenched approximately 9 Gyr  ago and is now found in the most massive halo (∼ 1014 M⊙) in  TNG50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Both of these subhalos stay inside the subbox across their  life time making it possible to compare their full histories in the higher  cadence outputs to that of the hundred full box snapshot outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  B1 Selection of central stars  We show in Figure B1 the energy and angular momentum distribution  of stars in the galaxy’s centers at z = 0 by selecting particles based on  their current radius as well as on their energies according to Equation  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The selection was made with 𝑟cut = 500 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We also show their  peri- and apocenters which we calculated by recording their radii  from all 16 subbox outputs between the last full box snapshot, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  z = 0 (SnapNum 99), and the one prior to that, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' ≈ 136 Myr  (SnapNum 98) before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We then found all minima and maxima and  took the average respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Some particles selected by their instantaneous radius at z = 0 have  large energies and are hence able to move away from the center to  much larger radii, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' they are not actually spending the majority of  their orbital time within our selected spherical aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This is also  reflected by their larger (up to 10 kpc) apocenters, whereas particles  selected by their energy have time-averaged peri- and apocenters not  larger than 1 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Even though that is larger than our selected 𝑟cut  value, probably due to our simplifying assumptions in calculating  𝐸cut, we argue that this selection gives a much cleaner selection of  stars actually belonging to the center without interloping particles on  much more eccentric orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Both selection criteria yield a compara-  ble number of stars, with around 21000 stars for SubfindID 537941  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='6% of total amount of stars) and ∼ 40000 stars for SubfindID 35  (∼ 15% of total amount of stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  B2 Definition of birth radii of stellar particles  In Figure B2 we show the difference between the center position of  the two galaxies once taken from the high cadence subbox outputs  (the most bound stellar particle)12 and once from the SubhaloPos  12 As there is in fact no subhalo information from Subfind available for the  subboxes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' we start off with the interpolated center values from the full box  snapshots and then recalculate the center in a 5 kpc box around that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' recenter  and recalculate the center again in a 5 kpc box around that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We verified that this  MNRAS 000, 1–34 (2022)  28  Boecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Figure B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Selection of stellar particles belonging to a galaxy’s center, left column for SubfindID 537941 and right column for SubfindID 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Blue points  show stellar particles that have radii smaller than 𝑟cut = 500 pc and orange points where selected based on their energies being smaller than 𝐸cut according to  Equation 1, also with 𝑟cut = 500 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Top row: Angular momentum of selected particles as a function of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The dashed lines emphasizes 𝐸cut, whereas the  two dotted lines show Lz = ±𝑟cut𝑣circ(𝑟cut).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Grey points show all other stellar particles belonging to the respective galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Bottom row: Pericenter of stellar  particles versus their apocenter time averaged from subbox outputs between 0 (full box snapshot 99) and 136 Myr (full box snapshot 98).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Stellar particles purely  selected on their radius have a wider distribution in their energies and are hence able to move much further outside our selected spherical volume of 500 pc,  which is also reflected by their larger apocenters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  argument from the full box snapshots, which we interpolated onto  the finer time sampling of the subbox outputs using a cubic spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  It is evident that deviations of around 2 − 5 kpc appear, specifically  at times of pericenter passages of either satellites merging with the  host (SubfindID 537941) or of the galaxy itself around its host  (SubfindID 35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  These deviations of a galaxy’s center position are enough to  severely miss-classify migrated and in-situ stars, when their in-  stantaneous birth positions, as provided by the simulations output  (BirthPos), are used in conjunction with the interpolated subhalo  center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For example, in Figure B3 we show histograms of migrated  and in-situ stars selected according to their instantaneous birth po-  sition using the proper center from the subbox and the interpolated  one from the full box snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For both galaxies the number of mi-  gives the correct center by inspecting the galaxies by eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We note however  that this approach will not yield correct results in a general black box fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  grated stars doubles when the interpolated center is used compared  to the correct center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Ideally, we would like to apply our analysis to as many galaxies as  possible, but as only a handful of galaxies reside inside the subboxes  during their whole life time, we need an alternative measure for  classifying them into migrated and in-situ stars that can be applied  to the full box of TNG50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We therefore take the particle’s position  at the full box snapshot it first appeared in as its “birth” position and  compare it to the instantaneous one with the proper centering, also  shown in Figure B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We see that the shapes of their histograms as  well as their percentages match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Of course, their classification is not  identical, as particles move between their exact formation time and  the time of when the snapshot was taken, however this measure seems  to be more accurate than using the instantaneous birth positions with  the interpolated center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Finally, we compare the instantaneous birth radii using the correct  center of the migrated and in-situ stars (classified with the instanta-  neous positions using the correct center) with birth radii calculated  MNRAS 000, 1–34 (2022)  SubfindID 537941  E<Ecut (21910)  r<rcut (21256)  150  100  [km/s kpc]  50  0  50  100  150  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='00 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='75 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='50 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='25 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='00  Energy [105 (km/s)2]  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  [odx]  E<Ecut (21910)  r<rcut (21256)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1  10  30  (Apocentero≤t≤136Myr〉[kpc]SubfindID 35  E<Ecut (42137)  r<rcut (37125)  150-  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  [km/s kpc]  50  0  50  100  150  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='50 -16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='25 -16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='00 -15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='75 -15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='50  Energy [105 (km/s)2j  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  [kpc]  E<Ecut (42137)  r<rcut (37125)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1  10  30  <Apocentero≤t≤136Myr>[kpc]Centers of TNG50 Galaxies  29  0  2  4  6  8  10  12  14  Lookback Time [Gyr]  4  2  0  2  4  10  100  [kpc]  SubfindID 537941  Distance to two  most massive mergers  Centersubbox  fullbox  x  y  z  0  2  4  6  8  10  12  14  Lookback Time [Gyr]  10  5  0  5  10  100  1000  [kpc]  SubfindID 35  Distance to host  Centersubbox  fullbox  x  y  z  Figure B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Differences in galaxy centering between full box and sub-  box snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Comparison between the interpolated (cubic spline) center  position from the 100 full box snapshots (using SubhaloPos from the sub-  halo catalogue) and the center position (most bound stellar particle) from  the higher cadence subbox outputs for two example galaxies (blue shaded  lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The top panel shows SubfindID 537941, a Milky Way like galaxy  and the bottom panel shows SubfindID 35, a ∼ 1010 M⊙, quenched galaxy  that is now a satellite of the most massive halo in TNG50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We see that the  interpolated center of the galaxies starts to deviate significantly from the true  center position when there are pericenter passages from satellites that merge  with galaxy or when the galaxy itself is a satellite and approaches its host  (thick orange lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  by the other two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The birth radii determined from the full  box snapshots scatter around the one-to-one relation, whereas the  ones with the interpolated center do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Hence, we conclude that  applying the migrated and in-situ classification based on their birth  snapshot position to the whole TNG50 box seems to provide us with  similar knowledge about their origin as if we would have used their  instantaneous birth position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  APPENDIX C: THE TOTAL EX-SITU STELLAR MASS  FRACTION OF TNG50  In Figure C1 we show the total ex-situ stellar mass fraction as a  function of total stellar mass for TNG50 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' They are further  divided into star forming and quenched galaxies according to their  distance to the star forming main sequence at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Most importantly, we see that the total ex-situ stellar mass fraction  does not change worryingly, when comparing the fiducial definition  of ex-situ stars according to Rodriguez-Gomez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' (2016) as well as  our definition, where we exclude stars from accreted satellites classi-  fied as clumps (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' SubhaloFlag = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Even though the clumps do  not play an important role for the definition of the ex-situ fraction,  they are abundant in TNG50 (see Appendix E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Only due to our defi-  nition, we could confirm the migration of these clumps contributing  significantly to the build-up of the galaxy center for galaxies above  1011 M⊙ in stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Additionally, Figure C1 shows that star forming galaxies have  a higher total ex-situ fraction on average than quenched galaxies  across all stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Until galaxy stellar masses of 1010 M⊙  the ex-situ fraction stays roughly constant with values around 4-5%  and 3% for star forming and quenched galaxies respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' There  is however a large scatter associated with the ex-situ fraction for  galaxies in this mass regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Above 1010 M⊙ the ex-situ fraction  sharply increases with galaxy stellar mass reaching approximately  50% for galaxy between 1011 − 1012 M⊙ for TNG50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The scatter  decreases accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We have checked this exact relation with the  lower resolution run TNG50-2 as well as TNG100 and obtained  similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Figure C2 shows the mass-size relation for TNG50 galaxies col-  ored according their relative ex-situ fraction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' if they have high  or low ex-situ fractions with respect to the average typical for their  respective stellar mass (Merritt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2020, following).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Galaxies with  stellar masses ≲ 5×1010 M⊙ and above average ex-situ fractions are  on average more extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This median trend is not observed for the  high-mass end, however compact galaxies at 1011 M⊙ have almost  exclusively below average ex-situ fractions in agreement with other  studies (see Davison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2020, for EAGLE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Merritt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2020, for  TNG100;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2021, for TNG50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  APPENDIX D: RESOLUTION CONVERGENCE TESTS  To test numerical convergence we perform the exact same analy-  sis from the main text for three different resolution realizations of  TNG50: TNG50-1 (flagship), TNG50-2 and TNG50-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Throughout,  the galaxy sample selection is the same as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2 except we  omit cutting galaxies with less then one hundred stellar particles in  their centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  For reference, the mass resolution of TNG50-2 and TNG50-3  is 8 and 64 times and the gravitational softening length is 2 and  4 times worse than TNG50-1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The latter translates to  physical sizes of 288 pc (TNG50-1), 576 pc (TNG50-2) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='152 kpc  (TNG50-3) for collisionless particles at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  In the top row of Figure D1 we show the results of the central stellar  mass of the in-situ, migrated and ex-situ stars as a function of total  dynamical mass for all three resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We see that the total stellar  mass in the central 500 pc of TNG50 galaxies is converging at all halo  masses, meaning that the distance between TNG50-1 and TNG50-2  (∼ 1 dex) is around 50% smaller than the distance between TNG50-2  and TNG50-3 (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 dex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The same is true when looking at the  central mass for just the in-situ stars, even though the converging  MNRAS 000, 1–34 (2022)  30  Boecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Figure B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Effect of erroneous galaxy centering on the classification of migrated stellar particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Left panels: Histograms of particles being classified  as migrated (orange) and in-situ (pink) using three different approaches: the instantaneous position at birth with the proper centering from the subbox outputs  (thick solid line), the same but with the interpolated center (thin dashed line) as well as the position of particles taken from the full box snapshots they first  appeared in (thin solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Right panels: Comparing the values of the instantaneous birth radii with the centering from the subbox outputs with the ones with  the interpolated center (fainter, smaller points) as well as the radii from the full box snapshots in which the particles first appeared in (bolder, bigger points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The one-to-one relation is also shown (black dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The top panel shows SubfindID 537941, a Milky Way like galaxy and the bottom panel shows  SubfindID 35, a ∼ 1010 M⊙, quenched galaxy that fell into the most massive halo of TNG50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Using the instantaneous birth position with the interpolated center  leads to a wrong classification of migrated and in-situ stars, whereas using the positions from the full box snapshots at birth gives comparable results to the  instantaneous birth positions with the proper centering in the actual number of particles in the two categories as well as the values of their birth radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  MNRAS 000, 1–34 (2022)  SubfindID 537941  1000  5  Migrated  Instantenous (proper center): 21 %  les  From Snapshot: 20 %  Particle  800-  Instantenous (interpolated center): 43 %  600  of  Number  400-  200-  1  2500  Insitu  Instantenous (proper center): 79 %  8  From Snapshot: 80 %  2000  Instantenous (interpolated center): 57 %  arti  Instantenous  Number  1000  500  From Snapshot  Instantenous (interpolated center)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='02  14  12  10  8  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='02  6  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1  5  Age [Gyr]  Birth Radius (see legend) 「kpciSubfindID 35  2000  5  Migrated  Instantenous (proper center): 26 %  les  Birth Radius (proper center) [kpc]  From Snapshot: 26 %  1500  Partic  Instantenous (interpolated center): 51 %  1000  Number  500-  0  2500  Insitu  Instantenous (proper center): 73 %  8  From Snapshot: 74 %  2000  Instantenous (interpolated center): 49 %  Instantenous  Number  1000-  500  From Snapshot  Instantenous (interpolated center)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='02  14  12  10  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='02  8  6  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1  5  Age [Gyr]  Birth Radius (see legend) [kpc]Centers of TNG50 Galaxies  31  Figure C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Total ex-situ stellar mass fraction versus total stellar mass in  TNG50 at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The solid lines show the fiducial ex-situ fraction as classified  by methods in Rodriguez-Gomez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' (2016), whereas the dashed lines  show total ex-situ fractions excluding spurious (SubhaloFlag = 0) galaxies  or “clumps” (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The faint bands depict the 16th and 84th  percentiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' There is no significant difference between the two classifications  regarding the total ex-situ stellar mass fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We also divide by star-forming  (blue lines) and quenched galaxies (red lines) at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The star forming  galaxies have higher ex-situ fractions on average at fixed stellar mass than  quenched galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  of the lines becomes much less obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For the migrated stars it  looks like the central mass is better converged for galaxies with total  dynamical masses below 1012 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' However, when comparing the  central stellar mass fraction of in-situ and migrated stars (bottom row  of Figure D1), the convergence behaviour seems to be much more  complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' For galaxies outside masses of 1012±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5 M⊙, the central  stellar mass fractions seem to be converging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Overall though, the  central migrated mass becomes on average larger than the in-situ  mass with decreasing resolution for all galaxy masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  This is due to the fact that the numerical resolution also influ-  ences which stars are classified as in-situ and migrated, which is a  consequence of how (spatial and mass) resolution effects star for-  mation and feedback in the TNG model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Better resolution allows  for higher gas densities and better spatial sampling of the gas cells,  which produces galaxies with higher stellar mass and more compact  sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Thus, in an absolute sense more stellar mass resides overall  in the center of TNG50 galaxies for higher resolution runs, however  differentially more stellar mass comes from outside the fixed 500 pc  aperture for lower resolution runs (see also Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Interest-  ingly, the complexity in the central in-situ and migrated fractions  seen around 1012 M⊙ coincides with the complexity in the conver-  gence behaviour of the mass-size relation in TNG50 (see Pillepich  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2019, Figure B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  For the absolute stellar mass of the central ex-situ stars the start of  convergence is not yet apparent, but the fractions is clearly converg-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Again, the ex-situ mass in the center increases with increasing  resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Additionally, the minimum galaxy mass that exhibits any  ex-situ stellar mass in the center is extended towards lower mass  galaxies for increasing resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Again, this behaviour is a conse-  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  fexsitu  109  1010  1011  1012  Total Stellar Mass [M ]  10  1  2  50  3D Stellar Half Mass Radius [kpc]  TNG50  z = 0  fexsitu  < fexsitu  Figure C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Connection between the galaxy mass-size relation and total  ex-situ fractions in TNG50 at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The 3D half mass radius against total  stellar mass for 4344 galaxies in TNG50 (M★, tot > 5 × 108 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The points  are color-coded according to the relative ex-situ fraction Δfexsitu indicating  whether the total ex-situ fraction of a given galaxy above or below the average  at fixed galaxy stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Median mass-size relations are shown separately  for galaxies having above (solid black line) and below (dashed black line)  average ex-situ fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Below ∼ 5 × 1010 M⊙, galaxies with high relative  ex-situ fractions are on average more extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  quence of the complex interplay between mass and spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Subhalos that were either not sufficiently resolved with enough stellar  particles or even remained dark at lower resolution are able to form  enough stars at higher resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' As a consequence, galaxies that  become accreted are not only more massive in stellar mass, but also  more abundant, especially at the low mass end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Thus, the ex-situ stel-  lar mass is higher in general and also contributes at the lower galaxy  masses when resolution is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Numerical resolution has also  been shown to impact the disruption of merging satellite galaxies,  which could lead to overmerging, especially in the outskirts of galax-  ies (see van den Bosch & Ogiya 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Merritt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Hence,  a higher resolution will result in more accreted stellar material in  the center of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Furthermore, the resolution of TNG50 will  impact the ability to resolve the dynamics of its resolution elements  (stars, clumps, gas cells, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' ), including dynamical friction, at scales  much below the spatial resolution of the simulation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' certainly at  the tens of pc scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Even though TNG50 is a tremendous improvement in resolution  for cosmological box simulations, we still need a higher resolution  to fully assess the convergence behaviour of the absolute central ex-  situ stellar mass (see Grand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' 2021, for a study of the satellite  galaxy population of a highly resolved Milky Way like galaxy in a  cosmological context and comparison to lower resolution models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  MNRAS 000, 1–34 (2022)  Iotal Exsitu Stellar Mass Fraction  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='00  TNG50  Z=0  fiducial  without clumps  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='10  Star Forming  Quenched  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='01  109  1010  1011  1012  Total Stellar Mass [Mo]32  Boecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content="  1011  1012  1013  Total Dynamical Mass [M ]  104  105  106  107  108  109  1010  Central Stellar Mass [M ]  z = 0  All  Insitu  Migrated 'Smooth+Clumpy'  Exsitu  TNG50-1  TNG50-2  TNG50-3  1011  1012  1013  Total Dynamical Mass [M ]  z = 0  TNG50-1  TNG50-2  TNG50-3  1011  1012  1013  Total Dynamical Mass [M ]  z = 0  TNG50-1  TNG50-2  TNG50-3  1011  1012  1013  Total Dynamical Mass [M ]  z = 0  TNG50-1  TNG50-2  TNG50-3  1011  1012  1013  Total Dynamical Mass [M ]  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content="0  Central Stellar Mass Fraction  z = 0  Insitu  Migrated 'Smooth+Clumpy'  Exsitu  TNG50-1  TNG50-2  TNG50-3  1011  1012  1013  Total Dynamical Mass [M ]  z = 0  TNG50-1  TNG50-2  TNG50-3  1011  1012  1013  Total Dynamical Mass [M ]  10  4  10  3  10  2  10  1  100  z = 0  TNG50-1  TNG50-2  TNG50-3  Figure D1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Influence of the numerical resolution on the central (500 pc) in-situ, migrated and ex-situ stars of TNG50 galaxies at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Top panel: Median  trends of the central stellar mass as a function of a galaxy’s total dynamical mass for all (all), in-situ (pink), migrated (orange) and ex-situ (blue) stars and  different numerical resolution realizations of the same cosmological volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The thicker the line the better the numerical resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Shaded areas show the  16th and 84th percentiles for the highest resolution run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Bottom panel: Instead of the absolute stellar mass we now show the central stellar mass fraction of the  in-situ, migrated and ex-situ stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The central ex-situ mass faction is converging, whereas the behaviour is more complex for the in-situ and migrated fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  APPENDIX E: CLUMPS IN TNG50  Here, we show some examples and properties of clumps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' sub-  halos detected by the Subfind algorithm, but which were formed  not through the collapse of a dark matter halo, that are present in  TNG50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Instead these clumps form in the gaseous disk of galaxies  or fragments of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Particularly, we could observe qualitatively that  significant clump formation predominantly occurs during galaxy in-  teractions at z > 1, such as mergers or fly-bys, but it can also take  place when a galaxy evolves in isolation (see also Di Matteo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  2008, for a similar finding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  We present a visual example of such clumps at the top of Figure  E1 for a galaxy that lives in an environment with many galaxy inter-  actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Clumps embedded in the stellar disk of that galaxy clearly  exhibit a gaseous counterpart, whereas clumps further out and not  within the disk are only seen in the stellar surface mass density (at  the time of inspection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The clumps visible in both the stars and  gas are clearly distributed along the spiral (or tidal) arms of that  galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' All of them migrate to the center and deposit stars (and also  gas) there (marked by orange arrows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' In that process some of them  (marked by lighter orange arrows) merge with other clumps, which  we reconstructed with the merger trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The statistics on clumps in TNG50 shown on the bottom left hand  side of Figure E1 reveals that around 36% of all TNG50 galax-  ies posses clumps at any point in their lifetime of which a fourth  have clumps that migrate to the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Above 5 × 1010 M⊙ in stel-  lar mass all galaxies exhibit clumps at some point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The amount of  clumps per galaxy starts to rise above one for galaxy stellar masses  larger than 1010 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' On average a Milky Way mass galaxy has five  clumps, whereas the most massive galaxies in TNG50 can have up  to a hundred clumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The number of clumps that actually migrate  to a galaxy’s center starts to rise at higher galaxy masses at around  5 × 1010 M⊙ reaching an average of about 5 migrated clumps per  galaxy at the high mass end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We remind the reader that these and  following numbers do not account for clumps that merged with other  clumps, thus the true number of all clumps ever formed is actually  even higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  We also show the distribution of four properties of the clumps in  TNG50 on the bottom right hand side of Figure E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Each property  is shown for all clumps at the time of formation, for all clumps that  migrated at the time of formation and for all clumps that migrated at  the time they arrive at their galaxy’s center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  We see that clumps are formed with a broad distribution of stellar  masses with a peak at 108 M⊙13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Only clumps that are formed with  such stellar masses or higher are able to migrate to the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The  13 We caution the reader to not trust the few clumps that have stellar masses  of around 1010 M⊙, which likely originate from switching between the host  galaxy and the actual clump during the halo/subhalo finding process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  MNRAS 000, 1–34 (2022)  Centers of TNG50 Galaxies  33  16 kpc  TNG50  z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='46  Stars  16 kpc  TNG50  z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='46  Gas  106  107  108  109  [M /kpc2]  106  107  108  gas [M /kpc2]  SubfindID 220595  Figure E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Example and summary statistics of clumps in TNG50 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Top panel: Stellar (left, full projection) and gas (right, thin slice with |𝑧 | ≤ 5 kpc)  surface mass density of a TNG50 galaxy (SubfindID 220595) in face-on projection exhibiting clumps at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Clumps migrating to the center (pink dot)  are marked with orange arrows, whereas clumps not belonging to the galaxy are marked with white arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Clumps with lighter orange arrows will first merge  with other clumps (darker orange arrows), before arriving at the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Lower panel, left: Distribution of host galaxy stellar masses (top) displaying clumps in  general (blue) and clumps that migrate to the center (orange) at any time, compared to the total galaxy sample (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Numbers inside the brackets display the  total amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' All galaxies in our sample above ∼ 5 × 1010 M⊙ exhibit clumps at some point in their life time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The median number of clumps that ever existed per  galaxy as a function of host galaxy stellar mass are also shown (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The shaded area shows the 16th and 84th percentile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The number of clumps arriving at  a galaxy’s center is about a decade lower than the total amount of clumps formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Note however that the true number of all clumps ever formed is higher due  to clump-clump mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Lower panel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' right: Distribution of certain clump properties (blue solid line: all clumps at the time of formation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' orange solid line:  clumps that migrate to the center at the time of formation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' light orange dashed-dotted line: clumps that migrated to the center at the time of arrival at the center):  from left to right: total stellar mass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' gas fraction (ratio of gas mass to total baryonic mass),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' formation time (solid lines)/time taken to arrive at the center (dashed  line) and distance from the host galaxy of the clumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  MNRAS 000, 1–34 (2022)  All Galaxies (2531)  Galaxies with clumps (916)  Galaxies with clumps  All Clumps  Migrated Clumps  Migrated Clumps  migrating to center (235)  at Formation  at Formation  arriving at Center  103  TNG50  TNG50  TNG50  103  Z=0  z=0  Z=0  Clumps  Galaxies  102  TT  # 10  TT  10  #  108  1010  104  106  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='0  Total Stellar Mass  Gas Fraction  of Clumps [Mo]  of Clumps  TNG50  z=O  TNG50  TNG50  All Clumps  103  z=0  z=0  LL  per  (12885)  Clumps  Migrated  Clumps  Clumps  102  (648)  #  10  =  #  LLLI  TTTT  T  TT  TTTTT  TTTTT  109  1010  1011  24681012  1012  214  10  102  0  103  Total Stellar Mass  Lookback Time [Gyr]  Distance from Host [kpc  of Host Galaxy [Mo]34  Boecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  stellar mass that they then actually deposit at the center is a flattened  out distribution all the way down to a few stellar particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Thus  clumps can suffer significant stellar mass loss while travelling to the  galaxy center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Even though there is a broader peak of clumps formed with high  gas fractions (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='6 − 1), the gas fraction distribution of clumps that  migrate to the center is significantly flatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Thus, a high gas fraction  is not necessarily an indication of whether a clump is able to migrate  to the center or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  Furthermore, the clumps are formed all throughout cosmic time in  TNG50, with a slight increase towards younger lookback times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' The  number of clumps that migrated with formation times younger than  6 Gyr ago drops compared to clumps formed at older ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This is  understandable as it takes a few Gyr for the clumps to migrate to the  center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Most clumps need around 1 Gyr to do so, but there is a long  tail towards higher migration times up until 6 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  The peak distance from the host galaxy, where clumps form, is  around 20 kpc and flattens out towards smaller distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Whereas  the number of migrated clumps drop sharply after that distance, the  distance of clumps formed overall extends all the way until several  hundred kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Hence, clumps in TNG50 can form in the halo of  galaxies, possibly in gaseous tidal or ram-pressure-stripped tails and  gas fragments during galaxy interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' We empathize here that the  clumps formed at such large distances are not part of any satellite  galaxy that then merged with the host (at least according to the  Subfind algorithm and the merger trees).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' This is because we identify  clumps as subhalos with SubhaloFlag = 0 that directly merged onto  the main progenitor branch of the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content=' Hence, clumps that  are part of satellite galaxies that are then brought in by merging with  the host galaxy, are also not accounted for in our statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  We provide further discussion and comparison to clumps in other  simulations in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
+page_content='  MNRAS 000, 1–34 (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQf6i75/content/2301.11942v1.pdf'}
diff --git a/xdE3T4oBgHgl3EQfOwnl/content/tmp_files/2301.04397v1.pdf.txt b/xdE3T4oBgHgl3EQfOwnl/content/tmp_files/2301.04397v1.pdf.txt
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+TBV Radar SLAM – trust but verify loop candidates
+Daniel Adolfsson, Mattias Karlsson, Vladim´ır Kubelka, Martin Magnusson, Henrik Andreasson
+Abstract— Robust SLAM in large-scale environments re-
+quires fault resilience and awareness at multiple stages, from
+sensing and odometry estimation to loop closure. In this work,
+we present TBV (Trust But Verify) Radar SLAM, a method
+for radar SLAM that introspectively verifies loop closure candi-
+dates. TBV Radar SLAM achieves a high correct-loop-retrieval
+rate by combining multiple place-recognition techniques: tightly
+coupled place similarity and odometry uncertainty search, cre-
+ating loop descriptors from origin-shifted scans, and delaying
+loop selection until after verification. Robustness to false con-
+straints is achieved by carefully verifying and selecting the most
+likely ones from multiple loop constraints. Importantly, the
+verification and selection are carried out after registration when
+additional sources of loop evidence can easily be computed. We
+integrate our loop retrieval and verification method with a fault-
+resilient odometry pipeline within a pose graph framework. By
+evaluating on public benchmarks we found that TBV Radar
+SLAM achieves 65% lower error than the previous state of the
+art. We also show that it’s generalizing across environments
+without needing to change any parameters.
+I. INTRODUCTION
+Robust localization is key to enabling safe and reliable
+autonomous systems. Achieving robustness requires careful
+design at multiple stages of a localization pipeline, from
+environment-tolerant sensing and pose estimation, to place
+recognition and pose refinement. At each stage, a localization
+and mapping pipeline should be designed for fault awareness
+to detect failures and fault resilience to mitigate failures as
+they inevitably occur. Today, active exteroceptive sensors
+such as lidar and radar are suitable when robust uninterrupted
+localization is required. Of these two, radar perception is
+significantly less affected when operating within dusty envi-
+ronments or under harsh weather conditions. It is currently
+debated how the sensing properties affect localization and
+mapping performance [1], [2]. Compared to lidar, limited
+work focuses on robust and accurate radar SLAM, and none
+of the existing methods include introspective fault detection.
+In this letter, we propose TBV (Trust But Verify) Radar
+SLAM – a 2D pose-graph localization and mapping pipeline
+which integrates fault resilience and fault-aware robustness
+at multiple stages. A radar-only odometry front-end adds
+pose nodes to the graph. In parallel, a robust loop-detection
+module adds loop closure constraints such that the SLAM
+back-end can optimize the graph to correct drift. TBV Radar
+SLAM uses radar odometry (further only odometry), place
+descriptors, and scan alignment measures to retrieve, verify,
+and select between loop constraints. We achieve a high
+correct-loop-retrieval rate by combining: a tightly coupled
+The authors are with the MRO lab of the AASS research centre at ¨Orebro
+University, Sweden. E-mail: firstname.lastname@oru.se
+This work has received funding...
+Verified
+VERIFY LOOP CANDIDATES
+Trusted
+ODOMETRY
+ALIGNMENT
+SCAN CONTEXT
+Ground Truth,        Est. Trajectory,        Loop Closure
+Query
+Fig. 1: Overview and demonstration of TBV Radar SLAM
+https://tinyurl.com/TBVRadarSLAM
+place similarity and odometry uncertainty search, creating
+place descriptors computed from origin-shifted scans, and by
+delaying loop selection until after verification. Robustness,
+with a high loop detection rate, is achieved by unifying
+the process of place recognition and verification. Rather
+than rejecting candidates early during place recognition, we
+register and verify multiple loop constraints in parallel. Our
+verification combines place similarity, odometry consistency,
+and an alignment quality assessment automatically learned
+from odometry and scans. We integrate our loop retrieval and
+verification module with a robust method for radar odometry,
+into a full-fledged SLAM pipeline visualized in Fig. 1. The
+contributions of this letter are:
+• A combination of techniques for a high rate of correct
+loop retrievals, including: coupled place similarity and
+odometry uncertainty search, creating place descriptors
+from origin-shifted scans, and selection between multi-
+ple competing loop constraints.
+• A unified loop retrieval and verification step that jointly
+considers place similarity, odometry uncertainty, and
+alignment quality after registration. Multiple loop can-
+didates are retrieved, registered, and verified.
+• We integrate these techniques with a robust odometry
+estimator into a SLAM framework that pushes state
+of the art in radar SLAM accuracy while generalizing
+between environments without parameter tuning.
+II. RELATED WORK
+Early methods for radar SLAM employ filtering [3] and
+landmark graph SLAM [4]. We instead use pose-graph
+SLAM based on odometry and loop constraints obtained by
+arXiv:2301.04397v1  [cs.RO]  11 Jan 2023
+
+       Loop retrieval and verification (Sec.III.B & D) 
+    
+    Place recognition (Sec.III.B)
+    Pose graph optimization 
+    (Sec.III.E)
+    Verification of loop candidates (Sec.III.D)
+Loop
+registration
+    Odometry estimation (Sec.III.A)
+CFEAR 
+Radar odometry
+(Sec.III.A)
+Loop
+verification
+Pose graph
+Loop  
+constraints
+Odometry 
+constraints
+Optimizer
+Descriptor  
+generation
+(Sec.III.C.2)
+Radar 
+Scan Context
+(Sec.III.C.1&3)
+loop  
+candidates
+Registered 
+Scan  
+pairs
+Learning of 
+alignment
+verification 
+(Sec.III.C) 
+Input radar data
+Radar scans
+scans database
+Fig. 2: Overview of TBV Radar SLAM. The main contribution is the Loop retrieval and verification module.
+registering scans. Holder et al. [5] proposed a pose graph
+SLAM based on automotive radar, using GLARE [6] for
+place recognition. Their method uses gyroscope and wheel
+odometry to provide an initial alignment guess to Iterative
+Closest Point (ICP) for scan matching. Our method uses a
+pose graph back-end but relies solely on a spinning radar.
+Recent progress in the development of spinning 2D radar
+has quickly inspired numerous works on radar odometry
+estimation [1], [7]–[19], place recognition [11], [20]–[25],
+topological localization [2], [21], [25] localization in over-
+head imagery [26], [27] and SLAM [1], [10], [28]. Most
+similar to ours is the preliminary work by Wang et a. [28],
+and the seminal work on radar SLAM by Hong et al. [1].
+Hong et al. [1] estimates odometry and registers loop con-
+straints using KLT tracker keypoints. For place recognition,
+they use adaptive thresholding to filter data, and M2DP [29]
+to compute and match descriptors. The trajectory is corrected
+using pose graph optimization. Our work brings a larger
+focus on the introspective verification of loop constraints.
+Martini et al. [21] proposed a teach-and-repeat localization
+framework using a hierarchical approach. First, a place
+candidate is retrieved via nearest neighbor search within
+a learned metric space using the method by Saftescu et
+al. [20]. Second, sensor pose is estimated (without the need
+for an initial guess) by finding and registering the (globally)
+optimal set of landmark matches as described by Cen et
+al. [7]. Unlike [21], we use a fast local registration method,
+motivated by the access of an initial alignment guess from
+the place recognition module. However, we carefully verify
+registration success before accepting new loop constraints.
+Verification of pose estimates is essential for safety. E.g.,
+such tests could have prevented a reported accident that
+occurred during an automated driving test [30]. Holder
+et al. [5] verify the detected loop candidates using radar
+data by the condition that ICP residuals cannot exceed a
+set threshold. Rather than verifying loops as a final step
+(separated from loop detection), we unify loop retrieval
+and verification. Some works [21], [31] use the landmark
+matching compatibility matrix [7] to assess quality. Martini
+et al. [21] reject place candidates based on the quality
+score of the matrix. Aldera et al. [31] learn detection of
+pose estimate failures from features extracted from the
+eigenvectors of the compatibility matrix. Training labels are
+provided by an external ground truth system. We build on
+the method CorAl [32] to detect false loop candidates or
+constraints. In the lidar domain, Behley et al. [33] accept
+only reappearing loop candidates that are consistent with
+odometry over multiple consecutive scans, we instead focus
+on verification using only individual scans.
+Some methods attempt to measure how observations
+would constrain registration in the current scene; a low
+level of constraints in one direction suggests registration
+could be unstable. The measure has been used to predict the
+risk of registration failure [34], [35], abstain from closing
+loops when risk is high [1], or prioritize the inclusion of
+loop closures accordingly [36]. We instead use odometry
+uncertainty as a prior, which we combine with additional
+loop evidence computed after registration. Finally, a range of
+methods aims at verifying pose estimates by detecting point
+cloud misalignment [30], [32], [37]. We fuse misalignment
+detection [32] together with place similarity and odometry
+uncertainty to achieve robust unified loop detection and
+verification.
+III. TBV RADAR SLAM
+An overview of TBV Radar SLAM is presented in Fig. 2.
+In this section, we detail the components including CFEAR
+Radar Odometry (Sec. III-A), place recognition (Sec. III-
+B), verification of loop candidates (Sec. III-D), and pose
+graph optimization (Sec. III-E). The self-supervised training
+of alignment verification (Sec. III-C) runs once during a
+learning phase and is not required for online SLAM.
+A. CFEAR Radar Odometry
+We use the radar odometry pipeline CFEAR Radar Odom-
+etry [17] (specifically the CFEAR-3 configuration). This
+method takes raw polar radar sweeps as input and estimates
+sensor pose and velocity. As an intermediate step, the method
+computes a set of radar peaks P t and a set of oriented surface
+points Mt at time t. These representations will be referred
+to in the later stages of our pipeline. CFEAR filters the radar
+data by keeping the k-strongest range bins per azimuth. The
+filtered point set is compensated for motion distortion and
+transformed into Cartesian space, - this is also the point set
+from which we extract radar peaks (P t). A grid method is
+then used to compute a set of oriented surface points Mt
+from the filtered set. Odometry is estimated by finding the
+pose xt in SE(2) which minimizes the sum of surface point
+distances between the current scan Mt and the |K| latest
+keyframes MK jointly as
+f(MK, Mt, xt) =
+�
+k∈K
+�
+∀{i,j}∈Ccorr
+wi,jρ
+�
+g(mk
+j , mt
+i, xt)
+�
+, (1)
+where wi,j are surface point similarity weights, ρ is the
+Huber loss function, and g is the pairwise distance between
+surface points in the correspondence set Ccorr. A keyframe
+k ∈ K is created when the distance to the previous exceeds
+1.5 m. This technique reduces drift and removes excessive
+scans acquired when the robot remains stationary. Upon
+
+creation of a new keyframe, odometry constraints are created
+from the relative alignment between the current pose and the
+latest keyframe. Odometry constraints are added to Codom
+and used to correct trajectory (via pose graph optimization)
+as detailed in Sec. III-E.
+B. Place recognition
+We attempt to unify the stages of loop closure, from
+detection to constraint verification. Accordingly, we retrieve,
+register, verify multiple candidates, of which one can is
+selected. For that reason, we do not discard potential loop
+candidates based on place similarity. Instead, we trust multi-
+ple candidates to be meaningful until verified. At this point,
+we select the verifiably best loop candidate, if such exist.
+1) Scan Context: We build upon the place recognition
+method Scan Context by Giseop Kim et al. [38], [39]. Their
+method detects loops and relative orientation by matching
+the query (q) scan with candidates ({c}) stored in a database
+using scan descriptors. Scenes are encoded by their polar
+representations into 2D descriptors Iring×sec. The 2D de-
+scriptor is in turn used to create a rotation-invariant 1D
+descriptor (ring key) via a ring encoding function. Loops
+are detected via a two-step search: First, the query 1D
+ring key is matched with the top 10 candidates via a fast
+nearest neighbor (NN) search. Second, for each of the top
+candidates, a sparse optimization finds the relative rotation
+that minimizes a distance metric dsc(Iq, Ic): the sum of
+cosine distances between descriptors columns. The candidate
+c with the lowest score (dsc) which does not exceed the fixed
+threshold τ is selected as loop candidate.
+c = argmin
+c∈C
+dsc(Iq, Ic), s.t. dsc < τ.
+(2)
+In
+our
+implementation,
+query
+descriptors
+are
+created,
+matched, and stored in the database for each keyframe rather
+than for each scan.
+2) Descriptor generation: As described in [28], [40],
+a raw polar representation such as those produced by a
+spinning 2D radar, can be used directly as a Scan Context
+descriptor. However, we believe that doing so poses multiple
+challenges, including sensor noise, motion distortion, scene
+dynamics and translation sensitivity. Thus, we create our
+descriptor from multiple filtered and motion-compensated
+scans. Conveniently, such processed scans are already pro-
+vided by the CFEAR. We aggregate the peak detections from
+keyframe q and its two neighbours in the odometry frame.
+Having the radar scan represented as a sparse point cloud
+in Cartesian space allows us to address translation sensitivity
+in place recognition by applying the data augmentation step
+(Augmented PC) from [39] before computing place descrip-
+tors. We perform data augmentation by shifting the sensor
+origin, i.e. by transforming P ˆq with ±2 and ±4 m lateral
+translation offsets. The original, and the 4 augmented point
+clouds, are each used to compute and match descriptors, after
+which the best matching pair of query/candidate is selected.
+Note that by using the aggregated sparse point cloud, rather
+than the dense raw radar scan, we can efficiently compute all
+augmentations and corresponding descriptors. As such, the
+main computational load from the augmentation technique
+is due to matching of additional descriptors and not the
+computation of these. The descriptor itself is created by
+populating the Scan Context I with radar intensity readings.
+Specifically, for each grid cell I(i, j) we sum the intensity
+of all coinciding point intensities (of radar peaks) divided by
+1000. Empty cells are set to I(i, j) = −1, which we found
+increased descriptiveness compared to I(i, j) = 0.
+3) Coupled odometry/appearance matching: When re-
+trieving loop candidates, odometry information can be used
+to filter unlikely candidates. This could be done by rejecting
+unlikely loop constraints. For example, if the likelihood
+of the loop constraint xq,c
+loop (given the estimated odometry
+trajectory vc:q
+odom between c and q ) is close to zero:
+p(xq,c
+loop |vc:q
+odom) ≈ 0.
+(3)
+While this strategy may provide higher tolerance to spatial
+aliasing by rejecting false positives, it does not provide
+means to detect the correct candidate under such circum-
+stances. For that reason, we propose a coupled place similar-
+ity / odometry uncertainty search, which combines Eq. 2 and
+Eq. 3. Candidates are thus selected jointly by the similarity of
+appearance dsc(Iq, Ic) and the similarity of odometry dq,c
+odom:
+c = argmin
+c∈C
+dsc(Iq, Ic) + dq,c
+odom,
+dq,c
+odom = 1 − p(xq,c
+loop |vc:q
+odom) .
+(4)
+We estimate p(xq,c
+loop |vc:q
+odom) = exp (− t2
+err
+2σ2 ).
+terr = max(||transl(xq) − transl(xc)|| − ϵ, 0)
+dist(vc:q
+odom)
+.
+(5)
+Here, transl is the translation component, ϵ is the expected
+maximum spacing between loop candidates (fixed to ϵ = 5
+i.e. slightly higher than the lateral augmentation distance),
+and dist(vc:q
+odom) is the traversed distance between the query
+and loop candidate estimated by the odometry. Note that
+terr quantifies the relative final position error, thus σ can be
+chosen according to expected odometry quality to penalize
+unlikely loops. We refrained, however, from making strong
+assumptions on odometry quality, and fixed σ = 0.05; i.e., a
+pessimistic assumption of 5% relative translation error.
+Note that the two-step search of Scan Context requires
+that odometry uncertainty is integrated already in the 1D
+NN search. We do this by extending all 1D descriptors (of
+size ring = 40) with odometry similarity scores (dodom) as
+an extra element. (dodom) is scaled with a factor (ring/4)
+to balance odometry uncertainty and appearance similarity.
+C. Automatic learning of alignment verification
+To improve loop closure verification, we build upon the
+system CorAl [32] which learns to detect alignment errors
+between two registered scans. CorAl allows us to determine
+if a loop constraint is correct by formulating loop constraint
+verification as a misalignment detection problem. Specifi-
+cally, a loop (between scan nr q and c) is verified as correct
+only if the scans are correctly aligned via the relative align-
+ment xq,c
+loop. During the learning phase, CorAl automatically
+
+generates labeled training data. The input to CorAl is pairs of
+odometry estimates, radar peak detections (P), and computed
+sets of oriented surface points (M). These entities are used
+to extract alignment quality residuals from which alignment
+quality can be assessed. After the learning phase, CorAl can
+verify loops by detecting alignment errors (caused e.g. by
+heavy motion distortion or incorrect convergence). CorAl
+also aids in distinguishing between places that differ by small
+geometric details. We generate training data by repeating the
+following process for each pair of consecutive keyframes.
+Positive training labels (yaligned = true) and training data
+Xquality are computed using the scan alignment provided by
+the odometry. For each pair, the alignment quality measures
+in Eq. 6 are extracted. Negative training labels (yaligned =
+false) and training data are extracted similarly. However,
+before extracting the alignment quality, an error is induced in
+the alignment in both translation and rotation. This allows us
+to learn the detection of different types of errors. Specifically,
+we distribute 12 translation errors symmetrically in either the
+positive or negative x or y-axis. We use 4 small (±0.5 m), 4
+medium (±1 m) and 4 large (±2 m) errors. To these errors,
+we additionally induce a clockwise rotation with matching
+rotation errors: small (0.5◦), medium (2◦) or large (15◦).
+Note that the class ratio 1:12, between positive to negative
+training is alleviated during learning by assigning weights
+according to the inverse of class frequency.
+1) Alignment measure: We extract the following align-
+ment measures between each pair of scans:
+Xquality = [Hj Hs Ho Cf Co Ca 1]T .
+(6)
+The joint entropy (Hj) and separate entropy (Hs) are average
+per-point differential entropies, extracted from point cloud
+pairs of radar peak detections (Pq, Pc). These metrics are
+described in-depth in [32]. We complement these measures
+with a measure of overlap Ho: (Hoverlap), defined as the
+portion of peaks in Pq or Pc with a neighboring point within
+the radius r in the other point cloud.
+In this work, we combine these CorAl measures with
+additional ones, extracted from (Mq, Mc), i.e. from pairs of
+scans represented as oriented surface points. The measures
+are obtained from the registration cost(Eq. 1), but with a
+single keyframe and with the point-to-line cost function
+(gP 2L [17]). Note that these measures are already computed
+during the final iteration of the registration, and this step
+brings little computational overhead. Specifically, from Eq. 1
+we reuse Cf: f(Mq, Mc, xq,c), the number of correspon-
+dences (overlap) Co: |Ccorr|, and average surface point set
+size Ca: 1/2(|Mq| + |Mc|). The intuition of combining
+these quality measures is that the CorAl measures (which
+use small-region data association) are specialized in detecting
+small errors whereas the CFEAR measures are more suitable
+for larger alignment errors. We refer to [32] for details.
+2) Assessing alignment: Once training data has been
+computed, we train a logistic regression classifier
+palign = 1/(1 + e−dalign),
+(7a)
+dalign = βXquality,
+(7b)
+where β1×7 are the learned model parameters. We train
+on discrete alignment classification here as we consider all
+visible errors to be undesired. However, dalign is passed to
+our loop verification module rather than palign. We found
+dalign to be more suitable, as the sigmoid output palign is
+insensitive to alignment change close to 0 or 1.
+D. Verification of loop candidates
+We allow for multiple competing loop candidates ck per
+query q as illustrated in Fig. 1. Each of the Ncand = 3
+best matching pairs {(q, ck)} provided by the place recog-
+nition module is used to compute and verify potential loop
+constraints. A constraint is computed by finding the relative
+alignment xq,ck
+loop that minimizes Eq. 1 i.e. the distance be-
+tween correspondences, similarly to the odometry module.
+As an initial guess, we use the relative orientation provided
+by the place recognition module. If the loop candidate was
+retrieved from a match with an augmented query scan, the
+corresponding augmented lateral offset is used together with
+the rotation as an initial guess. Note that the local registration
+method is motivated by the access to an initial guess, required
+for convergence. After registration, we extract and assess
+the alignment quality dalign = βXq,ck
+quality following the
+procedure in Sec. (III-C.1&III-C.2). Each constraint is finally
+verified by combining the Scan Context distance (dsc) with
+odometry uncertainty (dodom) and alignment quality (dalign)
+with a logistic regression classifier
+yq,ck
+loop =
+1
+1 + e−ΘX
+q,ck
+loop , s.t. yq,ck
+loop > yth,
+Xq,ck
+loop = [dodom dsc dalign 1]T .
+(8)
+The model parameters Θ can be learned via ground truth
+loop labels, or simply tuned as the 4 parameters have intuitive
+meaning. yth is the sensitivity threshold – we rarely observe
+false positives when fixed to 0.9.
+We investigated two strategies for selecting loop closures
+after a successful verification: (i) We select the first candidate
+retrieved from the place recognition module (Ncand = 1)
+– the lowest Scan Context distance score dsc. (ii) We use
+Ncand = 3 candidates and select the best candidate according
+to our verifier – the highest probability yq,ck
+loop. The intuition
+for strategy (i) is that the first retrieved place candidate is of-
+ten the best guess, without considering registration. However,
+there are cases where one of the latter candidates is preferred.
+For example, registration of query and the first candidate
+may fail, or subtle scene differences that distinguish places
+can be hard to detect until a more thorough local analysis of
+alignment has been carried out. Thus, selecting the verifiably
+better loop constraint candidate is desired. We compare these
+two strategies in Sec. IV-B ( T.6 and
+T.8) Once a loop
+constraint has been verified, the loop is added to Cloop.
+E. Pose Graph Optimization
+We correct the odometry by solving a sparse least squares
+optimization problem. We do so by minimizing Eq. 9, using
+
+the odometry and loop constraints Codom, Cloop:
+J(Y) =
+�
+(i,j)∈Codom
+eT
+i,jC−1
+odomei,j
+�
+��
+�
+odometry constraints
++
+�
+(i,j)∈Cloop
+ρ(eT
+i,jC−1
+loopei,j)
+�
+��
+�
+loop constraints
+.
+(9)
+Y = [y1 y2...yn] is the vector of optimization parameters,
+e=yi,j − xi,j is the difference between parameter and con-
+straint, C is the covariance matrix. We used two strategies to
+compute covariance; fixed: computed from registration error
+statistics C = diag([vxx = 1e-2, vyy = 1e-2, vθθ = 1e-3]);
+dynamic: obtained from the Hessian, approximated from the
+Jacobian (C = (H)−1 ≈ (JT J)−1) of the registration cost
+(Eq. 1). Note that the dynamically obtained C was tuned by
+a factor γ to provide realistic uncertainties, discussed in [9].
+Loop constraints are scaled by an additional optional factor
+for (5e-5). This factor retains the high odometry quality
+through pose graph optimization, and alleviates the need for
+a more accurate but computationally more expensive multi-
+scan loop registration
+Finally,
+we
+solve
+argminY J(Y)
+using
+Levenberg-
+Marquardt. Note that we do not need a robust back-end
+to mitigate outlier constraints (such as dynamic covariance
+scaling [41] or switchable loop constraints [42]).
+IV. EVALUATION
+We evaluate our method on the Oxford [43] and
+MulRan [40] datasets. Both datasets were collected by
+driving a car with a roof-mounted radar. The Oxford dataset
+contains 30 repetitions of a 10 km urban route. The MulRan
+dataset has a wider mix of routes; from structured urban to
+partly feature poor areas such as open fields and bridge cross-
+ings. In MulRan, places are generally revisited in the same
+driving direction. This is not the case in Oxford – where
+loop closure is significantly harder. From these datasets,
+we selected the previously most evaluated sequences, see
+Tab. (I&II). The radars used are Navtech CTS350-X, con-
+figured with 4.38 cm resolution in the Oxford dataset, and
+CIR204-H, with 5.9 cm resolution in the MulRan dataset.
+We use standard parameters for CFEAR-3 [17], CorAl [32],
+and Scan Context [38], except where explicitly mentioned.
+A. Run-time performance
+After the initial generation of training data, the full
+pipeline including odometry and loop closure runs in real-
+time. Pose graph optimization runs in a separate thread, either
+continuously as new verified loop constraints are computed,
+or once at the end. Run-time performance is as follows (mea-
+sured on an i7-6700K CPU). Odometry: 37 ms; Generation of
+training data: 236 ms/keyframe pair; Pose graph optimization
+(once in the end with only odometry as prior): 992 ms @ 4k
+keyframes; Loop closure (128 ms @ Ncand = 3); Descriptor
+generation: 1 ms/keyframe; Detect loops: 25 ms/keyframe;
+Registration: 6 ms/candidate; Verification 19 ms/candidate.
+B. Ablation study – loop detection
+In this section, we evaluate the effect of the various aspects
+of our loop detection pipeline. Loops are classified as correct
+if the difference between estimated alignment and ground
+truth does not exceed 4 m or 2.5◦. This error limit was set
+slightly higher than the largest pose error that we found in
+the ground truth at loop locations. While we would like to
+demonstrate the full capability of our system by analysis
+of smaller errors, we found ground truth accuracy to be
+a limiting factor. Note that the ground truth quality has
+previously been discussed as a limitation [13].
+We compare the impact of each technique summarized
+below. Note that a later technique in the list either adapts or
+replaces the former technique.
+T.1 Radar Scan Context: Polar radar data is, without pre-
+processing, directly downsampled into a Scan Context
+descriptor [40]. We found the OpenCV interpolation op-
+tion INTER AREA to yield the best results. Verification
+includes only place similarity dsc.
+T.2 Aggregated point cloud map: We instead create the Scan
+Context descriptor by aggregating motion-compensated
+peak detections from multiple scans (Sec. III-B.2).
+T.3 Origin augmentation: Additional augmentations (origin-
+shifted copies with lateral offsets) are matched. Out of
+these, the best match is selected (Sec. III-B.2).
+T.4 Alignment loop verification: Verification includes align-
+ment quality dalign from Sec. III-D.
+T.5 Odometry decoupled: Verification includes dodom.
+T.6 Odometry coupled: dodom is embedded into the loop
+retrieval search as described in Sec. III-B.3.
+T.7 Separate verification: Instead of unified verification,
+loops are verified separately by alignment (dalign).
+T.8 Multiple candidate selection: Based on item T.6.
+Ncand = 3 competing candidates {(q, ck)} are used
+(previously 1). The most likely loop (yq,ck
+loop) is selected.
+The impact is presented in Fig. 4a for the eight Oxford
+(a) T.1
+(b) T.2
+(c) T.3
+(d) T.4
+(e) T.5
+(f) T.6
+(g) T.8
+Fig. 3: Detected loops in the Oxford dataset using the methods in Sec. IV-B. All potential loops are colored and ordered
+according to success: red (dangerous failure – False Positive), orange (safe failure – False Negative), blue (safe failure – False
+Negative), green (success – True Positive). Blue-colored loops differ compared to orange in that the suggested candidates
+are actually correct, but no loop is predicted as likelihood doesn’t exceed the decision boundary: (yq,ck
+loop < yth).
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Recall
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Precision
+oxford
+1) Radar Scan Context
+2) Aggregated point cloud map
+3) Origin augmentation
+4) Alignment loop verification
+5) Odometry decoupled
+6) Odometry coupled
+7) Cascaded classifier
+8) Multiple candidate selection
+(a) Oxford
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Recall
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Precision
+Mulran
+1) Radar Scan Context
+2) Aggregated point cloud map
+3) Origin augmentation
+4) Alignment loop verification
+5) Odometry decoupled
+6) Odometry coupled
+7) Cascaded classifier
+8) Multiple candidate selection
+(b) MulRan
+Fig. 4: Loop closure performance over all sequences.
+sequences, and Fig. 4b for the nine MulRan sequences. Loop
+detections are visualized in Fig. 3 for the Oxford sequence
+16-13-09 with yth = 0.9. The raw Scan Context ( T.1)
+achieves higher recall compared to the sparse local map (T.2).
+The difference is larger in MulRan, where scans are acquired
+in the same moving direction, and motion distortion is less
+of a challenge. Additionally, we noted that the difference
+is highest within the feature-poor Riverside sequences.
+This suggests that maintaining information-rich radar data is
+largely advantageous compared to using a sparse local map,
+especially when features are scarce. Note however that our
+local mapping technique is primarily motivated by the need
+for a sparse Cartesian point cloud for efficient augmentation.
+Oxford is more challenging compared to MulRan as
+a majority of the revisits occur in opposite road lanes
+and directions. However, the augmentation technique (T.3)
+allows the detection of additional candidates with higher
+lateral displacement, and as expected, increases the highest
+top recall, yet at a cost of lower precision. This loss of
+precision can however be alleviated via alignment loop
+verification (T.4). The improvement is larger in Oxford
+and we believe the more structured scenes are favorable
+for alignment analysis. The decoupled odometry approach
+(T.5), which extends verification by including odometry
+uncertainty, gives a higher tolerance to false positives. At this
+point, the decision boundary can be chosen such that almost
+all candidates are correctly classified. Unified verification is
+preferred over separate verification (T.7). Selecting the can-
+didate with the highest probability (T.8), rather than the first
+place recognition candidate (T.6) yields a clear improvement
+in Oxford. We believe this improvement is because our
+alignment quality aids in distinguishing between places and
+detecting registration failures, especially in structured scenes.
+C. SLAM performance - comparative evaluation
+We compare TBV Radar SLAM to previous meth-
+ods for radar, lidar, and visual SLAM within the Ox-
+ford and Mulran dataset. We primarily compare methods
+over full trajectories i.e. Absolute Trajectory Error (ATE)
+ATERMSE =
+�
+1
+n
+�i=n
+i=1 ||transl(xest
+i
+) − transl(xgt
+i ))||2.
+Additionally, we provide the KITTI odometry metric [48],
+which computes the relative error between 100-800 m,
+e.g. error over a shorter distance. ATE metrics for method
+Type/modality
+Method
+Evaluation
+10-12-32T raining
+16-13-09
+17-13-26
+18-14-14
+18-15-20
+10-11-46
+16-11-53
+18-14-46
+Mean
+ATE
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+ATE
+SLAM/camera
+ORB-SLAM2 [44]
+[45]
+7.96
+7.59
+7.61
+24.63
+12.17
+7.30
+3.54
+9.72
+10.07
+Odometry/Radar
+CFEAR-3 (odometry used)
+7.29
+23.32
+15.58
+20.95
+20.02
+16.87
+15.47
+28.58
+18.51
+SLAM/Radar
+RadarSLAM-Full [1]
+[45]
+9.59
+11.18
+5.84
+21.21
+7.74
+13.78
+7.14
+6.01
+10.31
+SLAM/Radar
+MAROAM [28]
+From author
+13.76
+6.95
+8.36
+10.34
+10.96
+12.42
+12.51
+7.71
+10.38
+SLAM/Radar
+TBV Radar SLAM-dyn-cov-T.8 (ours)
+4.22
+4.30
+3.37
+4.04
+4.27
+3.38
+4.98
+4.27
+4.10
+SLAM/Radar
+TBV Radar SLAM-T.8 (ours)
+4.07
+4.04
+3.58
+3.79
+3.83
+3.14
+4.39
+4.33
+3.90
+Drift
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Drift
+SLAM/Lidar
+SuMa (Lidar - SLAM) [46]
+[45]
+1.1/0.3p
+1.2/0.4p
+1.1/0.3p
+0.9/0.1p
+1.0/0.2p
+1.1/0.3p
+0.9/0.3p
+1.0/0.1p
+1.03/0.3p
+Odometry/Radar
+CFEAR-3-S50 [17]
+[17]
+1.05/0.34
+1.08/0.34
+1.07/0.36
+1.11/0.37
+1.03/0.37
+1.05/0.36
+1.18/0.36
+1.11/0.36
+1.09/0.36
+Odometry/Radar
+CFEAR-3 (odometry used)
+1.20/0.36
+1.24/0.40
+1.23/0.39
+1.35/0.42
+1.24/0.41
+1.22/0.39
+1.39/0.40
+1.39/0.44
+1.28/0.40
+SLAM/Radar
+RadarSLAM-Full [1]
+[45]
+1.98/0.6
+1.48/0.5
+1.71/0.5
+2.22/0.7
+1.77/0.6
+1.96/0.7
+1.81/0.6
+1.68/0.5
+1.83/0.6
+SLAM/Radar
+MAROAM-Full [28]
+[28]
+1.63/0.46
+1.83/0.56
+1.49/0.47
+1.54/0.47
+1.61/0.50
+1.55/0.53
+1.78/0.54
+1.55/0.50
+1.62/0.50
+SLAM/Radar
+TBV Radar SLAM-T.8 (ours)
+1.17/0.35
+1.15/0.35
+1.06/0.35
+1.12/0.37
+1.09/0.36
+1.18/0.35
+1.32/0.36
+1.10/0.36
+1.15/0.36
+TABLE I: Top: Absolute Trajectory error (ATE) [m], Bottom: Drift (% translation error / deg/100 m) on the Oxford
+Radar RobotCar dataset [43]. Methods marked with p finished prematurely. Methods for Radar SLAM are shadowed.
+0
+200
+400
+600
+800
+1000
+1200
+x (m)
+−800
+−600
+−400
+−200
+0
+200
+400
+y (m)
+2019-01-10-12-32-52-radar-oxford-10k
+Ground truth
+TBV SLAM-8
+CFEAR-3
+(a) 10-12-32
+0
+200
+400
+600
+800
+1000
+1200
+x (m)
+−800
+−600
+−400
+−200
+0
+200
+400
+y (m)
+2019-01-16-13-09-37-radar-oxford-10k
+Ground truth
+TBV SLAM-8
+CFEAR-3
+(b) 16-13-09
+0
+200
+400
+600
+800
+1000
+1200
+x (m)
+−800
+−600
+−400
+−200
+0
+200
+400
+y (m)
+2019-01-17-13-26-39-radar-oxford-10k
+Ground truth
+TBV SLAM-8
+CFEAR-3
+(c) 17-13-26
+0
+200
+400
+600
+800
+1000
+x (m)
+−800
+−600
+−400
+−200
+0
+200
+400
+y (m)
+2019-01-10-11-46-21-radar-oxford-10k
+Ground truth
+TBV SLAM-8
+CFEAR-3
+(d) 10-11-46
+0
+200
+400
+600
+800
+1000
+x (m)
+−800
+−600
+−400
+−200
+0
+200
+400
+y (m)
+2019-01-18-14-14-42-radar-oxford-10k
+Ground truth
+TBV SLAM-8
+CFEAR-3
+(e) 18-14-14
+0
+200
+400
+600
+800
+1000
+1200
+x (m)
+−800
+−600
+−400
+−200
+0
+200
+400
+y (m)
+2019-01-18-15-20-12-radar-oxford-10k
+Ground truth
+TBV SLAM-8
+CFEAR-3
+(f) 18-15-20
+0
+200
+400
+600
+800
+1000
+1200
+x (m)
+−800
+−600
+−400
+−200
+0
+200
+400
+y (m)
+2019-01-16-11-53-11-radar-oxford-10k
+Ground truth
+TBV SLAM-8
+CFEAR-3
+(g) 16-11-53
+0
+200
+400
+600
+800
+1000
+1200
+x (m)
+−800
+−600
+−400
+−200
+0
+200
+400
+y (m)
+2019-01-18-14-46-59-radar-oxford-10k
+Ground truth
+TBV SLAM-8
+CFEAR-3
+(h) 18-14-46
+Fig. 5: Oxford trajectories using the proposed method TBV Radar SLAM, compared with CFEAR-3 [17] (odometry only)
+and Ground truth. Initial and final pose are marked with × and □. Trajectories can be directly compared to [10], [17].
+
+Type/Modality
+Method
+Evaluation
+KAIST01
+KAIST02
+KAIST03
+DCC01
+DCC02
+DCC03
+RIV.01
+RIV.02
+RIV.03
+Mean
+ATE
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+ATE
+SLAM/Lidar
+SuMa Full [46]
+[1]
+38.70
+31.90
+46.00
+13.50
+17.80
+29.60
+-
+-
+-
+22.90
+Odometry/Lidar
+KISS-ICP (odometry) [47]
+17.40
+17.40
+17.40
+15.16
+15.16
+15.16
+49.02
+49.02
+49.02
+27.2
+Odometry/Radar
+CFEAR-3 [17] (odometry used)
+7.53
+7.58
+12.21
+6.39
+3.67
+5.40
+6.45
+3.87
+19.44
+8.06
+SLAM/Radar
+RadarSLAM Full [45]
+[1]
+6.90
+6.00
+4.20
+12.90
+9.90
+3.90
+9.00
+7.00
+10.70
+7.80
+SLAM/Radar
+MAROAM Full [28]
+[28]
+-
+-
+-
+-
+5.81
+-
+-
+4.85
+-
+-
+SLAM/Radar
+TBV SLAM-T.8-dyn-cov (ours)
+1.71
+1.42
+1.52
+5.41
+3.29
+2.61
+2.66
+2.49
+2.52
+2.63
+SLAM/Radar
+TBV SLAM-T.8-cov (ours)
+1.66
+1.39
+1.50
+5.44
+3.32
+2.66
+2.61
+2.36
+1.48
+2.49
+Drift
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Drift
+SLAM/Lidar
+SuMa Full [46]
+2.9/0.8
+2.64/0.6
+2.17/0.6
+2.71/0.4
+4.07/0.9
+2.14/0.6
+1.66/0.6P
+1.49/0.5P
+1.65/0.4P
+2.38/0.5
+Odometry/Lidar
+KISS-ICP (odometry) [47]
+2.28/0.68
+2.28/0.68
+2.28/0.68
+2.34/0.64
+2.34/0.64
+2.34/0.64
+2.89/0.64
+2.89/0.64
+2.89/0.64
+2.5/0.65
+Odometry/Radar
+CFEAR-3-s50 [17]
+[17]
+1.48/0.65
+1.51/0.63
+1.59/0.75
+2.09/0.55
+1.38/0.47
+1.26/0.47
+1.62/0.62
+1.35/0.52
+1.19/0.37
+1.50/0.56
+Odometry/Radar
+CFEAR-3 [17] (odometry used)
+1.59/0.66
+1.62/0.66
+1.73/0.78
+2.28/0.54
+1.49/0.46
+1.47/0.48
+1.59/0.63
+1.39/0.51
+1.41/0.40
+1.62/0.57
+SLAM/Radar
+RadarSLAM-Full [45]
+[1]
+1.75/0.5
+1.76/0.4
+1.72/0.4
+2.39/0.4
+1.90/0.4
+1.56/0.2
+3.40/0.9
+1.79/0.3
+1.95/0.5
+2.02/0.4
+SLAM/Radar
+TBV SLAM-T.8 (ours)
+1.01/0.30
+1.03/0.30
+1.08/0.35
+2.01/0.27
+1.35/0.25
+1.14/0.22
+1.25/0.35
+1.09/0.30
+0.99/0.18
+1.22/0.28
+TABLE II: Top: Absolute Trajectory error (ATE), Bottom: Drift (% translation error / deg/100 m) on the MulRan dataset [40].
+−300
+−200
+−100
+0
+100
+200
+300
+x (m)
+−600
+−500
+−400
+−300
+−200
+−100
+0
+100
+y (m)
+KAIST01
+Ground truth
+TBV SLAM-8
+CFEAR-3
+(a) KAIST01
+−100
+0
+100
+200
+300
+400
+500
+x (m)
+−600
+−500
+−400
+−300
+−200
+−100
+0
+y (m)
+KAIST02
+Ground truth
+TBV SLAM-8
+CFEAR-3
+(b) KAIST02
+−100
+0
+100
+200
+300
+400
+500
+600
+x (m)
+−600
+−500
+−400
+−300
+−200
+−100
+0
+y (m)
+KAIST03
+Ground truth
+TBV SLAM-8
+CFEAR-3
+(c) KAIST03
+−100
+0
+100
+200
+300
+400
+x (m)
+−500
+−400
+−300
+−200
+−100
+0
+y (m)
+DCC01
+Ground truth
+TBV SLAM-8
+CFEAR-3
+(d) DCC01
+0
+100
+200
+300
+400
+500
+x (m)
+−500
+−400
+−300
+−200
+−100
+0
+100
+y (m)
+DCC02
+Ground truth
+TBV SLAM-8
+CFEAR-3
+(e) DCC02
+0
+100
+200
+300
+400
+500
+x (m)
+−500
+−400
+−300
+−200
+−100
+0
+y (m)
+DCC03
+Ground truth
+TBV SLAM-8
+CFEAR-3
+(f) DCC03
+0
+100
+200
+x (m)
+0
+200
+400
+600
+800
+1000
+1200
+1400
+1600
+y (m)
+Riverside01
+Ground truth
+TBV SLAM-8
+CFEAR-3
+(g) RIV.01
+0
+200
+x (m)
+−500
+−250
+0
+250
+500
+750
+1000
+1250
+y (m)
+Riverside02
+Ground truth
+TBV SLAM-8
+CFEAR-3
+(h) RIV.02
+−200
+0
+x (m)
+−750
+−500
+−250
+0
+250
+500
+750
+y (m)
+Riverside03
+Ground truth
+TBV SLAM-8
+CFEAR-3
+(i) RIV.03
+(j) VolvoCE - forest
+(k) Kvarntorp - mine
+Fig. 6: TBV Radar SLAM, CFEAR-3 [17] (odometry only), and Ground truth. First (×) and final(□) pose are indicated.
+MAROAM [28] was kindly computed and provided by Wang
+et al. for this letter. We tuned our parameters on the Oxford
+sequence 10-12-32 and evaluated the performance of
+SLAM on all other Oxford and MulRan sequences.
+The estimated trajectories are depicted in Fig. 5 and
+Fig. 6(a-i). We found that TBV effortlessly closes loops
+and corrects odometry in all sequences. ATE is substantially
+corrected over the full trajectory, with slightly reduced drift
+(Tab. I & Tab. II). TBV outperforms previous methods for
+radar SLAM in terms of ATE and drift over all sequences.
+Hence, we conclude that our method improves the state
+of the art in radar SLAM. Surprisingly, we did not observe
+any improvement from using dynamic covariance (dyn-cov)
+compared to fixed. The Hessian-approximated covariance oc-
+casionally under- or over-estimates the odometry uncertainty
+[49] and thus deteriorates the optimization process.
+D. Generalization to off-road environments
+Finally, we tested TBV on the sequences Kvarntorp
+and VolvoCE from the Diverse ORU Radar Dataset [16],
+see footnote for a demo1. Kvarntorp is an underground
+mine with partly feature poor sections, while VolvoCE is a
+mixed environment with forest and open fields. Trajectories
+are visualized in Fig. 6.(j-k). We found that TBV was able
+to produce smooth and globally consistent maps, through
+1ORU dataset download: https://tinyurl.com/radarDataset.
+Demo video: https://tinyurl.com/TBV-KvarntorpVolvo.
+substantially different environments, including challenging
+road conditions – without any parameter changes.
+V. CONCLUSIONS
+We proposed TBV Radar SLAM – a real-time method
+for robust and accurate large-scale SLAM using a spin-
+ning 2D radar. We showed that loop candidate retrieval
+can be largely improved by origin-shifting, coupled place
+similarity/odometry uncertainty search, and selecting the
+most likely loop constraint as proposed by our verification
+model. A high level of loop robustness was achieved by
+carefully verifying loop constraints based on multiple sources
+of information, such as place similarity, consistency with
+odometry uncertainty, and alignment quality assessed after
+registration. We evaluated TBV on two public datasets and
+demonstrated a substantial improvement to the state of the art
+in radar SLAM, making radar an attractive option to lidar for
+robust and accurate localization. Quantitative and qualitative
+experiments demonstrated a high level of generalization
+across environments. Some findings in our ablation study
+suggest that heavy filtering is undesired as it discards details
+that are important for place recognition. Thus, in the future,
+we will explore building detailed and dense representations
+of scenes, fully utilizing the geometric information richness,
+uniquely provided by spinning FMCW radar.
+
+REFERENCES
+[1] Z. Hong, Y. Petillot, A. Wallace, and S. Wang, “RadarSLAM: A
+robust simultaneous localization and mapping system for all weather
+conditions,” IJRR, vol. 41, no. 5, pp. 519–542, 2022.
+[2] K. Burnett, Y. Wu, D. J. Yoon, A. P. Schoellig, and T. D. Barfoot,
+“Are we ready for radar to replace lidar in all-weather mapping and
+localization?” RAL, vol. 7, no. 4, pp. 10 328–10 335, 2022.
+[3] J. W. Marck, A. Mohamoud, E. vd Houwen, and R. van Heijster,
+“Indoor radar SLAM: A radar application for vision and GPS denied
+environments,” in 2013 EuRAD, 2013, pp. 471–474.
+[4] F. Schuster, C. G. Keller, M. Rapp, M. Haueis, and C. Curio,
+“Landmark based radar SLAM using graph optimization,” in ITSC,
+2016, pp. 2559–2564.
+[5] M. Holder, S. Hellwig, and H. Winner, “Real-time pose graph SLAM
+based on radar,” in IV, 2019, pp. 1145–1151.
+[6] M. Himstedt, J. Frost, S. Hellbach, H.-J. B¨ohme, and E. Maehle,
+“Large scale place recognition in 2d lidar scans using geometrical
+landmark relations,” in IROS, 2014, pp. 5030–5035.
+[7] S. H. Cen and P. Newman, “Precise ego-motion estimation with
+millimeter-wave radar under diverse and challenging conditions,” in
+ICRA, 2018, pp. 6045–6052.
+[8] S. H. Cen and P. Newman, “Radar-only ego-motion estimation in
+difficult settings via graph matching,” in ICRA, 2019, pp. 298–304.
+[9] D. Barnes, R. Weston, and I. Posner, “Masking by moving: Learning
+distraction-free radar odometry from pose information,” in CoRL, ser.
+CoRL, L. P. Kaelbling, D. Kragic, and K. Sugiura, Eds., vol. 100.
+PMLR, 30 Oct–01 Nov 2020, pp. 303–316.
+[10] Z. Hong, Y. Petillot, and S. Wang, “RadarSLAM: Radar based large-
+scale SLAM in all weathers,” in IROS, 2020, pp. 5164–5170.
+[11] D. Barnes and I. Posner, “Under the radar: Learning to predict robust
+keypoints for odometry estimation and metric localisation in radar,”
+in ICRA, 2020, pp. 9484–9490.
+[12] Y. S. Park, Y. S. Shin, and A. Kim, “PhaRaO: Direct radar odometry
+using phase correlation,” in ICRA, 2020, pp. 2617–2623.
+[13] P.-C. Kung, C.-C. Wang, and W.-C. Lin, “A normal distribution
+transform-based radar odometry designed for scanning and automotive
+radars,” 2021.
+[14] K. Burnett, A. P. Schoellig, and T. D. Barfoot, “Do we need to
+compensate for motion distortion and doppler effects in spinning radar
+navigation?” IEEE RAL, vol. 6, no. 2, pp. 771–778, 2021.
+[15] K. Burnett, D. J. Yoon, A. P. Schoellig, and T. D. Barfoot, “Radar
+odometry combining probabilistic estimation and unsupervised feature
+learning,” 2021.
+[16] D. Adolfsson, M. Magnusson, A. Alhashimi, A. J. Lilienthal, and
+H. Andreasson, “CFEAR radarodometry - conservative filtering for
+efficient and accurate radar odometry,” in IROS, 2021, pp. 5462–5469.
+[17] D. Adolfsson, M. Magnusson, A. Alhashimi, A. J. Lilienthal, and
+H. Andreasson, “Lidar-level localization with radar? The CFEAR
+approach to accurate, fast and robust large-scale radar odometry in
+diverse environments,” T-RO, pp. 1–20, 2022.
+[18] R. Aldera, M. Gadd, D. De Martini, and P. Newman, “What goes
+around: Leveraging a constant-curvature motion constraint in radar
+odometry,” RAL, vol. 7, no. 3, pp. 7865–7872, 2022.
+[19] R. Weston, M. Gadd, D. De Martini, P. Newman, and I. Posner, “Fast-
+mbym: Leveraging translational invariance of the fourier transform for
+efficient and accurate radar odometry,” in ICRA, 2022, pp. 2186–2192.
+[20] S. Saftescu, M. Gadd, D. De Martini, D. Barnes, and P. Newman,
+“Kidnapped radar: Topological radar localisation using rotationally-
+invariant metric learning,” in ICRA, 2020, pp. 4358–4364.
+[21] D. De Martini, M. Gadd, and P. Newman, “kRadar++: Coarse-to-fine
+FMCW scanning radar localisation,” Sensors, vol. 20, no. 21, 2020.
+[22] D. Williams, M. Gadd, D. De Martini, and P. Newman, “Fool Me
+Once: Robust Selective Segmentation via Out-of-Distribution Detec-
+tion with Contrastive Learning,” 2021.
+[23] M. Gadd, D. De Martini, and P. Newman, “Look around you:
+Sequence-based radar place recognition with learned rotational invari-
+ance,” in 2020 IEEE/ION (PLANS), 2020, pp. 270–276.
+[24] H. Yin, X. Xu, Y. Wang, and R. Xiong, “Radar-to-lidar: Heterogeneous
+place recognition via joint learning,” Frontiers in Robotics and AI,
+2021.
+[25] H. Yin, R. Chen, Y. Wang, and R. Xiong, “Rall: End-to-end radar
+localization on lidar map using differentiable measurement model,”
+IEEE TITS, vol. 23, no. 7, pp. 6737–6750, 2022.
+[26] T. Y. Tang, D. De Martini, D. Barnes, and P. Newman, “RSL-net:
+Localising in satellite images from a radar on the ground,” RAL, vol. 5,
+no. 2, pp. 1087–1094, 2020.
+[27] T. Y. Tang, D. D. Martini, S. Wu, and P. Newman, “Self-supervised
+learning for using overhead imagery as maps in outdoor range sensor
+localization,” IJRR, vol. 40, no. 12-14, pp. 1488–1509, 2021.
+[28] D. Wang, Y. Duan, X. Fan, C. Meng, J. Ji, and Y. Zhang, “Maroam:
+Map-based radar slam through two-step feature selection,” 2022.
+[Online]. Available: https://arxiv.org/abs/2210.13797
+[29] L. He, X. Wang, and H. Zhang, “M2dp: A novel 3d point cloud
+descriptor and its application in loop closure detection,” in IROS.
+IEEE, 2016, pp. 231–237.
+[30] N. Akai, Y. Akagi, T. Hirayama, T. Morikawa, and H. Murase, “De-
+tection of localization failures using markov random fields with fully
+connected latent variables for safe lidar-based automated driving,” T-
+ITS, vol. 23, no. 10, pp. 17 130–17 142, 2022.
+[31] R. Aldera, D. D. Martini, M. Gadd, and P. Newman, “What could go
+wrong? introspective radar odometry in challenging environments,” in
+2019 IEEE (ITSC), 2019, pp. 2835–2842.
+[32] D. Adolfsson, M. Castellano-Quero, M. Magnusson, A. J. Lilienthal,
+and H. Andreasson, “CorAl: Introspection for robust radar and lidar
+perception in diverse environments using differential entropy,” RAS,
+p. 104136, 2022.
+[33] J. Behley and C. Stachniss, “Efficient surfel-based SLAM using 3D
+laser range data in urban environments,” in Robotics: Science and
+Systems, vol. 2018, 2018, p. 59.
+[34] S. Nobili, G. Tinchev, and M. Fallon, “Predicting alignment risk to
+prevent localization failure,” in ICRA, 2018, pp. 1003–1010.
+[35] M. Castellano Quero, T. P. Kucner, and M. Magnusson, “Alignability
+maps for ensuring high-precision localization,” in 13th IROS Workshop
+on Planning, Perception, Navigation for Intelligent Vehicles.
+[36] C. E. Denniston, Y. Chang, A. Reinke, K. Ebadi, G. S. Sukhatme,
+L. Carlone, B. Morrell, and A.-a. Agha-mohammadi, “Loop closure
+prioritization for efficient and scalable multi-robot SLAM,” 2022.
+[Online]. Available: https://arxiv.org/abs/2205.12402
+[37] H. Almqvist, M. Magnusson, T. P. Kucner, and A. J. Lilienthal,
+“Learning to detect misaligned point clouds,” JFR, vol. 35, no. 5,
+pp. 662–677, 2018.
+[38] G. Kim and A. Kim, “Scan context: Egocentric spatial descriptor for
+place recognition within 3D point cloud map,” in IROS, Oct. 2018.
+[39] G. Kim, S. Choi, and A. Kim, “Scan context++: Structural place recog-
+nition robust to rotation and lateral variations in urban environments,”
+IEEE Transactions on Robotics, vol. 38, no. 3, pp. 1856–1874, 2022.
+[40] G. Kim, Y. S. Park, Y. Cho, J. Jeong, and A. Kim, “Mulran:
+Multimodal range dataset for urban place recognition,” in ICRA, Paris,
+May 2020.
+[41] P. Agarwal, G. D. Tipaldi, L. Spinello, C. Stachniss, and W. Burgard,
+“Robust map optimization using dynamic covariance scaling,” in 2013
+IEEE ICRA, 2013, pp. 62–69.
+[42] N. S¨underhauf and P. Protzel, “Switchable constraints for robust pose
+graph SLAM,” in IROS, 2012, pp. 1879–1884.
+[43] D. Barnes, M. Gadd, P. Murcutt, P. Newman, and I. Posner, “The
+oxford radar robotcar dataset: A radar extension to the oxford robotcar
+dataset,” in ICRA, 2020, pp. 6433–6438.
+[44] R. Mur-Artal and J. D. Tard´os, “ORB-SLAM2: An open-source SLAM
+system for monocular, stereo, and RGB-D cameras,” IEEE T-RO,
+vol. 33, no. 5, pp. 1255–1262, 2017.
+[45] Z. Hong, Y. Petillot, A. Wallace, and S. Wang, “Radar SLAM: A
+robust SLAM system for all weather conditions,” 2021.
+[46] J. Behley and C. Stachniss, “Efficient surfel-based SLAM using 3D
+laser range data in urban environments,” in Systems (RSS), 2018.
+[47] I. Vizzo, T. Guadagnino, B. Mersch, L. Wiesmann, J. Behley,
+and
+C.
+Stachniss,
+“KISS-ICP:
+In
+Defense
+of
+Point-to-Point
+ICP – Simple, Accurate, and Robust Registration If Done the
+Right
+Way,”
+arXiv
+preprint,
+2022.
+[Online].
+Available:
+https:
+//arxiv.org/pdf/2209.15397.pdf
+[48] A. Geiger, P. Lenz, and R. Urtasun, “Are we ready for autonomous
+driving? the KITTI vision benchmark suite,” in Pattern Recognition
+(CVPR), 2012.
+[49] D. Landry, F. Pomerleau, and P. Gigu`ere, “Cello-3d: Estimating the
+covariance of icp in the real world,” in ICRA, 2019, pp. 8190–8196.
+
diff --git a/xdE3T4oBgHgl3EQfOwnl/content/tmp_files/load_file.txt b/xdE3T4oBgHgl3EQfOwnl/content/tmp_files/load_file.txt
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@@ -0,0 +1,1476 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf,len=1475
+page_content='TBV Radar SLAM – trust but verify loop candidates  Daniel Adolfsson, Mattias Karlsson, Vladim´ır Kubelka, Martin Magnusson, Henrik Andreasson  Abstract— Robust SLAM in large-scale environments re-  quires fault resilience and awareness at multiple stages, from  sensing and odometry estimation to loop closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' In this work,  we present TBV (Trust But Verify) Radar SLAM, a method  for radar SLAM that introspectively verifies loop closure candi-  dates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' TBV Radar SLAM achieves a high correct-loop-retrieval  rate by combining multiple place-recognition techniques: tightly  coupled place similarity and odometry uncertainty search, cre-  ating loop descriptors from origin-shifted scans, and delaying  loop selection until after verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Robustness to false con-  straints is achieved by carefully verifying and selecting the most  likely ones from multiple loop constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Importantly, the  verification and selection are carried out after registration when  additional sources of loop evidence can easily be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We  integrate our loop retrieval and verification method with a fault-  resilient odometry pipeline within a pose graph framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' By  evaluating on public benchmarks we found that TBV Radar  SLAM achieves 65% lower error than the previous state of the  art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We also show that it’s generalizing across environments  without needing to change any parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' INTRODUCTION  Robust localization is key to enabling safe and reliable  autonomous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Achieving robustness requires careful  design at multiple stages of a localization pipeline, from  environment-tolerant sensing and pose estimation, to place  recognition and pose refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' At each stage, a localization  and mapping pipeline should be designed for fault awareness  to detect failures and fault resilience to mitigate failures as  they inevitably occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Today, active exteroceptive sensors  such as lidar and radar are suitable when robust uninterrupted  localization is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Of these two, radar perception is  significantly less affected when operating within dusty envi-  ronments or under harsh weather conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' It is currently  debated how the sensing properties affect localization and  mapping performance [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Compared to lidar, limited  work focuses on robust and accurate radar SLAM, and none  of the existing methods include introspective fault detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  In this letter, we propose TBV (Trust But Verify) Radar  SLAM – a 2D pose-graph localization and mapping pipeline  which integrates fault resilience and fault-aware robustness  at multiple stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' A radar-only odometry front-end adds  pose nodes to the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' In parallel, a robust loop-detection  module adds loop closure constraints such that the SLAM  back-end can optimize the graph to correct drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' TBV Radar  SLAM uses radar odometry (further only odometry), place  descriptors, and scan alignment measures to retrieve, verify,  and select between loop constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We achieve a high  correct-loop-retrieval rate by combining: a tightly coupled  The authors are with the MRO lab of the AASS research centre at ¨Orebro  University, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' E-mail: firstname.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='lastname@oru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='se  This work has received funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  Verified  VERIFY LOOP CANDIDATES  Trusted  ODOMETRY  ALIGNMENT  SCAN CONTEXT  Ground Truth,        Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Trajectory,        Loop Closure  Query  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 1: Overview and demonstration of TBV Radar SLAM  https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='com/TBVRadarSLAM  place similarity and odometry uncertainty search, creating  place descriptors computed from origin-shifted scans, and by  delaying loop selection until after verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Robustness,  with a high loop detection rate, is achieved by unifying  the process of place recognition and verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Rather  than rejecting candidates early during place recognition, we  register and verify multiple loop constraints in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Our  verification combines place similarity, odometry consistency,  and an alignment quality assessment automatically learned  from odometry and scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We integrate our loop retrieval and  verification module with a robust method for radar odometry,  into a full-fledged SLAM pipeline visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' The  contributions of this letter are:  A combination of techniques for a high rate of correct  loop retrievals, including: coupled place similarity and  odometry uncertainty search, creating place descriptors  from origin-shifted scans, and selection between multi-  ple competing loop constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  A unified loop retrieval and verification step that jointly  considers place similarity, odometry uncertainty, and  alignment quality after registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Multiple loop can-  didates are retrieved, registered, and verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  We integrate these techniques with a robust odometry  estimator into a SLAM framework that pushes state  of the art in radar SLAM accuracy while generalizing  between environments without parameter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' RELATED WORK  Early methods for radar SLAM employ filtering [3] and  landmark graph SLAM [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We instead use pose-graph  SLAM based on odometry and loop constraints obtained by  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='04397v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='RO]  11 Jan 2023  Loop retrieval and verification (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='B & D)  Place recognition (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='B)  Pose graph optimization  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='E)  Verification of loop candidates (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='D)  Loop  registration  Odometry estimation (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='A)  CFEAR  Radar odometry  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='A)  Loop  verification  Pose graph  Loop  constraints  Odometry  constraints  Optimizer  Descriptor  generation  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='2)  Radar  Scan Context  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='1&3)  loop  candidates  Registered  Scan  pairs  Learning of  alignment  verification  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='C)  Input radar data  Radar scans  scans database  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 2: Overview of TBV Radar SLAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' The main contribution is the Loop retrieval and verification module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  registering scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Holder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' [5] proposed a pose graph  SLAM based on automotive radar, using GLARE [6] for  place recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Their method uses gyroscope and wheel  odometry to provide an initial alignment guess to Iterative  Closest Point (ICP) for scan matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Our method uses a  pose graph back-end but relies solely on a spinning radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  Recent progress in the development of spinning 2D radar  has quickly inspired numerous works on radar odometry  estimation [1], [7]–[19], place recognition [11], [20]–[25],  topological localization [2], [21], [25] localization in over-  head imagery [26], [27] and SLAM [1], [10], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Most  similar to ours is the preliminary work by Wang et a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' [28],  and the seminal work on radar SLAM by Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' [1] estimates odometry and registers loop con-  straints using KLT tracker keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' For place recognition,  they use adaptive thresholding to filter data, and M2DP [29]  to compute and match descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' The trajectory is corrected  using pose graph optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Our work brings a larger  focus on the introspective verification of loop constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  Martini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' [21] proposed a teach-and-repeat localization  framework using a hierarchical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' First, a place  candidate is retrieved via nearest neighbor search within  a learned metric space using the method by Saftescu et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Second, sensor pose is estimated (without the need  for an initial guess) by finding and registering the (globally)  optimal set of landmark matches as described by Cen et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Unlike [21], we use a fast local registration method,  motivated by the access of an initial alignment guess from  the place recognition module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' However, we carefully verify  registration success before accepting new loop constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  Verification of pose estimates is essential for safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=',  such tests could have prevented a reported accident that  occurred during an automated driving test [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Holder  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' [5] verify the detected loop candidates using radar  data by the condition that ICP residuals cannot exceed a  set threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Rather than verifying loops as a final step  (separated from loop detection), we unify loop retrieval  and verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Some works [21], [31] use the landmark  matching compatibility matrix [7] to assess quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Martini  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' [21] reject place candidates based on the quality  score of the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Aldera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' [31] learn detection of  pose estimate failures from features extracted from the  eigenvectors of the compatibility matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Training labels are  provided by an external ground truth system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We build on  the method CorAl [32] to detect false loop candidates or  constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' In the lidar domain, Behley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' [33] accept  only reappearing loop candidates that are consistent with  odometry over multiple consecutive scans, we instead focus  on verification using only individual scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  Some methods attempt to measure how observations  would constrain registration in the current scene;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' a low  level of constraints in one direction suggests registration  could be unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' The measure has been used to predict the  risk of registration failure [34], [35], abstain from closing  loops when risk is high [1], or prioritize the inclusion of  loop closures accordingly [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We instead use odometry  uncertainty as a prior, which we combine with additional  loop evidence computed after registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Finally, a range of  methods aims at verifying pose estimates by detecting point  cloud misalignment [30], [32], [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We fuse misalignment  detection [32] together with place similarity and odometry  uncertainty to achieve robust unified loop detection and  verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' TBV RADAR SLAM  An overview of TBV Radar SLAM is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  In this section, we detail the components including CFEAR  Radar Odometry (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' III-A), place recognition (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' III-  B), verification of loop candidates (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' III-D), and pose  graph optimization (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' III-E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' The self-supervised training  of alignment verification (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' III-C) runs once during a  learning phase and is not required for online SLAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' CFEAR Radar Odometry  We use the radar odometry pipeline CFEAR Radar Odom-  etry [17] (specifically the CFEAR-3 configuration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' This  method takes raw polar radar sweeps as input and estimates  sensor pose and velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' As an intermediate step, the method  computes a set of radar peaks P t and a set of oriented surface  points Mt at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' These representations will be referred  to in the later stages of our pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' CFEAR filters the radar  data by keeping the k-strongest range bins per azimuth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' The  filtered point set is compensated for motion distortion and  transformed into Cartesian space, - this is also the point set  from which we extract radar peaks (P t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' A grid method is  then used to compute a set of oriented surface points Mt  from the filtered set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Odometry is estimated by finding the  pose xt in SE(2) which minimizes the sum of surface point  distances between the current scan Mt and the |K| latest  keyframes MK jointly as  f(MK, Mt, xt) =  �  k∈K  �  ∀{i,j}∈Ccorr  wi,jρ  �  g(mk  j , mt  i, xt)  �  , (1)  where wi,j are surface point similarity weights, ρ is the  Huber loss function, and g is the pairwise distance between  surface points in the correspondence set Ccorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' A keyframe  k ∈ K is created when the distance to the previous exceeds  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='5 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' This technique reduces drift and removes excessive  scans acquired when the robot remains stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Upon  creation of a new keyframe, odometry constraints are created  from the relative alignment between the current pose and the  latest keyframe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Odometry constraints are added to Codom  and used to correct trajectory (via pose graph optimization)  as detailed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' III-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Place recognition  We attempt to unify the stages of loop closure, from  detection to constraint verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Accordingly, we retrieve,  register, verify multiple candidates, of which one can is  selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' For that reason, we do not discard potential loop  candidates based on place similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Instead, we trust multi-  ple candidates to be meaningful until verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' At this point,  we select the verifiably best loop candidate, if such exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  1) Scan Context: We build upon the place recognition  method Scan Context by Giseop Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' [38], [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Their  method detects loops and relative orientation by matching  the query (q) scan with candidates ({c}) stored in a database  using scan descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Scenes are encoded by their polar  representations into 2D descriptors Iring×sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' The 2D de-  scriptor is in turn used to create a rotation-invariant 1D  descriptor (ring key) via a ring encoding function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Loops  are detected via a two-step search: First, the query 1D  ring key is matched with the top 10 candidates via a fast  nearest neighbor (NN) search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Second, for each of the top  candidates, a sparse optimization finds the relative rotation  that minimizes a distance metric dsc(Iq, Ic): the sum of  cosine distances between descriptors columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' The candidate  c with the lowest score (dsc) which does not exceed the fixed  threshold τ is selected as loop candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  c = argmin  c∈C  dsc(Iq, Ic), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' dsc < τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  (2)  In  our  implementation,  query  descriptors  are  created,  matched, and stored in the database for each keyframe rather  than for each scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  2) Descriptor generation: As described in [28], [40],  a raw polar representation such as those produced by a  spinning 2D radar, can be used directly as a Scan Context  descriptor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' However, we believe that doing so poses multiple  challenges, including sensor noise, motion distortion, scene  dynamics and translation sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Thus, we create our  descriptor from multiple filtered and motion-compensated  scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Conveniently, such processed scans are already pro-  vided by the CFEAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We aggregate the peak detections from  keyframe q and its two neighbours in the odometry frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  Having the radar scan represented as a sparse point cloud  in Cartesian space allows us to address translation sensitivity  in place recognition by applying the data augmentation step  (Augmented PC) from [39] before computing place descrip-  tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We perform data augmentation by shifting the sensor  origin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' by transforming P ˆq with ±2 and ±4 m lateral  translation offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' The original, and the 4 augmented point  clouds, are each used to compute and match descriptors, after  which the best matching pair of query/candidate is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  Note that by using the aggregated sparse point cloud, rather  than the dense raw radar scan, we can efficiently compute all  augmentations and corresponding descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' As such, the  main computational load from the augmentation technique  is due to matching of additional descriptors and not the  computation of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' The descriptor itself is created by  populating the Scan Context I with radar intensity readings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  Specifically, for each grid cell I(i, j) we sum the intensity  of all coinciding point intensities (of radar peaks) divided by  1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Empty cells are set to I(i, j) = −1, which we found  increased descriptiveness compared to I(i, j) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  3) Coupled odometry/appearance matching: When re-  trieving loop candidates, odometry information can be used  to filter unlikely candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' This could be done by rejecting  unlikely loop constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' For example, if the likelihood  of the loop constraint xq,c  loop (given the estimated odometry  trajectory vc:q  odom between c and q ) is close to zero:  p(xq,c  loop |vc:q  odom) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  (3)  While this strategy may provide higher tolerance to spatial  aliasing by rejecting false positives, it does not provide  means to detect the correct candidate under such circum-  stances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' For that reason, we propose a coupled place similar-  ity / odometry uncertainty search, which combines Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 2 and  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Candidates are thus selected jointly by the similarity of  appearance dsc(Iq, Ic) and the similarity of odometry dq,c  odom:  c = argmin  c∈C  dsc(Iq, Ic) + dq,c  odom,  dq,c  odom = 1 − p(xq,c  loop |vc:q  odom) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  (4)  We estimate p(xq,c  loop |vc:q  odom) = exp (− t2  err  2σ2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  terr = max(||transl(xq) − transl(xc)|| − ϵ, 0)  dist(vc:q  odom)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  (5)  Here, transl is the translation component, ϵ is the expected  maximum spacing between loop candidates (fixed to ϵ = 5  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' slightly higher than the lateral augmentation distance),  and dist(vc:q  odom) is the traversed distance between the query  and loop candidate estimated by the odometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Note that  terr quantifies the relative final position error, thus σ can be  chosen according to expected odometry quality to penalize  unlikely loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We refrained, however, from making strong  assumptions on odometry quality, and fixed σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=', a  pessimistic assumption of 5% relative translation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  Note that the two-step search of Scan Context requires  that odometry uncertainty is integrated already in the 1D  NN search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We do this by extending all 1D descriptors (of  size ring = 40) with odometry similarity scores (dodom) as  an extra element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' (dodom) is scaled with a factor (ring/4)  to balance odometry uncertainty and appearance similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Automatic learning of alignment verification  To improve loop closure verification, we build upon the  system CorAl [32] which learns to detect alignment errors  between two registered scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' CorAl allows us to determine  if a loop constraint is correct by formulating loop constraint  verification as a misalignment detection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Specifi-  cally, a loop (between scan nr q and c) is verified as correct  only if the scans are correctly aligned via the relative align-  ment xq,c  loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' During the learning phase, CorAl automatically  generates labeled training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' The input to CorAl is pairs of  odometry estimates, radar peak detections (P), and computed  sets of oriented surface points (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' These entities are used  to extract alignment quality residuals from which alignment  quality can be assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' After the learning phase, CorAl can  verify loops by detecting alignment errors (caused e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' by  heavy motion distortion or incorrect convergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' CorAl  also aids in distinguishing between places that differ by small  geometric details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We generate training data by repeating the  following process for each pair of consecutive keyframes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  Positive training labels (yaligned = true) and training data  Xquality are computed using the scan alignment provided by  the odometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' For each pair, the alignment quality measures  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 6 are extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Negative training labels (yaligned =  false) and training data are extracted similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' However,  before extracting the alignment quality, an error is induced in  the alignment in both translation and rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' This allows us  to learn the detection of different types of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Specifically,  we distribute 12 translation errors symmetrically in either the  positive or negative x or y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We use 4 small (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='5 m), 4  medium (±1 m) and 4 large (±2 m) errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' To these errors,  we additionally induce a clockwise rotation with matching  rotation errors: small (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='5◦), medium (2◦) or large (15◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  Note that the class ratio 1:12, between positive to negative  training is alleviated during learning by assigning weights  according to the inverse of class frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  1) Alignment measure: We extract the following align-  ment measures between each pair of scans:  Xquality = [Hj Hs Ho Cf Co Ca 1]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  (6)  The joint entropy (Hj) and separate entropy (Hs) are average  per-point differential entropies, extracted from point cloud  pairs of radar peak detections (Pq, Pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' These metrics are  described in-depth in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We complement these measures  with a measure of overlap Ho: (Hoverlap), defined as the  portion of peaks in Pq or Pc with a neighboring point within  the radius r in the other point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  In this work, we combine these CorAl measures with  additional ones, extracted from (Mq, Mc), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' from pairs of  scans represented as oriented surface points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' The measures  are obtained from the registration cost(Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 1), but with a  single keyframe and with the point-to-line cost function  (gP 2L [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Note that these measures are already computed  during the final iteration of the registration, and this step  brings little computational overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Specifically, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 1  we reuse Cf: f(Mq, Mc, xq,c), the number of correspon-  dences (overlap) Co: |Ccorr|, and average surface point set  size Ca: 1/2(|Mq| + |Mc|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' The intuition of combining  these quality measures is that the CorAl measures (which  use small-region data association) are specialized in detecting  small errors whereas the CFEAR measures are more suitable  for larger alignment errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We refer to [32] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  2) Assessing alignment: Once training data has been  computed, we train a logistic regression classifier  palign = 1/(1 + e−dalign),  (7a)  dalign = βXquality,  (7b)  where β1×7 are the learned model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We train  on discrete alignment classification here as we consider all  visible errors to be undesired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' However, dalign is passed to  our loop verification module rather than palign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We found  dalign to be more suitable, as the sigmoid output palign is  insensitive to alignment change close to 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Verification of loop candidates  We allow for multiple competing loop candidates ck per  query q as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Each of the Ncand = 3  best matching pairs {(q, ck)} provided by the place recog-  nition module is used to compute and verify potential loop  constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' A constraint is computed by finding the relative  alignment xq,ck  loop that minimizes Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' the distance be-  tween correspondences, similarly to the odometry module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  As an initial guess, we use the relative orientation provided  by the place recognition module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' If the loop candidate was  retrieved from a match with an augmented query scan, the  corresponding augmented lateral offset is used together with  the rotation as an initial guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Note that the local registration  method is motivated by the access to an initial guess, required  for convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' After registration, we extract and assess  the alignment quality dalign = βXq,ck  quality following the  procedure in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' (III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='1&III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Each constraint is finally  verified by combining the Scan Context distance (dsc) with  odometry uncertainty (dodom) and alignment quality (dalign)  with a logistic regression classifier  yq,ck  loop =  1  1 + e−ΘX  q,ck  loop , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' yq,ck  loop > yth,  Xq,ck  loop = [dodom dsc dalign 1]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  (8)  The model parameters Θ can be learned via ground truth  loop labels, or simply tuned as the 4 parameters have intuitive  meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' yth is the sensitivity threshold – we rarely observe  false positives when fixed to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  We investigated two strategies for selecting loop closures  after a successful verification: (i) We select the first candidate  retrieved from the place recognition module (Ncand = 1)  – the lowest Scan Context distance score dsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' (ii) We use  Ncand = 3 candidates and select the best candidate according  to our verifier – the highest probability yq,ck  loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' The intuition  for strategy (i) is that the first retrieved place candidate is of-  ten the best guess, without considering registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' However,  there are cases where one of the latter candidates is preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  For example, registration of query and the first candidate  may fail, or subtle scene differences that distinguish places  can be hard to detect until a more thorough local analysis of  alignment has been carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Thus, selecting the verifiably  better loop constraint candidate is desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We compare these  two strategies in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' IV-B ( T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='6 and  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='8) Once a loop  constraint has been verified, the loop is added to Cloop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Pose Graph Optimization  We correct the odometry by solving a sparse least squares  optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We do so by minimizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 9, using  the odometry and loop constraints Codom, Cloop:  J(Y) =  �  (i,j)∈Codom  eT  i,jC−1  odomei,j  �  ��  �  odometry constraints  +  �  (i,j)∈Cloop  ρ(eT  i,jC−1  loopei,j)  �  ��  �  loop constraints  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  (9)  Y = [y1 y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='yn] is the vector of optimization parameters,  e=yi,j − xi,j is the difference between parameter and con-  straint, C is the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We used two strategies to  compute covariance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' fixed: computed from registration error  statistics C = diag([vxx = 1e-2, vyy = 1e-2, vθθ = 1e-3]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  dynamic: obtained from the Hessian, approximated from the  Jacobian (C = (H)−1 ≈ (JT J)−1) of the registration cost  (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Note that the dynamically obtained C was tuned by  a factor γ to provide realistic uncertainties, discussed in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  Loop constraints are scaled by an additional optional factor  for (5e-5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' This factor retains the high odometry quality  through pose graph optimization, and alleviates the need for  a more accurate but computationally more expensive multi-  scan loop registration  Finally,  we  solve  argminY J(Y)  using  Levenberg-  Marquardt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Note that we do not need a robust back-end  to mitigate outlier constraints (such as dynamic covariance  scaling [41] or switchable loop constraints [42]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' EVALUATION  We evaluate our method on the Oxford [43] and  MulRan [40] datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Both datasets were collected by  driving a car with a roof-mounted radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' The Oxford dataset  contains 30 repetitions of a 10 km urban route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' The MulRan  dataset has a wider mix of routes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' from structured urban to  partly feature poor areas such as open fields and bridge cross-  ings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' In MulRan, places are generally revisited in the same  driving direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' This is not the case in Oxford – where  loop closure is significantly harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' From these datasets,  we selected the previously most evaluated sequences, see  Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' (I&II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' The radars used are Navtech CTS350-X, con-  figured with 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='38 cm resolution in the Oxford dataset, and  CIR204-H, with 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='9 cm resolution in the MulRan dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  We use standard parameters for CFEAR-3 [17], CorAl [32],  and Scan Context [38], except where explicitly mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Run-time performance  After the initial generation of training data, the full  pipeline including odometry and loop closure runs in real-  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Pose graph optimization runs in a separate thread, either  continuously as new verified loop constraints are computed,  or once at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Run-time performance is as follows (mea-  sured on an i7-6700K CPU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Odometry: 37 ms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Generation of  training data: 236 ms/keyframe pair;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Pose graph optimization  (once in the end with only odometry as prior): 992 ms @ 4k  keyframes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Loop closure (128 ms @ Ncand = 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Descriptor  generation: 1 ms/keyframe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Detect loops: 25 ms/keyframe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  Registration: 6 ms/candidate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Verification 19 ms/candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Ablation study – loop detection  In this section, we evaluate the effect of the various aspects  of our loop detection pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Loops are classified as correct  if the difference between estimated alignment and ground  truth does not exceed 4 m or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' This error limit was set  slightly higher than the largest pose error that we found in  the ground truth at loop locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' While we would like to  demonstrate the full capability of our system by analysis  of smaller errors, we found ground truth accuracy to be  a limiting factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Note that the ground truth quality has  previously been discussed as a limitation [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  We compare the impact of each technique summarized  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Note that a later technique in the list either adapts or  replaces the former technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='1 Radar Scan Context: Polar radar data is, without pre-  processing, directly downsampled into a Scan Context  descriptor [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We found the OpenCV interpolation op-  tion INTER AREA to yield the best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Verification  includes only place similarity dsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='2 Aggregated point cloud map: We instead create the Scan  Context descriptor by aggregating motion-compensated  peak detections from multiple scans (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='3 Origin augmentation: Additional augmentations (origin-  shifted copies with lateral offsets) are matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Out of  these, the best match is selected (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='4 Alignment loop verification: Verification includes align-  ment quality dalign from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' III-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='5 Odometry decoupled: Verification includes dodom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='6 Odometry coupled: dodom is embedded into the loop  retrieval search as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='7 Separate verification: Instead of unified verification,  loops are verified separately by alignment (dalign).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='8 Multiple candidate selection: Based on item T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  Ncand = 3 competing candidates {(q, ck)} are used  (previously 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' The most likely loop (yq,ck  loop) is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  The impact is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 4a for the eight Oxford  (a) T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='1  (b) T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='2  (c) T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='3  (d) T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='4  (e) T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='5  (f) T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='6  (g) T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='8  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 3: Detected loops in the Oxford dataset using the methods in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' IV-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' All potential loops are colored and ordered  according to success: red (dangerous failure – False Positive), orange (safe failure – False Negative), blue (safe failure – False  Negative), green (success – True Positive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Blue-colored loops differ compared to orange in that the suggested candidates  are actually correct, but no loop is predicted as likelihood doesn’t exceed the decision boundary: (yq,ck  loop < yth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='0  Recall  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='0  Precision  oxford  1) Radar Scan Context  2) Aggregated point cloud map  3) Origin augmentation  4) Alignment loop verification  5) Odometry decoupled  6) Odometry coupled  7) Cascaded classifier  8) Multiple candidate selection  (a) Oxford  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='0  Recall  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='0  Precision  Mulran  1) Radar Scan Context  2) Aggregated point cloud map  3) Origin augmentation  4) Alignment loop verification  5) Odometry decoupled  6) Odometry coupled  7) Cascaded classifier  8) Multiple candidate selection  (b) MulRan  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 4: Loop closure performance over all sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  sequences, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 4b for the nine MulRan sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Loop  detections are visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 3 for the Oxford sequence  16-13-09 with yth = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' The raw Scan Context ( T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='1)  achieves higher recall compared to the sparse local map (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  The difference is larger in MulRan, where scans are acquired  in the same moving direction, and motion distortion is less  of a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Additionally, we noted that the difference  is highest within the feature-poor Riverside sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  This suggests that maintaining information-rich radar data is  largely advantageous compared to using a sparse local map,  especially when features are scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Note however that our  local mapping technique is primarily motivated by the need  for a sparse Cartesian point cloud for efficient augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  Oxford is more challenging compared to MulRan as  a majority of the revisits occur in opposite road lanes  and directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' However, the augmentation technique (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='3)  allows the detection of additional candidates with higher  lateral displacement, and as expected, increases the highest  top recall, yet at a cost of lower precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' This loss of  precision can however be alleviated via alignment loop  verification (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' The improvement is larger in Oxford  and we believe the more structured scenes are favorable  for alignment analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' The decoupled odometry approach  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='5), which extends verification by including odometry  uncertainty, gives a higher tolerance to false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' At this  point, the decision boundary can be chosen such that almost  all candidates are correctly classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Unified verification is  preferred over separate verification (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Selecting the can-  didate with the highest probability (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='8), rather than the first  place recognition candidate (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='6) yields a clear improvement  in Oxford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We believe this improvement is because our  alignment quality aids in distinguishing between places and  detecting registration failures, especially in structured scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' SLAM performance - comparative evaluation  We compare TBV Radar SLAM to previous meth-  ods for radar, lidar, and visual SLAM within the Ox-  ford and Mulran dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We primarily compare methods  over full trajectories i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Absolute Trajectory Error (ATE)  ATERMSE =  �  1  n  �i=n  i=1 ||transl(xest  i  ) − transl(xgt  i ))||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  Additionally, we provide the KITTI odometry metric [48],  which computes the relative error between 100-800 m,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' error over a shorter distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' ATE metrics for method  Type/modality  Method  Evaluation  10-12-32T raining  16-13-09  17-13-26  18-14-14  18-15-20  10-11-46  16-11-53  18-14-46  Mean  ATE  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  ATE  SLAM/camera  ORB-SLAM2 [44]  [45]  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='96  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
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+page_content='61  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
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+page_content='17  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='30  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='54  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='72  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='07  Odometry/Radar  CFEAR-3 (odometry used)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='29  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='32  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='58  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='95  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='02  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
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+page_content='58  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='51  SLAM/Radar  RadarSLAM-Full [1]  [45]  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
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+page_content='31  SLAM/Radar  MAROAM [28]  From author  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='76  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
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+page_content='38  SLAM/Radar  TBV Radar SLAM-dyn-cov-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='8 (ours)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
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+page_content='10  SLAM/Radar  TBV Radar SLAM-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='8 (ours)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
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+page_content='  Type/Modality  Method  Evaluation  KAIST01  KAIST02  KAIST03  DCC01  DCC02  DCC03  RIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
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+page_content='03  Mean  ATE  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  ATE  SLAM/Lidar  SuMa Full [46]  [1]  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
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+page_content='(g) RIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
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+page_content='03  (j) VolvoCE - forest  (k) Kvarntorp - mine  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 6: TBV Radar SLAM, CFEAR-3 [17] (odometry only), and Ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' First (×) and final(□) pose are indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  MAROAM [28] was kindly computed and provided by Wang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' for this letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We tuned our parameters on the Oxford  sequence 10-12-32 and evaluated the performance of  SLAM on all other Oxford and MulRan sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  The estimated trajectories are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 5 and  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 6(a-i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We found that TBV effortlessly closes loops  and corrects odometry in all sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' ATE is substantially  corrected over the full trajectory, with slightly reduced drift  (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' I & Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' TBV outperforms previous methods for  radar SLAM in terms of ATE and drift over all sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  Hence, we conclude that our method improves the state  of the art in radar SLAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Surprisingly, we did not observe  any improvement from using dynamic covariance (dyn-cov)  compared to fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' The Hessian-approximated covariance oc-  casionally under- or over-estimates the odometry uncertainty  [49] and thus deteriorates the optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Generalization to off-road environments  Finally, we tested TBV on the sequences Kvarntorp  and VolvoCE from the Diverse ORU Radar Dataset [16],  see footnote for a demo1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Kvarntorp is an underground  mine with partly feature poor sections, while VolvoCE is a  mixed environment with forest and open fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Trajectories  are visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='(j-k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We found that TBV was able  to produce smooth and globally consistent maps, through  1ORU dataset download: https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='com/radarDataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  Demo video: https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='com/TBV-KvarntorpVolvo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  substantially different environments, including challenging  road conditions – without any parameter changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' CONCLUSIONS  We proposed TBV Radar SLAM – a real-time method  for robust and accurate large-scale SLAM using a spin-  ning 2D radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We showed that loop candidate retrieval  can be largely improved by origin-shifting, coupled place  similarity/odometry uncertainty search, and selecting the  most likely loop constraint as proposed by our verification  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' A high level of loop robustness was achieved by  carefully verifying loop constraints based on multiple sources  of information, such as place similarity, consistency with  odometry uncertainty, and alignment quality assessed after  registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' We evaluated TBV on two public datasets and  demonstrated a substantial improvement to the state of the art  in radar SLAM, making radar an attractive option to lidar for  robust and accurate localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Quantitative and qualitative  experiments demonstrated a high level of generalization  across environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Some findings in our ablation study  suggest that heavy filtering is undesired as it discards details  that are important for place recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Thus, in the future,  we will explore building detailed and dense representations  of scenes, fully utilizing the geometric information richness,  uniquely provided by spinning FMCW radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  REFERENCES  [1] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Hong, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Petillot, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Wallace, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Wang, “RadarSLAM: A  robust simultaneous localization and mapping system for all weather  conditions,” IJRR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 41, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 519–542, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [2] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Burnett, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Wu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Yoon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Schoellig, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Barfoot,  “Are we ready for radar to replace lidar in all-weather mapping and  localization?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' RAL, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 7, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 10 328–10 335, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Marck, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Mohamoud, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' vd Houwen, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' van Heijster,  “Indoor radar SLAM: A radar application for vision and GPS denied  environments,” in 2013 EuRAD, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 471–474.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [4] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Schuster, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Keller, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Rapp, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Haueis, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Curio,  “Landmark based radar SLAM using graph optimization,” in ITSC,  2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 2559–2564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Holder, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Hellwig, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Winner, “Real-time pose graph SLAM  based on radar,” in IV, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 1145–1151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Himstedt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Frost, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Hellbach, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' B¨ohme, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Maehle,  “Large scale place recognition in 2d lidar scans using geometrical  landmark relations,” in IROS, 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 5030–5035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Cen and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Newman, “Precise ego-motion estimation with  millimeter-wave radar under diverse and challenging conditions,” in  ICRA, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 6045–6052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Cen and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Newman, “Radar-only ego-motion estimation in  difficult settings via graph matching,” in ICRA, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 298–304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [9] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Barnes, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Weston, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Posner, “Masking by moving: Learning  distraction-free radar odometry from pose information,” in CoRL, ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  CoRL, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Kaelbling, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Kragic, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Sugiura, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  PMLR, 30 Oct–01 Nov 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 303–316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [10] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Hong, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Petillot, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Wang, “RadarSLAM: Radar based large-  scale SLAM in all weathers,” in IROS, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 5164–5170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [11] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Barnes and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Posner, “Under the radar: Learning to predict robust  keypoints for odometry estimation and metric localisation in radar,”  in ICRA, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 9484–9490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [12] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Park, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Shin, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Kim, “PhaRaO: Direct radar odometry  using phase correlation,” in ICRA, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 2617–2623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [13] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Kung, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Wang, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Lin, “A normal distribution  transform-based radar odometry designed for scanning and automotive  radars,” 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [14] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Burnett, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Schoellig, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Barfoot, “Do we need to  compensate for motion distortion and doppler effects in spinning radar  navigation?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' IEEE RAL, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 6, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 771–778, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [15] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Burnett, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Yoon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Schoellig, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Barfoot, “Radar  odometry combining probabilistic estimation and unsupervised feature  learning,” 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [16] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Adolfsson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Magnusson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Alhashimi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Lilienthal, and  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Andreasson, “CFEAR radarodometry - conservative filtering for  efficient and accurate radar odometry,” in IROS, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 5462–5469.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [17] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Adolfsson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Magnusson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Alhashimi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Lilienthal, and  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Andreasson, “Lidar-level localization with radar?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' The CFEAR  approach to accurate, fast and robust large-scale radar odometry in  diverse environments,” T-RO, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 1–20, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [18] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Aldera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Gadd, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' De Martini, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Newman, “What goes  around: Leveraging a constant-curvature motion constraint in radar  odometry,” RAL, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 7, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 7865–7872, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [19] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Weston, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Gadd, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' De Martini, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Newman, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Posner, “Fast-  mbym: Leveraging translational invariance of the fourier transform for  efficient and accurate radar odometry,” in ICRA, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 2186–2192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [20] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Saftescu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Gadd, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' De Martini, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Barnes, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Newman,  “Kidnapped radar: Topological radar localisation using rotationally-  invariant metric learning,” in ICRA, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 4358–4364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [21] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' De Martini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Gadd, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Newman, “kRadar++: Coarse-to-fine  FMCW scanning radar localisation,” Sensors, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 20, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 21, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [22] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Williams, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Gadd, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' De Martini, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Newman, “Fool Me  Once: Robust Selective Segmentation via Out-of-Distribution Detec-  tion with Contrastive Learning,” 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [23] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Gadd, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' De Martini, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Newman, “Look around you:  Sequence-based radar place recognition with learned rotational invari-  ance,” in 2020 IEEE/ION (PLANS), 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 270–276.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [24] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Yin, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Wang, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Xiong, “Radar-to-lidar: Heterogeneous  place recognition via joint learning,” Frontiers in Robotics and AI,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [25] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Yin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Wang, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Xiong, “Rall: End-to-end radar  localization on lidar map using differentiable measurement model,”  IEEE TITS, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 23, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 6737–6750, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [26] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Tang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' De Martini, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Barnes, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Newman, “RSL-net:  Localising in satellite images from a radar on the ground,” RAL, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 5,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 1087–1094, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [27] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Tang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Martini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Wu, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Newman, “Self-supervised  learning for using overhead imagery as maps in outdoor range sensor  localization,” IJRR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 40, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 12-14, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 1488–1509, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [28] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Duan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Fan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Meng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Ji, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Zhang, “Maroam:  Map-based radar slam through two-step feature selection,” 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='org/abs/2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='13797  [29] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Wang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Zhang, “M2dp: A novel 3d point cloud  descriptor and its application in loop closure detection,” in IROS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  IEEE, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 231–237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [30] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Akai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Akagi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Hirayama, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Morikawa, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Murase, “De-  tection of localization failures using markov random fields with fully  connected latent variables for safe lidar-based automated driving,” T-  ITS, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 23, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 17 130–17 142, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [31] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Aldera, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Martini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Gadd, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Newman, “What could go  wrong?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' introspective radar odometry in challenging environments,” in  2019 IEEE (ITSC), 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 2835–2842.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [32] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Adolfsson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Castellano-Quero, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Magnusson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Lilienthal,  and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Andreasson, “CorAl: Introspection for robust radar and lidar  perception in diverse environments using differential entropy,” RAS,  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 104136, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [33] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Behley and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Stachniss, “Efficient surfel-based SLAM using 3D  laser range data in urban environments,” in Robotics: Science and  Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 2018, 2018, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [34] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Nobili, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Tinchev, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Fallon, “Predicting alignment risk to  prevent localization failure,” in ICRA, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 1003–1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [35] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Castellano Quero, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Kucner, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Magnusson, “Alignability  maps for ensuring high-precision localization,” in 13th IROS Workshop  on Planning, Perception, Navigation for Intelligent Vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [36] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Denniston, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Chang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Reinke, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Ebadi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Sukhatme,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Carlone, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Morrell, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Agha-mohammadi, “Loop closure  prioritization for efficient and scalable multi-robot SLAM,” 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='org/abs/2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='12402  [37] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Almqvist, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Magnusson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Kucner, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Lilienthal,  “Learning to detect misaligned point clouds,” JFR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 35, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 5,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 662–677, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [38] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Kim and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Kim, “Scan context: Egocentric spatial descriptor for  place recognition within 3D point cloud map,” in IROS, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [39] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Choi, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Kim, “Scan context++: Structural place recog-  nition robust to rotation and lateral variations in urban environments,”  IEEE Transactions on Robotics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 38, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 1856–1874, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [40] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Kim, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Park, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Cho, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Jeong, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Kim, “Mulran:  Multimodal range dataset for urban place recognition,” in ICRA, Paris,  May 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [41] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Agarwal, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Tipaldi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Spinello, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Stachniss, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Burgard,  “Robust map optimization using dynamic covariance scaling,” in 2013  IEEE ICRA, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 62–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [42] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' S¨underhauf and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Protzel, “Switchable constraints for robust pose  graph SLAM,” in IROS, 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 1879–1884.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [43] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Barnes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Gadd, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Murcutt, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Newman, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Posner, “The  oxford radar robotcar dataset: A radar extension to the oxford robotcar  dataset,” in ICRA, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 6433–6438.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [44] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Mur-Artal and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Tard´os, “ORB-SLAM2: An open-source SLAM  system for monocular, stereo, and RGB-D cameras,” IEEE T-RO,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 33, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 1255–1262, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [45] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Hong, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Petillot, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Wallace, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Wang, “Radar SLAM: A  robust SLAM system for all weather conditions,” 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [46] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Behley and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Stachniss, “Efficient surfel-based SLAM using 3D  laser range data in urban environments,” in Systems (RSS), 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [47] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Vizzo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Guadagnino, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Mersch, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Wiesmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Behley,  and  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  Stachniss,  “KISS-ICP:  In  Defense  of  Point-to-Point  ICP – Simple, Accurate, and Robust Registration If Done the  Right  Way,”  arXiv  preprint,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  Available:  https:  //arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='org/pdf/2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='15397.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='pdf  [48] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Geiger, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Lenz, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Urtasun, “Are we ready for autonomous  driving?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' the KITTI vision benchmark suite,” in Pattern Recognition  (CVPR), 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content='  [49] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Landry, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Pomerleau, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' Gigu`ere, “Cello-3d: Estimating the  covariance of icp in the real world,” in ICRA, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
+page_content=' 8190–8196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQfOwnl/content/2301.04397v1.pdf'}
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+Object as Query: Equipping Any 2D Object Detector with 3D Detection Ability
+Zitian Wang1
+Zehao Huang2
+Jiahui Fu1
+Naiyan Wang2
+Si Liu1
+1Institute of Artificial Intelligence, Beihang University
+2TuSimple
+{wangzt.kghl,zehaohuang18,winsty}@gmail.com
+{jiahuifu,liusi}@buaa.edu.cn
+Abstract
+3D object detection from multi-view images has drawn
+much attention over the past few years. Existing methods
+mainly establish 3D representations from multi-view im-
+ages and adopt a dense detection head for object detec-
+tion, or employ object queries distributed in 3D space to
+localize objects. In this paper, we design Multi-View 2D
+Objects guided 3D Object Detector (MV2D), which can be
+equipped with any 2D object detector to promote multi-view
+3D object detection. Since 2D detections can provide valu-
+able priors for object existence, MV2D exploits 2D detec-
+tor to generate object queries conditioned on the rich im-
+age semantics. These dynamically generated queries en-
+able MV2D to detect objects in larger 3D space without in-
+creased computational cost and show a strong capability of
+localizing 3D objects. For the generated queries, we design
+a sparse cross attention module to force them to focus on
+the features of specific objects, which reduces the compu-
+tational cost and suppresses interference from noises. The
+evaluation results on the nuScenes dataset demonstrate the
+dynamic object queries and sparse feature aggregation do
+not harm 3D detection capability. MV2D also exhibits a
+state-of-the-art performance among existing methods. We
+hope MV2D can serve as a new baseline for future research.
+1. Introduction
+Camera-based 3D object detection in unconstrained real-
+world scenes has drawn much attention over the past few
+years. Early monocular 3D object detection methods [1,
+15, 27, 33, 37, 38, 42] typically build their framework fol-
+lowing the 2D object detection pipeline. The 3D location
+and attributes of objects are directly predicted from a sin-
+gle view image. Though these methods have achieved great
+progress, they are incapable of utilizing the geometric con-
+figuration of surrounding cameras and multi-view image
+correspondences, which are pivotal for the 3D position of
+objects in the real world. Moreover, adapting these meth-
+ods to the multi-view setting relies on sophisticated cross-
+(a) original image
+(b) fixed queries
+(c) 2D detection
+(d) generated queries from 2D detection
+Figure 1. Motivation of MV2D. The 3D detector with fixed object
+queries (fixed queries means the queries are invariant for different
+inputs) might miss some objects, e.g., traffic cones (b), which are
+however successfully detected by a 2D detector (c). If generating
+object queries based on 2D detector, a 3D detector can produce
+more precise locations.
+camera post-processing, which further causes degraded ef-
+ficiency and efficacy. To handle these problems, recent re-
+searchers [14,19,20,24,39] propose to directly localize ob-
+jects in 3D world space based on multi-view images, pro-
+viding a new paradigm for vision-based 3D object detec-
+tion.
+According to the representation of fused features, current
+multi-view 3D object detection methods can be mainly di-
+vided into two streams: dense 3D methods and sparse query
+methods. Concretely, dense 3D methods render multi-view
+features into 3D space, such as Bird’s-Eye-View (BEV) fea-
+ture space [4, 14, 19, 20] or voxel feature space [18, 32].
+However, since the computational costs are squarely pro-
+portional to the range of 3D space, they inevitably cannot
+scale up to large-scale scenarios [7]. Alternatively, query-
+based methods [24,39] adopt learnable object queries to ag-
+gregate features from multi-view images and predict object
+bounding boxes based on query features. Although fixed
+number of object queries avoid computational cost explod-
+ing with 3D space, the query number and position relied on
+1
+arXiv:2301.02364v1  [cs.CV]  6 Jan 2023
+
+Fused RADARs
+LIDAR_TOP
+CAM_FRONT
+CAM_FRONT_RIGHT
+CAMBACKRIGHT
+CAM_BACK
+CAM_BACK_LEFT
+CAM_FRONT_LEFTFused RADARs
+LIDAR_TOP
+CAM_FRONT
+CAM_FRONT_RIGHT
+CAMBACKRIGHT
+CAM_BACK
+CAM BACK LEFT
+CAM_FRONT_LEFTAAFAEempirical prior may cause false positive or undetected ob-
+jects in dynamic scenarios.
+In this paper, we seek a more reliable way to generate
+object queries dynamically.
+The motivation comes from
+the rapid developments of 2D object detection methods
+[10, 21, 31, 34] which can generate high quality 2D bound-
+ing boxes for object localization in image space. Fortu-
+nately, 3D detectors are usually built upon 2D detection
+backbones. One natural idea is to turn each 2D detection
+into one query for the following 3D detection task. In this
+way, we design Multi-View 2D Objects guided 3D Object
+Detector (MV2D), which could equip any 2D detector with
+3D detection ability, and the 3D detector could harvest all
+the advancements from 2D detection field.
+Given the input multi-view images, we first obtain 2D
+detection results from a 2D detector and then generate a dy-
+namic object query for each 2D bounding box. Instead of
+aggregating features from all regions in the multi-view in-
+puts, one object query is forced to focus on one specific ob-
+ject. To this end, we propose an efficient correlated feature
+selection method based on the 2D detection results and cam-
+era configurations. Then the dynamically generated object
+queries, together with their 3D position embedded corre-
+lated features, are input to a transformer decoder with sparse
+cross-attention layers. Lastly, the updated object queries
+predict the final 3D bounding boxes.
+Thanks to the powerful 2D detection performances, the
+dynamic queries generated from 2D detection results can
+cover most objects that appeared in the images, leading to
+higher precision and recall, especially for small and distant
+objects compared with fixed queries. As shown in Figure 1,
+the objects that a fixed query-based 3D detector might miss
+can be recalled in MV2D with the help of a 2D object detec-
+tor. Theoretically, since 3D object queries stem from 2D de-
+tection results, our method can benefit from all off-the-shelf
+excellent 2D detector improvements. Our contributions can
+be summarized as:
+• We propose a framework named MV2D that can be
+equipped with any 2D object detector to promote
+multi-view 3D object detection. Specifically, MV2D
+generates sparse dynamic object queries based on the
+results of 2D object detector.
+• We demonstrate that sparse dynamic object queries
+and sparse aggregation from certain relevant regions in
+multi-view images do not sacrifice any detection per-
+formance.
+• We test MV2D on the standard nuScenes dataset, and
+it achieves state-of-the-art performance among single
+frame methods.
+2. Related Work
+2.1. 2D Object Detection
+2D object detection aims to predict and localize objects
+in 2D images, which is a fundamental problem in computer
+vision. The RCNN series [9,10,21,31] propose a two-stage
+pipeline for 2D object detection. These methods first gen-
+erate a set of region proposals likely to contain objects in
+the image, then make a second stage decision on the object
+classification and bounding box regression. Another group
+of researchers investigate to detect object in a single-stage
+pipeline [22, 30, 34], aiming to provide faster detectors for
+deployment. Recently, DETR [3] introduces a transformer
+encoder-decoder architecture into 2D object detection and
+formulates object detection as a set prediction problem. In
+DETR, a set of learnable object queries are adopted to in-
+teract with image features and responsible for detection ob-
+jects. Deformable DETR [44] improves DETR by intro-
+ducing multi-scale deformable attention. Besides, it trans-
+fers the vanilla DETR into a two-stage framework, where
+region proposals are firstly generated based on encoder fea-
+tures and then sent to the decoder as object queries for fur-
+ther refinement.
+2.2. Vision-based 3D Object Detection
+The goal of vision-based 3D object detection is to pre-
+dict 3D bounding boxes from camera images. Many previ-
+ous [1, 15, 28, 33, 37, 38] works perform this task based on
+the single-view image setting. Concretely, these methods
+mostly focus on instance depth estimation and use an ex-
+tra depth prediction module to complement 2D detectors on
+depth information. However, when dealing with multi-view
+images surrounding the ego vehicle, these methods need to
+detect objects from each view and project them to the same
+3D space with cumbersome NMS post-processing to merge
+results.
+Recently, some works attempt to directly detect objects
+in 3D space from multi-view images. One branch of these
+methods lift 2D image into 3D space, then conduct detec-
+tion based on the 3D representations.
+ImVoxelNet [32]
+builds a 3D voxelized space and samples image features
+from multi-view to obtain the voxel representation. BEV-
+Former [20] leverages dense BEV queries to project and
+aggregate features from multi-view images by deformable
+attention [44]. BEVDet and BEVDepth [14, 19] adopt the
+Lift-Splat module [29] to transform multi-view image fea-
+tures into the BEV representation based on the predicted
+depth distribution. Although such 3D space representations
+are conducive to unifying multi-view images, the memory
+consumption and computational cost would increase rapidly
+with the enlarging of the detection range in 3D space.
+Another way of such works adopts learnable object
+queries to aggregate image features and predict objects fol-
+2
+
+Multi-View Images
+2D Detector
+Feat Extractor
+2D boxes
+weight sharing
+RoI align
+RoI feats
+Dynamic Query 
+Generator 
+cam params
+object queries
+Self Attention
+Sparse Cross 
+Attention
+FFN
+×𝐿 layers
+Head
+3D predictions
+…
+…
+𝑁 object queries
+RoI features
+Correlated Feature Selection
+Cross Attention
+…
+𝑖th object query
+𝑖th correlated features
+×𝑁 groups
+Q
+K, V
+…
+𝑁 object queries
+3D PE
+…
+Figure 2. The framework of the proposed MV2D. Given the input multi-view images, a 2D detector is used to obtain per-view 2D detection
+results. Meanwhile, image feature maps are extracted by a feature extractor. The parameter weights of the backbone network of 2D
+detector and feature extractor can be shared. Then the RoI features are extracted through RoI-align for detected 2D bounding boxes. Based
+on the RoI features, RoI positions and camera parameters, MV2D initializes a set of object queries using a dynamic query generator.
+These generated object queries and RoI features with 3D PE (3D position embedding) [24] are input to a decoder to update query features.
+Compared to vallina transformer decoder, the decoder in MV2D employ sparse cross attention where each object query only interact with
+its correlated features. Lastly, a prediction head is applied on the updated object queries to generate 3D detection results.
+lowing the DETR [3] framework. DETR3D [39] gener-
+ates 3D reference points from object queries and projects
+them to multi-camera images by camera parameters. Then
+the query features are refined by the corresponding point
+features sampling from images.
+In contrast to estab-
+lishing fixed mapping through 3D-to-2D query project-
+ing, some methods learn the flexible mapping via attention
+mechanisms [24, 41]. PETR [24] introduces 3D position-
+aware image features and learns the flexible mapping be-
+tween query and image features by global cross-attention.
+Nonetheless, these approaches usually require dense object
+queries distributed in 3D space to ensure sufficient recall of
+objects.
+In this paper, we propose a 2D objects guided framework
+for multi-view 3D object detection. Based on the 2D detec-
+tor, our method can count on much sparser queries to recall
+the objects and eliminate the interference of noises and dis-
+tractors.
+3. Method
+3.1. Overview
+The overall pipeline of MV2D is shown in Figure 2.
+Given N multi-view images I = {Iv | 0 ≤ v < N},
+image feature maps F = {Fv | 0 ≤ v < N} are first
+extracted from the input images using a backbone network,
+where Fv ∈ RHf ×W f ×C is the extracted feature map from
+the v-th image. To get 2D object detection, a 2D object
+detector, i.e., Faster R-CNN [31], is applied to all the in-
+put images, resulting in a set of 2D object bounding boxes
+B = {Bv | 0 ≤ v < N}, where Bv ∈ RMv×4 represents
+the predicted 2D bounding boxes in the v-th image and Mv
+is the number of detected boxes. In practice, the weights of
+the backbone of 2D detector can also be used for subsequent
+feature extraction.
+Different from common methods in multi-view 3D ob-
+ject detection which adopt fixed object queries across the
+whole dataset [24,39], MV2D generates object queries con-
+ditioned on the input multi-view images. Given the pre-
+dicted 2D bounding boxes and extracted image features,
+MV2D employs RoI-Align [11] to extract 2D object fea-
+tures.
+Then the 2D object features together with corre-
+sponding bounding boxes and camera parameters are in-
+put to the dynamic object query generator to produce ob-
+ject queries. For each object query, relevant image features
+in multi-view images are selected based on 2D detection
+results and camera parameters through a feature selection
+module. Then the object queries interact with each other
+and integrate information from correlated object features it-
+eratively through transformer decoder layers [35]. Finally, a
+simple feed forward network (FFN) head is used to generate
+3D object predictions with updated features.
+3.2. Dynamic Object Query Generation
+With the help of an effective 2D object detector, the ob-
+ject existences are well demonstrated and the object loca-
+tions are constrained within certain image regions, thus pro-
+viding valuable priors for localizing objects in 3D space.
+To generate object queries from 2D detection results, we
+propose a dynamic query generator as shown in Figure 3.
+The dynamic query generator derives a 3D reference point
+pref ∈ R3 in 3D world space for each RoI. Then the ob-
+ject query is generated from pref by a positional encoding
+layer [35].
+3
+
+𝐾
+𝑥!"#, 𝑦!"#
+𝑥!$%, 𝑦!$%
+Camera Parameters
+MLP
+Conv
+Pool
+𝐾!
+"
+𝑝!"
+𝑝#$%
+Coordinate
+Transformation
+RoI Features
+𝑅|t
+Figure 3. Dynamic object query generator.
+Specifically, given the 2D object detection results B and
+image feature maps F from all the images, we first ex-
+tract object RoI features O through RoI-Align, where Ov ∈
+RMv×Hroi×W roi×C are the RoI features corresponding to
+2D object bounding boxes Bv:
+Ov = RoI-Align(Fv, Bv),
+0 ≤ v < N.
+(1)
+The RoI features Ov contain sufficient object appearance
+information to infer the object center locations in image
+space. However, further estimating the object depths di-
+rectly from RoI features is difficult. As the object region
+in original image space is rescaling into a fixed sized RoI,
+the geometric information contained in original images are
+missed.
+Taking this into consideration, we apply equivalent cam-
+era intrinsic transformation to each RoI, such that the rescal-
+ing operation conducted on different RoIs is equivalent to
+perspective projection with different camera parameters.
+Thus a point in the RoI coordinate system can be trans-
+formed to 3D world space with equivalent camera intrinsic:
+p3d = [R|t]−1(Kroi)−1 proi,
+(2)
+where proi ∈ R4 (homogeneous coordinate) represents the
+point in RoI coordinate system, p3d ∈ R4 represents the
+point in 3D world space, Kroi is the equivalent camera in-
+trinsic of that RoI and [R|t] is the camera extrinsic.
+Denote the RoI size is Hroi × W roi (e.g., 7 × 7), the
+i-th 2D object bounding box in the v-th view is Bi
+v =
+(xi
+min, yi
+min, xi
+max, yi
+max), and the original camera intrin-
+sic matrix is:
+Kv =
+�
+��
+fx
+0
+ox
+0
+0
+fy
+oy
+0
+0
+0
+1
+0
+0
+0
+0
+1
+�
+��.
+(3)
+The equivalent camera intrinsic for i-th RoI can be formu-
+lated as:
+Ki
+v =
+�
+��
+fx ∗ rx
+0
+(ox − xi
+min) ∗ rx
+0
+0
+fy ∗ ry
+(oy − yi
+min) ∗ ry
+0
+0
+0
+1
+0
+0
+0
+0
+1
+�
+��,
+(4)
+where rx = W roi/(xi
+max − xi
+min), ry = Hroi/(yi
+max −
+yi
+min).
+Since the equivalent camera intrinsic matrix contains the
+geometric property of the camera and object, we adopt a
+small network to implicitly encode the object location pi
+v ∈
+R4 based on the RoI feature oi
+v and the equivalent camera
+intrinsic Ki
+v:
+pi
+v = H(MLP(Pool(Conv(oi
+v)); Ki
+v)),
+(5)
+where (; ) means feature concatenation and H(·) represents
+homogeneous transformation.
+The implicitly encoded object location pi
+v, which can be
+seen as the 3D object location in RoI coordinate system,
+is then transformed into 3D world space using equivalent
+camera intrinsic and camera extrinsic as in Equation 2. Con-
+sequently, the transformed 3D location serves as reference
+point pref for localizing object in 3D world space.
+By this transformation, MV2D generates a set of dy-
+namic reference points Pref = {Pref,v ∈ RMv×3 | 0 ≤
+v < N} based on 2D detections and camera configurations.
+These reference points are then passed through a 3D po-
+sitional encoding layer to initialize a set of object queries
+Q = {Qv ∈ RMv×C | 0 ≤ v < N}.
+3.3. Relevant Object Feature Selection
+Each object is only captured in a sub-region within the
+input multi-view images. To precisely depict a certain ob-
+ject, an object query should focus on all the relevant image
+regions that cover the target object, and discard irrelevant
+regions that might mislead the 3D localization of the target
+object.
+To this end, we propose to select relevant features for
+each object query’s updating. Since 2D object detector can
+predict convincing 2D object proposals, the detected object
+bounding boxes imply which region contains the most dis-
+tinctive information about an object. So we consider two
+parts of the relevant features of an object: (1) the 2D bound-
+ing box bi
+v from where the object query is generated; (2) the
+bounding boxes in other views that corresponds to bi
+v.
+In this paper, we develop an efficient method to as-
+sociate the bounding boxes in other views for a given
+query. As shown in Figure 4, we create a 3D meshgrid
+G ∈ RW roi×Hroi×D×4 for each RoI. We denote gx,y,z =
+(x × dz, y × dz, dz, 1) as each point in the meshgrid, where
+(x, y) is the coordinate in the RoI, dz ∈ {d0, d1, . . . , dD−1}
+is a set of predifined depth values. The meshgrid of RoI
+is then transformed into the coordinate system of the w-th
+view:
+gi
+v→w;x,y,z = KwTv→w(Ki
+v)−1gx,y,z,
+(6)
+where gi
+v→w;x,y,z is the transformed meshgrid point in the
+w-th view and Tv→w is the coordinate transformation ma-
+trix from the v-th view to the w-th view.
+4
+
+: uncorrelated boxes
+: correlated boxes
+: minimum bounding box 
+: query box
+Figure 4. Illustration of relevant region selection. Each query box
+generates a discretized camera frustum from 3D meshgrid. The
+camera frustum is then projected to another view’s pixel coordi-
+nate to calculate a minimum bounding box. Then correlated box
+is selected based on IoU with the minimum bounding box.
+According to the transformed meshgrid Gi
+v→w, a min-
+imum bounding box bi
+v→w in the image space of the wth
+view can be calculated. Then the detected box that has high-
+est Intersection-over-Union (IoU) with bi
+v→w (if greater
+than 0) is selected as correlated box in the w-th view:
+bi
+corr,v→w = arg max
+bj
+w∈Bw
+IoU(bi
+v→w, bj
+w).
+(7)
+Then, only the RoI features of bi
+v and {bi
+corr,v→w | 0 ≤
+w < N, w ̸= v} will be adopted to update object query qi
+v.
+We will describe the details in the following section.
+3.4. Decoder with Sparse Cross Attention
+The generated object queries interact with their relevant
+features through a DETR-like transformer decoder [3, 24].
+3D position embedding is provided for the relevant features
+following PETR [24]. The difference in our decoder lies in
+the sparse cross attention layer as shown in Figure 2. In-
+stead of using the whole multi-view feature maps to con-
+struct keys and values shared by all the queries, MV2D
+only uses the relevant features to construct their own keys
+and values for each object query. This design not only con-
+tributes a compact set of keys and values, but also prevents
+the object queries from being interfered by background
+noises and distractors.
+Lastly, we apply classification head and regression head
+composed of MLPs to the updated object queries to predict
+the final 3D object detection results.
+3.5. Loss Functions
+In our implementation, the 2D detector and the 3D detec-
+tor are jointly trained in MV2D, with the backbone weights
+shared across both detectors. The 2D object detection loss
+L2d is directly taken from 2D detectors. As for 3D ob-
+ject detection loss, we follow previous works [24, 39] to
+use Hungarian algorithm [16] for label assignment. Focal
+loss [22] and L1 loss are adopted for classification and box
+regression respectively. The 3D object detection loss can be
+summarized as:
+L3d = λcls3d · Lcls3d + Lreg3d.
+(8)
+The overall loss function of MV2D is:
+L = L2d + λ3d · L3d,
+(9)
+where λ3d is the weight term to balance 2D detection and
+3D detection losses, set to 0.1 in our experiments.
+4. Experiments
+4.1. Datasets and Metrics
+We validate the effectiveness of MV2D on the large-
+scale nuScenes dataset [2]. NuScenes contains 1000 driving
+sequences, which are divided into 700 samples for training,
+150 samples for validation, and 150 samples for testing, re-
+spectively. Each sequence is approximately 20-second long,
+including annotated 3D bounding boxes from 10 categories
+of sampled key frames. Each sample consists of 6 surround-
+view images with 1600 × 900 resolution, which provide
+360◦ horizontal FOV in total. We submit test set results to
+the online server for official evaluation. Other experiments
+are based on the val set.
+4.2. Implementation Details
+Training and evaluation
+All the models are trained using
+AdamW [26] optimizer with a weight decay of 0.01. Co-
+sine annealing policy [25] is adopted and the initial learn-
+ing rate is set to 2 × 10−4. Since MV2D generates object
+queries from 2D detection results, we pretrain the 2D detec-
+tors on nuImages [2] to provide appropriate initialization.
+Since nuScenes dataset does not provide 2D bounding box
+annotations, we generate the box labels from 3D bounding
+boxes following [40]. We employ image-space augmenta-
+tion (e.g., random flip, crop and resize) and BEV-space aug-
+mentation [14] during training. If not specified, the models
+are trained for 24 epochs without CBGS [43]. No test time
+augmentation is used during inference. Our implementation
+is based on MMDetection3D [5].
+Network architecture
+Faster-RCNN [31] is adopted as
+2D detector in our experiments.
+The 2D score thresh-
+old and Non-maximum Suppression (NMS) IoU thresh-
+old are set to 0.05 and 0.6 respectively. We use ResNet-
+50, ResNet-101 [12] and VoVNetV2 [17] as backbone
+networks. ResNet-50 and ResNet-101 are equipped with
+deformable convolution [6] in the 3rd and 4th stages.
+VoVNetV2 is initialized from a DD3D checkpoint trained
+with extra data [28]. RoI-align is applied on the P4 stage of
+5
+
+Method
+Backbone Resolution NDS↑ mAP↑ mATE↓ mASE↓ mAOE↓ mAVE↓ mAAE↓
+DETR3D [39]
+Res-50
+1600 × 900 0.373 0.302
+0.811
+0.282
+0.493
+0.979
+0.212
+BEVDepth∗ [19]
+Res-50
+704 × 256
+0.367 0.315
+0.702
+0.271
+0.621
+1.042
+0.315
+PETR [24]
+Res-50
+1408 × 512 0.403 0.339
+0.748
+0.273
+0.539
+0.907
+0.203
+MV2D‡
+Res-50
+1408 × 512 0.433 0.388
+0.697
+0.271
+0.479
+0.952
+0.208
+FCOS3D† [37]
+Res-101
+1600 × 900 0.415 0.343
+0.725
+0.263
+0.422
+1.292
+0.153
+PGD† [38]
+Res-101
+1600 × 900 0.428 0.369
+0.683
+0.260
+0.439
+1.268
+0.185
+DETR3D† [39]
+Res-101
+1600 × 900 0.425 0.346
+0.773
+0.268
+0.383
+0.842
+0.216
+BEVDepth∗ [19]
+Res-101
+1408 × 512 0.408 0.376
+0.659
+0.267
+0.543
+1.059
+0.335
+BEVFormer-S† [20]
+Res-101
+1600 × 900 0.448 0.375
+0.725
+0.272
+0.391
+0.802
+0.200
+PETR† [24]
+Res-101
+1600 × 900 0.442 0.370
+0.711
+0.267
+0.383
+0.865
+0.201
+MV2D‡
+Res-101
+1600 × 900 0.451 0.406
+0.685
+0.265
+0.446
+0.912
+0.215
+Table 1. 3D object detection results on nuScenes val set. †: the model is initialized from a FCOS3D backbone. ‡: the model is
+pretrained on nuImages. ∗: the model uses extra modal data (e.g., point clouds) during training.
+Method
+Backbone
+NDS↑ mAP↑ mATE↓ mASE↓ mAOE↓ mAVE↓ mAAE↓
+FCOS3D† [37]
+Res-101
+0.428 0.358
+0.690
+0.249
+0.452
+1.434
+0.124
+PGD† [38]
+Res-101
+0.448 0.386
+0.626
+0.245
+0.451
+1.509
+0.127
+BEVFormer-S† [20]
+Res-101
+0.462 0.409
+0.650
+0.261
+0.439
+0.925
+0.147
+PETR† [24]
+Res-101
+0.455 0.391
+0.647
+0.251
+0.433
+0.933
+0.143
+MV2D‡
+Res-101
+0.462 0.427
+0.640
+0.257
+0.457
+0.997
+0.162
+BEVDet [14]
+Swin-Base
+0.488 0.424
+0.524
+0.242
+0.373
+0.950
+0.148
+M2BEV [40]‡
+ResNeXt-101 0.474 0.429
+0.583
+0.254
+0.376
+1.053
+0.190
+DETR3D§ [39]
+V2-99
+0.479 0.412
+0.641
+0.255
+0.394
+0.845
+0.133
+BEVFormer-S§ [20]
+V2-99
+0.495 0.435
+0.589
+0.254
+0.402
+0.842
+0.131
+PETR§ [24]
+V2-99
+0.504 0.441
+0.593
+0.249
+0.383
+0.808
+0.132
+MV2D§‡
+V2-99
+0.514 0.463
+0.542
+0.247
+0.403
+0.857
+0.127
+Table 2. 3D detection results on nuScenes test set. †: the model is initialized from a FCOS3D checkpoint. ‡: the model is pretrained
+on nuImages. §: the model is initialized from a DD3D [28] checkpoint trained with external data.
+FPN [21] with RoI size 7×7 for dynamic object query gen-
+eration and correlated feature selection. The decoder con-
+tains 6 transformer decoder layers, followed by MLP head
+to produce classification and regression results.
+4.3. Comparison with State-of-the-arts
+We compare the MV2D performance with state-of-the-
+art methods on nuScenes val set and test set under single
+frame setting. The results are shown in Table 1 and Table 2.
+Table 1 shows comparison on nuScenes val set. From the
+table, MV2D achieves higher performace with both ResNet-
+50 and ResNet-101 backbone, especially on mAP. The sig-
+nificant improvement on mAP suggests MV2D has a strong
+capability at localizing objects in 3D space. In comparison
+with multi-view 3D object detection methods, MV2D with
+ResNet-101 outperforms BEV based methods BEVFormer-
+S [20] by 3.1% and 0.3% on mAP and NDS, and outper-
+forms BEVDepth [19] with extra depth supervision by 3.0%
+and 4.3% on mAP and NDS. In comparison with query
+based methods, MV2D outperforms the best performing
+PETR [24] by 3.6% and 0.9% on mAP and NDS.
+Table 2 shows the comparison on nuScenes test set.
+We use both the training and validation splits for training
+MV2D. As can be seen from the table, MV2D with ResNet-
+101 backbone achieves 42.7% mAP and 46.2% NDS, which
+outperforms existing methods by 1.8% on mAP. For MV2D
+with VoVNetV2, we train it for 90 epochs without CBGS
+(which has almost the same amount of samples as training
+24 epochs with CBGS) for better performance. In that case,
+MV2D achieves 46.3% mAP and 51.4% NDS, which im-
+proves mAP by 2.2% compared to PETR. Note that our
+method gets larger mAVE compared to state-of-the-arts,
+which is the main cause of degraded NDS.
+4.4. Ablation Study
+In this section, we provide a detailed analysis on the core
+design choices of MV2D. All the experiments are based
+on ResNet-50 backbone. For fair comparison, we design a
+6
+
+Figure 5. Precision-recall curves of different classes under a 2D object detection setting. The models still produce 3D object predictions,
+while the 2D bounding boxes in each image are generated by projecting 3D bounding boxes using camera parameters. Precisoin & recall
+are calculated with 2D IoU threshold 0.5 in nuScenes val set. The blue dash line represents the baseline method (fixed object queries). The
+red solid line represents our method (generated object queries).
+#
+object query
+key & value
+mAP↑
+NDS↑
+1
+fixed (×900)
+all feats
+36.2
+39.9
+2
+fixed (×300)
+all feats
+36.1
+39.4
+3
+fixed (×1500)
+all feats
+36.6
+40.6
+4
+generated (×165)
+all feats
+38.0
+41.2
+5
+generated (×165)
+all RoIs
+37.9
+41.1
+6
+generated (×165)
+correlated RoIs
+38.8
+43.3
+Table 3. Ablation study on the object query and key & value in the
+cross attention layers. The number of generated queries is calcu-
+lated on average.
+baseline method also trained with a 2D detector and initial-
+ized from a nuImages pretrained checkpoint. This method
+uses a fixed number of learnable object queries to aggre-
+gate information from the whole multi-view feature maps
+as in PETR [24]. As in Table 3, we denote the design in #1
+which follows PETR to use 900 object queries as baseline
+in our experiments.
+Generated object queries vs. fixed object queries
+We
+first compare the generated object queries with fixed object
+queries. From #1 to #3 in Table 3, it can be observed that
+in the case of fixed queries, a larger query amount can im-
+prove performance since the fixed query based methods rely
+on densely placed object queries to localize objects. How-
+ever, the computational cost and memory consumption also
+rise with the growth of query density. According to #4,
+when replacing the fixed queries by dynamically generated
+queries, mAP and NDS are improved by 1.8% and 1.3%
+#
+2D NMS IoU
+2D score thr
+RoI size
+mAP↑
+NDS↑
+1
+0.5
+0.05
+7
+38.2
+42.9
+2
+0.6
+0.05
+7
+38.8
+43.3
+3
+0.7
+0.05
+7
+37.4
+41.1
+4
+0.6
+0.1
+7
+38.5
+43.2
+5
+0.6
+0.2
+7
+38.0
+42.9
+6
+0.6
+0.05
+14
+38.6
+43.1
+Table 4. Ablation study on RoI configurations.
+respectively. Note that the number of generated queries is
+only around 165 on average, which is much less than that of
+fixed queries. This demonstrates the object queries gener-
+ated by 2D detector produce higher-quality object location
+hypotheses even with smaller query amount.
+Effectiveness of correlated features
+We also validate the
+effectiveness of using only correlated features for object
+query updating. In Table 3, #4 represents using the whole
+multi-view feature maps for each object query, #5 repre-
+sents using all the RoI features for each object query, while
+#6 represents using only the correlated RoI features for
+each object query.
+As shown by the results, simply re-
+placing the whole feature maps by all RoI features dose not
+bring performance gain. Nevertheless, using only the corre-
+lated RoI features for object query updating as introduced in
+Section 3.3 can improve mAP and NDS by 0.8% and 2.1%
+respectively. This result verifies the effectiveness of object
+query to aggregate information in a specified foreground re-
+gion.
+7
+
+CAR
+TRUCK
+TRAILER
+BUS
+CONSTRUCTION_VEHICLE
+1.0
+1.0 -
+1.0
+1.0
+1.0
+ baseline
+.-- baseline
+-- baseline
+baseline
+ours
+ours
+ours
+ours
+ours
+0.8
+0.8 -
+0.8
+0.8
+0.2 -
+0.2 -
+0.2 -
+0.2 -
+0.2 -
+←0
+0.0 +
++0
+0.0 +
++0
+0.0
+1.0
+0.0
+0.4
+0.6
+1.0
+0.4
+0.6
+0.8
+1.0
+0.0
+0.4
+0.6
+0.4
+0.6
+0.8
+0.0
+0.2
+0.8
+0.2
+0.2
+0.8
+0.0
+0.2
+1.0
+0.2
+0.4
+0.6
+0.8
+1.0
+recall
+recall
+recall
+recall
+recall
+BICYCLE
+MOTORCYCLE
+PEDESTRIAN
+TRAFFIC_CONE
+BARRIER
+1.0
+1.0
+1.0
+1.0
+1.0
+baseline
+baseline
+baseline
+ baseline
+ baseline
+ours
+ ours
+-ours
+ours
+ours
+0.8 -
+- 80
+0.8 -
+0.8 -
+0.6 -
+0.2 -
+0.2 
+0.2 -
+0.2 -
+0.2
+0.0
+0
+0.0 -
+0.0-
++ 0°
+1.0
+0.8
+0.2
+0.8
+1.0
+0.4
+0.6
+1.0
+0.4
+0.6
+0.8
+1.0
+0.4
+0.0
+0.4
+0.6
+0.0
+0.2
+0.4
+0.6
+0.0
+0.2
+0.8
+0.0
+0.2
+0.0
+0.2
+0.6
+0.8
+1.0
+recall 
+recall
+recall
+recall
+recall: ground truth centers
+: reference points
+(a) baseline
+(b) MV2D
+Figure 6. Illustration of different kinds of object queries. The
+top row (a) represents the baseline method (fixed object queries)
+and the bottom row (b) represents our method (generated object
+queries). We visualize ground truth center locations and the refer-
+ence points in BEV space. The samples are taken from nuScenes
+val set. We suggest readers view this figure with zooming in.
+Influence of RoI configuration
+In this part, we study the
+influence of different RoI configurations in MV2D. We ex-
+periment with different 2D detection score thresholds, 2D
+NMS IoU thresholds, and RoI-align sizes. The results are
+listed in Table 4. As can be seen from the table, the perfor-
+mance of MV2D is robust to different RoI settings.
+4.5. Qualitative Analysis
+In this section, we analyze how might a 2D detector pro-
+mote the performance of multi-view 3D object detection. To
+verify if the object queries generated from 2D detector can
+better localize objects with the help of image semantics, we
+compare the 2D object detection performance of the base-
+line method (fixed object queries) and our method (gener-
+ated object queries) by perspective projection. Specifically,
+we project the 3D bounding boxes into each image, then cal-
+culate the minimum 2D bounding boxes as 2D results. The
+performance is evaluated between the projected 2D bound-
+ing boxes of predictions and ground truths. We set the IoU
+threshold to 0.5 for true positive assignment.
+In Figure 5, we draw the precision-recall curves of
+10 classes in nuScenes.
+As seen from the figure, our
+method achieves higher precision and recall compared to
+the baseline method, especially for the classes with rela-
+tively smaller size. This demonstrates that 2D object detec-
+tors can make the most of image semantics to discover ob-
+Figure 7. Comparison under different numbers of decoder layers.
+jects from image space, which can serve as solid evidence of
+object existence to be exploited by 3D detectors. For com-
+plementary, we provide the visualization of different kinds
+of queries in Figure 6. Though the generated queries are
+much sparser compared to the fixed queries, they are mostly
+distributed around the objects.
+Moreover, we compare the 3D detection performance of
+baseline and our method under different numbers of de-
+coder layers. As shown in Figure 7, baseline with fixed
+queries rely on a heavy decoder to refine the query features.
+In contrast, our method with generated queries and sparse
+cross attention shows a faster convergence on mAP and
+NDS. This result demonstrates that better initialized queries
+and constrained search regions might help to reduce the am-
+biguity for localizing objects.
+5. Conclusion
+In this paper, we propose Multi-View 2D Objects guided
+3D Object Detector (MV2D) for multi-view 3D object de-
+tection. In our framework, we utilizes 2D objects as sparse
+queries and adopt a sparse cross attention module to limit
+the attention region of queries.
+In our experiments, we
+demonstrate promising results on nuScenes dataset with our
+proposed sparse dynamic object queries and sparse aggre-
+gation strategies. Our framework can equip any 2D detec-
+tor with 3D detection ability, and we believe the insight of
+utilizing the 2D objects as guidance can further inspire the
+design of multi-view 3D object detection methods.
+Limitations
+Since MV2D generates object queries from
+2D detections, an object might be missed if the 2D detector
+fails to recall it. Besides, it requires careful design of tem-
+poral correlation for MV2D under multi-frame setting. We
+look forward to further extending MV2D in the temporal
+dimension in the future.
+8
+
+0.45
+0.40
+38.8%
+38.8%
+38.9%
+38.9%
+38.8%
+38.1%
+?
+36.2%
+36.2%
+35.9%
+35.4%
+mAP
+?
+0.35
+33.2%
+29.7%
+0.30
+X
+mAP (ours)
+ mAP (baseline)
+?
+0.25
+1
+2
+3
+4
+5
+6
+number of decoder layers
+0.45
+43.4%
+43.2%
+43.3%
+43.0%
+42.5%
+?
+41.7%
+39.9%
+39.6%
+0.40
+39.0%
+38.5%
+36.3%
+NDS
+0.35
+32.2%
+X
+0.30
+NDS (ours)
+NDS (baseline)
+0.25
+1
+2
+3
+4
+5
+6
+number of decoder layersReferences
+[1] Garrick Brazil and Xiaoming Liu. M3D-RPN: monocular 3d
+region proposal network for object detection. In ICCV, 2019.
+1, 2
+[2] Holger Caesar, Varun Bankiti, Alex H. Lang, Sourabh Vora,
+Venice Erin Liong, Qiang Xu, Anush Krishnan, Yu Pan, Gi-
+ancarlo Baldan, and Oscar Beijbom.
+nuscenes: A multi-
+modal dataset for autonomous driving. In CVPR, 2020. 5,
+11
+[3] Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas
+Usunier, Alexander Kirillov, and Sergey Zagoruyko. End-to-
+end object detection with transformers. In ECCV, 2020. 2,
+3, 5
+[4] Shaoyu Chen, Xinggang Wang, Tianheng Cheng, Qian
+Zhang, Chang Huang, and Wenyu Liu. Polar parametrization
+for vision-based surround-view 3d detection. arXiv preprint
+arXiv:2206.10965, 2022. 1
+[5] MMDetection3D Contributors.
+MMDetection3D: Open-
+MMLab next-generation platform for general 3D object
+detection.
+https://github.com/open-mmlab/
+mmdetection3d, 2020. 5
+[6] Jifeng Dai, Haozhi Qi, Yuwen Xiong, Yi Li, Guodong
+Zhang, Han Hu, and Yichen Wei. Deformable convolutional
+networks. In ICCV, 2017. 5, 11
+[7] Lue Fan, Feng Wang, Naiyan Wang, and Zhaoxiang Zhang.
+Fully sparse 3d object detection. In NeurIPS, 2022. 1
+[8] Zheng Ge, Songtao Liu, Feng Wang, Zeming Li, and Jian
+Sun. Yolox: Exceeding yolo series in 2021. arXiv preprint
+arXiv:2107.08430, 2021. 11
+[9] Ross Girshick. Fast R-CNN. In ICCV, 2015. 2, 11
+[10] Ross Girshick, Jeff Donahue, Trevor Darrell, and Jitendra
+Malik. Rich feature hierarchies for accurate object detection
+and semantic segmentation. In CVPR, 2014. 2, 11
+[11] Kaiming He, Georgia Gkioxari, Piotr Doll´ar, and Ross Gir-
+shick. Mask R-CNN. In ICCV, 2017. 3, 11
+[12] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.
+Deep residual learning for image recognition.
+In CVPR,
+2016. 5
+[13] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.
+Deep residual learning for image recognition.
+In CVPR,
+2016. 11
+[14] Junjie Huang, Guan Huang, Zheng Zhu, and Dalong Du.
+BEVDet: High-performance multi-camera 3d object detec-
+tion in bird-eye-view.
+arXiv preprint arXiv:2112.11790,
+2021. 1, 2, 5, 6
+[15] Jason Ku, Alex D. Pon, and Steven L. Waslander. Monocular
+3d object detection leveraging accurate proposals and shape
+reconstruction. In CVPR, 2019. 1, 2
+[16] Harold W Kuhn. The hungarian method for the assignment
+problem. Naval research logistics quarterly, 1955. 5
+[17] Youngwan Lee, Joong-Won Hwang, Sangrok Lee, Yuseok
+Bae, and Jongyoul Park. An energy and gpu-computation
+efficient backbone network for real-time object detection. In
+CVPRW, 2019. 5
+[18] Yanwei Li, Yilun Chen, Xiaojuan Qi, Zeming Li, Jian Sun,
+and Jiaya Jia.
+Unifying voxel-based representation with
+transformer for 3d object detection. In NeurIPS, 2022. 1
+[19] Yinhao Li, Zheng Ge, Guanyi Yu, Jinrong Yang, Zengran
+Wang, Yukang Shi, Jianjian Sun, and Zeming Li. BEVDepth:
+Acquisition of reliable depth for multi-view 3d object detec-
+tion. arXiv preprint arXiv:2206.10092, 2022. 1, 2, 6
+[20] Zhiqi Li, Wenhai Wang, Hongyang Li, Enze Xie, Chong-
+hao Sima, Tong Lu, Qiao Yu, and Jifeng Dai. BEVFormer:
+Learning bird’s-eye-view representation from multi-camera
+images via spatiotemporal transformers. In ECCV, 2022. 1,
+2, 6
+[21] Tsung-Yi Lin, Piotr Doll´ar, Ross Girshick, Kaiming He,
+Bharath Hariharan, and Serge Belongie.
+Feature pyramid
+networks for object detection. In CVPR, 2017. 2, 6, 11
+[22] Tsung-Yi Lin, Priya Goyal, Ross Girshick, Kaiming He, and
+Piotr Doll´ar. Focal loss for dense object detection. In ICCV,
+2017. 2, 5, 11
+[23] Shu Liu, Lu Qi, Haifang Qin, Jianping Shi, and Jiaya Jia.
+Path aggregation network for instance segmentation.
+In
+CVPR, 2018. 11
+[24] Yingfei Liu, Tiancai Wang, Xiangyu Zhang, and Jian Sun.
+PETR: Position embedding transformation for multi-view 3d
+object detection. In ECCV, 2022. 1, 3, 5, 6, 7
+[25] Ilya Loshchilov and Frank Hutter. SGDR: stochastic gradient
+descent with warm restarts. In ICLR, 2017. 5
+[26] Ilya Loshchilov and Frank Hutter. Decoupled weight decay
+regularization. In ICLR, 2019. 5
+[27] Arsalan Mousavian, Dragomir Anguelov, John Flynn, and
+Jana Kosecka. 3d bounding box estimation using deep learn-
+ing and geometry. In CVPR, 2017. 1
+[28] Dennis Park, Rares Ambrus, Vitor Guizilini, Jie Li, and
+Adrien Gaidon.
+Is pseudo-lidar needed for monocular 3d
+object detection? In ICCV, 2021. 2, 5, 6
+[29] Jonah Philion and Sanja Fidler. Lift, splat, shoot: Encoding
+images from arbitrary camera rigs by implicitly unprojecting
+to 3d. In ECCV, 2020. 2
+[30] Joseph Redmon, Santosh Divvala, Ross Girshick, and Ali
+Farhadi. You only look once: Unified, real-time object de-
+tection. In CVPR, 2016. 2
+[31] Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun.
+Faster R-CNN: Towards real-time object detection with re-
+gion proposal networks. In NeurIPS, 2015. 2, 3, 5, 11
+[32] Danila Rukhovich, Anna Vorontsova, and Anton Konushin.
+Imvoxelnet: Image to voxels projection for monocular and
+multi-view general-purpose 3d object detection. In WACV,
+2022. 1, 2
+[33] Andrea Simonelli, Samuel Rota Bul`o, Lorenzo Porzi,
+Manuel Lopez-Antequera, and Peter Kontschieder. Disen-
+tangling monocular 3d object detection. In ICCV, 2019. 1,
+2
+[34] Zhi Tian, Chunhua Shen, Hao Chen, and Tong He. FCOS:
+Fully convolutional one-stage object detection.
+In ICCV,
+2019. 2
+[35] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszko-
+reit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia
+Polosukhin. Attention is all you need. In NeurIPS, 2017. 3
+[36] Chien-Yao Wang, Hong-Yuan Mark Liao, Yueh-Hua Wu,
+Ping-Yang Chen, Jun-Wei Hsieh, and I-Hau Yeh. Cspnet: A
+new backbone that can enhance learning capability of CNN.
+In CVPR Workshops, 2020. 11
+9
+
+[37] Tai Wang, Xinge Zhu, Jiangmiao Pang, and Dahua Lin.
+FCOS3D: fully convolutional one-stage monocular 3d object
+detection. In ICCVW, 2021. 1, 2, 6
+[38] Tai Wang, Xinge Zhu, Jiangmiao Pang, and Dahua Lin.
+Probabilistic and geometric depth: Detecting objects in per-
+spective. In CoRL, 2021. 1, 2, 6
+[39] Yue Wang, Vitor Guizilini, Tianyuan Zhang, Yilun Wang,
+Hang Zhao, and Justin Solomon. DETR3D: 3d object detec-
+tion from multi-view images via 3d-to-2d queries. In CoRL,
+2021. 1, 3, 5, 6
+[40] Enze Xie, Zhiding Yu, Daquan Zhou, Jonah Philion, Anima
+Anandkumar, Sanja Fidler, Ping Luo, and Jose M Alvarez.
+M2BEV: Multi-camera joint 3d detection and segmentation
+with unified birds-eye view representation. arXiv preprint
+arXiv:2204.05088, 2022. 5, 6
+[41] Brady Zhou and Philipp Kr¨ahenb¨uhl. Cross-view transform-
+ers for real-time map-view semantic segmentation. In CVPR,
+2022. 3
+[42] Xingyi Zhou, Dequan Wang, and Philipp Kr¨ahenb¨uhl. Ob-
+jects as points. CoRR, abs/1904.07850, 2019. 1
+[43] Benjin Zhu, Zhengkai Jiang, Xiangxin Zhou, Zeming Li, and
+Gang Yu. Class-balanced grouping and sampling for point
+cloud 3d object detection. arXiv preprint arXiv:1908.09492,
+2019. 5
+[44] Xizhou Zhu, Weijie Su, Lewei Lu, Bin Li, Xiaogang Wang,
+and Jifeng Dai. Deformable DETR: deformable transformers
+for end-to-end object detection. In ICLR, 2021. 2
+10
+
+A. Experiments with More Architectures
+We provide more experiments with different 2D detec-
+tors and feature extractors in this section. The 2D detec-
+tor part of MV2D is pretrained on nuImages [2], then the
+2D detector part and 3D detector part are jointly trained
+on nuScenes train set. 3D object detection performance
+and model latency are evaluated on nuScenes val set [2].
+For model latency, we only consider the latency of the net-
+work forward pass and ignore the pre-processing and post-
+processing time (e.g., image loading). The latency is eval-
+uated on a single NVIDIA RTX 3090 GPU with batch size
+1. We provide the detailed model architectures below.
+A.1. Model Architecture
+2D detector
+We choose 3 kinds of 2D detectors, including
+Faster R-CNN [31], a single-stage anchor-based 2D detec-
+tor RetinaNet [22], and a single-stage anchor-free 2D detec-
+tor YOLOX [8].
+Feature extractor
+We choose 2 kinds of feature extractor,
+including ResNet-50 [13] with FPN [21] and CSPDarkNet-
+S [8, 36] with PAN [23].
+ResNet-50 is equipped with
+deformable convolution [6] in the 3rd and 4th stages.
+CSPDarkNet-S with PAN is a lightweight feature extractor
+to validate the generalization ability of MV2D with small
+models. Moreover, the output channels of CSPDarkNet-S
+with PAN is 128, while that of ResNet-50 with FPN is 256.
+Feature pyramid
+For models with CSPDarkNet-S back-
+bone, the feature pyramid is built to produce feature maps
+with downsample stride {8, 16, 32}.
+For models with
+ResNet-50 backbone and Faster R-CNN detector, the fea-
+ture pyramid is built to produce feature maps with down-
+sample stride {4, 8, 16, 32, 64}. For models with ResNet-
+50 backbone and RetinaNet/YOLOX detector, the feature
+pyramid is built to produce feature maps with downsample
+stride {8, 16, 32, 64, 128}.
+A.2. Latency and Performance Comparison
+We equip MV2D with different 2D detectors and feature
+extractors, then evaluate their latency and performance on
+nuScenes val set. The results are listed in Table 5 and Ta-
+ble 6. Both models use an input resolution of 1408×512.
+Feat Latency includes the latency of backbone and neck
+(feature pyramid). 2D Det Latency includes the latency of
+2D detection part (e.g., RPN [9] and R-CNN [10]). Head
+Latency includes the latency of RoI-align [11], dynamic
+query generator and decoder.
+As
+demonstrated
+in
+the
+results,
+MV2D
+with
+CSPDarkNet-S has lower latency compared to ResNet-50.
+Specifically, when using YOLOX as 2D detector, MV2D
+with CSPDarkNet-S has overall latency of 60ms (16.6
+FPS), while achieving 32.2% mAP and 35.9% NDS. When
+replacing YOLOX with a stronger 2D detector Faster
+R-CNN, the 2D Det Latency increases by about 14ms, and
+performance also improves by 1.9% and 1.7% on mAP
+and NDS respectively. The difference in Head Latency is
+mainly caused by the number of 2D proposals generated
+by different 2D detectors. As for MV2D with ResNet-50
+backbone, we implement 3 kinds of 2D detectors and
+compare their performances. Among these 2D detectors,
+Faster R-CNN achieves higher mAP and NDS with a higher
+overall latency of 168ms (5.9 FPS). These experiments
+show that MV2D can generalize well to other feature
+extractors and 2D detectors.
+11
+
+Feat Extractor
+2D Detector
+Resolution
+Feat Latency
+2D Det Latency
+Head Latency
+Latency ↓
+FPS↑
+CSPDarkNet-S
+YOLOX
+1408×512
+22ms
+10ms
+28ms
+60ms
+16.6
+CSPDarkNet-S
+Faster R-CNN
+1408×512
+22ms
+24ms
+31ms
+77ms
+12.9
+Res-50
+YOLOX
+1408×512
+73ms
+22ms
+31ms
+126ms
+7.9
+Res-50
+RetinaNet
+1408×512
+73ms
+50ms
+38ms
+161ms
+6.2
+Res-50
+Faster R-CNN
+1408×512
+86ms
+48ms
+34ms
+168ms
+5.9
+Table 5. Latency comparison on nuScenes val set.
+Feat Extractor
+2D Detector
+Resolution
+mAP ↑
+NDS↑
+mATE
+mASE
+mAOE
+mAVE
+mAAE
+CSPDarkNet-S
+YOLOX
+1408×512
+0.322
+0.359
+0.745
+0.272
+0.700
+1.213
+0.302
+CSPDarkNet-S
+Faster R-CNN
+1408×512
+0.341
+0.376
+0.738
+0.276
+0.662
+1.199
+0.266
+Res-50
+YOLOX
+1408×512
+0.372
+0.412
+0.689
+0.269
+0.578
+1.008
+0.206
+Res-50
+RetinaNet
+1408×512
+0.376
+0.413
+0.711
+0.276
+0.544
+1.084
+0.229
+Res-50
+Faster R-CNN
+1408×512
+0.388
+0.433
+0.697
+0.271
+0.479
+0.952
+0.208
+Table 6. Performance comparison on nuScenes val set.
+12
+
diff --git a/yNE0T4oBgHgl3EQfcgCd/content/tmp_files/load_file.txt b/yNE0T4oBgHgl3EQfcgCd/content/tmp_files/load_file.txt
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@@ -0,0 +1,877 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf,len=876
+page_content='Object as Query: Equipping Any 2D Object Detector with 3D Detection Ability  Zitian Wang1  Zehao Huang2  Jiahui Fu1  Naiyan Wang2  Si Liu1  1Institute of Artificial Intelligence, Beihang University  2TuSimple  {wangzt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='kghl,zehaohuang18,winsty}@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='com  {jiahuifu,liusi}@buaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='cn  Abstract  3D object detection from multi-view images has drawn  much attention over the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Existing methods  mainly establish 3D representations from multi-view im-  ages and adopt a dense detection head for object detec-  tion, or employ object queries distributed in 3D space to  localize objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In this paper, we design Multi-View 2D  Objects guided 3D Object Detector (MV2D), which can be  equipped with any 2D object detector to promote multi-view  3D object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Since 2D detections can provide valu-  able priors for object existence, MV2D exploits 2D detec-  tor to generate object queries conditioned on the rich im-  age semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' These dynamically generated queries en-  able MV2D to detect objects in larger 3D space without in-  creased computational cost and show a strong capability of  localizing 3D objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' For the generated queries, we design  a sparse cross attention module to force them to focus on  the features of specific objects, which reduces the compu-  tational cost and suppresses interference from noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The  evaluation results on the nuScenes dataset demonstrate the  dynamic object queries and sparse feature aggregation do  not harm 3D detection capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' MV2D also exhibits a  state-of-the-art performance among existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' We  hope MV2D can serve as a new baseline for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Introduction  Camera-based 3D object detection in unconstrained real-  world scenes has drawn much attention over the past few  years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Early monocular 3D object detection methods [1,  15, 27, 33, 37, 38, 42] typically build their framework fol-  lowing the 2D object detection pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The 3D location  and attributes of objects are directly predicted from a sin-  gle view image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Though these methods have achieved great  progress, they are incapable of utilizing the geometric con-  figuration of surrounding cameras and multi-view image  correspondences, which are pivotal for the 3D position of  objects in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Moreover, adapting these meth-  ods to the multi-view setting relies on sophisticated cross-  (a) original image  (b) fixed queries  (c) 2D detection  (d) generated queries from 2D detection  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Motivation of MV2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The 3D detector with fixed object  queries (fixed queries means the queries are invariant for different  inputs) might miss some objects, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=', traffic cones (b), which are  however successfully detected by a 2D detector (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' If generating  object queries based on 2D detector, a 3D detector can produce  more precise locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  camera post-processing, which further causes degraded ef-  ficiency and efficacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' To handle these problems, recent re-  searchers [14,19,20,24,39] propose to directly localize ob-  jects in 3D world space based on multi-view images, pro-  viding a new paradigm for vision-based 3D object detec-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  According to the representation of fused features, current  multi-view 3D object detection methods can be mainly di-  vided into two streams: dense 3D methods and sparse query  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Concretely, dense 3D methods render multi-view  features into 3D space, such as Bird’s-Eye-View (BEV) fea-  ture space [4, 14, 19, 20] or voxel feature space [18, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  However, since the computational costs are squarely pro-  portional to the range of 3D space, they inevitably cannot  scale up to large-scale scenarios [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Alternatively, query-  based methods [24,39] adopt learnable object queries to ag-  gregate features from multi-view images and predict object  bounding boxes based on query features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Although fixed  number of object queries avoid computational cost explod-  ing with 3D space, the query number and position relied on  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='02364v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='CV]  6 Jan 2023  Fused RADARs  LIDAR_TOP  CAM_FRONT  CAM_FRONT_RIGHT  CAMBACKRIGHT  CAM_BACK  CAM_BACK_LEFT  CAM_FRONT_LEFTFused RADARs  LIDAR_TOP  CAM_FRONT  CAM_FRONT_RIGHT  CAMBACKRIGHT  CAM_BACK  CAM BACK LEFT  CAM_FRONT_LEFTAAFAEempirical prior may cause false positive or undetected ob-  jects in dynamic scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  In this paper, we seek a more reliable way to generate  object queries dynamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  The motivation comes from  the rapid developments of 2D object detection methods  [10, 21, 31, 34] which can generate high quality 2D bound-  ing boxes for object localization in image space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Fortu-  nately, 3D detectors are usually built upon 2D detection  backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' One natural idea is to turn each 2D detection  into one query for the following 3D detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In this  way, we design Multi-View 2D Objects guided 3D Object  Detector (MV2D), which could equip any 2D detector with  3D detection ability, and the 3D detector could harvest all  the advancements from 2D detection field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Given the input multi-view images, we first obtain 2D  detection results from a 2D detector and then generate a dy-  namic object query for each 2D bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Instead of  aggregating features from all regions in the multi-view in-  puts, one object query is forced to focus on one specific ob-  ject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' To this end, we propose an efficient correlated feature  selection method based on the 2D detection results and cam-  era configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Then the dynamically generated object  queries, together with their 3D position embedded corre-  lated features, are input to a transformer decoder with sparse  cross-attention layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Lastly, the updated object queries  predict the final 3D bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Thanks to the powerful 2D detection performances, the  dynamic queries generated from 2D detection results can  cover most objects that appeared in the images, leading to  higher precision and recall, especially for small and distant  objects compared with fixed queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' As shown in Figure 1,  the objects that a fixed query-based 3D detector might miss  can be recalled in MV2D with the help of a 2D object detec-  tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Theoretically, since 3D object queries stem from 2D de-  tection results, our method can benefit from all off-the-shelf  excellent 2D detector improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Our contributions can  be summarized as:  We propose a framework named MV2D that can be  equipped with any 2D object detector to promote  multi-view 3D object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Specifically, MV2D  generates sparse dynamic object queries based on the  results of 2D object detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  We demonstrate that sparse dynamic object queries  and sparse aggregation from certain relevant regions in  multi-view images do not sacrifice any detection per-  formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  We test MV2D on the standard nuScenes dataset, and  it achieves state-of-the-art performance among single  frame methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Related Work  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 2D Object Detection  2D object detection aims to predict and localize objects  in 2D images, which is a fundamental problem in computer  vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The RCNN series [9,10,21,31] propose a two-stage  pipeline for 2D object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' These methods first gen-  erate a set of region proposals likely to contain objects in  the image, then make a second stage decision on the object  classification and bounding box regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Another group  of researchers investigate to detect object in a single-stage  pipeline [22, 30, 34], aiming to provide faster detectors for  deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Recently, DETR [3] introduces a transformer  encoder-decoder architecture into 2D object detection and  formulates object detection as a set prediction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In  DETR, a set of learnable object queries are adopted to in-  teract with image features and responsible for detection ob-  jects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Deformable DETR [44] improves DETR by intro-  ducing multi-scale deformable attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Besides, it trans-  fers the vanilla DETR into a two-stage framework, where  region proposals are firstly generated based on encoder fea-  tures and then sent to the decoder as object queries for fur-  ther refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Vision-based 3D Object Detection  The goal of vision-based 3D object detection is to pre-  dict 3D bounding boxes from camera images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Many previ-  ous [1, 15, 28, 33, 37, 38] works perform this task based on  the single-view image setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Concretely, these methods  mostly focus on instance depth estimation and use an ex-  tra depth prediction module to complement 2D detectors on  depth information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' However, when dealing with multi-view  images surrounding the ego vehicle, these methods need to  detect objects from each view and project them to the same  3D space with cumbersome NMS post-processing to merge  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Recently, some works attempt to directly detect objects  in 3D space from multi-view images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' One branch of these  methods lift 2D image into 3D space, then conduct detec-  tion based on the 3D representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  ImVoxelNet [32]  builds a 3D voxelized space and samples image features  from multi-view to obtain the voxel representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' BEV-  Former [20] leverages dense BEV queries to project and  aggregate features from multi-view images by deformable  attention [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' BEVDet and BEVDepth [14, 19] adopt the  Lift-Splat module [29] to transform multi-view image fea-  tures into the BEV representation based on the predicted  depth distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Although such 3D space representations  are conducive to unifying multi-view images, the memory  consumption and computational cost would increase rapidly  with the enlarging of the detection range in 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Another way of such works adopts learnable object  queries to aggregate image features and predict objects fol-  2  Multi-View Images  2D Detector  Feat Extractor  2D boxes  weight sharing  RoI align  RoI feats  Dynamic Query  Generator  cam params  object queries  Self Attention  Sparse Cross  Attention  FFN  ×𝐿 layers  Head  3D predictions  …  …  𝑁 object queries  RoI features  Correlated Feature Selection  Cross Attention  …  𝑖th object query  𝑖th correlated features  ×𝑁 groups  Q  K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' V  …  𝑁 object queries  3D PE  …  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The framework of the proposed MV2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Given the input multi-view images, a 2D detector is used to obtain per-view 2D detection  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Meanwhile, image feature maps are extracted by a feature extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The parameter weights of the backbone network of 2D  detector and feature extractor can be shared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Then the RoI features are extracted through RoI-align for detected 2D bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Based  on the RoI features, RoI positions and camera parameters, MV2D initializes a set of object queries using a dynamic query generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  These generated object queries and RoI features with 3D PE (3D position embedding) [24] are input to a decoder to update query features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Compared to vallina transformer decoder, the decoder in MV2D employ sparse cross attention where each object query only interact with  its correlated features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Lastly, a prediction head is applied on the updated object queries to generate 3D detection results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  lowing the DETR [3] framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' DETR3D [39] gener-  ates 3D reference points from object queries and projects  them to multi-camera images by camera parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Then  the query features are refined by the corresponding point  features sampling from images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  In contrast to estab-  lishing fixed mapping through 3D-to-2D query project-  ing, some methods learn the flexible mapping via attention  mechanisms [24, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' PETR [24] introduces 3D position-  aware image features and learns the flexible mapping be-  tween query and image features by global cross-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Nonetheless, these approaches usually require dense object  queries distributed in 3D space to ensure sufficient recall of  objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  In this paper, we propose a 2D objects guided framework  for multi-view 3D object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Based on the 2D detec-  tor, our method can count on much sparser queries to recall  the objects and eliminate the interference of noises and dis-  tractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Method  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Overview  The overall pipeline of MV2D is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Given N multi-view images I = {Iv | 0 ≤ v < N},  image feature maps F = {Fv | 0 ≤ v < N} are first  extracted from the input images using a backbone network,  where Fv ∈ RHf ×W f ×C is the extracted feature map from  the v-th image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' To get 2D object detection, a 2D object  detector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=', Faster R-CNN [31], is applied to all the in-  put images, resulting in a set of 2D object bounding boxes  B = {Bv | 0 ≤ v < N}, where Bv ∈ RMv×4 represents  the predicted 2D bounding boxes in the v-th image and Mv  is the number of detected boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In practice, the weights of  the backbone of 2D detector can also be used for subsequent  feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Different from common methods in multi-view 3D ob-  ject detection which adopt fixed object queries across the  whole dataset [24,39], MV2D generates object queries con-  ditioned on the input multi-view images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Given the pre-  dicted 2D bounding boxes and extracted image features,  MV2D employs RoI-Align [11] to extract 2D object fea-  tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Then the 2D object features together with corre-  sponding bounding boxes and camera parameters are in-  put to the dynamic object query generator to produce ob-  ject queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' For each object query, relevant image features  in multi-view images are selected based on 2D detection  results and camera parameters through a feature selection  module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Then the object queries interact with each other  and integrate information from correlated object features it-  eratively through transformer decoder layers [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Finally, a  simple feed forward network (FFN) head is used to generate  3D object predictions with updated features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Dynamic Object Query Generation  With the help of an effective 2D object detector, the ob-  ject existences are well demonstrated and the object loca-  tions are constrained within certain image regions, thus pro-  viding valuable priors for localizing objects in 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  To generate object queries from 2D detection results, we  propose a dynamic query generator as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  The dynamic query generator derives a 3D reference point  pref ∈ R3 in 3D world space for each RoI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Then the ob-  ject query is generated from pref by a positional encoding  layer [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  3  𝐾  𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' "#, 𝑦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' "#  𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='$%, 𝑦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='$%  Camera Parameters  MLP  Conv  Pool  𝐾!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  "  𝑝!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='"  𝑝#$%  Coordinate  Transformation  RoI Features  𝑅|t  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Dynamic object query generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Specifically, given the 2D object detection results B and  image feature maps F from all the images, we first ex-  tract object RoI features O through RoI-Align, where Ov ∈  RMv×Hroi×W roi×C are the RoI features corresponding to  2D object bounding boxes Bv:  Ov = RoI-Align(Fv, Bv),  0 ≤ v < N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  (1)  The RoI features Ov contain sufficient object appearance  information to infer the object center locations in image  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' However, further estimating the object depths di-  rectly from RoI features is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' As the object region  in original image space is rescaling into a fixed sized RoI,  the geometric information contained in original images are  missed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Taking this into consideration, we apply equivalent cam-  era intrinsic transformation to each RoI, such that the rescal-  ing operation conducted on different RoIs is equivalent to  perspective projection with different camera parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Thus a point in the RoI coordinate system can be trans-  formed to 3D world space with equivalent camera intrinsic:  p3d = [R|t]−1(Kroi)−1 proi,  (2)  where proi ∈ R4 (homogeneous coordinate) represents the  point in RoI coordinate system, p3d ∈ R4 represents the  point in 3D world space, Kroi is the equivalent camera in-  trinsic of that RoI and [R|t] is the camera extrinsic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Denote the RoI size is Hroi × W roi (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=', 7 × 7), the  i-th 2D object bounding box in the v-th view is Bi  v =  (xi  min, yi  min, xi  max, yi  max), and the original camera intrin-  sic matrix is:  Kv =  �  ��  fx  0  ox  0  0  fy  oy  0  0  0  1  0  0  0  0  1  �  ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  (3)  The equivalent camera intrinsic for i-th RoI can be formu-  lated as:  Ki  v =  �  ��  fx ∗ rx  0  (ox − xi  min) ∗ rx  0  0  fy ∗ ry  (oy − yi  min) ∗ ry  0  0  0  1  0  0  0  0  1  �  ��,  (4)  where rx = W roi/(xi  max − xi  min), ry = Hroi/(yi  max −  yi  min).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Since the equivalent camera intrinsic matrix contains the  geometric property of the camera and object, we adopt a  small network to implicitly encode the object location pi  v ∈  R4 based on the RoI feature oi  v and the equivalent camera  intrinsic Ki  v:  pi  v = H(MLP(Pool(Conv(oi  v));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Ki  v)),  (5)  where (;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' ) means feature concatenation and H(·) represents  homogeneous transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  The implicitly encoded object location pi  v, which can be  seen as the 3D object location in RoI coordinate system,  is then transformed into 3D world space using equivalent  camera intrinsic and camera extrinsic as in Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Con-  sequently, the transformed 3D location serves as reference  point pref for localizing object in 3D world space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  By this transformation, MV2D generates a set of dy-  namic reference points Pref = {Pref,v ∈ RMv×3 | 0 ≤  v < N} based on 2D detections and camera configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  These reference points are then passed through a 3D po-  sitional encoding layer to initialize a set of object queries  Q = {Qv ∈ RMv×C | 0 ≤ v < N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Relevant Object Feature Selection  Each object is only captured in a sub-region within the  input multi-view images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' To precisely depict a certain ob-  ject, an object query should focus on all the relevant image  regions that cover the target object, and discard irrelevant  regions that might mislead the 3D localization of the target  object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  To this end, we propose to select relevant features for  each object query’s updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Since 2D object detector can  predict convincing 2D object proposals, the detected object  bounding boxes imply which region contains the most dis-  tinctive information about an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' So we consider two  parts of the relevant features of an object: (1) the 2D bound-  ing box bi  v from where the object query is generated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' (2) the  bounding boxes in other views that corresponds to bi  v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  In this paper, we develop an efficient method to as-  sociate the bounding boxes in other views for a given  query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' As shown in Figure 4, we create a 3D meshgrid  G ∈ RW roi×Hroi×D×4 for each RoI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' We denote gx,y,z =  (x × dz, y × dz, dz, 1) as each point in the meshgrid, where  (x, y) is the coordinate in the RoI, dz ∈ {d0, d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' , dD−1}  is a set of predifined depth values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The meshgrid of RoI  is then transformed into the coordinate system of the w-th  view:  gi  v→w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='x,y,z = KwTv→w(Ki  v)−1gx,y,z,  (6)  where gi  v→w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='x,y,z is the transformed meshgrid point in the  w-th view and Tv→w is the coordinate transformation ma-  trix from the v-th view to the w-th view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  4  : uncorrelated boxes  : correlated boxes  : minimum bounding box  : query box  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Illustration of relevant region selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Each query box  generates a discretized camera frustum from 3D meshgrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The  camera frustum is then projected to another view’s pixel coordi-  nate to calculate a minimum bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Then correlated box  is selected based on IoU with the minimum bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  According to the transformed meshgrid Gi  v→w, a min-  imum bounding box bi  v→w in the image space of the wth  view can be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Then the detected box that has high-  est Intersection-over-Union (IoU) with bi  v→w (if greater  than 0) is selected as correlated box in the w-th view:  bi  corr,v→w = arg max  bj  w∈Bw  IoU(bi  v→w, bj  w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  (7)  Then, only the RoI features of bi  v and {bi  corr,v→w | 0 ≤  w < N, w ̸= v} will be adopted to update object query qi  v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  We will describe the details in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Decoder with Sparse Cross Attention  The generated object queries interact with their relevant  features through a DETR-like transformer decoder [3, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  3D position embedding is provided for the relevant features  following PETR [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The difference in our decoder lies in  the sparse cross attention layer as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In-  stead of using the whole multi-view feature maps to con-  struct keys and values shared by all the queries, MV2D  only uses the relevant features to construct their own keys  and values for each object query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' This design not only con-  tributes a compact set of keys and values, but also prevents  the object queries from being interfered by background  noises and distractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Lastly, we apply classification head and regression head  composed of MLPs to the updated object queries to predict  the final 3D object detection results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Loss Functions  In our implementation, the 2D detector and the 3D detec-  tor are jointly trained in MV2D, with the backbone weights  shared across both detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The 2D object detection loss  L2d is directly taken from 2D detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' As for 3D ob-  ject detection loss, we follow previous works [24, 39] to  use Hungarian algorithm [16] for label assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Focal  loss [22] and L1 loss are adopted for classification and box  regression respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The 3D object detection loss can be  summarized as:  L3d = λcls3d · Lcls3d + Lreg3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  (8)  The overall loss function of MV2D is:  L = L2d + λ3d · L3d,  (9)  where λ3d is the weight term to balance 2D detection and  3D detection losses, set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='1 in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Experiments  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Datasets and Metrics  We validate the effectiveness of MV2D on the large-  scale nuScenes dataset [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' NuScenes contains 1000 driving  sequences, which are divided into 700 samples for training,  150 samples for validation, and 150 samples for testing, re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Each sequence is approximately 20-second long,  including annotated 3D bounding boxes from 10 categories  of sampled key frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Each sample consists of 6 surround-  view images with 1600 × 900 resolution, which provide  360◦ horizontal FOV in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' We submit test set results to  the online server for official evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Other experiments  are based on the val set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Implementation Details  Training and evaluation  All the models are trained using  AdamW [26] optimizer with a weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Co-  sine annealing policy [25] is adopted and the initial learn-  ing rate is set to 2 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Since MV2D generates object  queries from 2D detection results, we pretrain the 2D detec-  tors on nuImages [2] to provide appropriate initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Since nuScenes dataset does not provide 2D bounding box  annotations, we generate the box labels from 3D bounding  boxes following [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' We employ image-space augmenta-  tion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=', random flip, crop and resize) and BEV-space aug-  mentation [14] during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' If not specified, the models  are trained for 24 epochs without CBGS [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' No test time  augmentation is used during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Our implementation  is based on MMDetection3D [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Network architecture  Faster-RCNN [31] is adopted as  2D detector in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  The 2D score thresh-  old and Non-maximum Suppression (NMS) IoU thresh-  old are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='05 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='6 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' We use ResNet-  50, ResNet-101 [12] and VoVNetV2 [17] as backbone  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' ResNet-50 and ResNet-101 are equipped with  deformable convolution [6] in the 3rd and 4th stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
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+page_content='127  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 3D detection results on nuScenes test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' †: the model is initialized from a FCOS3D checkpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' ‡: the model is pretrained  on nuImages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' §: the model is initialized from a DD3D [28] checkpoint trained with external data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  FPN [21] with RoI size 7×7 for dynamic object query gen-  eration and correlated feature selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The decoder con-  tains 6 transformer decoder layers, followed by MLP head  to produce classification and regression results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Comparison with State-of-the-arts  We compare the MV2D performance with state-of-the-  art methods on nuScenes val set and test set under single  frame setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The results are shown in Table 1 and Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Table 1 shows comparison on nuScenes val set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' From the  table, MV2D achieves higher performace with both ResNet-  50 and ResNet-101 backbone, especially on mAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The sig-  nificant improvement on mAP suggests MV2D has a strong  capability at localizing objects in 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In comparison  with multi-view 3D object detection methods, MV2D with  ResNet-101 outperforms BEV based methods BEVFormer-  S [20] by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='1% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='3% on mAP and NDS, and outper-  forms BEVDepth [19] with extra depth supervision by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='0%  and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='3% on mAP and NDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In comparison with query  based methods, MV2D outperforms the best performing  PETR [24] by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='6% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='9% on mAP and NDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Table 2 shows the comparison on nuScenes test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  We use both the training and validation splits for training  MV2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' As can be seen from the table, MV2D with ResNet-  101 backbone achieves 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='7% mAP and 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='2% NDS, which  outperforms existing methods by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='8% on mAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' For MV2D  with VoVNetV2, we train it for 90 epochs without CBGS  (which has almost the same amount of samples as training  24 epochs with CBGS) for better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In that case,  MV2D achieves 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='3% mAP and 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='4% NDS, which im-  proves mAP by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='2% compared to PETR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Note that our  method gets larger mAVE compared to state-of-the-arts,  which is the main cause of degraded NDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Ablation Study  In this section, we provide a detailed analysis on the core  design choices of MV2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' All the experiments are based  on ResNet-50 backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' For fair comparison, we design a  6  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Precision-recall curves of different classes under a 2D object detection setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The models still produce 3D object predictions,  while the 2D bounding boxes in each image are generated by projecting 3D bounding boxes using camera parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Precisoin & recall  are calculated with 2D IoU threshold 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='5 in nuScenes val set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The blue dash line represents the baseline method (fixed object queries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The  red solid line represents our method (generated object queries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  #  object query  key & value  mAP↑  NDS↑  1  fixed (×900)  all feats  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='2  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='9  2  fixed (×300)  all feats  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='1  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='4  3  fixed (×1500)  all feats  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='6  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='6  4  generated (×165)  all feats  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='0  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='2  5  generated (×165)  all RoIs  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='9  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='1  6  generated (×165)  correlated RoIs  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='8  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='3  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Ablation study on the object query and key & value in the  cross attention layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The number of generated queries is calcu-  lated on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  baseline method also trained with a 2D detector and initial-  ized from a nuImages pretrained checkpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' This method  uses a fixed number of learnable object queries to aggre-  gate information from the whole multi-view feature maps  as in PETR [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' As in Table 3, we denote the design in #1  which follows PETR to use 900 object queries as baseline  in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Generated object queries vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' fixed object queries  We  first compare the generated object queries with fixed object  queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' From #1 to #3 in Table 3, it can be observed that  in the case of fixed queries, a larger query amount can im-  prove performance since the fixed query based methods rely  on densely placed object queries to localize objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' How-  ever, the computational cost and memory consumption also  rise with the growth of query density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' According to #4,  when replacing the fixed queries by dynamically generated  queries, mAP and NDS are improved by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='8% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='3%  #  2D NMS IoU  2D score thr  RoI size  mAP↑  NDS↑  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='05  7  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='2  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='9  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='05  7  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='8  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='3  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
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+page_content='2  7  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='0  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='9  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='05  14  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='6  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='1  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Ablation study on RoI configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Note that the number of generated queries is  only around 165 on average, which is much less than that of  fixed queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' This demonstrates the object queries gener-  ated by 2D detector produce higher-quality object location  hypotheses even with smaller query amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Effectiveness of correlated features  We also validate the  effectiveness of using only correlated features for object  query updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In Table 3, #4 represents using the whole  multi-view feature maps for each object query, #5 repre-  sents using all the RoI features for each object query, while  #6 represents using only the correlated RoI features for  each object query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  As shown by the results, simply re-  placing the whole feature maps by all RoI features dose not  bring performance gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Nevertheless, using only the corre-  lated RoI features for object query updating as introduced in  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='3 can improve mAP and NDS by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='8% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='1%  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' This result verifies the effectiveness of object  query to aggregate information in a specified foreground re-  gion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  7  CAR  TRUCK  TRAILER  BUS  CONSTRUCTION_VEHICLE  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
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+page_content='-- baseline  -- baseline  baseline  ours  ours  ours  ours  ours  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
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+page_content='0  recall  recall  recall  recall  recall: ground truth centers  : reference points  (a) baseline  (b) MV2D  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Illustration of different kinds of object queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The  top row (a) represents the baseline method (fixed object queries)  and the bottom row (b) represents our method (generated object  queries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' We visualize ground truth center locations and the refer-  ence points in BEV space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The samples are taken from nuScenes  val set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' We suggest readers view this figure with zooming in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Influence of RoI configuration  In this part, we study the  influence of different RoI configurations in MV2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' We ex-  periment with different 2D detection score thresholds, 2D  NMS IoU thresholds, and RoI-align sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The results are  listed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' As can be seen from the table, the perfor-  mance of MV2D is robust to different RoI settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Qualitative Analysis  In this section, we analyze how might a 2D detector pro-  mote the performance of multi-view 3D object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' To  verify if the object queries generated from 2D detector can  better localize objects with the help of image semantics, we  compare the 2D object detection performance of the base-  line method (fixed object queries) and our method (gener-  ated object queries) by perspective projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Specifically,  we project the 3D bounding boxes into each image, then cal-  culate the minimum 2D bounding boxes as 2D results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The  performance is evaluated between the projected 2D bound-  ing boxes of predictions and ground truths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' We set the IoU  threshold to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='5 for true positive assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  In Figure 5, we draw the precision-recall curves of  10 classes in nuScenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  As seen from the figure, our  method achieves higher precision and recall compared to  the baseline method, especially for the classes with rela-  tively smaller size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' This demonstrates that 2D object detec-  tors can make the most of image semantics to discover ob-  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Comparison under different numbers of decoder layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  jects from image space, which can serve as solid evidence of  object existence to be exploited by 3D detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' For com-  plementary, we provide the visualization of different kinds  of queries in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Though the generated queries are  much sparser compared to the fixed queries, they are mostly  distributed around the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Moreover, we compare the 3D detection performance of  baseline and our method under different numbers of de-  coder layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' As shown in Figure 7, baseline with fixed  queries rely on a heavy decoder to refine the query features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  In contrast, our method with generated queries and sparse  cross attention shows a faster convergence on mAP and  NDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' This result demonstrates that better initialized queries  and constrained search regions might help to reduce the am-  biguity for localizing objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Conclusion  In this paper, we propose Multi-View 2D Objects guided  3D Object Detector (MV2D) for multi-view 3D object de-  tection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In our framework, we utilizes 2D objects as sparse  queries and adopt a sparse cross attention module to limit  the attention region of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  In our experiments, we  demonstrate promising results on nuScenes dataset with our  proposed sparse dynamic object queries and sparse aggre-  gation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Our framework can equip any 2D detec-  tor with 3D detection ability, and we believe the insight of  utilizing the 2D objects as guidance can further inspire the  design of multi-view 3D object detection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Limitations  Since MV2D generates object queries from  2D detections, an object might be missed if the 2D detector  fails to recall it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Besides, it requires careful design of tem-  poral correlation for MV2D under multi-frame setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' We  look forward to further extending MV2D in the temporal  dimension in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
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+page_content='4%  mAP  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
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+page_content='25  1  2  3  4  5  6  number of decoder layersReferences  [1] Garrick Brazil and Xiaoming Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' M3D-RPN: monocular 3d  region proposal network for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In ICCV, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  1, 2  [2] Holger Caesar, Varun Bankiti, Alex H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Lang, Sourabh Vora,  Venice Erin Liong, Qiang Xu, Anush Krishnan, Yu Pan, Gi-  ancarlo Baldan, and Oscar Beijbom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  nuscenes: A multi-  modal dataset for autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In CVPR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 5,  11  [3] Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas  Usunier, Alexander Kirillov, and Sergey Zagoruyko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' End-to-  end object detection with transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In ECCV, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 2,  3, 5  [4] Shaoyu Chen, Xinggang Wang, Tianheng Cheng, Qian  Zhang, Chang Huang, and Wenyu Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Polar parametrization  for vision-based surround-view 3d detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' arXiv preprint  arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='10965, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 1  [5] MMDetection3D Contributors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  MMDetection3D: Open-  MMLab next-generation platform for general 3D object  detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='com/open-mmlab/  mmdetection3d, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 5  [6] Jifeng Dai, Haozhi Qi, Yuwen Xiong, Yi Li, Guodong  Zhang, Han Hu, and Yichen Wei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Deformable convolutional  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In ICCV, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 5, 11  [7] Lue Fan, Feng Wang, Naiyan Wang, and Zhaoxiang Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Fully sparse 3d object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In NeurIPS, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 1  [8] Zheng Ge, Songtao Liu, Feng Wang, Zeming Li, and Jian  Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Yolox: Exceeding yolo series in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' arXiv preprint  arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='08430, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 11  [9] Ross Girshick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Fast R-CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In ICCV, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 2, 11  [10] Ross Girshick, Jeff Donahue, Trevor Darrell, and Jitendra  Malik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Rich feature hierarchies for accurate object detection  and semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In CVPR, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 2, 11  [11] Kaiming He, Georgia Gkioxari, Piotr Doll´ar, and Ross Gir-  shick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Mask R-CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In ICCV, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 3, 11  [12] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Deep residual learning for image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  In CVPR,  2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 5  [13] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Deep residual learning for image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  In CVPR,  2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 11  [14] Junjie Huang, Guan Huang, Zheng Zhu, and Dalong Du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  BEVDet: High-performance multi-camera 3d object detec-  tion in bird-eye-view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  arXiv preprint arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='11790,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 1, 2, 5, 6  [15] Jason Ku, Alex D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Pon, and Steven L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Waslander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Monocular  3d object detection leveraging accurate proposals and shape  reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In CVPR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 1, 2  [16] Harold W Kuhn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The hungarian method for the assignment  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Naval research logistics quarterly, 1955.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 5  [17] Youngwan Lee, Joong-Won Hwang, Sangrok Lee, Yuseok  Bae, and Jongyoul Park.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' An energy and gpu-computation  efficient backbone network for real-time object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In  CVPRW, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 5  [18] Yanwei Li, Yilun Chen, Xiaojuan Qi, Zeming Li, Jian Sun,  and Jiaya Jia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Unifying voxel-based representation with  transformer for 3d object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In NeurIPS, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 1  [19] Yinhao Li, Zheng Ge, Guanyi Yu, Jinrong Yang, Zengran  Wang, Yukang Shi, Jianjian Sun, and Zeming Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' BEVDepth:  Acquisition of reliable depth for multi-view 3d object detec-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' arXiv preprint arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='10092, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 1, 2, 6  [20] Zhiqi Li, Wenhai Wang, Hongyang Li, Enze Xie, Chong-  hao Sima, Tong Lu, Qiao Yu, and Jifeng Dai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' BEVFormer:  Learning bird’s-eye-view representation from multi-camera  images via spatiotemporal transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In ECCV, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 1,  2, 6  [21] Tsung-Yi Lin, Piotr Doll´ar, Ross Girshick, Kaiming He,  Bharath Hariharan, and Serge Belongie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Feature pyramid  networks for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In CVPR, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 2, 6, 11  [22] Tsung-Yi Lin, Priya Goyal, Ross Girshick, Kaiming He, and  Piotr Doll´ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Focal loss for dense object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In ICCV,  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 2, 5, 11  [23] Shu Liu, Lu Qi, Haifang Qin, Jianping Shi, and Jiaya Jia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Path aggregation network for instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  In  CVPR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 11  [24] Yingfei Liu, Tiancai Wang, Xiangyu Zhang, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  PETR: Position embedding transformation for multi-view 3d  object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In ECCV, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 1, 3, 5, 6, 7  [25] Ilya Loshchilov and Frank Hutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' SGDR: stochastic gradient  descent with warm restarts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In ICLR, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 5  [26] Ilya Loshchilov and Frank Hutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Decoupled weight decay  regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In ICLR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 5  [27] Arsalan Mousavian, Dragomir Anguelov, John Flynn, and  Jana Kosecka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 3d bounding box estimation using deep learn-  ing and geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In CVPR, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 1  [28] Dennis Park, Rares Ambrus, Vitor Guizilini, Jie Li, and  Adrien Gaidon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Is pseudo-lidar needed for monocular 3d  object detection?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In ICCV, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 2, 5, 6  [29] Jonah Philion and Sanja Fidler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Lift, splat, shoot: Encoding  images from arbitrary camera rigs by implicitly unprojecting  to 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In ECCV, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 2  [30] Joseph Redmon, Santosh Divvala, Ross Girshick, and Ali  Farhadi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' You only look once: Unified, real-time object de-  tection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In CVPR, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 2  [31] Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Faster R-CNN: Towards real-time object detection with re-  gion proposal networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In NeurIPS, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 2, 3, 5, 11  [32] Danila Rukhovich, Anna Vorontsova, and Anton Konushin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Imvoxelnet: Image to voxels projection for monocular and  multi-view general-purpose 3d object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In WACV,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 1, 2  [33] Andrea Simonelli, Samuel Rota Bul`o, Lorenzo Porzi,  Manuel Lopez-Antequera, and Peter Kontschieder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Disen-  tangling monocular 3d object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In ICCV, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 1,  2  [34] Zhi Tian, Chunhua Shen, Hao Chen, and Tong He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' FCOS:  Fully convolutional one-stage object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  In ICCV,  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 2  [35] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszko-  reit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia  Polosukhin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Attention is all you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In NeurIPS, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 3  [36] Chien-Yao Wang, Hong-Yuan Mark Liao, Yueh-Hua Wu,  Ping-Yang Chen, Jun-Wei Hsieh, and I-Hau Yeh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Cspnet: A  new backbone that can enhance learning capability of CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  In CVPR Workshops, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 11  9  [37] Tai Wang, Xinge Zhu, Jiangmiao Pang, and Dahua Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  FCOS3D: fully convolutional one-stage monocular 3d object  detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In ICCVW, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 1, 2, 6  [38] Tai Wang, Xinge Zhu, Jiangmiao Pang, and Dahua Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Probabilistic and geometric depth: Detecting objects in per-  spective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In CoRL, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 1, 2, 6  [39] Yue Wang, Vitor Guizilini, Tianyuan Zhang, Yilun Wang,  Hang Zhao, and Justin Solomon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' DETR3D: 3d object detec-  tion from multi-view images via 3d-to-2d queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In CoRL,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 1, 3, 5, 6  [40] Enze Xie, Zhiding Yu, Daquan Zhou, Jonah Philion, Anima  Anandkumar, Sanja Fidler, Ping Luo, and Jose M Alvarez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  M2BEV: Multi-camera joint 3d detection and segmentation  with unified birds-eye view representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' arXiv preprint  arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='05088, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 5, 6  [41] Brady Zhou and Philipp Kr¨ahenb¨uhl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Cross-view transform-  ers for real-time map-view semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In CVPR,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 3  [42] Xingyi Zhou, Dequan Wang, and Philipp Kr¨ahenb¨uhl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Ob-  jects as points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' CoRR, abs/1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='07850, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 1  [43] Benjin Zhu, Zhengkai Jiang, Xiangxin Zhou, Zeming Li, and  Gang Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Class-balanced grouping and sampling for point  cloud 3d object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' arXiv preprint arXiv:1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='09492,  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 5  [44] Xizhou Zhu, Weijie Su, Lewei Lu, Bin Li, Xiaogang Wang,  and Jifeng Dai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Deformable DETR: deformable transformers  for end-to-end object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' In ICLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 2  10  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Experiments with More Architectures  We provide more experiments with different 2D detec-  tors and feature extractors in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The 2D detec-  tor part of MV2D is pretrained on nuImages [2], then the  2D detector part and 3D detector part are jointly trained  on nuScenes train set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 3D object detection performance  and model latency are evaluated on nuScenes val set [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  For model latency, we only consider the latency of the net-  work forward pass and ignore the pre-processing and post-  processing time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=', image loading).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The latency is eval-  uated on a single NVIDIA RTX 3090 GPU with batch size  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' We provide the detailed model architectures below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Model Architecture  2D detector  We choose 3 kinds of 2D detectors, including  Faster R-CNN [31], a single-stage anchor-based 2D detec-  tor RetinaNet [22], and a single-stage anchor-free 2D detec-  tor YOLOX [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Feature extractor  We choose 2 kinds of feature extractor,  including ResNet-50 [13] with FPN [21] and CSPDarkNet-  S [8, 36] with PAN [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  ResNet-50 is equipped with  deformable convolution [6] in the 3rd and 4th stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  CSPDarkNet-S with PAN is a lightweight feature extractor  to validate the generalization ability of MV2D with small  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Moreover, the output channels of CSPDarkNet-S  with PAN is 128, while that of ResNet-50 with FPN is 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Feature pyramid  For models with CSPDarkNet-S back-  bone, the feature pyramid is built to produce feature maps  with downsample stride {8, 16, 32}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  For models with  ResNet-50 backbone and Faster R-CNN detector, the fea-  ture pyramid is built to produce feature maps with down-  sample stride {4, 8, 16, 32, 64}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' For models with ResNet-  50 backbone and RetinaNet/YOLOX detector, the feature  pyramid is built to produce feature maps with downsample  stride {8, 16, 32, 64, 128}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Latency and Performance Comparison  We equip MV2D with different 2D detectors and feature  extractors, then evaluate their latency and performance on  nuScenes val set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The results are listed in Table 5 and Ta-  ble 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Both models use an input resolution of 1408×512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Feat Latency includes the latency of backbone and neck  (feature pyramid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' 2D Det Latency includes the latency of  2D detection part (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=', RPN [9] and R-CNN [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Head  Latency includes the latency of RoI-align [11], dynamic  query generator and decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  As  demonstrated  in  the  results,  MV2D  with  CSPDarkNet-S has lower latency compared to ResNet-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Specifically, when using YOLOX as 2D detector, MV2D  with CSPDarkNet-S has overall latency of 60ms (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='6  FPS), while achieving 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='2% mAP and 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='9% NDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' When  replacing YOLOX with a stronger 2D detector Faster  R-CNN, the 2D Det Latency increases by about 14ms, and  performance also improves by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='9% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='7% on mAP  and NDS respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' The difference in Head Latency is  mainly caused by the number of 2D proposals generated  by different 2D detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' As for MV2D with ResNet-50  backbone, we implement 3 kinds of 2D detectors and  compare their performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Among these 2D detectors,  Faster R-CNN achieves higher mAP and NDS with a higher  overall latency of 168ms (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='9 FPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' These experiments  show that MV2D can generalize well to other feature  extractors and 2D detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  11  Feat Extractor  2D Detector  Resolution  Feat Latency  2D Det Latency  Head Latency  Latency ↓  FPS↑  CSPDarkNet-S  YOLOX  1408×512  22ms  10ms  28ms  60ms  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='6  CSPDarkNet-S  Faster R-CNN  1408×512  22ms  24ms  31ms  77ms  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='9  Res-50  YOLOX  1408×512  73ms  22ms  31ms  126ms  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='9  Res-50  RetinaNet  1408×512  73ms  50ms  38ms  161ms  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='2  Res-50  Faster R-CNN  1408×512  86ms  48ms  34ms  168ms  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='9  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content=' Latency comparison on nuScenes val set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='  Feat Extractor  2D Detector  Resolution  mAP ↑  NDS↑  mATE  mASE  mAOE  mAVE  mAAE  CSPDarkNet-S  YOLOX  1408×512  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
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+page_content='359  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='745  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='272  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='700  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='213  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='302  CSPDarkNet-S  Faster R-CNN  1408×512  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
+page_content='341  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
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+page_content='266  Res-50  YOLOX  1408×512  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
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+page_content='412  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
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+page_content='206  Res-50  RetinaNet  1408×512  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
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+page_content='229  Res-50  Faster R-CNN  1408×512  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
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+page_content=' Performance comparison on nuScenes val set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE0T4oBgHgl3EQfcgCd/content/2301.02364v1.pdf'}
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